Valter MorettiDepartment of Mathematics

University of Trento

Fundamental MathematicalStructures of Quantum Theory

Lecture notes for MSc Courses in Mathematics and Physics

Lecture notes authored by Valter Moretti and freely downloadable from the web pagehttp://www.science.unitn.it/∼moretti/dispense.html and are licensed under a Creative Commons

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Contents

Aim and structure of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 General Phenomenology of the Quantum World and Elementary Formalism 81.1 Physics of Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.1 When is a physical system quantum? . . . . . . . . . . . . . . . . . . . . . 81.1.2 Basic properties of quantum systems . . . . . . . . . . . . . . . . . . . . . 9

1.2 Elementary Quantum Formalism: The finite dimensional case . . . . . . . . . . . 111.2.1 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.2 Composite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 A first look to the infinite dimensional case, CCRs and quantization procedures . 171.3.1 L2(R, dx) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.2 L2(Rn, dnx) model and Heisenberg inequalities . . . . . . . . . . . . . . . 20

2 Observables and states in general Hilbert spaces: The Spectral Theory 242.1 Hilbert Spaces: A summary of fundamental facts . . . . . . . . . . . . . . . . . . 24

2.1.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1.2 Orthogonality and Hilbert bases . . . . . . . . . . . . . . . . . . . . . . . 262.1.3 Two notions of Hilbert direct orthogonal sum . . . . . . . . . . . . . . . . 282.1.4 Tensor product of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Classes of (especially unbounded) operators in Hilbert spaces . . . . . . . . . . . 302.2.1 Operators and abstract algebras . . . . . . . . . . . . . . . . . . . . . . . 302.2.2 Adjoint operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.3 Closed and closable operators . . . . . . . . . . . . . . . . . . . . . . . . . 352.2.4 Types of operators relevant in quantum theory . . . . . . . . . . . . . . . 382.2.5 Interplay of Ker, Ran, ∗, and ⊥ . . . . . . . . . . . . . . . . . . . . . . . 422.2.6 Criteria for (essential) selfadjointness . . . . . . . . . . . . . . . . . . . . . 43

2.3 Basic on spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.3.1 Resolvent and spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.3.2 Spectra of some types of operators . . . . . . . . . . . . . . . . . . . . . . 53

2.4 Integration of projector-valued measures . . . . . . . . . . . . . . . . . . . . . . . 562.4.1 Orthogonal projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4.2 Projector-valued measures (PVM) . . . . . . . . . . . . . . . . . . . . . . 58

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2.4.3 Integral of bounded functions . . . . . . . . . . . . . . . . . . . . . . . . . 692.4.4 Integral of generally unbounded functions . . . . . . . . . . . . . . . . . . 73

2.5 Spectral Decomposition of Selfadjoint Operators . . . . . . . . . . . . . . . . . . 752.5.1 The spectral theorem for selfadjoint generally unbounded operators . . . 762.5.2 Some technically relevant consequences of the spectral theorem . . . . . . 842.5.3 Joint spectral measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.5.4 Measurable functional calculus . . . . . . . . . . . . . . . . . . . . . . . . 89

2.6 Elementary Quantum Formalism: A rigorous approach . . . . . . . . . . . . . . . 932.6.1 Elementary formalism for the infinite dimensional case . . . . . . . . . . . 932.6.2 Commuting spectral measures . . . . . . . . . . . . . . . . . . . . . . . . . 962.6.3 A first look to time evolution of quantum states . . . . . . . . . . . . . . 992.6.4 A first look to (continuous) symmetries and conserved quantities . . . . . 102

2.7 Some Basic Operator Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042.8 Appendix. On the spectral theorem and joint spectral measures again . . . . . . 106

2.8.1 Continuous functional calculus for bounded selfadjoint operators and C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

2.8.2 Existence of the spectral measure for bounded selfadjoint operators . . . . 1102.8.3 Spectral theorem for normal operators in B(H) . . . . . . . . . . . . . . . 1132.8.4 Existence of the spectral measure for unbounded selfadjoint operators . . 1162.8.5 Existence of joint spectral measures . . . . . . . . . . . . . . . . . . . . . 117

3 Fundamental Quantum Structures in Hilbert Space 1223.1 Lattices in CM and QM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.1.1 A different viewpoint on classical mechanics . . . . . . . . . . . . . . . . . 1223.1.2 The notion of lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.2 The non-Boolean Logic of QM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1273.2.1 The lattice of quantum elementary observables . . . . . . . . . . . . . . . 1273.2.2 Part of CM is hidden in QM . . . . . . . . . . . . . . . . . . . . . . . . . 1293.2.3 A reason why observables are selfadjoint operators . . . . . . . . . . . . . 131

3.3 Recovering the Hilbert space structure: The “coordinatization” problem . . . . . 1323.4 Quantum states as probability measures and Gleason’s Theorem . . . . . . . . . 135

3.4.1 Probability measures over L (H) . . . . . . . . . . . . . . . . . . . . . . . 1353.4.2 Polar decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.4.3 The two-sided ∗-ideal of compact operators . . . . . . . . . . . . . . . . . 1403.4.4 Trace-class operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453.4.5 The mathematical notion of quantum state and Gleason’s theorem . . . . 1513.4.6 Physical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1573.4.7 Post measurement states: Meaning of Luders-von Neumann’s postulate . 1593.4.8 General interplay of quantum observables and quantum states . . . . . . 1613.4.9 Non existence of dispersion-free quantum probability measures . . . . . . 166

3.5 Appendix. On L (H) and trace-class operators again. . . . . . . . . . . . . . . . . 1663.5.1 Proof of some technical results about L (H). . . . . . . . . . . . . . . . . 166

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3.5.2 Proof of some technical results about trace-class operators. . . . . . . . . 169

4 Von Neumann algebras of Observables and Superselection Rules 1724.1 Introduction to von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . 172

4.1.1 The mathematical notion of von Neumann algebra . . . . . . . . . . . . . 1734.1.2 Unbounded selfadjoint operators affiliated to a von Neumann algebra . . 1764.1.3 Lattices of orthogonal projectors of von Neumann algebras and factors . . 1794.1.4 Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1824.1.5 Von Neumann algebra associated to a PVM . . . . . . . . . . . . . . . . . 183

4.2 Von Neumann algebras of observables . . . . . . . . . . . . . . . . . . . . . . . . 1834.2.1 The von Neumann algebra of a quantum system . . . . . . . . . . . . . . 1844.2.2 Maximal set of compatible observables and preparation of vector states . 185

4.3 Superselection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894.3.1 Abelian case: superselection charges and coherent sectors . . . . . . . . . 1894.3.2 Global gauge group formulation and non-Abelian case . . . . . . . . . . . 1934.3.3 Quantum states in the presence of Abelian superselection rules . . . . . . 1954.3.4 Quantum states in the general case of R ( B(H). . . . . . . . . . . . . . . 199

4.4 Appendix. On the von Neumann algebra of a PVM again . . . . . . . . . . . . . 200

5 Quantum Symmetries 2035.1 Quantum Symmetries according to Kadison and Wigner . . . . . . . . . . . . . . 203

5.1.1 Wigner, Kadison symmetries and ortho-automorphism symmetries . . . . 2045.1.2 Theorems by Wigner, Kadison and Dye . . . . . . . . . . . . . . . . . . . 2075.1.3 Action of symmetries on observables and their physical interpretation . . 209

5.2 Groups of quantum symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2115.2.1 Unitary(-projective) representations of groups of quantum symmetries . . 2115.2.2 Representations including anti unitary operators . . . . . . . . . . . . . . 2135.2.3 Unitary-projective representations of Lie groups and Bargmann’s theorem 2145.2.4 Inequivalent unitary-projective representations and superselection rules . 2175.2.5 Continuous unitary-projective and unitary representations of R . . . . . . 2205.2.6 Stone’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2225.2.7 Time evolution, Heisenberg picture and quantum Noether theorem . . . . 226

5.3 More on strongly-continuous unitary representations of Lie groups . . . . . . . . 2315.3.1 Strongly continuous unitary representations . . . . . . . . . . . . . . . . . 2325.3.2 From Garding’s space to Nelson’s theorem . . . . . . . . . . . . . . . . . . 2335.3.3 Selfadjoint version of Stone - von Neumann - Mackey Theorem . . . . . . 2405.3.4 Pauli’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

6 A very quick Introduction to the Algebraic Formulation 2436.1 Algebraic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

6.1.1 The GNS reconstruction theorem . . . . . . . . . . . . . . . . . . . . . . . 2476.1.2 Pure states and irreducible representations . . . . . . . . . . . . . . . . . 248

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6.2 Symmetries and Algebraic Formulation . . . . . . . . . . . . . . . . . . . . . . . . 2506.2.1 Symmetries and symmetry spontaneous breakdown . . . . . . . . . . . . . 2506.2.2 Groups of symmetries in algebraic approach . . . . . . . . . . . . . . . . . 252

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Aim and structure of this work

This work faithfully reflects the lectures originally prepared for the MSc six-month course Math-ematical Physics: Quantum Relativistic Theories that I have held at the University of Trento inthe academic year 2017-2018. Overall author’s intention is to present in the first five chaptersthe rigorous advanced technology to formalize and develop important general physical ideas ofquantum theories formulated in Hilbert spaces into a relatively concise and self-contained way.The last chapter offers a quick view on the C∗-algebra formulation stating relevant propositionsAs a matter of fact the reader is introduced to the beautiful interconnection between logic,lattice theory, general probability theory, and general spectral theory including the basic the-ory of von Neumann algebras, naturally arising in the study of the mathematical machineryof quantum theories. This book should be appreciated by mathematicians who whish to learnthe fundamental advanced mathematical technology at the base of quantum theories and byphysicists who want to elevate their standpoint on the general mathematical structure of quan-tum theories especially in Hilbert space. Many examples and solved exercises accompany themathematical statements – almost all carefully demonstrated in the first five chapters – andphysical motivations for every mathematical notion are presented and discussed.

Some ideological overlap exists with the content of [Mor18] which is however a much moremathematically complete book of more than 950 pages and not suitable for a Master course.The proofs appearing in these lectures are therefore almost always original and different fromthose of [Mor18], because autonomously developed to reflect the relative conciseness of this work.

The book is organized as follows.In Chapter 1, we quickly review some elementary facts and properties, either of physical or

mathematical nature, proper of Quantum Systems, without fully entering into the mathematicaldetails, but pointing out some technical issues the naive approach shows up.

Chapter 2 establishes the fundamental mathematical corpus of these lectures. This part isdevoted to present technical definitions and results of spectral analysis in complex Hilbert spaces,including the classic theorems about spectral decomposition of (generally unbounded) selfadjointoperators and bounded normal operators and the so-called measurable functional calculus. Theproofs of these theorems are complete and self-contained (part of them is however developed ina technical detour placed inside of an appendix to chapter 2). The introduced mathematicaltechnology is finally exploited to restate into a rigorous way the elementary formulation of QMin infinite-dimensional Hilbert spaces

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Within Chapter 3, we pass to analyse the mathematical structure of QM from a finer andadvanced viewpoint, adopting the framework based on orthomodular lattices’ theory. Thisapproach permits one to justify some basic assumptions of QM, like the mathematical natureof the observables represented in terms of selfadjoint operators and the quantum states viewedas trace class operators. Quantum Theory form this perspective turns out to the probabilitymeasure theory on the non-Boolean lattice L (H) of elementary observables. A key tool of thatanalysis is the theorem by Gleason characterising the notion of probability measure on L (H) interms of certain trace-class operators whose basic theory is introduced as part of the theory ofcompact operators. Some of the proofs are confined in a final technical appendix.

In Chapter 4, we focus on the algebra of observables in the presence of superselection rules,after having introduced the mathematical notion of von Neumann algebra discussing some of thebasic mathematical properties and their physical significance. Several physical technical resultsare stated and proved with the help of von Neumann algebra technology. The notion of factorand, e.g., maximal set of compatible observables and its use in the preparation of quantumstates, the idea of superselection rules and the notion of Gauge group are in particular analysed.

Chapter 5 is devoted to offer the idea of quantum symmetry, illustrated in terms of Wignerand Kadison theorems. Some basic mathematical facts about groups of quantum symmetries arediscussed, especially in relation with the problem of their unitarisation. Bargmann’s conditionis stated. The particular case of a strongly continuous one-parameter unitary group is focusedin some more detail, mentioning von Neumann’s theorem and the celebrated Stone theorem,remarking its use to describe the time evolution of quantum systems. A quantum formulation ofNoether theorem ends this part. The last part of this chapter aims to introduce some elementaryresults about continuous unitary representations of Lie groups, discussing in particular a theoremby Nelson which proposes sufficient conditions for lifting a (anti)selfadjoint representation of aLie algebra to a unitary representation of the unique simply connected Lie group associated tothat Lie algebra.

Chapter 6 closes the work focussing on elementary ideas and results of the so called (C∗-)algebraic formulation of quantum theories discussing also some important notions like sponta-neous breakdown of symmetry.

General mathematical prerequisites to fully understands the proofs of these lectures areabstract measure theory [Coh80, Rud86] and elementary notions of complex Hilbert space theory[Rud86, Mor18] and Fourier-Plancherel transform. Some already acquired elementary notionsabout Quantum Mechanics at undergraduate level would be preferable, but not strictly necessary.To properly understand Chapter 5, the reader should posess a basic knowledge of the theory ofLie groups and their representations (see [Mor18] for a summary of notions relevant in physics,[NaSt82] and [Var84] for classical treatises stressing on the analytic structire of Lie groups,[HiNe13] for a modern complete treatise on the subject).As modern general references on mathematical foundations of quantum theories we suggest[Tes14, Lan17, Mor18].

I am very grateful to Antonio Lorenzin who helped me correct many mistakes of variousnature affecting this work and to Alex Strohmaier for useful technical discussions.

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Chapter 1

General Phenomenology of theQuantum World and ElementaryFormalism

1.1 Physics of Quantum Systems

We quickly review in this section the most relevant common features of quantum systems.For a concise and smart physical introduction to Quantum Mechanics the interested reader canprofitably consult [SaTu94], where Dirac’s formulation of QM is discussed within a modern smartview without paying attention to mathematical rigour. Our intention is to initially present anelementary mathematical formulation of the ideas which will be improved in the next chapters,after having introduced the appropriate mathematical technology.

1.1.1 When is a physical system quantum?

Quantum Mechanics can be roughly defined as the physics of microscopic world (elementaryparticles, atoms, molecules). That realm is characterized by a universal constant indicated byh and known as Planck constant. An associated constant – nowadays of more frequent use –is called the reduced Planck constant,

~ :=h

2π= 1.054571726× 10−34J · s .

The physical dimensions of h (or ~) are those of an action, i.e. energy × time. An elementarybut effective check on the appropriateness of a quantum physical description for a physicalsystem under consideration consists of comparing the value of some characteristic action of thesystem with ~. As an example, focus on a macroscopic pendulum (say, length ∼ 1m, mass ∼ 1kgmaximal speed ∼ 1ms−1), multiplying the maximal kinetic energy and the period of oscillations,we find a typical action of ∼ 2Js >> h. In this situation, quantum physics is expected to be

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definitiely inappropriate, exactly as we actually experience every days. Instead referring to thehydrogen electron orbiting around its proton, the first ionization energy multiplied with theorbital temporal period of rotation (computed using the classical formula with a value of theradius of the order of 1 A) gives a typical action of the order of h. Here quantum mechanics isnecessary.

1.1.2 Basic properties of quantum systems

A triple of features proper of Quantum Mechanics (QM) which seem to be very different fromproperties of Classical Mechanics (CM) is listed below. These remarkable general propertiesconcern the physical quantities of physical systems. In QM physical quantities are called ob-servables.

(1) Randomness. When we perform a measurement of an observable of a quantum system,the outcomes appear to be stochastic: Performing measurements of the same observableA on completely identical systems prepared in the same physical state, one generally findsdifferent outcomes a, a′, a′′ . . ..Referring to the standard interpretation of the formalism of QM (see [SEP] for a niceup-to-date account on the various interpretations), randomness of measurement outcomesshould not be considered as due to an incomplete knowledge of the state of the system asit happens, for instance, in Classical Statistical Mechanics. Randomness is not epistemic,but it is ontological. It is a fundamental property of quantum systems.On the other hand, QM permits one to compute the probability distribution of all theoutcomes of a given observable, once the state of the system is known.Moreover, it is always possible to prepare a state ψa where a certain observable A is definedand takes its value a. That is, repeated measurements of A give rise to the same value awith probability 1. (Notice that we can perform simultaneous measurements on identicalsystems all prepared in the state ψa, or we can perform different subsequent measurementson the same system in the state ψa. In the second case, these measurements have to beperformed very close to each other in time to prevent the state of the system from evolvingin view of Schrodinger evolution as said after (3) below.) Such states, where observabletake definite values, cannot be prepared for all observables simultaneously as discussed in(2) below.

(2) Compatible and Incompatible Observables. The second noticeable feature of QM isthe existence of incompatible observables. Differently from CM, there are physical quanti-ties which cannot be measured simultaneously since there is no physical instrument capableto do it. If an observable A is defined in a given state ψ – i.e. it attains a precise value awith probability 1 in case of a measurement – an observable B incompatible with A turnsout to be not defined in the state ψ – i.e. it may attain several different values b, b′, b′′ . . .,none with probability 1, in case of measurement. So, if performing a measurement of B,we generally obtain a spectrum of values described by a distribution of frequencies aspre-announced in (1) above if identifying these frequencies with corresponding a priori

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probabilities.Incompatibility is symmetric – A is incompatible with B if and only if B is incompatiblewith A – though it is not transitive.Compatible observables also exist and, by definition, they can be measured simultaneously.The component x of the position of a particle and the component y of the momentum ofthat particle are an example, referring to the rest space of a given inertial reference frame.A popular case of incompatible observables is a pair of canonically conjugated observableslike the position X and the momentum P of a particle both along the same fixed axisof a reference frame. Here a lower bound for the product of the standard deviations –resp. ∆Xψ, ∆Pψ – exists for the outcomes of the measurements of these observables in agiven state ψ. These measurements have to be performed on different identical systemsall prepared in the same state ψ. The lower bound is independent from the state and isencoded in the popular mathematical formula of the Heisenberg principle (a theorem inthe modern formulations):

∆Xψ∆Pψ ≥ ~/2 , (1.1)

where Planck constant shows up.

(3) Collapse of the State. Measurements of QM generally change the state of the system giv-ing rise to a post-measurement state from the state on which the measurement is performed.(We are here considering quite idealized measurement procedures, because measurementprocedures are very often destructive.) Assuming ψ is the initial state, immediately afterthe measurement of an observable A obtaining the value a among a plethora of possiblevalues a, a′, a′′, . . ., the state settles in ψ′ generally different form ψ. Referring to ψ′, theprobabilities of the outcomes of A change to 1 for the outcome a and 0 for all other possibleoutcomes. In this sense A becomes defined in ψ′.If performing repeated and alternated measurements of a pair of incompatible observables,A, B, the outcomes disturb each other: If the first outcome of A is a, after a measurementof B, a subsequent measurement of A gives a′ 6= a in general. Instead, if A and B arecompatible, the outcomes of subsequent measurements do not disturb each other.Also in CM there are measurements that, in practice, mutually disturb and modify thestate of the system. It is however ideally possible to decrease the disturbance arbitrarily,making it negligible. In QM it is not always possible as for instance witnessed by (1.1).

Two types of time evolution of the state of a system exist in QM. One is that due to the dynamicsand is encoded in the famous Schrodinger equation we shall encounter shortly. It is nothing buta quantum version of classical Hamiltonian evolution [Erc15]. The other is the sudden changeof the state due to the measurement procedure of an observable, outlined in (3). The collapseof the state (or wavefunction) of the system.The physical nature of the latter type of evolution still nowadays remains the source of ananimated debate in the scientific community of physicists and philosophers of Science. Severalattempts exist to reduce the collapse of the state to a dynamical evolution of the whole physicalsystem including also the measurement instruments and the environment by means of a de-

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coherence processes [SEP, BGJ00]. None of these approaches seem however to be completelysatisfactory up to now [Lan17].

1.2 Elementary Quantum Formalism: The finite dimensionalcase

Remark 1.1. Unless explicitly differently stated, we adopt a physical unit system such that~ = 1 throughout the work.

We add here some further technical details to the presented picture. We intend to show howpractically (1)-(3) should be mathematically interpreted (swapping (2) and (3) for our conve-nience). A large fraction of the rest of the paper aims to make technically precise, justify andwidely develop these ideas from a mathematically more advanced viewpoint.To mathematically simplify this introductory discussion, throughout this chapter, excluding Sec-tion 1.3, we suppose that H always indicates a finite dimensional complex vector space endowedwith a Hermitian scalar product, denoted by 〈·|·〉. The linear entry is the second one. Withthe said H, L(H) is the complex algebra of operators A : H → H. We remind the reader that,if A ∈ L(H) with H finite dimensional, the adjoint operator, A∗ ∈ L(H), is the unique linearoperator satisfying

〈A∗x|y〉 = 〈x|Ay〉 for all x, y ∈ H. (1.2)

A is selfadjoint when A = A∗. As a consequence,

〈Ax|y〉 = 〈x|Ay〉 for all x, y ∈ H. (1.3)

As 〈·|·〉 is linear in the second argument and antilinear in the first argument, we evidently havethat all eigenvalues of a selfadjoint operator A must be real.

Our axioms on the mathematical description of quantum systems are listed below.

1. A quantum mechanical system S is associated to a complex vector space H (finite dimen-sional for now) endowed with a Hermitian scalar product 〈·|·〉;

2. observables are described by selfadjoint operators A on H;

3. states are equivalence classes of unit vectors ψ ∈ H, with ψ ∼ ψ′ iff ψ = eiaψ′ for somea ∈ R.

Remark 1.2.(a) States are therefore one-to-one with the elements of the complex projective space PH.

The states we consider within this introduction are actually called pure states. A more generalnotion will be introduced later.

(b) H is an elementary version of a complex Hilbert space: it is automatically complete inview of its finite dimensionality.

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(c) Since dim(H) < +∞, every selfadjoint operator A ∈ L(H) admits a spectral decomposi-tion

A =∑

a∈σ(A)

aP (A)a , (1.4)

where σ(A) is the finite set of eigenvalues – which must be real as A is selfadjoint – and P(A)a is

the orthogonal projector onto the eigenspace associated to a. Notice that PaPa′ = 0 if a 6= a′ aseigenvectors with different eigenvalue are orthogonal.

Let us see how the mathematical assumptions 1-3 allows us to set the physical properties ofquantum systems (1)-(3) into a mathematically defined form.

(1) Randomness: The eigenvalues of an observable A are physically interpreted as thepossible values of the outcomes of a measurement of A.Given a state, represented by the unit vector ψ ∈ H, the probability to obtain a ∈ σ(A) for A is

µ(A)ψ (a) := ||P (A)

a ψ||2 .

Going along with this interpretation, the expectation value of A in the state ψ, results to be

〈A〉ψ :=∑

a∈σ(A)

aµ(A)ψ (a) = 〈ψ|Aψ〉 .

Hence the identity holds〈A〉ψ = 〈ψ|Aψ〉 . (1.5)

Similarly, the standard deviation ∆Aψ results to be

∆A2ψ :=

∑a∈σ(A)

(a− 〈A〉ψ)2µ(A)ψ (a) = 〈ψ|A2ψ〉 − 〈ψ|Aψ〉2 . (1.6)

Remark 1.3.(a) We stress that the arbitrary phase affecting the unit vector ψ ∈ H (eiaψ and ψ represent

the same quantum state for every a ∈ R) is actually armless,(b) If A is an observable and f : R→ R is given, f(A) is interpreted as an observable whose

values are f(a) if a ∈ σ(a): Taking (1.4) into account,

f(A) :=∑

a∈σ(A)

f(a)P (A)a . (1.7)

For polynomials f(x) =∑nk=0 akx

k, it results f(A) =∑nk=0 akA

k as expected. The selfadjointoperator A2 can naturally be interpreted this way as the natural observable whose values area2 when a ∈ σ(A). With this interpretation, the last term in (1.6) reads if taking (1.5) intoaccount,

∆A2ψ = 〈A2〉ψ − 〈A〉2ψ = 〈(A− 〈A〉ψI)2〉ψ = 〈ψ|(A− 〈A〉ψI)2ψ〉 . (1.8)

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(3) Collapse of the state: Let a be the outcome of the (idealized) measurement of A when

the state is represented by ψ. The post-measurement state is given by the unit vector

ψ′ :=P

(A)a ψ

||P (A)a ψ||

. (1.9)

Remark 1.4. This formula is meaningless if µ(A)ψ (a) = 0 as expected. Yet, the arbitrary

phase affecting ψ does not lead to troubles, due to the linearity of P(A)a .

(2) Compatible and Incompatible Observables: Observables A and B are compatible– i.e. they can be simultaneously measured – if and only if the associated operators commute:

AB −BA = 0 .

Since H is finite-dimension, the observables A and B are compatible if and only if the associatedspectral projectors commute as well (the proof is elementary):

P (A)a P

(B)b = P

(B)b P (A)

a a ∈ σ(A) , b ∈ σ(B) .

In particular,

||P (A)a P

(B)b ψ||2 = ||P (B)

b P (A)a ψ||2

has the natural interpretation of the probability to obtain the outcomes a and b for a simul-taneous measurement of A and B. If conversely A and B are incompatible, it may happenthat

||P (A)a P

(B)b ψ||2 6= ||P (B)

b P (A)a ψ||2 .

Sticking to the case of A and B incompatible, exploiting (1.9),

||P (A)a P

(B)b ψ||2 =

∣∣∣∣∣∣∣∣∣∣∣∣P (A)a

P(B)b ψ

||P (B)b ψ||

∣∣∣∣∣∣∣∣∣∣∣∣2

||P (B)b ψ||2 (1.10)

has the natural meaning of the probability of obtaining first b and next a in a subsequent mea-surement of B and A.

Remark 1.5.(a) In general, the role of A and B in (1.10) cannot be swapped because generally P

(A)a P

(B)b 6=

P(B)b P

(A)a when A and B are incompatible. The measurement procedures “disturb each other”

as already discussed.(b) The interpretation of (1.10) as probability of subsequent measurements is consistent also

if A and B are compatible. Here, the probability of obtaining first b and next a in a subsequentmeasurement of B and A is identical to the probability of measuring a and b simultaneously. In

13

turn, it coincides with the probability of obtaining first a and next b in subsequent measurementsof A and B

(c) A is always compatible with itself. Moreover P(A)a P

(A)a = P

(A)a just due to the defini-

tion of projector. This fact has the immediate consequence that if we obtain a measuring A so

that the state immediately after the measurement is represented by ψa = ||P (A)a ψ||−1ψ, it will

remain ψa even after other subsequent measurements of A and the outcome will result to bealways a. Versions of this phenomenon, especially regarding the decay of unstable particles, areexperimentally confirmed and it is called the quantum Zeno effect.

Example 1.1. An electron admits a triple of internal observables, Sx, Sy, Sz, known as thethree components of the spin. Very roughly speaking, we can think of the spin as the angularmomentum of the particle referred to a reference frame always at rest with the centre of theparticle and carrying its axes parallely to the ones of the reference frame of the laboratory, wherethe electron moves. In view of its peculiar properties, the spin cannot actually have a completeclassical corresponding and thus that interpretation is eventually untenable. For instance, onecannot “stop” the spin of a particle or change the constant value of S2 = S2

x + S2y + S2

z : It is agiven property of the particle like the mass. The electron spin is described within an internalHilbert space Hs, which has dimension 2 so that it can be identified with C2. The three spinobservables, up to a factor where we occasionally re-introducing the constant ~, correspond tothe three well known Pauli matrices

Sx =~2σx , Sy =

~2σy , Sz =

~2σz . (1.11)

Above,

σx =

ñ0 11 0

ô, σy =

ñ0 −ii 0

ô, σz =

ñ1 00 −1

ô. (1.12)

Notice that [Sa, Sb] 6= 0 if a 6= b implying that the components of the spin are incompatibleobservables. In fact, one has

[Sx, Sy] = i~Szand this identity holds also cyclically permuting the three indices. These commutation relationsare the same as for the observables Lx,Ly,Lz describing the angular momentum referred to thelaboratory system, which possess classical corresponding (we shall return on these observablesin Example 5.6). So, differently from CM, the observables describing the components of theangular momentum are incompatible and cannot be simultaneously measured. However thefailure of the compatibility is related to the appearance of ~ on the right-hand side of

[Lx, Ly] = i~Lz .

That number is extremely small when compared with macroscopic scales. This is the definitereason why the incompatibility of Lx and Lz is pratically unobservable for macroscopic systems.Direct inspection proves that σ(Sa) = ±~/2. Similarly σ(La) = n~ | n ∈ Z. Therefore,

14

differently from CM, the values of the angular momentum components are discrete in QM.Yet notice that the difference of two closest values is extremely small if compared with typicalvalues of the angular momentum of macroscopic systems. This is the practical reason why thisdiscreteness macroscopically disappears.

1.2.1 Time evolution

Few words about the time evolution is necessary now, a wider discussion will take place later inthis work.Among the class of observables of a quantum system described in a given inertial reference frame,an observable H called the (quantum) Hamiltonian plays a fundamental role. We are assuminghere that the system interacts with a stationary physical environment and we refer everythingto the rest space of an inertial system. The one-parameter group of unitary operators associatedto H (exploiting (1.7) to explain the notation)

Ut := e−itH :=∑

h∈σ(H)

e−ithP(H)h , t ∈ R (1.13)

describes the time evolution of quantum states as follows. Let the state at time t = 0 berepresented by the unit vector ψ ∈ H, at the generic time t the state is represented by

ψt = Utψ .

(The vector ψt has norm 1 as due to describe states, since Ut is norm preserving it being unitary.)Taking (1.13) into account, this identity is equivalent to

idψtdt

= Hψt . (1.14)

Equation (1.14) is nothing but a form of the celebrated Schrodinger equation. If the environmentis not stationary, a more complicated description can be given where H is replaced by a classof Hamiltonian (selfadjoint) operators parametrized in time, H(t), with t ∈ R. This timedependence accounts for the time evolution of the external system interacting with our quantumsystem. In that case, it is simply assumed that the time evolution of states is again provided bythe equation above where H is replaced by H(t):

idψtdt

= H(t)ψt . (1.15)

This equation permits one to define a two-parameter groupoid of unitary operators U(t2, t1),where t2, t1 ∈ R, such that

ψt2 = U(t2, t1)ψt1 , t2, t1 ∈ R .

The groupoid structure arises from the following identities: U(t, t) = I and U(t3, t2)U(t2, t1) =U(t3, t2) and U(t2, t1)−1 = U(t2, t1)∗ = U(t1, t2).

15

In our elementary case where H is finite dimensional, Dyson’s formula holds with the simplehypothesis that the map R 3 t 7→ Ht ∈ L(H) is continuous (adopting any topology compatiblewith the vector space structure of L(H)) [Mor18]

U(t2, t1) =+∞∑n=0

(−i)n

n!

∫ t2

t1

· · ·∫ t2

t1

T [H(τ1) · · ·H(τn)] dτ1 · · · dτn .

Above, we define T [H(τ1) · · ·H(τn)] = H(τπ(1)) · · ·H(τπ(n)), where the bijective function π :1, . . . , n → 1, . . . , n is any permutation with τπ(1) ≥ · · · ≥ τπ(n).

1.2.2 Composite systems

If a quantum system S consists of two parts, S1 and S2, respectively described in the Hilbertspaces H1 and H2, it is assumed that the whole system is described in the space H1⊗H2 equippedwith the unique Hermitian scalar product 〈·|·〉 such that 〈ψ1⊗ψ2|φ1⊗φ2〉 = 〈ψ1|φ1〉1〈ψ2|φ2〉2 (inthe infinite dimensional case H1⊗H2 is the Hilbert completion of the afore-mentioned algebraictensor product).If H1 ⊗ H2 is the space of a composite system S as before and A1 represents an observable forthe part S1, it is naturally identified with the selfadjoint operator A1⊗ I2 defined in H1⊗H2. Asimilar statement holds swapping 1 and 2. Notice that σ(A1⊗ I2) = σ(A1) as one easily proves.(The result survives the extension to the infinite dimensional case.)

Remark 1.6.(a) Composite systems are in particular systems made of many (either identical or not)

particles. If we have a pair of particles respectively described in the Hilbert space H1 and H2,the full system is described in H1⊗H2. Notice that the dimension of the final space is the productof the dimension of the component spaces. In CM the system would instead be described ina space of phases which is the Cartesian product of the two spaces of phases. In that casethe dimension would be the sum, rather than the product, of the dimensions of the componentspaces.

(b) H1 ⊗ H2 contains the so-called entangled states. They are states represented by vectorsnot factorized as ψ1 ⊗ ψ2, but they are linear combinations of such vectors. Suppose the wholestate is represented by the entangled state

Ψ =1√2

(ψa ⊗ φ+ ψa′ ⊗ φ′

),

where A1ψa = aψa and A1ψa′ = a′ψa′ for a certain observable A1 of the part S1 of the totalsystem. Performing a measurement of A1 on S1, due to the collapse of state phenomenon, weautomatically act on the whole state and on the part describing S2. As a matter of fact, up tonormalization, the state of the full system after the measurement of A1 will be ψa ⊗ φ if theoutcome of A1 is a, or it will be ψa′ ⊗ φ′ if the outcome of A1 is a′. It could happen that thetwo measurement apparatuses, respectively measuring S1 and S2, are localized very far in the

16

physical space. Therefore acting on S1 by measuring A1, we “instantaneously” produce a changeof S2 which can be seen performing measurements on it, even if the measurement apparatus ofS2 is very far from the one of S1. This seems to be in explicit contradiction with the localitypostulate of Relativity that a maximal speed exists, the one of light, for propagating physicalinformation. After the popular analysis due to Bell, improving the original one by Einstein,Podolsky and Rosen, the phenomenon has been experimentally observed. Locality is truly vio-lated, but in a such subtle way connected to the stochastic nature of outcomes of measurementswhich does not permit superluminal propagation of physical information. Non-locality of QMis nowadays widely accepted as a real and fundamental feature of Nature [Ghi07, SEP, Lan17].

Example 1.2. An electron possesses an electric charge in addition to the spin. That isanother internal quantum observable, Q, with two values ±e, where e = 1.602176565× 10−19Cis the value elementary electrical charge. So there are two types of electrons. Proper electrons,whose internal state of charge is an eigenvector of Q with eigenvalue −e and positrons, whoseinternal state of charge is a eigenvector of Q with eigenvalue e. The simplest version of theinternal Hilbert space of the electrical charge is therefore Hc which1, again, is isomorphic to C2.With this representation Q = eσ3. The full Hilbert space of an electron must therefore includea factor Hs⊗Hc. Obviously this is by no means sufficient to describe an electron, since we mustinclude the observables describing at least the position of the electron.

1.3 A first look to the infinite dimensional case, CCRs and quan-tization procedures

All the introduced formalism, excluding technicalities we shall examine in the rest of the work,holds also for quantum systems whose complex vector space of the states is infinite dimensional.To extend the ideas treated in Section 1.2 to a more general setup, relaxing finite-dimensionalityof H, it seems natural to assume that H is complete with respect to the 〈·|·〉 norm. H istherefore a complex Hilbert space. In particular, completeness assures the existence of spectraldecompositions, generalizing (1.4) when referring to compact selfadjoint operators (e.g., see[Mor18]).

1.3.1 L2(R, dx) model

The most elementary example of a quantum system described in an infinite dimensional Hilbertspace is a quantum particle confined to stay along the real line R. In this case, the Hilbert spaceis H := L2(R, dx), dx denoting the standard Lebesgue measure on R. States are still representedby elements of PH, namely equivalence classes [ψ] of measurable functions ψ : R→ C with unitnorm, ||[ψ]|| =

∫R |ψ(x)|2dx = 1.

1As we shall say later, in view of a superselection rule not all normalized vectors of Hc represent (pure) states.

17

Remark 1.7. We have here a pair of different quotient procedures. ψ and ψ′ define thesame element [ψ] of L2(R, dx) iff ψ(x) − ψ′(x) 6= 0 on a zero Lebesgue measure set. Two unitvectors [ψ] and [φ] define the same state if [ψ] = eia[φ] for some a ∈ R.

Notation 1.1. In the rest of the paper we adopt the standard convention of many textbookson functional analysis denoting by ψ, instead of [ψ], the elements of spaces L2 and tacitly iden-tifying pair of functions which are different on a zero measure set.

The functions ψ defining (up to zero-measure set and phases) states, are called wavefunctions.There is a pair of fundamental observables describing our quantum particle moving in R. Oneis the position observable. The corresponding selfadjoint operator, denoted by X, is defined asfollows

(Xψ)(x) := xψ(x) , x ∈ R , ψ ∈ L2(R, dx) .

The other observable is the momentum which is indicated by P . Restoring ~ for the occasion,the momentum operator is

(Pψ)(x) := −i~dψ(x)

dx, x ∈ R , ψ ∈ L2(R, dx) .

We immediately face several mathematical issues with these, actually quite naive, definitions.Let us first focus on X. First of all, generally Xψ 6∈ L2(R, dx) even if ψ ∈ L2(R, dx). To fix theproblem, one could simply restrict the domain of X to the linear subspace of L2(R, dx)

D(X) :=

ßψ ∈ L2(R, dx)

∣∣∣∣ ∫R|xψ(x)|2dx < +∞

™. (1.16)

Even if〈Xψ|φ〉 = 〈ψ|Xφ〉 for all ψ, φ ∈ D(X), (1.17)

is valid, we cannot argue that X is properly selfadjoint because we have not yet given thedefinition of adjoint operator of an operator defined in a non-maximal domain in an infinite-dimensional Hilbert space. Identity (1.2) in an infinite-dimensiona Hilbert space does not definea (unique) operator X∗ without further technical requirements. (To comfort the reader, wepre-announce that X is truly selfadjoint with respect to a general definition we shall give inthe next chapter, when its domain is (1.16).) From a very practical viewpoint however, theidentity (1.17) implies that all eigenvalues of X must be real if any and this seems sufficient toadopt the standard interpretation of eigenvalues as outcomes of measurements of the observableX. Unfortunately life is not so easy: for every fixed x0 ∈ R there is no ψ ∈ L2(R, dx) withXψ = x0ψ and ψ 6= 0. (A function ψ satisfying Xψ = x0ψ must also satisfy ψ(x) = 0 if x 6= x0,due to the definition of X. Hence ψ = 0, as an element of L2(R, dx) just because x0 has zeroLebesgue measure!)All that seems to prevent the existence of a spectral decomposition of X like the one in (1.4),since X does not admit eigenvectors in L2(R, dx) (and a fortiori in D(X)).

18

The definition of P appears to suffer from even worse troubles. Its domain cannot be thewhole L2(R, dx) but should be restricted to a subset of differentiable functions with derivativein L2(R, dx). The weakest notion of differentiability we can assume is just weak differentiabilityleading to this domain candidate,

D(P ) :=

ψ ∈ L2(R, dx)

∣∣∣∣∣∣ ∃ w-dψ(x)

dx,

∫R

∣∣∣∣∣w-dψ(x)

dx

∣∣∣∣∣2

dx < +∞

. (1.18)

Above w-dψ(x)dx denotes the weak derivative of ψ2. As a matter of fact D(P ) coincides with the

Sobolev space H1(R).Again, without a precise definition of adjoint operator in an infinite dimensional Hilbert space(with non-maximal domain) we cannot say anything more precise about selfadjointness of Pwith that domain. (As before P turns out to be selfadjoint with respect to the general definitionwe shall give in the next chapter.)Passing to the Fourier-Plancherel transform, one finds again (it is not so easy to see it)

〈Pψ|φ〉 = 〈ψ|Pφ〉 for all ψ, φ ∈ D(P ), (1.19)

so that, eigenvalues are real if exist. However, exactly as X also P does not admit eigen-vectors. The naive eigenvectors with eigenvalue p ∈ R are functions proportional to the mapR 3 x 7→ eipx/~, which does not belong to L2(R, dx) nor D(P ). We will tackle all these issues inthe next chapter in a very general fashion.

Remark 1.8.(a) We observe that the space of Schwartz functions, S (R) 3 satisfies

S (R) ⊂ D(X) ∩D(P )

and furthermore S (R) is dense in L2(R, dx) and invariant under X and P : X(S (R)) ⊂ S (R)and P (S (R)) ⊂ S (R). This observation has many technical consequences will be discussedelsewhere in this work.

(b) Though we shall not pursue this approach within this work, we stress that X admits aset of eigenvectors if we extend the domain of X to the space S ′(R) of Schwartz distributionsin a standard way, taking (a) into account. If T ∈ S ′(R),

〈X(T ), f〉 := 〈T,X(f)〉 for every f ∈ S (R).

With this extension, the eigenvectors in S ′(R) of X with eigenvalues x0 ∈ R are the distribu-tions cδ(x − x0). This class of eigenvectors can be exploited to build a spectral decomposition

2f : R → C, defined up to zero-measure set, is the weak derivative of g ∈ L2(R, dx) if it holds∫R g

dhdxdx =

−∫R fhdx for every h ∈ C∞0 (R). If g is differentiable, its standard derivative coincide with the weak one.3S (Rn) is the vector space of the C∞ complex valued functions on Rn which, together with their derivatives

of all orders in every set of coordinate, decay faster than every negative integer power of |x| for |x| → +∞.

19

of X similar to that in (1.4).P analogously admits eigenvectors in S ′(R) with the same procedure. They are just the aboveexponential functions. Again, this class of eigenvectors can be used to construct a spectral de-composition of P like that in (1.4). The idea of this procedure can be traced back to Dirac [Dir30]and, in fact, something like ten years later it gave rise to the rigorous theory of distributions byL. Schwartz. The modern formulation of this approach to construct spectral decompositions ofselfadjoint operators was developed by Gelfand in terms of the so called rigged Hilbert spaces[GeVi64].

1.3.2 L2(Rn, dnx) model and Heisenberg inequalities

Referring to a quantum particle moving in Rn, whose Hilbert space is L2(Rn, dnx), one canintroduce observables Xk and Pk representing position and momentum with respect to the k-thaxis, k = 1, 2, . . . , n. These operators, which are defined analogously to the case n = 1, havedomains smaller than the full Hilbert space. We do not write the form of these domain (wherethe operators turn out to be properly selfadjoint referring to the general definition we shall statein the next chapter). We just mention the fact that all these operators admit S (Rn) as commoninvariant subspace included in their domains. Thereon

(Xkψ)(x) = xkψ(x) , (Pkψ)(x) = −i~∂ψ(x)

∂xk, ψ ∈ S (Rn) (1.20)

and so〈Xkψ|φ〉 = 〈ψ|Xkφ〉 , 〈Pkψ|φ〉 = 〈ψ|Pkφ〉 for all ψ, φ ∈ S (Rn), (1.21)

By direct inspection one easily proves that the canonical commutation relations (CCRs) holdwhen all the operators in the subsequent formulas are supposed to be restricted to S (Rn)

[Xh, Pk] = i~δhkI , [Xh, Xk] = 0 , [Ph, Pk] = 0 . (1.22)

We have introduced the commutator [A,B] := AB − BA of the operators A and B generallywith different domains, defined on a subspace where both compositions AB and BA makessense, S (Rn) in the considered case. Assuming that (1.5) and (1.8) are still valid for Xk andPk referring to ψ ∈ S (Rn), (1.22) easily leads to the Heisenberg uncertainty relations,

∆Xkψ∆Pkψ ≥~2, for ψ ∈ S (Rn) , ||ψ|| = 1 , k = 1, 2, . . . , n . (1.23)

Exercise 1.1.(1) Prove inequality (1.23) out of (1.22), referring to (1.5) and (1.8).

Solution. Using (1.5), (1.8) and the Cauchy-Schwarz inequality, one easily finds (we omitthe index k for simplicity),

∆Xψ∆Pψ = ||X ′ψ||||P ′ψ|| ≥ |〈X ′ψ|P ′ψ〉| .

20

where X ′ := X − 〈X〉ψI and P ′ := X − 〈X〉ψI. Next notice that

|〈X ′ψ|P ′ψ〉| ≥ |Im〈X ′ψ|P ′ψ〉| = 1

2|〈X ′ψ|P ′ψ〉 − 〈P ′ψ|X ′ψ〉|

Taking advantage from (1.21) and the definitions of X ′ and P ′ and exploiting (1.22),

|〈X ′ψ|P ′ψ〉 − 〈P ′ψ|X ′ψ〉| = |〈ψ|(X ′P ′ − P ′X ′)ψ〉| = |〈ψ|(XP − PX)ψ〉| = ~|〈ψ|ψ〉|

Since 〈ψ|ψ〉 = ||ψ||2 = 1 by hypotheses, (1.23) is proved. Obviously the open problem is tojustify the validity of (1.5) and (1.8) also in the infinite dimensional case. 2

(2) Prove that there are no operators Xk, Pk, for h, k = 1, 2, . . . , n, over a finite-dimensionalHilbert space H 6= 0 satisfying (1.22).

Solution. If operators and Hilbert space as declared existed, we would have

iδhk dim(H) = tr([Xh, Pk]) = tr(XhPk)− tr(PkXh) = tr(PkXh)− tr(PkXh) = 0 ,

and this is not possible for h = k since dim(H) > 0. 2

A philosophically remarkable consequence of the CCRs (1.22) is that they resemble the classicalcanonical commutation relations of the Hamiltonian variables qh and pk of the standard Poissonbrackets ·, ·P ,

qh, pkP = δhk , qh, qkP = 0 , ph, pkP = 0 . (1.24)

as soon as one identifies (i~)−1[·, ·] with ·, ·P . This fact, initially noticed by Dirac [Dir30],leads to the idea of “quantization” of a classical Hamiltonian theory [Erc15, Lan17].One starts from a classical system described on a symplectic manifold (Γ, ω), for instanceΓ = R2n equipped with the standard symplectic form as ω and considers the (real) Lie al-gebra (C∞(Γ,R), ·, ·P ). To “quantize” the system one looks for a map associating classicalobservables f ∈ C∞(Γ,R) to quantum observables Of , i.e. selfadjoint operators restricted4 to acommon invariant domain S of a certain Hilbert space H. (In case Γ = T ∗Q, H can be chosenas L2(Q, dµ) where µ is some natural measure.) The map f 7→ Of is expected to satisfy a setof conditions. The most important are written below

1. R-linearity;

2. Oı = I|S ;

3. Of,gP = −i~[Of , Og]

4. If (Γ, ω) is R2n equipped with the standard symplectic form, it must hold that Oxk = Xk|Sand Opk = Pk|S , k = 1, 2, . . . , n.

4The restriction should be such that it admits a unique selfadjoint extension. A sufficient requirement on Sis that every Of is essentially selfadjoint thereon, notion we shall discuss in the next chapter.

21

The penultimate requirement says that the map f 7→ Of transforms the real Lie algebra(C∞(Γ,R), ·, ·P ) into a real Lie algebra of operators whose Lie bracket is i~[Of , Og]. A mapfulfilling these constraints, in particular the third one, is possible if f , g are both functions ofonly the q or the p coordinates separately or if they are linear in them. But it is false in view ofGroenewold’s theorem, already if we consider elementary physical systems. The ultimate reasonof this obstructions due to the fact that the operators Pk, Xk do not commute, contrary to thefunctions pk, q

k which do. The problem can be solved, in the paradigm of the so-called Geo-metric Quantization (and more generally to the so-called Deformation Quantization) [Erc15], inparticular replacing (C∞(Γ,R), ·, ·P ) with a sub-Lie algebra (as large as possible). There areother remarkable procedures of “quantization” in the literature, we shall not insist on them anyfurther here [Erc15, Lan17].

Example 1.3.(1) The Hilbert space of an electron is given by the tensor product L2(R3, d3x)⊗ Hs ⊗ Hc.

(2) Consider a spinless particle in 3D with mass m > 0, whose potential energy is a bounded-below real function U ∈ C∞(R3) with polynomial growth. Classically, its Hamiltonian functionreads

h :=3∑

k=1

p2k

2m+ U(x) .

A brute force quantization procedure in L2(R3, d3x) consists of replacing every classical objectwith corresponding operators. It may make sense at most when there are no ordering ambiguitiesin translating functions like p2x, since classically p2x = pxp = xp2, but these identities are falseat quantum level. In our case these problems do not arise so that

H :=3∑

k=1

P 2k

2m+ U , (1.25)

where (Uψ)(x) := U(x)ψ(x), could be accepted as first quantum model of the Hamiltonianfunction of our system. The written operator is at least defined on S (R3), where it satisfies〈Hψ|φ〉 = 〈ψ|Hφ〉. The existence of selfadjoint extensions is a delicate issue (see [Mor18] andespecially [Tes14]) we shall not address here. Taking (1.20) into account, always on S (R3), oneimmediately finds

H := − ~2

2m∆ + U ,

where ∆ is the standard Laplace operator in R3. If we assume that the equation describing theevolution of the quantum system is again5 (1.14), in our case we find the known form of theSchrodinger equation,

i~dψtdt

= − ~2

2m∆ψt + Uψt ,

5A factor ~ has to be added in front of the left-hand side of (1.14) if we deal with a unit system where ~ 6= 1.

22

when ψτ ∈ S (R3) for τ varying in a neighbourhood of t (this requirement may be relaxed).Actually the meaning of the derivative on the left-hand side should be specified. We only sayhere that it is computed with respect to the natural topology of L2(R3, d3x).

23

Chapter 2

Observables and states in generalHilbert spaces: The Spectral Theory

The main goal of this chapter is to offer a suitable mathematical theory, sufficient to extendto the infinite dimensional case the elementary mathematical formulation of QM introducedin the previous chapter. As seen in Section 1.3, the main issue concerns the fact that, in theinfinite dimensional case, there are operators representing observables which do not have propereigenvalues and eigenvectors, like X and P . So, naive expansions as that in Eq.(1.4) cannot beliterally extended to the general case. These expansions, together with the interpretation of theeigenvalues as values attained by the observable associated with a selfadjoint operator, play acrucial role in the mathematical interpretation of the quantum phenomenology introduced inSect.1.1 and mathematically discussed in Sect.1.2. In particular, we need a precise definition ofselfadjoint operator and a result regarding a spectral decomposition in the infinite dimensionalcase. These tools are basic elements of the so called spectral theory in Hilbert spaces, literallyinvented by von Neumann in his famous book [Neu32] to give a rigorous form to QuantumMechanics and successively developed by various authors towards many different directions ofpure and applied mathematics. The same notion of abstract Hilbert space, as nowadays known,was born in the second chapter of that book, joining and elaborating previous mathematicalconstructions by Hilbert and Riesz. The remaining part of this chapter is devoted to introducethe reader to some elements of that formalism. Reference books are [Ped89, Rud91, Schm12,Tes14, Mor18].

2.1 Hilbert Spaces: A summary of fundamental facts

We henceforth assume that the reader be familiar with the basic definitions of the theory ofnormed, Banach and Hilbert spaces [Rud91, Mor18]. Notions like orthogonal of a set, Hilbertbasis (also called complete orthonormal systems) and that their properties and use are supposedto be well known [Rud91, Mor18]. We summarize here some basic results especially concerningorthogonal sets and Hilbert bases.

24

Remark 2.1. We only deal with complex Hilbert spaces: Even if we do not mention the wordcomplex explicitly, every considered Hilbert space is always supposed to be complex throughoutthe work.

2.1.1 Basic properties

Definition 2.1. A Hermitian scalar product over the complex vector space H is a map

〈·|·〉 : H× H→ C

such that, for a, b ∈ C and x, y, z ∈ H,

(i) 〈x|y〉 = 〈y|x〉,

(ii) 〈x|ay + bz〉 = a〈x|y〉+ b〈x|z〉,

(iii) 〈x|x〉 ≥ 0 with x = 0 if 〈x|x〉 = 0.

The space H is said to be a (complex) Hilbert space if is complete with respect to the natural

norm ||x|| :=»〈x|x〉, x ∈ H.

Remark 2.2. A closed subspace H0 of a Hilbert space H is a Hilbert space with respect tothe restriction of the structure, since it contains the limits of its Cauchy sequences.

Just in view of (semi)positivity of the scalar product and regardless the completeness property,the Cauchy-Schwartz inequality holds

|〈x|y〉| ≤ ||x|| ||y|| , x, y ∈ H .

Another elementary purely algebraic fact is the polarization identity concerning the Hermitianscalar product (here, H is not necessarily complete)

4〈x|y〉 = ||x+ y||2 − ||x− y||2 − i||x+ iy||2 + i||x− iy||2 for of x, y ∈ H, (2.1)

which immediately implies the following elementary result.

Proposition 2.1. If H is a complex vector space with Hermitian scalar product 〈 | 〉, a linearmap L : H→ H which is an isometry – ||Lx|| = ||x|| if x ∈ H – also preserves the scalar product– 〈Lx|Ly〉 = 〈x|y〉 for x, y ∈ H.

The converse proposition is obviously true.Similarly to the above identity we have another useful identity, for a linear map A : H→ H:

4〈x|Ay〉 = 〈x+ y|A(x+ y)〉 − 〈x− y|A(x− y)〉 − i〈x+ iy|A(x+ iy)〉

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+ i〈x− iy|A(x− iy)〉 for of x, y ∈ H. (2.2)

That in particular proves that the following elementary result is true.

Proposition 2.2. Let A : H→ H a linear map over the complex vector space H with Hermi-tian scalar product. If 〈x|Ax〉 = 0 for all x ∈ H, then A = 0.

Observe that this fact is not generally true if dealing with real vector spaces equipped with areal symmetric scalar product.We remind the reader the validity of a fundamental technical result (e.g., see [Rud91, Mor18]):

Theorem 2.1. [Riesz’ lemma]Let H be a Hilbert space. φ : H→ C is linear and continuous if and only if has the form

φ = 〈x| 〉

for some x ∈ H. The vector x is uniquely determined by φ.

2.1.2 Orthogonality and Hilbert bases

Notation 2.1. If M ⊂ H, M⊥ := y ∈ H | 〈y|x〉 = 0 ∀x ∈ M denotes the orthogonal ofM . When N ⊂M⊥ (that is evidently equivalent to M ⊂ N⊥), we write N ⊥M .

Evidently M⊥ is a closed subspace of H because the scalar product is continuous in both entries.⊥ enjoys several nice properties quite easy to prove (e.g., see [Rud91, Mor18]), in particular,

spanM = (M⊥)⊥ and H = spanM ⊕M⊥ (2.3)

where spanM indicates the set of finite linear combinations of vectors in M , the bar denotesthe topological closure and ⊕ denotes the direct sum of (orthogonal) subspaces. (We remind thereader that a vector space X is the direct sum of a pair of subspace X1,X2 – and we thereforewrite X = X1⊕X2 – if every x ∈ H can be decomposed as x = x1 +x2 for exactly a pair x1 ∈ X1

and x2 ∈ X2.)An elementary but important technical lemma is the following [Mor18].

Lemma 2.1. Let H be a Hilbert space. If xnn∈N ⊂ H is such that 〈xk|xh〉 = 0 if h 6= k,then the following facts are equivalent.

(a)∑+∞k=0 xn := limN→+∞

∑Nk=0 xn exists in H;

(b)∑+∞k=0 ||xn||2 < +∞.

If (a) and (b) hold, then+∞∑k=0

xn =+∞∑k=0

xf(n) ∈ H ,

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for every bijective map f : N→ N. In other words, the series in (a) can be re-ordered arbitrarilypreserving its sum.

Definition 2.2. A Hilbert basis N of a Hilbert space H is a set of orthonormal vectors(i.e. ||u|| = 1 and 〈u|v〉 = 0 for u, v ∈ N with u 6= v) such that if s ∈ H satisfies 〈s|u〉 = 0 forevery u ∈ H, then s = 0.

Hilbert bases always exist as a consequence of Zorn’s lemma. Notice that, as consequence of (2.3),

Proposition 2.3. A set of orthonormal vectors N ⊂ H is a Hilbert basis of the Hilbert spaceH if and only if spanN = H.

If M ⊂ H is a set of orthonormal vectors, the so-called Bessel inequality can be immediatelyproved

||x||2 ≥∑u∈M|〈u|x〉|2 for every x ∈ H .

Hilbert bases are exactly orthonormal sets saturating the inequality. In fact, a generalized ver-sion of Pythagorean theorem holds true.

Proposition 2.4. A set of orthonormal vectors N ⊂ H is a Hilbert basis of the Hilbert spaceH if and only if

||x||2 =∑u∈N|〈u|x〉|2 for every x ∈ H.

The sum appearing in the above decomposition of x is understood as the supremum of the sums∑u∈F |〈u|x〉|2 for every finite set F ⊂ N .

Remark 2.3.(a) If N is a Hilbert basis and x ∈ H, only a set at most countable of elements |〈u|x〉|2 with

u ∈ N do not vanish: Only a finite number of |〈u|x〉|2 can belong in [1,+∞) otherwise the sumdiverges, the same argument proves that only a finite number of |〈u|x〉|2 can belong in [1/2, 1),and also in [1/3, 1/2) and so on. In summary, since these disjoint sets whose union is [0,+∞) arecountably many, the number of non-vanishing elements |〈u|x〉|2 is finite or countable. The sum||x||2 =

∑u∈N |〈u|x〉|2 can be therefore interpreted and computed as a standard series summing

over the non-vanishing elements only. Furthermore, it can be re-ordered preserving the sumbecause the series absolutely converges.

(b) It turns out that all Hilbert bases of H have the same cardinality and H is separable,i.e. it admits a dense countable subset, if and only if H has an either finite our countable Hilbertbasis.

As a consequence of Lemma 2.1 and (a) in the remark above, if N ⊂ H is an Hilbert basis, the

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decompositions hold for every x ∈ H

x =∑u∈N〈u|x〉u . (2.4)

More precisely, in view of the fact that only a finite or countable set of elements 〈un|x〉 do notvanish, the sum is actually a finite sum or at most series limm→+∞

∑mn=0〈un|x〉un computed

with respect to the norm of H, where the order used to label the element un does not matter inview of Lemma 2.1. For this reason we do not indicate the order of summation.(2.4) and continuity of the scalar product immediately imply, for every couple x, y ∈ H,

〈x|y〉 =∑u∈N〈x|u〉〈u|y〉 (2.5)

The sum is absolutely convergent in a proper sense (by Cauchy-Schwartz inequality) this isanother direct reason why it can be re-ordered arbitrarily.

2.1.3 Two notions of Hilbert direct orthogonal sum

Hilbert structures can be constructed orthogonally summing a given family of Hilbert spaces.There are two cases (see, e.g., [Mor18]).

(1) The first case concerns the definition of Hilbert (direct orthogonal) sum of closed subspacesHjj∈J of a given Hilbert space H with Hj 6= 0 for every j ∈ J . Here J is a set with arbitrarycardinality and we suppose Hr ⊥ Hs when r 6= s. Let spanHjj∈J denote the set of finite linearcombinations of vectors in the spaces Hj when j ∈ J , the Hilbert direct orthogonal sum ofthe subspaces Hj is the closed subspace of H⊕

j∈JHj := spanHjj∈J .

Applying Proposition 2.3, it arises that, if Nj ⊂ Hj is a Hilbert basis of Hj , then ∪j∈JNj is aHilbert basis of

⊕j∈J Hj . Decomposing x ∈ ⊕j∈J Hj along every Nj , we have corresponding

elements xj ∈ Hj such that

∀x ∈⊕j∈J

Hj , ||x||2 =∑j∈J||xj ||2 .

Furthermore, by Lemma 2.1,

∀x ∈⊕j∈J

Hj , x =∑j∈J

xj , xj ∈ Hj for j ∈ J

where the sum is a series, since an at most countable number of xj do not vanish and it can bearbitrarly re-ordered preserving the sum. The sum is direct in the sense that every x ∈⊕j∈J Hjcan be decomposed uniquely into a sum of vectors xj ∈ Hj . Another decomposition of the samex, namely, x =

∑j∈J x

′j with x′j ∈ Hj for j ∈ J , would imply 0 = x − x =

∑j∈J(x′j − xj)

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and also, computing the norm using the fact that for different j we have orthogonal vectors,0 =

∑j∈J ||x′j − xj ||2 which implies x′j = xj for every j ∈ J .

(2) If Hjj∈J is a family of generic non-trivial Hilbert spaces, we can analogously define aHilbert space still denoted by

⊕j∈J Hj called Hilbert (direct orthogonal) sum of Hilbert

spaces Hjj∈J . To this end, consider the elements x = xjj∈J of the standard direct sum

of the complex vector spaces Hj whose norm ||x|| :=»∑

j∈J ||xj ||2j is finite. This defines aHilbert space structure over the said vectors when they are equipped with the scalar product〈x|x′〉 =

∑j∈J〈xj |x′j〉j with obvious notation.

The two definitions are evidently interrelated. Indeed, referring to the second defintion, (a)every Hj is a closed subspace of

⊕j∈J Hj , (b) referring to the overall scalar product it also holds

Hj ⊥ Hk if j 6= k, and (c)⊕j∈J Hj is also the Hilbert direct orthogonal sum these subspaces

according to the first definition.

2.1.4 Tensor product of Hilbert spaces

If Hjj=1,2,...,N is a finite family of Hilbert spaces (which are not necessarily subspaces of alarger Hilbert space), their Hilbert tensor product constructed as follows. First consider thestandard algebraic tensor product, we denote by H1 ⊗A · · · ⊗A HN . We can endow this spacewith the inner product uniquely induced by linearity and anti linearity from the requirement

〈x1 ⊗ · · · ⊗ xN |y1 ⊗ · · · ⊗ yN 〉 :=N∏j=1

〈xj |yj〉j for xj , yj ∈ Hj , j = 1, . . . , N . (2.6)

It is easy to prove [Mor18] that there is only one Hermitian scalar product induced by therequirement above on H1 ⊗A · · · ⊗A HN . The Hilbert tensor product H1 ⊗ · · · ⊗ HN of thefamily Hjj=1,2,...,N is the completion of H1⊗A · · · ⊗A HN with respect to the norm induced bythe unique scalar product extending (2.6).As a consequence of the given defintion, given the Hilbert bases Nj ⊂ Hj , the orthonormal set

u1 ⊗ · · · ⊗ uN | uj ∈ Nj , j = 1, . . . , N

is a Hilbert basis of H1 ⊗ · · · ⊗ HN [Mor18].

Remark 2.4. Considering Hilbert spaces L2(Xj , µj), for j = 1, . . . , N and where eachµj is σ-finite, the Hilbert space L2(X1 × · · · × XN , µ1 ⊗ · · · ⊗ µN ) turns out to be naturallyisomorphic to the Hilbert space L2(X1, µ1) ⊗ · · · ⊗ L2(XN , µN ) [Mor18]. The natural Hilbert-spaces isomorphism is the unique continuous linear extension of the map

L2(X1, µ1)⊗ · · ·⊗L2(XN , µN ) 3 f1⊗ · · · ⊗ fN 7→ f1 · · · fN ∈ L2(X1× · · · ×XN , µ1⊗ · · · ⊗µN ) ,

where f1 · · · fN is the pointwise product of the said functions:

(f1 · · · fN )(s1, . . . , sN ) := f1(s1) · · · fn(sN ) ,

if (s1, . . . , sn) ∈ X1 × · · · ×XN .

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2.2 Classes of (especially unbounded) operators in Hilbert spaces

Since our overall goal is to present some basic results of spectral analysis, useful in QM, a numberof preparatory notions on operators must be introduced.

2.2.1 Operators and abstract algebras

From now on, an operator is a linear map A : X → Y from a complex linear space X toanother linear space Y . In case Y = C, we say that A is a functional over X.

Focussing to the special case of Hilbert spaces H, an operator A in H always means a linearmap A : D(A) → H, whose domain, D(A) ⊂ H, is a subspace of H. In particular, I alwaysdenotes the identity operator defined on the whole space (D(I) = H)

I : H 3 x 7→ x ∈ H .

If A is an operator in H, Ran(A) := Ax | x ∈ D(A) is the image or range of A.

Notation 2.2. If A and B are operators in H

A ⊂ B means that D(A) ⊂ D(B) and B|D(A) = A,

where |S is the standard “restriction to S” symbol. We also adopt usual conventions regardingstandard domains of combinations of operators A,B:

(i) D(AB) := x ∈ D(B) |Bx ∈ D(A) is the domain of AB,(ii) D(A+B) := D(A) ∩D(B) is the domain of A+B,(ii) D(αA) = D(A) for α 6= 0 is the domain of αA.

With these definitons it is easy to prove that(1) (A+B) + C = A+ (B + C),(2) A(BC) = (AB)C,(3) A(B + C) = AB +BC,(4) (B + C)A ⊃ BA+ CA,(5) A ⊂ B and B ⊂ C imply A ⊂ C,(6) A ⊂ B and B ⊂ A imply A = B,(7) AB ⊂ BA implies A(D(B)) ⊂ D(B) if D(A) = H,(8) AB = BA implies D(B) = A−1(D(B)) if D(A) = H (so, A(D(B)) = D(B) is A is

surjective).To go on, we define some abstract algebraic structures naturally arising in the space of operatorson a Hilbert space.

Definition 2.3. Let A be an associative complex algebra A.

(1) A is a Banach algebra if it is a Banach space such that ||ab|| ≤ ||a|| ||b|| for a, b ∈ A. Anunital Banach algebra is a Banach algebra with unit multiplicative element 11, satisfying||11|| = 1.

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(2) A is an (unital) ∗-algebra if it is an (unital) algebra equipped with an anti linear mapA 3 a 7→ a∗ ∈ A, called involution, such that (a∗)∗ = a and (ab)∗ = b∗a∗ for a, b ∈ A.

(3) A is a (unital) C∗-algebra if it is a (unital) Banach algebra A which is also a ∗-algebraand ||a∗a|| = ||a||2 for a ∈ A.

A ∗-homomorphism from the ∗-algebra A to the the ∗-algebra B is an algebra homomorphismpreserving the involutions (and the unities if both present). A bijective ∗-homomorphism is called∗-isomorphism.

Exercise 2.1. Prove that 11∗ = 11 in a unital ∗-algebra and that ||a∗|| = ||a|| if a ∈ A whenA is a C∗-algebra.

Solution. From 11a = a11 = a and the definition of ∗, we immediately have a∗11∗ = 11∗a∗ =a∗. Since (b∗)∗ = b, we have found that b11∗ = 11∗b = b for every b ∈ A. Uniqueness of the unitimplies 11∗ = 11. Regarding the second property, ||a||2 = ||a∗a|| ≤ ||a∗|| ||a|| so that ||a|| ≤ ||a∗||.Everywhere replacing a for a∗ and using (a∗)∗, we also obtain ||a∗|| ≤ ||a||, so that ||a∗|| = ||a||. 2

We remind the reader that an operator A : X → Y , where X and Y are normed complex vectorspaces with resp. norms || · ||X and || · ||Y , is said to be bounded if

||Ax||Y ≤ b||x||X for some b ∈ [0,+∞) and all x ∈ X. (2.7)

As is well known [Rud91, Mor18], it turns out that

Proposition 2.5. An operator A : X → Y , with X,Y normed spaces, is continuous if andonly if it is bounded.

Proof. It is evident that A bounded is continuous because, for x, x′ ∈ X, ||Ax − Ax′||Y ≤b||x − x′||X . Conversely, if A is continuous then it is continuous for x = 0, so ||Ax||y ≤ ε forε > 0 if ||x||X < δ for δ > 0 sufficiently small. If ||x|| = δ/2 we therefore have ||Ax||Y < ε andthus, dividing by δ/2, we also find ||Ax′||Y < 2ε/δ, where ||x′||X = 1. Multiplying for λ > 0,||Aλx′||Y < 2λε/δ which can be re-written ||Ax||Y < 2 εδ ||x|| for every x ∈ X, proving that A isbounded.

For bounded operators it is possible to define the operator norm,

||A|| := sup06=x∈X

||Ax||Y||x||X

(= sup

x∈X, ||x||X=1||Ax||Y

).

It is easy to prove that this is a true norm on the complex vector spaces B(X,Y ) of boundedoperators T : X → Y , with X,Y complex normed spaces when defining the linear combination ofoperators αA+βB ∈ B(X,Y ) for α, β ∈ C and A,B ∈ B(X,Y ) by (αA+βB)x := αAx+βBx

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for every x ∈ X.

An important elementary technical result is stated in the following proposition.

Proposition 2.6. Let A : S → Y be a bounded operator defined on the subspace S ⊂ X,where X,Y are normed spaces with Y complete. If S is dense in X, then A can be continuouslyextended to a unique bounded operator A1 : X → Y . It holds ||A1|| = ||A||.

Proof. Uniqueness is obvious from continuity: if S 3 xn → x ∈ X and A1, A′1 are continuous

extensions, A1x − A′1x = limn→+∞A1xn − A′1xn = limn→+∞ 0 = 0. Let us construct a linearcontinuous extension. If x ∈ X, there is a sequence S 3 xn → x ∈ X since S is dense. Axnn∈Nis Cauchy because xnn∈N is Cauchy and ||Axn − Axm||Y ≤ ||A||||xn − xm||X . So the limitA1x := limn→+∞Axn exists because Y is complete. The limit does not depend on the sequence:if S 3 x′n → x, then ||Axn−Ax′n|| ≤ ||A|| ||xn−x′n|| → 0, so A1 is well defined. It is immediatelyproved that A1 is linear from linearity of A, hence A1 is an operator which extends A to thewhole X. By construction, ||A1x||Y = limn→+∞ ||Axn||Y ≤ limn→+∞ ||A||||xn||X ≤ ||A||||x||X ,so ||A1|| ≤ ||A||, in particular A1 is bounded. On the other hand

||A1|| = sup||A1x|| ||x||−1 | x ∈ X \ 0 ≥ sup||A1x|| ||x||−1 | x ∈ S \ 0

= sup||Ax|| ||x||−1 | x ∈ S \ 0 = ||A|| ,so that also ||A1|| ≥ ||A|| is valid, proving ||A1|| = ||A||.

Notation 2.3. From now on, B(H) := B(H,H) denotes the vector space of bounded opera-tors A : H→ H over the Hilbert space H.

B(H) acquires the structure of a unital Banach algebra: The complex vector space structure isthe standard one of operators, the associative algebra product is the composition of operatorswith unit given by I, and the norm being the above defined operator norm,

||A|| := sup06=x∈H

||Ax||||x||

.

This definition of ||A|| can be given also for an operator A : D(A) → H, if A is bounded andD(A) ⊂ H but D(A) 6= H. It immediately arises that

||Ax|| ≤ ||A|| ||x|| if x ∈ D(A).

As we already know, || · || is a norm over B(H). Furthermore it satisfies

||AB|| ≤ ||A|| ||B|| A,B ∈ B(H) .

It is also evident that ||I|| = 1. Actually B(H) is a Banach space so that B(H) is a unitalBanach algebra. In fact, a fundamental result is the following theorem.

Theorem 2.2. If H is a Hilbert space, B(H) is a Banach space with respect to the norm ofoperators.

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Proof. The only non-trivial property is completeness of B(H). Let us prove it. Consider aCauchy sequence Tnn∈N ⊂ B(H). We want to prove that there exists T ∈ B(H) whichsatisfies ||T − Tn|| → 0 as n → +∞. Let us define Tx := limn→+∞ Tx for every x ∈ H. Thelimit exists because Tnxn∈N is Cauchy from ||Tnx−Tmx|| ≤ ||Tn−Tm|| ||x||. Linearity of T iseasy to prove from linearity of every Tn. Next observe that ||Tx−Tmx|| = || limn Tnx−Tmx|| =limn ||Tnx − Tmx|| ≤ ε||x|| is m is sufficiently large. Assuming that T ∈ B(H), the foundinequality, dividing by ||x|| and taking the sup over x with ||x|| 6= 0 proves that ||T − Tm|| ≤ εand thus ||T − Tm|| → 0 for m → +∞ as wanted. The proof is over because T ∈ B(H) since||Tx|| ≤ ||Tx− Tmx||+ ||Tmx|| ≤ ε||x||+ ||Tm||||x|| and thus ||T || ≤ (ε+ ||Tm||) < +∞.

Remark 2.5. The result, with the same proof, is valid for the above defined complex vectorspace B(X,Y ), provided the normed space Y is || · ||Y -complete. In particular the topologicaldual of X, denoted by X∗ = B(X,C) with X complex normed spaces, is complete since C iscomplete.

Exercise 2.2. Prove that in a Hilbert space H 6= 0 there are no operators Xh, Pk ∈ B(H),h, k = 1, 2, . . . , n satisfying CCRs (1.22).

Solution. It is enough considering the case n = 1. Suppose that [X,P ] = iI (where we use~ = 1 without lack of generality) for X,P ∈ B(H). By induction, one immediatley sees that[X,P k] = kiP k−1 if k = 1, 2, . . .. Hence

k||P k−1|| = ||[X,P k]|| ≤ 2||X|| ||P k|| ≤ 2||X||||P ||||P k−1|| .

Dividing by ||P k−1|| (which cannot vanish otherwise we would have P k−2 = 0 from [X,P k−1] =(k − 1)iP k−2 and P = 0 by induction, which is forbidden since [X,P ] = iI 6= 0), we havek ≤ 2||X|| ||P || for every k = 1, 2, . . ., which is impossible because X,P ∈ B(H). 2

2.2.2 Adjoint operator

B(H) is more strongly a unital C∗-algebra, if we introduce the notion of adjoint of an operator.To this end, we have the following general definition concerning also unbounded operators de-fined on non-maximal domains.

Definition 2.4. Let A be a densely defined operator in the Hilbert space H. Define thesubspace of H,

D(A∗) := y ∈ H | ∃zy ∈ H s.t. 〈y|Ax〉 = 〈zy|x〉 ∀x ∈ D(A) .

The linear map A∗ : D(A∗) 3 y 7→ zy is called the adjoint operator of A.

Let us comment why the defintion is well posed. Above, zy is uniquely determined by y, sinceD(A) is dense. If both zy, z

′y satisfy 〈y|Ax〉 = 〈zy|x〉 and 〈y|Ax〉 = 〈z′y|x〉, then 〈zy − z′y|x〉 = 0

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for every x ∈ D(A). Taking a sequence D(A) 3 xn → zy − z′y, we conclude that ||zy − z′y|| = 0.Thus zy = z′y and A∗ : D(A∗) 3 y 7→ zy is a well-defined function. Next, by definition of D(A∗),we have that azy + bzy′ satisfies 〈ay + by′|Ax〉 = 〈azy + bzy′ |x〉 for y, y′ ∈ D(A∗) and a, b ∈ C inview of (anti)linearity of the scalar product, so that A∗ : D(A∗) 3 u 7→ zu is also linear.

Remark 2.6.(a) If D(A) is not dense, A∗ cannot be defined in general. As an example, consider M ( H

a closed subspace, so that M⊥ 6= 0. Define A : D(A) = M 3 x 7→ x ∈ H. If 0 6= y ∈ M⊥we have 〈y|Ax〉 = 〈y|x〉 = 0, so that we could say that y ∈ D(A∗) and A∗y = y. However itwould be wrong, since we also have, e.g., 〈y|Ax〉 = 0 = 〈2y|x〉 and therefore we could also defineA∗y = 2y. Actually the function A∗ cannot be defined here because it would associate manyvalues to a single element in its domain.

(b) By construction, we immediately have that

〈A∗y|x〉 = 〈y|Ax〉 for x ∈ D(A) and y ∈ D(A∗) .

Exercise 2.3. Prove that D(A∗) can equivalently be defined as the set (subspace) of y ∈ Hsuch that the linear functional D(A) 3 x 7→ 〈y|Ax〉 is continuous.

Solution. It is a simple application of Riesz’ lemma, after having uniquely extendedD(A) 3 x 7→ 〈y|Ax〉 to a continuous linear functional defined on D(A) = H by continuity.2

Remark 2.7.(a) If A is densely defined and A∗ is also densely defined then A ⊂ (A∗)∗. The proof

immediately follows form the definition of adjoint.(b) If A is densely defined and A ⊂ B then B∗ ⊂ A∗. The proof immediately follows form

the definition of adjoint.(c) If A ∈ B(H) then A∗ ∈ B(H) and (A∗)∗ = A. Moreover

||A∗||2 = ||A||2 = ||A∗A|| = ||AA∗|| .

(See Exercise 2.4.)(d) Directly from given definition of adjoint one has, for densely defined operators A,B on

H,A∗ +B∗ ⊂ (A+B)∗ and A∗B∗ ⊂ (BA)∗ .

Furthermore

A∗ +B∗ = (A+B)∗ and A∗B∗ = (BA)∗ , (2.8)

whenever B ∈ B(H) and A is densely defined.

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(e) From (c) and the last statement in (d) in particular, it is clear that B(H) is a unitalC∗-algebra with involution B(H) 3 A 7→ A∗ ∈ B(H).

Definition 2.5. If A is a (unital) ∗-algebra and H a Hilbert space, a ∗-representation onH is a ∗-homomorphism π : A → B(H) referring to the natural (unital) ∗-algebra structure ofB(H).

Exercise 2.4. Prove that A∗ ∈ B(H) if A ∈ B(H) and that, in this case (A∗)∗ = A,||A|| = ||A∗|| and ||A∗A|| = ||AA∗|| = ||A||2.

Solution. If A ∈ B(H), for every y ∈ H, the linear map H 3 x 7→ 〈y|Ax〉 is continu-ous (|〈y|Ax〉| ≤ ||y|| ||Ax|| ≤ ||y|| ||A|| ||x||), therefore Theorem 2.1 proves that there existsa unique zy,A ∈ H with 〈y|Ax〉 = 〈zy,A|x〉 for all x, y ∈ H. The map H 3 y 7→ zy,A is lin-ear as consequence of the said uniqueness and the anti linearity of the left entry of scalarproduct. The map H 3 y 7→ zy,A fits the definition of A∗, so it coincides with A∗ andD(A∗) = H. Since 〈A∗x|y〉 = 〈x|Ay〉 for x, y ∈ H implies (taking the complex conjugation)〈y|A∗x〉 = 〈Ay|x〉 for x, y ∈ H, we have (A∗)∗ = A. To prove that A∗ is bounded observethat ||A∗x||2 = 〈A∗x|A∗x〉 = 〈x|AA∗x〉 ≤ ||x|| ||A|| ||A∗x|| so that ||A∗x|| ≤ ||A|| ||x|| and||A∗|| ≤ ||A||. Using (A∗)∗ = A, we have ||A∗|| = ||A||. Regarding the last identity, it isevidently enough to prove that ||A∗A|| = ||A||2. First of all, ||A∗A|| ≤ ||A∗|| ||A|| = ||A||2,so that ||A∗A|| ≤ ||A||2. On the other hand ||A||2 = (sup||x||=1 ||Ax||)2 = sup||x||=1 ||Ax||2 =sup||x||=1〈Ax|Ax〉 = sup||x||=1〈x|A∗Ax〉 ≤ sup||x||=1 ||x||||A∗Ax|| = sup||x||=1 ||A∗Ax|| = ||A∗A||.We have found that ||A∗A|| ≤ ||A||2 ≤ ||A∗A|| so that ||A∗A|| = ||A||2. 2

Exercise 2.5. Prove that if A ∈ B(H), A∗ is bijective if and only if A is. In this case(A−1)∗ = (A∗)−1.

Solution. If A ∈ B(H) is bijective we have AA−1 = A−1A = I. Taking the adjointoperators, (A−1)∗A∗ = A∗(A−1)∗ = I∗ = I from (d) Remark 2.7, that implies (A−1)∗ = (A∗)−1

in particular for the uniqueness of the inverse operator. If A∗ is bijective, taking the adjoint of(A∗)−1A∗ = A∗(A∗)−1 = I, using (A∗)∗ = A, we have that A is bijective as well.

2.2.3 Closed and closable operators

Definition 2.6. Let A be an operator in the Hilbert space H.

(1) A is said to be closed if the graph of A, that is the set

G(A) := (x,Ax) ⊂ H× H | x ∈ D(A) ,

is closed in the product topology of H× H.

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(2) A is closable if it admits extensions made of closed operators. This is equivalent to sayingthat the closure of the graph of A is the graph of an operator, denoted by A, and calledthe closure of A.

(3) If A is closable, a subspace S ⊂ D(A) is called core for A if A|S = A

Referring to (2), we observe that, given an operator A, we can always define the closure of thegraph G(A) in H × H. In general this closure is not the graph of an operator, because therecould exist sequences D(A) 3 xn → x and D(A) 3 x′n → x such that Txn → y and Txn → y′

with y 6= y′. However, both pairs (x, y) and (x, y′) belong to G(A). If this is not the case – andthis is equivalent to condition (a) below when making use of linearity – G(A) is the graph ofan operator, denoted by A, that is closed by definition. Therefore A admits closed operatorialextensions: at least A. If, conversely, A admits extensions in terms of closed operators, theintersection of the (closed) graphs of all these extensions G(A) is still closed and it is the graphof an operator as well, which must coincide with A by definition.

Remark 2.8.(a) Directly from the definition and using linearity, A is closable if and only if there are no

sequences of elements xn ∈ D(A) such that xn → 0 and Axn → y with y 6= 0 as n→ +∞. SinceG(A) is the union of G(A) and its accumulation points in H× H and, if A is closable, it is alsothe graph of the operator A, we conclude that

(i) D(A) is made of the elements x ∈ H such that xn → x and Axn → yx for some sequencesxnn∈N ⊂ D(A) and some yx ∈ D(A)

(ii) Ax = yx.

(b) As a consequence of (a) one has that, if A is closable, then aA + bI is closable andaA+ bI = aA+ bI for every a, b ∈ C.N.B. This result generally fails if replacing I for some closable operator B.

(c) Directly from the definition, A is closed if and only if D(A) 3 xn → x ∈ H andAxn → y ∈ H imply both x ∈ D(A) and y = Ax.

An useful proposition is the following.

Proposition 2.7. Consider an operator A : D(A)→ H with D(A) dense in the Hilbert spaceH. The following facts hold.

(a) A∗ is closed.

(b) A is closable if and only if D(A∗) is dense and, in this case, A = (A∗)∗.

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Proof. Define on the Hermitian inner product ((x, y)|(x′y′)) := 〈x|x′〉+〈y|y′〉 on the vector spacedefined as the standard (external) direct sum of vector spaces, we henceforth denote by H⊕ H.It is immediately proved that H ⊕ H becomes a Hilbert space when equipped with that innerproduct. Next define the operator

τ : H⊕ H 3 (x, y) 7→ (−y, x) ∈ H⊕ H .

It is easy to check that τ ∈ B(H⊕ H) and also (referring the adjoint to the space H⊕ H),

τ∗ = τ−1 = −τ . (2.9)

Finally, by direct computation one sees that τ and ⊥ (referred to H ⊕ H with the said innerproduct) commute

τ(F⊥) = (τ(F ))⊥ if F ⊂ H⊕ H. (2.10)

Let us pass to prove (a). The following noticeable identity holds true for every operator A :D(A)→ H with D(A) dense in H (so that A∗ is defined)

G(A∗) = τ(G(A))⊥ . (2.11)

Since the right hand side is closed (it being the orthogonal space of a set), the graph of A∗ isclosed and A∗ is therefore closed by definition. To prove (2.11) observe that, in view of thedefinition of τ ,

τ(G(A))⊥ = (y, z) ∈ H⊕ H | ((y, z)|(−Ax, x)) = 0 ,∀x ∈ D(A) ,

that isτ(G(A))⊥ = (y, z) ∈ H⊕ H | 〈y|Ax〉 = 〈z|x〉 ,∀x ∈ D(A) .

Since A∗ exists, the pairs (y, z) ∈ τ(G(A))⊥ can be written (y,A∗y) according to the definitionof A∗. Therefore τ(G(A))⊥ = G(A∗) proving (a).(b) From the properties of ⊥ we immediately have G(D(A)) = (G(A)⊥)⊥. Since τ and ⊥

commute by (2.10), and ττ = −I (2.9),

G(D(A)) = −τ τ((G(A)⊥)⊥) = −τ(τ(G(A))⊥)⊥ = τ(τ(G(A))⊥)⊥ = τ(G(A∗))⊥ ,

where we have omitted the minus sign since it appears in front of a subspace that, by definition,is closed with respect multiplication with scalars and we made use of (2.11). Now suppose thatD(A∗) is dense so that (A∗)∗ exists. In this case, taking advantage of (2.11) again, we haveG(A) = G((A∗)∗). Notice that the right-hand side is the graph of an operator so we haveobtained that, if D(A∗) is dense, then A is closable. In this case, by definition of closure of anoperator, we also have A = (A∗)∗.Vice versa, suppose that A is closable, so that the operator A exists and G(A) = G(A). In thiscase τ(G(A∗))⊥ = G(A) is the graph of an operator and thus its graph cannot include pairs (0, y)with y 6= 0 by linearity. In other words, if (0, y) ∈ τ(G(A∗))⊥, then y = 0. This is the same assaying that ((0, y)|(−A∗x, x)) = 0 for all x ∈ D(A∗) implies y = 0. Summing up, 〈y|x〉 = 0 forall x ∈ D(A∗) implies y = 0. Since H = D(A∗)⊥ ⊕ (D(A∗)⊥)⊥ = D(A∗)⊥ ⊕D(A∗), we concludethat D(A∗) = H, that is D(A∗) is dense.

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An immediate corollary follows.

Corollary 2.1. Let A : D(A) → H an operator in the Hilbert space H. If both D(A) andD(A∗) are densely defined then

A∗ = A∗

= A∗ = (((A∗)∗)∗ .

The Hilbert space version of the closed graph theorem holds (e.g., see [Rud91, Mor18]).

Theorem 2.3. [Closed graph Theorem]Let A : H→ H be an operator, H being a Hilbert space. A is closed if and only if A ∈ B(H).

As an important corollary we have the Hibert space version of Banach’s bounded inverse theorem(e.g., see [Rud91, Mor18]).

Corollary 2.2. [Banach’s bounded inverse theorem]Let A : H→ H be an operator, H being a Hilbert space. If A is bijective and bounded its inverseis bounded.

Proof. The graph of A−1 : H → H is closed because A is bounded and a fortiori closed and itsgraph is the same as that of A−1. Theorem 2.3 imples that A−1 is bounded.

Exercise 2.6. Prove that, if B ∈ B(H) and A is a closed operator in H such that Ran(B) ⊂D(A), then AB ∈ B(H).

Solution. AB is well defined by hypothesis and D(AB) = H. Exploiting (c) in remark 2.8and continuity of B, one easily finds that AB is closed as well. Theorem 2.3 finally proves thatAB ∈ B(H). 2

2.2.4 Types of operators relevant in quantum theory

Definition 2.7. An operator A in the Hilbert space H is said to be

(0) Hermitian if 〈Ax|y〉 = 〈x|Ay〉 for x, y ∈ D(A),

(1) symmetric if it is densely defined and Hermitian, which is equivalent to say that A ⊂ A∗.

(2) selfadjoint if it is symmetric and A = A∗,

(3) essentially selfadjoint if it is symmetric and (A∗)∗ = A∗.

(4) unitary if A∗A = AA∗ = I,

(5) normal if it is closed, densely defined and AA∗ = A∗A.

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Remark 2.9.(a) If A is unitary then A,A∗ ∈ B(H). Furthermore an operator A : H→ H is unitary if and

only if it is surjective and norm preserving. (See the exercises 2.7 below). These operators arenothing but the automorphisms of the given Hilbert space. Considering two Hilbert spacesH,H′ isomorphisms are linear maps T : H → H′ which are isometric and surjecitve. Noticethat T also preserve the scalar products in view of Proposition 2.1.

(b) A selfdjoint operator A does not admit proper symmetric extensions and essentiallyselfadjoint operators admit only one selfadjoint extension. (See Proposition 2.8 below).

(c) A symmetric operator A is always closable because A ⊂ A∗ and A∗ is closed (Proposition2.7), moreover for that operator the following conditions are equivalent in view of Proposition2.7 and Corollary 2.1, as the reader immediately proves:

(i) (A∗)∗ = A∗ (A is essentially selfadjoint),

(ii) A = A∗,

(iii) A = (A)∗.

(d) Unitary and selfadjoint operators are cases of normal operators.

The pair of elementary results in on (essentially) selfadjoint operators stated in (b) are worthto be proved.

Proposition 2.8. Let A : D(A) → H be a densely defined operator in the Hilbert space H.The following facts are true.

(a) If A is selfadjoint, then A does not admit proper symmetric extensions.

(b) If A is essentially selfadjoint, then A admits a unique selfadjoint extension, and this ex-tension is A∗.

Proof. (a) Let B be a symmetric extension of A. A ⊂ B then B∗ ⊂ A∗ due to (b) in remark2.7. As A = A∗ we have B∗ ⊂ A ⊂ B. Since B ⊂ B∗, we conclude that A = B.(b) Let B be a selfadjoint extension of the essentially selfadjoint operator A, so that A ⊂ B.Therefore A∗ ⊃ B∗ = B and (A∗)∗ ⊂ B∗ = B. Since A is essentially selfadjoint, we have foundA∗ ⊂ B. Here A∗ is selfadjoint and B is symmetric because selfadjoint. (b) implies A∗ = B.That is, every selfadjoint extension of A coincides with A∗

Another elementary yet important result, helping understand why in QM observables are veryoften described by selfadjoint operators which are unbounded and defined in proper subspaces,is the following proposition.

Theorem 2.4. [Hellinger-Toepliz theorem]Let A be a selfadjoint operator in the Hilbert space H. A is bounded if and only if D(A) = H(thus A ∈ B(H)).

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Proof. Assume D(A) = H. As A = A∗, we have D(A∗) = H. Since A∗ is closed, Theorem 2.3implies the A∗(= A) is bounded. Conversely, if A = A∗ is bounded, since D(A) is dense, wecan continuously extend it to a bounded operator A1 : H → H. That extension, by continuity,trivially satisfies 〈A1x|y〉 = 〈x|A1y〉 for all x, y ∈ H, hence A1 is symmetric. Since A∗ = A ⊂A1 ⊂ A∗1, (a) in Proposition 2.8 implies A = A1.

Let us pass to focus on unitary operators. The relevance of unitary operators is evident from thefollowing proposition where it is proved that they preserve the nature of operators with respectto the Hermitian conjugation.

Proposition 2.9. Let U : H→ H be a unitary operator in the complex Hilbert space H and Aanother operator in H. The operator UAU∗ with domain U(D(A)) (resp. U∗AU with domainU∗(D(A))) is symmetric, selfadjoint, essentially selfadjoint, unitary, normal if A is respectivelysymmetric, selfadjoint, essentially selfadjoint, unitary, normal.

Proof. Since U∗ is unitary when U is and (U∗)∗ = U , it is enough to establish the thesis forUAU∗. First of all notice that D(UAU∗) = U(D(A)) is dense if D(A) is dense since U is bijectiveand isometric and U(D(A)) = H if D(A) = H because U is bijective. By direct inspection,applying the definition of adjoint operator, one sees that (UAU∗)∗ = UA∗U∗ and D((UAU∗)∗) =U(D(A∗)). Now, if A is symmetric A ⊂ A∗, then UAU∗ ⊂ UA∗U∗ = (UAU∗)∗, so that UAU∗

is symmetric as well. If A is selfadjoint A = A∗, then UAU∗ = UA∗U∗ = (UAU∗)∗, so thatUAU∗ is selfadjoint as well. If A is essentially selfadjoint it is symmetric and (A∗)∗ = A∗, sothat UAU∗ is symmetric and U(A∗)∗U∗ = UA∗U∗, that is (UA∗U∗)∗ = UA∗U∗, which means((UAU∗)∗)∗ = (UAU∗)∗, so that UA∗U∗ is essentially selfadjoint. If A is unitary, we have A∗A =AA∗ = I, so that UA∗AU∗ = UAA∗U∗ = UU∗ which, since U∗U = I = UU∗, is equivalentto UA∗U∗UAU∗ = UAU∗UA∗U∗ = U∗U = I, that is (UA∗U∗)UAU∗ = (UAU∗)UA∗U∗ = I.Hence UAU∗ is unitary as well. If A is normal, UAU∗ is normal too, with the same reasoningas in the unitary case.

Remark 2.10. With the same proof the proposition extends to the case of a linear mapU : H→ H′ which is isometric and surjective. With a little different demonstration the proposi-tion remains also when U : H→ H′ is anti linear – U(αx+ βy) = αUx+ βUy if α, β ∈ C andx, y ∈ H – isometric and surjective. We leave to the reader these straightforward generalizations.

Exercise 2.7.(1) Prove that, if A is unitary, then A,A∗ ∈ B(H).

Solution. It holds D(A) = D(A∗) = D(I) = H and ||Ax||2 = 〈Ax|Ax〉 = 〈x|A∗Ax〉 = ||x||2if x ∈ H, so that ||A|| = 1. Due to (c) in remark 2.7, A∗ ∈ B(H). 2

(2) Prove that the operator A : H→ H is unitary if and only if is surjective and norm preserving.

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Solution. If A is unitary ((3) Definition 2.7), it is evidently bijective, moreover as D(A∗) =H, ||Ax||2 = 〈Ax|Ax〉 = 〈x|A∗Ax〉 = 〈x|x〉 = ||x||2, so that A is also isometric. If A : H → H isisometric its norm is 1 and thus A ∈ B(H). Therefore A∗ ∈ B(H). The condition ||Ax||2 = ||x||2can be re-written 〈Ax|Ax〉 = 〈x|A∗Ax〉 = 〈x|x〉 and thus 〈x|(A∗A − I)x〉 = 0 for x ∈ H. Usingx = y± z and x = y± iz, the found identity implies 〈z|(A∗A− I)y〉 = 0 for all y, z ∈ H. Takingz = (A∗A−I)y, we finally have ||(A∗A−I)y|| = 0 for all y ∈ H and thus A∗A = I. In particular,A is injective as it admits the left inverse A∗. Since A is also surjective, it is bijective and thusits left inverse (A∗) is also a right inverse, that is AA∗ = I.

(3) Prove that, if A : H → H satisfies 〈x|Ax〉 ∈ R for all x ∈ H (and in particular if A ≥ 0,which means 〈x|Ax〉 ≥ 0 for all x ∈ H), then A∗ = A and A ∈ B(H).

Solution. We have 〈x|Ax〉 = 〈x|Ax〉 = 〈Ax|x〉 = 〈x|A∗x〉 where, as D(A) = H, the adjointA∗ is well defined everywhere on H. Thus 〈x|(A − A∗)x〉 = 0 for every x ∈ H. Using therex = y± z and x = y± iz we obtain 〈y|(A−A∗)z〉 = 0 for all y, z ∈ H. Choosing y = (A−A∗)z,we conclude that A = A∗. Theorem 2.4 ends the proof. 2

Example 2.1. The Fourier transform (see, e.g. [Rud91, Mor18]), F : S (Rn)→ S (Rn),defined as1

(Ff)(k) :=1

(2π)n/2

∫Rne−ik·xf(x)dnx (2.12)

(k · x being the standard Rn scalar product of k and x) is a bijective linear map with inverseF− : S (Rn)→ S (Rn),

(F−g)(x) :=1

(2π)n/2

∫Rneik·xg(k)dnk , (2.13)

so that

F F− = F− F = ıS (Rn) . (2.14)

It is possible to prove (e.g., [Rud91, Mor18]) that both F and F− preserve the scalar product

〈Ff |Fg〉 = 〈f |g〉 , 〈F−f |F−g〉 = 〈f |g〉 ∀f, g ∈ S (Rn) (2.15)

and thus they also preserve the norm of L2(Rn, dnx), in particular, ||F || = ||F−|| = 1. As aconsequence of Proposition 2.6, using the fact that S (Rn) is dense in L2(Rn, dnx) [Rud91], oneeasily proves that F and F− uniquely continuously extend to bounded operators, respectively,F : L2(Rn, dnx) → L2(Rn, dnk) and F− : L2(Rn, dnk) → L2(Rn, dnx) such that F−1 = F−because also (2.14) trivially extends to L2(Rn, dnx) by continuity. Since the scalar product iscontinuous, from (2.15) we finally obtain

〈Ff |Fg〉 = 〈f |g〉 , 〈F−f |F−g〉 = 〈f |g〉 ∀f, g ∈ L2(Rn, dnx) . (2.16)

1In QM, adopting units with ~ 6= 1, k · x has to be replaced for k·x~ and (2π)n/2 for (2π~)n/2.

41

Summing up, F is a linear isometric surjective map form L2(Rn, dnx) to L2(Rn, dnx), thus aunitary operator, and F−, with the same properties, is its left and right inverse. The map F isthe Fourier-Plancherel (unitary) operator.

2.2.5 Interplay of Ker, Ran, ∗, and ⊥

To go on, we establish two useful technical facts which will turn out to be useful several timesin the rest of this paper.

Proposition 2.10. If A : D(A) → H is a densely defined operator in the Hilbert space H,then

Ker(A∗) = Ran(A)⊥ , Ker(A) ⊂ Ran(A∗)⊥ , (2.17)

where the inclusion becomes an identity if A ∈ B(H).

Proof. Consider the identity arising from the definition of adjoint operator,

〈A∗y|x〉 = 〈y|Ax〉 , ∀x ∈ D(A) , ∀y ∈ D(A∗) . (2.18)

If y ∈ Ker(A∗), then 〈y|Ax〉 = 0 for all x ∈ D(A) due to (2.18), so that y ∈ Ran(A)⊥. If,conversely, y ∈ Ran(A)⊥, then 〈y|Ax〉 = 0 for all x ∈ D(A). This means that y ∈ D(A∗), bydefinition of D(A∗), and A∗y = 0. We have proved that Ker(A∗) = Ran(A)⊥. Regarding thesecond inclusion, if x ∈ Ker(A), we have from (2.18) that 〈A∗y|x〉 = 0 for every y ∈ D(A∗) andtherefore x ∈ Ran(A∗)⊥. We have established that Ker(A) ⊂ Ran(A∗)⊥. To conclude, observethat the requirement x ∈ Ran(A∗)⊥ entails from (2.18) that 〈y|Ax〉 = 0 for every y ∈ D(A∗)provided x ∈ D(A). If A ∈ B(H), then x ∈ H belongs to D(A) = H and 〈y|Ax〉 = 0 is also validfor every y ∈ D(A∗) = H and thus Ax = 0. So that Ker(A) ⊃ Ran(A∗)⊥ is also true.

Notice that if for a densely defined operator A also D(A∗) is dense, then the first identity impliesKer(A∗∗) = Ran(A∗)⊥ namely, from Proposition 2.7, we can write a strong version of (2.17),

Ker(A∗) = Ran(A)⊥ , Ker(A) = Ran(A∗)⊥ . (2.19)

Replacing A for A − λI in (2.17) with λ ∈ C, we find the following useful relations in spectraltheory,

Ker(A∗ − λI) = [Ran(A− λI)]⊥ , Ker(A− λI) ⊂ [Ran(A∗ − λI)]⊥ (2.20)

where, again, the inclusion becomes an identity if A ∈ B(H) or also if A is closable provided Ais replaced by A in the second formula.

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2.2.6 Criteria for (essential) selfadjointness

Let us briefly introduce some commonly used mathematical technology to study (essential)selfadjointness of symmetric operators. If A is a densely defined symmetric operator in theHilbert space H, define the deficiency indices [ReSi80, Rud91, Schm12, Tes14, Mor18]

n± := dimH± (cardinal numbers in general), where H± := Ker(A∗ ± iI).

Proposition 2.11. If A is a symmetric operator in the Hilbert space H, the following holds.(a) The following facts are equivalent,

(i) A is selfadjoint,

(ii) n+ = n− = 0 and A is closed,

(iii) Ran(A± iI) = H.

(b) The following facts are equivalent,

(i) A is essentially selfadjoint,

(ii) n+ = n− = 0.

(iii) Ran(A± iI) = H.

Proof. (a) Assume (i) A = A∗. Then A is closed because A∗ is closed. Furthermore, if A∗x± ±ix = 0 then 〈x|A∗x〉 = ±i||x±||2. But 〈x±|A∗x±〉 = 〈x±|Ax±〉 is real so that the only possibilityis ||x±|| = 0 and n± = 0. We have proved that (i) implies (ii). Let us prove that (ii) implies (iii).Suppose that the symmetric perator A is closed and n± = 0. The latter condition explicitlyreads Ker(A∗ ± iI) = 0 which, in turn, means that Ran(A± iI) is dense in H due to (2.20).Since A ± iI is closed because A is closed, we more strongly have (iii) Ran(A ± iI) = Hbecause Ran(A ± iI) is closed as well. Indeed, suppose that Axn + ixn → y. From ||xn||2 ≤||Axn||2 + ||xn||2 = ||Axn+ ixn||2, that is valid because A ⊂ A∗, we infer that xnn∈N is Cauchyand thus xn → x ∈ H. Since A+iI is closed, x ∈ D(A+iI) and y = (A+iI)x as wanted. The caseof A− iI is identical. To conclude, we prove that (iii) entails (i) A∗ = A. Since A is symmetricit is enough establishing that D(A∗) ⊂ D(A). Take y ∈ D(A∗). Since Ran(A ± iI) = H, wemust have A∗y ± iy = Ax± ± ix± for a couple x+, x− ∈ D(A). Since A∗D(A)= A, it therefore

holds (A∗ ± iI)(y − x±) = 0. But we know that Ker(A∗ ± iI) = Ran(A ± iI)⊥ = 0, so thaty = x± ∈ D(A), concluding the proof of (a).(b) If, according to (i), A is essentially selfadjoint, then A∗ is selfadjoint A∗∗ = A∗, so that(ii) is valid from (ii)(a). (ii) is equivalent to (iii) in view of (2.20). To conclude, it is enoughdemonstrating that (ii) entails that the closure (it exists because A∗ ⊃ A) A is selfadjoint. Infact, this is equivalent to our thesis (i) from (c) Remark 2.9. Since A is symmetric, we can use(a). We know that A

∗= A∗ from Corollary 2.1. Since A∗ satisfies (ii) by hypothesis, A

∗satisfies

(a)(ii) and A is closed, hence it is selfadjoint because (a)(ii) implies (a)(i).

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Remark 2.11. It is possible to prove that a symmetric operator A admits selfadjoint ex-tensions if and only if n+ = n−. These extension are labelled by means of n+ parameters[Tes14, Mor18].

In view of Remark 2.11, a sufficient condition due to von Neumann for that a symmetric op-erator admits selfadjoint extensions is the following one, where a conjugation is an isometricsurjective anti linear map with CC = I.

Proposition 2.12. If A : D(A)→ H is a symmetric operator in the Hilbert space H and thereis a conjugation C : H→ H such that CA ⊂ AC, then A admits selfadjoint extensions.

Proof. Using the definition of A∗ and D(A∗) and observing that (from the polarization identity(2.1)) 〈Cy|Cx〉 = 〈y|x〉, the condition AC ⊂ CA implies the condition CA∗ ⊂ A∗C . There-fore, remembering that CC = I, we have that A∗x = ±ix if and only if A∗Cx = C(±ix) = ∓iCx.Since C preserves orthogonality and norm of vectors, it transforms a Hilbert basis of H+ into aHilbert basis of H− and vice versa. We conclude that n+ = n−. The thesis finally arises formRemark 2.11.

Taking C as the standard conjugation of functions in L2(Rn, dnx), this result proves in particularthat all operators in QM of the Schordinger form as (1.25) admit selfadjoint extensions whendefined on dense domains.

Exercise 2.8. Relying on Proposition 2.11 and Remark 2.11, prove that a symmetric oper-ator that admits a unique selfadjoint extension is necessarily essentially selfadjoint.

Solution. By Remark 2.11, n+ = n− if the operator admits selfadjoint extensions. Further-more, if n± 6= 0 there are many selfadjoint extension again for Remark 2.11. The only possibilityfor the uniqueness of the selfadjoint extension is n± = 0. Proposition 2.11 implies that A isessentially selfadjoint. 2

A very useful criterion to establish the essentially selfadjointness of a symmetric operator is dueto Nelson. It relies upon an important definition.

Definition 2.8. Let A be an operator in the complex Hilbert space H.If ψ ∈ ∩n∈ND(An) satisfies

+∞∑n=0

tn

n!||Anψ|| < +∞ for some t > 0,

then ψ is said to be an analytic vector of A.

44

We can state Nelson’s criterion here [ReSi80, Mor18].

Theorem 2.5. [Nelson’s criterion] Let A be a symmetric operator in the Hilbert space H,A is essentially selfadjoint if D(A) contains a dense set D of analytic vectors or – which isequivalent –a set D of analytic vectors whose finite span is dense in H.

The above equivalence is due to the fact that a finite linear combination of analytic vector is ananalytic vector as well, the proof being elementary. We have the following evident corollary dueto the fact that the (finite) span of a Hilbert basis is dense and that, if Aψ = aψ, then

+∞∑n=0

tn

n!||Anψ|| =

+∞∑n=0

antn

n!||ψ|| = eat||ψ|| < +∞ for some t ∈ R

Corollary 2.3. If A is a symmetric operator admitting a Hilbert basis of eigenvectors inD(A), then A is essentially selfadjoint.

Example 2.2.(1) For m ∈ 1, 2, . . . , n, consider the operators X ′m and X ′′m in L2(Rn, dnx) with dense domainsD(X ′m) = C∞0 (Rn), D(X ′′m) = S (Rn) for x ∈ Rn and, for ψ, φ in the respective domains,

(X ′mψ)(x) := xmψ(x) , (X ′′mφ)(x) := xmφ(x) ,

where xmis the m-th component of x ∈ Rn. Both operators are symmetric but not selfadjoint.They admit selfadjoint extensions because they commute with the standard complex conjugationof functions (see Proposition 2.12). It is furthermore possible to prove that both operators areessentially selfadjoint as follows. First define the k-axis position operator Xm in L2(Rn, dnx)with domain

D(Xm) :=

ßψ ∈ L2(Rn, dnx)

∣∣∣∣ ∫Rn|xmψ(x)|2dkn

™and

(Xmψ)(x) := xmψ(x) , x ∈ Rn . (2.21)

Just by applying the definition of adjoint one sees that X∗m = Xm so that Xm is selfdjoint[Mor18]. Again applying the definition of adjoint, one sees (see below) that X ′m

∗ = X ′′m∗ = Xm

where we know that the last one is selfadjoint. By definition, X ′m and X ′′m are therefore essen-tially selfadjoint. By (b) in proposition 2.8, X ′m and X ′′m admit a unique selfadjoint extensionwhich must coincide with Xm itself. We conclude that C∞0 (Rn) and S (Rn) are cores (Def. 2.6)for the m-axis position operator.Let us prove that X ′m

∗ = Xm, the proof for X ′′m∗ is identical. By direct inspection one eas-

ily sees that X ′m∗ ⊂ Xm. Let us prove the converse inclusion. We have that φ ∈ D(X ′m

∗) ifand only if there exists ηφ ∈ L2(Rn, dnx) such that

∫φ(x)xmψ(x)dx =

∫ηφ(x)ψ(x)dx, that is∫

(φ(x)xm − ηφ(x))ψ(x)dx = 0, for every ψ ∈ C∞0 (Rn). Fix a compact K ⊂ Rn of the form

45

[a, b]n, obviously the function K 3 x 7→ φ(x)xm−ηφ(x) is L2(K, dx) (we could not prove the sameresult if K were replaced by Rn). Since we can L2(K)-approximate that function with a sequenceof ψn ∈ C∞0 (Rn;C) such that supp(ψn) ⊂ K, we conclude that

∫K |φ(x)xm − ηφ(x)|2dx = 0,

so that K 3 x 7→ φ(x)xm − ηφ(x) is zero a.e.. Since K = [a, b]n was arbitrary, we infer thatRn 3 x 7→ φ(x)xm = ηφ(x) a.e.. In particular, both φ and Rn 3 x 7→ xmφ(x) are L2(Rn, dx) (thelatter because it is a.e. identical to ηφ ∈ L2(Rn, dx)), namely, D(X ′m

∗) 3 φ implies φ ∈ D(Xm).This proves that D(X ′m

∗) ⊂ D(Xm) and consequently X ′m∗ ⊂ Xm as wanted.

(2) For m ∈ 1, 2, . . . , n, the k-axis momentum operator, Pm, is obtained from the positionoperator using the Fourier-Plancherel unitary operator F introduced in example 2.1.

D(Pm) :=

ßψ ∈ L2(Rn, dnx)

∣∣∣∣ ∫Rn|km(Fψ)(k)|2dnk

™and

(Pmψ)(x) := (F−1KmFψ)(x) , x ∈ Rn . (2.22)

Above Km is the m-axis position operator just written for functions (in L2(Rn, dnk)) whosevariable, for pure convenience, is denoted by k instead of x. Indicating by ψ these functions (asis customary in textbooks of quantum physics) we haveÄ

Kmψä

(k) := kmψ(k) k ∈ Rn . (2.23)

Since Km is selfadjoint, Pm is selfadjoint as well, as established in Proposition 2.9 as a conse-quence of the fact that F is unitary.It is possible to give a more explicit form to Pm if restricting its domain. Taking ψ ∈ C∞0 (Rn) ⊂S (Rn) or directly ψ ∈ S (Rn), F reduces to the standard integral Fourier transform (2.12)with inverse (2.13). Using these integral expressions we easily obtain

(Pmψ)(x) = (F−1KmFψ)(x) = −i ∂

∂xmψ(x) (2.24)

because in S (Rn), which is invariant under the Fourier (and inverse Fourier) integral transfor-mation, ∫

Rneik·xkm(Fψ)(k)dnk = −i ∂

∂xm

∫Rneik·x(Fψ)(k)dnk .

This way leads us to consider the operators P ′m and P ′′m in L2(Rn, dnx) with

D(P ′m) = C∞0 (Rn) , D(P ′′m) = S (Rn)

and, for x ∈ Rn and ψ, φ in the respective domains,

(P ′mψ)(x) := −i ∂

∂xmψ(x) , (P ′′mφ)(x) := −i ∂

∂xmφ(x) .

46

Both operators are symmetric as one can easily prove by integrating by parts, but not selfadjoint.They admit selfadjoint extensions because they commute with the conjugation (Cψ)(x) = ψ(−x)(see Proposition 2.12). It is furthermore possible to prove that both operators are essentiallyselfadjoint by direct use of Proposition 2.11 [Mor18]. However we already know that P ′′m is es-sentially selfadjoint as it coincides with the essentially selfadjoint operator F−1K ′′mF beacauseS (Rn) is invariant under F .The unique selfadjoint extension of both operators turns out to be Pm. We conclude thatC∞0 (Rn) and S (Rn) are cores for the m-axis momentum operator.With the given definitions of selfadjoint operators Xk and Pk, S (Rn) turns out to be an invari-ant domain and thereon the CCRs (1.22) hold rigorously.As a final remark to conclude, we say that, if n = 1, D(P ) coincides to the already introduceddomain (1.18). In that domain P is nothing but the weak derivative times the factor −i.

(3) The most elementary example of application of Nelson’s criterion is in L2([0, 1], dx). Consider

A = − d2

dx2with dense domain D(A) given by the functions in C∞([0, 1]) such that ψ(0) = ψ(1)

and dψdx (0) = dψ

dx (1). A is symmetric thereon as it arises immediately from integration by parts,in particular its domain is dense since it includes the Hilbert basis of exponentials ei2πnx, n ∈ Z,which are eigenvectors of A. Thus A withthe said domain is also essentially selfadjoint.

(4) A more interesting case is the Hamiltonian operator of the harmonic oscillator, H[SaTu94] obtained as follows. One starts by

H0 = − 1

2m

d2

dx2+mω2

2x2

with D(H0) := S (R). Above, x2 is the multiplicative operator and m,ω > 0 are constants.To go on, we start by defining a triple of operators, A,A†,N : S (R)→ L2(R, dx) as

A† :=

…mω

2~

Åx− ~

mω

d

dx

ã, A :=

…mω

2~

Åx+

~mω

d

dx

ã, N := A†A .

These operators have the same domain which is also invariant:

A(S (R)) ⊂ S (R) , A†(S (R)) ⊂ S (R) , N(S (R)) ⊂ S (R) .

Using integration by parts, it is also easy to see that A† ⊂ A∗, A ⊂ (A†)∗, and that N isHermitian an also symmetric because S (R) is dense in L2(R, dx). By direct computation andexploiting the given definitions, one immediately sees that

H0 = ~ÅA†A+

1

2I

ã= ~ÅN +

1

2I

ã.

Finally we have the commutation relations, where both sised are evaluated on S (R),

[A,A†] = I . (2.25)

47

Supposing that there exists ψ0 ∈ S (R) such that

||ψ0|| = 1 , Aψ0 = 0 . (2.26)

Starting form (2.25) and using an inductive procedure on the vectors

ψn :=1√n!

(A†)nψ0 , (2.27)

it is quite easy to prove that (e.g., see [Mor18] for details), for n,m = 0, 1, 2, . . ., the relationshold

Aψn =√nψn−1 , A†ψn =

√n+ 1ψn+1 , 〈ψn|ψm〉 = δnm . (2.28)

Finally, the ψn are eigenvectors of H0 (and N) since

H0ψn = ~ωÅA†Aψn +

1

2ψn

ã= ~ω

ÅA†√nψn−1 +

1

2ψn

ã= ~ω

Å√n

2ψn +

1

2ψn

ã= ~ω

Ån+

1

2

ãψn . (2.29)

As a consequence, ψnn∈N is an orthonormal set of vectors. This set is actually a Hilbert basisbecause (1) a solution in S (R) of (2.26) exist (and is unique):

ψ0(x) =1

π1/4√se−

x2

2s2 , s :=

~mω

,

(2) up to a rescaling of the argument, the first identity in (2.28) is a well-known recurrencerelation of the Hilbert basis Hnn∈N of L2(R, dx) made of Schwartz’ functions known as Hermitefunctions, and ψ0(x) =

√sH0(x/s) (the factor

√s is necessary to preserve the L2 normalization

of ψ0 as a consequence of the change of the argument x→ x/s)

ψn(x) =√sHn(x/s) , Hn(x) :=

1√2nπ1/2n!

Åx− d

dx

ãne−x

2/2 , n = 0, 1, . . .

Exploiting Nelson’s criterion, we conclude that the symmetric operator H0 is essentially selfad-joint in D(H0) = S (R) and H := H0 = H∗0 , because H0 admits a Hilbert basis of eigenvectorswith corresponding eigenvalues ~ω(n+ 1

2).

2.3 Basic on spectral theory

Our goal is to extend (1.4) to a formula valid in the infinite dimensional case. As we shall seeshortly, passing to the infinite dimensional case, the sum is replaced by an integral and σ(A)

48

must be enlarged with respect to the pure set of eigenvalues of A. This is because, as alreadynoticed in the first chapter, there are operators which should be decomposed with the prescrip-tion (1.4) but they do not have eigenvalues, though they play a crucial role in QM.

Notation 2.4. If A : D(A) → H is injective, A−1 indicates its inverse when the co-domainof A is restricted to Ran(A). In other words, A−1 : Ran(A)→ D(A).

2.3.1 Resolvent and spectrum

The definition of spectrum of the operator A : D(A)→ H extends the notion of set of eigenvalues.The eigenvalues of A are the numbers λ ∈ C such that (A−λI)−1 does not exist. When passingto infinite dimensions, topological issues take place. As a matter of fact, even if (A − λI)−1

exists, it may be bounded or unbounded and its domain Ran(A−λI) may or may not be dense.These features permit us to define a suitable extension of the notion of set of eigenvalues.

Definition 2.9. Let A be an operator in the Hilbert space H. The resolvent set of A isthe subset of C,

ρ(A) := λ ∈ C | (A− λI) is injective, Ran(A− λI) = H , (A− λI)−1is bounded

The spectrum of A is the complement σ(A) := C \ ρ(A) and it is given by the union of thefollowing pairwise disjoint three parts:

(i) the point-spectrum, σp(A), where A−λI not injective (σp(A) is the set of eigenvaluesof A),

(ii) the continuous spectrum, σc(A), where A − λI injective, Ran(A− λI) = H and (A −λI)−1 not bounded,

(iii) the residual spectrum, σr(A), where A− λI injective and Ran(A− λI) 6= H.

If λ ∈ ρ(A), the the operator

Rλ(A) := (A− λI)−1 : Ran(A− λI)→ D(A)

is called the resolvent operator of A.

The following technically elementary fact provides a notion of approximated eigenvector of anelement of the continuous spectrum of an operator. Even if proper eigenvectors do not exist,they can be arbitrarily approximated.

Proposition 2.13. Let A : D(A)→ H an operator in the Hilbert space H and λ ∈ σc(A). Forevery ε > 0 there is xε ∈ D(A) such that ||Axε − λxε|| < ε but ||xε|| = 1.

49

Proof. Since λ ∈ σc(A), we have that (A − λI)−1 : Ran(A − λI) → D(A) is not bounded.Therefore, for every ε > 0 there is yε ∈ Ran(A − λI) with yε 6= 0 such that ||(A − λI)−1yε|| >ε−1||yε||. By construction, we can write yε = (A − λI)zε for some zε ∈ D(A) \ 0, so that||(A − λI)−1(A − λI)zε|| > ε−1||(A − λI)zε||. In other words, ε||zε|| > ||Azε − λzε||. It is nowevident that xε := ||zε||−1zε satisfies the thesis.

Observe that the written property is also valid (a) if λ ∈ σp(A), simply choosing xε as a λ-eigenvector independent form ε, and also (b) for λ ∈ σr(A) in the case (A − λI)−1 is notbounded. For this reason, it is sometimes convenient to decompose σ(A) differently when deal-ing with operators admitting residual spectrum (this is not the case for normal operators, aswe shall see shortly). The approximated point spectrum σap(A) is made of λ ∈ σ(A)such that, for every ε > 0, there exists xε ∈ D(A) with ||Axε − λxε|| < ε and ||xε|| = 1 (includ-ing the case of Ker(A−λI) = 0). The residual pure spectrum σrp(A) is just σ(A)\σap(A).

In Hilbert spaces, the spectrum and the resolvent of an operator are invariant under unitaryoperators and, more generally, under isomorphisms or anti isomorphisms of Hilbert spaces. Infact, we have the following elementary result whch can be proved immediately by applying thedefinitions and the basic properties of isometric surjective linear (or anti linear) maps.

Proposition 2.14. If U : H→ H′ is an isometric surjective linear (or anti linear) map formthe Hilbert space H onto the Hilbert space H′ and A is any operator in the Hilbert space H, itholds that σ(UAU−1) = σ(A) and, in particular,

σp(UAU∗) = σp(A) , σc(UAU

−1) = σc(A) , σr(UAU−1) = σr(A) . (2.30)

A technically important proposition concerns the notions of resolvent and spectrum in the par-ticular case where the involved operator is closed. Here things strongly simplify.

Proposition 2.15. Let A : D(A) → H be a closed operator in the Hilbert space H or, inparticular, A ∈ B(H). It turns out that λ ∈ ρ(A) if and only if A− λI admits an inverse whichbelongs to B(H), in particular Ran(A− λI) = H.

Proof. If (A − λI)−1 ∈ B(H), it must be Ran(A− λI) = Ran(A − λI) = H and (A − λI)−1 isbounded, so that λ ∈ ρ(A) by definition. Let us prove the converse. Suppose that λ ∈ ρ(A). Weknow that (A−λI)−1 is defined on the dense domain Ran(A−λI) and is bounded. To conclude,it is therefore enough proving that y ∈ H implies y ∈ Ran(A − λI). To this end, notice that ify ∈ H = Ran(A− λI), then y = limn→+∞(A− λI)xn for some xn ∈ D(A− λI). The sequenceof xn converges. Indeed, H is complete and xnn∈N is Cauchy as (1) xn = (A − λI)−1yn, (2)||xn − xm|| ≤ ||(A − λI)−1|| ||yn − ym||, and (3) yn → y. To end the proof, we observe that,A−λI is closed since A is such ((b) in remark 2.8). It must consequently be ((c) in remark 2.8)x = limn→+∞ xn ∈ D(A− λI) and y = (A− λI)x ∈ Ran(A− λI).

50

Remark 2.12.(a) As a consequence of the established result, if dealing with closed operators A : D(A)→ H

or operators A ∈ B(H), the definition of resolvent simplifies:

ρ(A) := λ ∈ C | ∃(A− λI)−1 ∈ B(H) .

Some textbooks start with this definition form scratch. In these cases, since all operators(A− λI)−1 have the same domain H when λ ∈ ρ(A), Rµ(A)−Rλ(A) is everywhere defined.

(b) Actually the thesis of Proposition 2.15 can be stated into an even stronger form. In fact,if A is closed, A − λI is closed and is such its inverse (A − λI)−1 (the have the same graph).So if this operator is everywhere defined on H, it is also automatically bounded in view of theclosed graph theorem. So we have an alternate formulation we state below.

Proposition 2.16. Let A : D(A) → H be a closed operator in the Hilbert space H or, inparticular, A ∈ B(H). It turns out that λ ∈ ρ(A) if and only if A− λI is a bijection from D(A)to the whole H.

The definitions of resolvent and spectrum can be extended as they stand to the case where His replaced by a complex Banach space [Rud91, Mor18]. Even more generally, it can be givenreplacing operators for elements af an abstract unital Banach algebra A.

Definition 2.10. If A is a unital Banach algebra, the resolvent of a ∈ A is made of allλ ∈ C such that (a− λ11) admits inverse in A denoted by Rλ(a) ∈ A. The spectrum of a ∈ Ais σ(a) := C \ ρ(a).

No finer spectral decompositions are made.

If A is closed, the so-called resolvent identity is true, which is also evidently valid for unitalBanach algebras replacing Rz(A) for Rz(a).

Proposition 2.17. Let A : D(A)→ H be a closed operator (or more strongly A ∈ B(H)) inthe Hilbert space H and µ, λ ∈ ρ(A). It holds that

Rµ(A)−Rλ(A) = (µ− λ)Rµ(A)Rλ(A) , (2.31)

and the above indentity is called resolvent identity.

Proof. First, we have Rλ(A)(A − λI) = I D(A) and (A − µI)Rµ(A) = I. As a consequence,Rλ(A)(A − λI)Rµ(A) = Rµ(A) and Rλ(A)(A − µI)Rµ(A) = Rλ(A). Taking the difference ofthe equations we produce (2.31).

We prove that in the special case of A ∈ B(A) we can assert that ρ(A) 6= ∅. The same statementand proof applies to unital Banach algebras.

51

Proposition 2.18. Let H be a Hilbert space and A ∈ B(H), then λ ∈ ρ(A) if |λ| > ||A||, soσ(A) is bounded by ||A||.

Proof. The series Sλ := −∑+∞n=0 λ

−(n+1)An (where A0 := I) converges in the operator norm ofB(H) for |λ| > ||A|| since it is dominated by the complex series

∑+∞n=0 |λ|−(n+1)||A||n and B(H)

is a Banach space. Furthermore

Sλ(A− λI) = (A− λI)Sλ =+∞∑n=0

Ä−λ−(n+1)An+1 + λ−nAn

ä= I ,

so that Sλ = Rλ(A) and λ ∈ ρ(A).

There are some general properties of the spectrum and the resolvent set which deserve attentionbecause they are relevant for applications to QM. The most important properties are stated inthe following proposition. Both statements extend with the same proof to the case of boundedoperators over Banach spaces or general unital Banach algebras A simply replacing 〈x|Rλ(A)y〉for f(Rλ(a)) with f ∈ A∗ (the topological dual of A).

Proposition 2.19. Let A : D(A) → H be a closed operator in the Hilbert space H, thefollowing facts are true.

(a) ρ(A) is open, σ(A) is closed and ρ(A) 3 λ 7→ 〈x|Rλ(A)y〉 ∈ C is holomorphic for everyx, y ∈ H if ρ(A) 6= ∅.

(b) if A ∈ B(H), then

(i) σ(A) 6= ∅,

(ii) ρ(A) 6= ∅.

(iii) σ(A) is compact.

Proof. Let us start from (b). Statement (ii) has been already proved in Proposition 2.18 andthis proves also (iii) provided (i) holds. (i) is established studying the function ρ(A) 3 λ 7→fxy(λ) := 〈y|(A − λI)−1x〉 ∈ C for every fixed x, y ∈ H. Using again the expansion in theproof of Proposition 2.18, we have fxy(λ) = −∑+∞

n=0 λ−(n+1)〈y|Anx〉. The series in the right-

hand side, for |λ| > |λ0|, is dominated by the series of constants∑+∞n=0 λ

−(n+1)0 ||A||n||x||||y||

which converges as |λ0| > ||A||, therefore the series of fxy converges absolutely and uniformly inλ ∈ C | |λ| > |λ0|. Exploiting dominated convergence theorem, we conclude that fxy(λ) → 0as |λ| → +∞. fxy is holomorphic because it is uniform limit of holomorphic functions (useMorera’s theorem). If it were ρ(A) = C, Liouville theorem would imply that fxy is constantfor every y, x ∈ H and that fxy(λ) = 0 everywhere in view of the computed limit. Hence(A− λI)−1 = 0, would not make sense for an invertible operator. We conclude that ρ(A) 6= C,so that σ(A) 6= ∅. Dealing with abstract Banach algebras A 3 a, the function fxy has to be

52

replaced by F (λ) = f((a− λ11)−1) for every element f of the topological dual A∗ and the proofgoes on similarly.(a) Suppose that λ0 ∈ ρ(A) and consider λ ∈ C with |λ−λ0| < ||Rλ0(A)||−1. We therefore have

A−λI = [(λ0−λ)I+(A−λ0I)] = (A−λ0I)[(λ−λ0)Rλ0(A)+I] = Rλ0(A)−1[(λ−λ0)Rλ0(A)+I],

so that(A− λI)−1 = [(λ− λ0)Rλ0(A) + I]−1Rλ0(A)

provided [(λ − λ0I)Rλ0(A) + I]−1 exists. With the same argument exploited in the proof ofProposition 2.18, we have that, for |λ− λ0| < ||Rλ0(A)||−1,

[(λ− λ0)Rλ0(A) + I]−1 =+∞∑n=0

(λ0 − λ)nRλ0(A)n . (2.32)

We have demonstrated that every point λ0 ∈ ρ(A) admits an open neighbourhood where Rλ(A)exists, we can therefore argue that ρ(A) ⊂ C is open and its complement σ(A) is closed. Since,if ρ(A) 6= ∅, the function ρ(A) 3 λ 7→ 〈x|(A− λI)−1y〉 admits a Taylor expansion around everyλ ∈ ρ(A) trivially constructed out of (2.32), the function is holomorphic.

Remark 2.13. If A ∈ B(H) and A is normal, then the spectral radius identity holds

sup|λ| | λ ∈ σ(A) = ||A|| , (2.33)

the spectral radius of A being, by definition, the left-hand side of (2.33). We shall derive thisidentity for selfadjoint operators as an immediate consequence of the spectral theorem. However,in Proposition 2.40 we shall establish its general version for normal operators independently fromthe spectral theorem. This identity is valid also replacing A for an element a of an abstractunital C∗-algebra which is normal: a∗a = aa∗.

2.3.2 Spectra of some types of operators

We are in a position to state and prove some general properties of the spectra of selfadjoint andunitary operators.

Proposition 2.20. Let A : D(A) → H be a densely defined in the Hilbert space H. Thefollowing facts hold true.

(a) If A is selfadjoint, then σ(A) ⊂ R.

(b) If A is unitary, then σ(A) ⊂ T := z ∈ C | |z| = 1.

(c) If A is normal (in particular selfadjoint or unitary), the following further facts hold, wherethe bar in (ii) and (iii) denotes the complex conjugation of the elements of the consideredsets.

53

(i) σr(A) = σr(A∗) = ∅,

(ii) σp(A) = σp(A∗), in particular for x 6= 0, Ax = λx if and only if A∗x = λx,

(iii) σc(A) = σc(A∗).

(d) If A is normal (in particular selfadjoint or unitary), then eigenvectors with different deigen-values are ortogonal.

Proof. (a) Suppose λ = µ+ iν with ν 6= 0 and let us prove λ ∈ ρ(A). If x ∈ D(A),

〈(A− λI)x|(A− λI)x〉 = 〈(A− µI)x|(A− µI)x〉+ ν2〈x|x〉+ iν[〈Ax|x〉 − 〈x|Ax〉] .

The last summand vanishes for A is selfadjoint. Hence

||(A− λI)x|| ≥ |ν| ||x|| .

With a similar argument we obtain

||(A− λI)x|| ≥ |ν| ||x|| .

The operators A− λI and A− λI are injective, and ||(A− λI)−1|| ≤ |ν|−1, where (A− λI)−1 :Ran(A− λI)→ D(A). Notice that, from (2.20),

Ran(A− λI)⊥

= [Ran(A− λI)]⊥ = Ker(A∗ − λI) = Ker(A− λI) = 0 ,

where the last equality makes use of the injectivity of A−λI. Summarising: A−λI in injective,

(A − λI)−1 bounded and Ran(A− λI)⊥

= 0, i.e. Ran(A − λI) is dense in H; thereforeλ ∈ ρ(A), by definition of resolvent set.(b) Suppose that λ ∈ C and |λ| 6= 1, we want to prove that λ ∈ ρ(A). If x ∈ H = D(A) we have

〈(A− λI)x|(A− λI)x〉 = 〈Ax|Ax〉+ |λ|2〈x|x〉 − 2Re(λ〈Ax|x〉) .

In other words, using 〈Ax|Ax〉 = 〈x|x〉 = ||x||2 and |〈Ax|x〉| ≤ ||x||2||A|| = ||x||2,

||(A− λI)x||2 ≥ (1 + |λ|2)||x||2 − 2|λ|||x||2 = (1 + |λ|2 − 2|λ|)||x||2 .

Summing up, we have proved that ||(A− λI)x||2 ≥ (1− |λ|)2||x||2.As in (a), since (1−|λ|)2 6= 0, this inequality implies thatKer(A−λI) = 0, that ||(A−λI)−1|| ≤(1−|λ|)−1, and that Ran(A−λI) is dense because Ran(A− λI)

⊥= Ker(A∗−λI) = 0, where,

in the last identity we have noticed that A∗ is unitary as A is unitary and |λ| = |λ| 6= 1, so thatthe previous argument applies.(c) First of all observe that for a normal operator A it holds that Ker(A) = Ker(A∗). Indeed,if x ∈ Ker(A), then Ax = 0 and thus A∗Ax = A∗0 = 0, so that by definition of normaloperator, AA∗x = A∗Ax = 0, in particular x ∈ D(A∗), and therefore 〈x|AA∗x〉 = 0. As aconsequence, ||A∗x||2 = 〈A∗x|A∗x〉 = 〈x|AA∗x〉 = 0 so that x ∈ Ker(A∗). Suppose, conversely,

54

that x ∈ Ker(A∗). Thus A∗x = 0 and AA∗x = A0 = 0. Using normality, A∗Ax = AA∗x = 0. Inparticular, since normal operators are closed by definition, x ∈ D(A) = D(A) = D((A∗)∗) andtherefore 〈x|A∗Ax〉 = 0 means 〈(A∗)∗x|Ax〉 = 〈Ax|Ax〉 = 0 which is nothing but ||Ax||2 = 0, sothat x ∈ Ker(A).Let us pass to prove that (i) σr(A) = ∅. Suppose λ ∈ σ(A), but λ 6∈ σp(A). Then A−λI must beinjective implying Ker(A−λI) = 0. Since A−λI is normal if A is such (in particular is closedfrom (b) in remark 2.8), we also conclude that Ker(A∗− λI) = Ker(A− λI) = 0. Therefore,[Ran(A − λI)]⊥ = Ker(A∗ − λI) = 0 due to (2.20), and Ran(A− λI) = H. Consequentlyλ ∈ σc(A) and no complex number of σ(A) is allowed to belong to σr(A). Observing that A∗

is normal if A is normal, we conclude that also σr(A∗) = ∅. The identity (ii) σp(A) = σp(A∗)

immediately arises from Ker(A−λI) = Ker(A∗−λI), using (2.19) and noticing that the involvedoperators are closed. The same argument used above to prove that Ker(A) = Ker(A∗) appliedto A−λI and A∗−λI proves that ||(A−λI)x|| = 0 if and only if ||(A∗−λI)x|| = 0 completing theproof of (ii). The proof of (iii), σc(A) = σc(A∗), is more involved. Suppose that λ ∈ σc(A) thenKer(A−λI) must be trivial – so that also Ker(A∗−λI) is trivial and (A∗−λI)−1 exists – andthe inverse (A−λI)−1 is an element of B(H) due to Proposition 2.15 since normal operators areclosed by definition. From (A−λI)−1(A−λI) = I|D(A), taking advantage of (2.8), we also have

(A∗−λI)(A−λI)−1∗ = I|∗D(A) = I. In particular (A∗−λI)(A−λI)−1∗|Ran(A∗−λI) = I|Ran(A∗−λI).

Since we already know that (A∗ − λI) is bijective from D(A∗) to Ran(A∗ − λI), we concludethat

(A− λI)−1∗|Ran(A∗−λI) = (A∗ − λI)−1

because the inverse element is unique. In particular, the right-hand side is bounded since the left-hand side is bounded. We conclude that λ ∈ σc(A) implies λ ∈ σc(A∗). Repeating the argumentstarting from A∗ and observing that (A∗ − λI)∗ = A− λI, we conclude that λ ∈ σc(A∗) implies

λ = λ ∈ σc(A), concluding the proof of (iii).(d) If λ 6= µ and Au = λu, Av = µv, then µ〈u|v〉 = 〈u|Av〉 = 〈A∗u|v〉 = λ〈u|v〉, so that(µ− λ)〈u|v〉 = 0 which is possible only if 〈u|v〉 = 0 because µ− λ 6= 0.

Example 2.3. The m-axis position operator Xm in L2(Rn, dnx) introduced in (1) Example2.2 satisfies

σ(Xm) = σc(Xm) = R . (2.34)

The proof can be obtained as follows. First observe that σ(Xm) ⊂ R since the operator isselfadjoint (Proposition 2.20). However σp(Xm) = ∅ as observed in Section 1.3 and σr(Xm) = ∅because Xm is selfadjoint (Proposition 2.20). Let us study the possibility that r ∈ R belongs toρ(Xm). If no r ∈ R belongs to ρ(Xm), we must conclude that σ(Xm) = σc(Xm) = R.Suppose that, for some r ∈ R, (Xm−rI)−1 exists and is bounded. If ψ ∈ D(Xm−rI) = D(Xm)with ||ψ|| = 1 we have ||ψ|| = ||(Xm−rI)−1(Xm−rI)ψ|| and thus ||ψ|| ≤ ||(Xm−rI)−1|| ||(Xm−rI)ψ||. Therefore

||(Xm − rI)−1|| ≥ 1

||(Xm − rI)ψ||.

55

For every fixed ε > 0, it is simply constructed ψ ∈ D(Xm) with ||ψ|| = 1 and ||(Xm−rI)ψ|| < ε.Assuming m = 1, it is sufficient considering the class of sets [r − 1/k, r + 1/k] × Rn−1 and acorresponding sequence of functions ψk ∈ C∞0 (Rn,C) such that supp(ψk) ⊂ [r − 1/k, r + 1/k]×Rn−1 and

∫Rn |ψk|2dnx = 1. In this case, for k → +∞

0 ≤ ||(Xm − rI)ψ||2 ≤∫Rn|x1 − r|2|ψ(x)|2dnx ≤ 4

k2

∫Rn|ψ(x)|2dnx =

4

k2→ 0 .

Therefore (Xm− rI)−1 cannot be bounded so that r ∈ σ(Xm), more precisely r ∈ σc(Xm) sinceno other possibility is allowed.In view of Proposition 2.14, we also conclude that

σ(Pm) = σc(Pm) = R , (2.35)

just because the momentum operator Pm is related to the position one by means of a unitaryoperator given by the Fourier-Plancherel operator F as discussed in (2) Example 2.2.

2.4 Integration of projector-valued measures

We introduce in this section the most important technical tool of the spectral theory, the notionof projector-valued measure, with fundamental consequences in the interpretation of quantumtheories. Before we do it, we prove some important elementary fact on orthogonal projectors.

2.4.1 Orthogonal projectors

Definition 2.11. Let H be a Hilbert space. P ∈ B(H) is called orthogonal projectorwhen PP = P and P ∗ = P . L (H) denotes the set of orthogonal projectors of H.

A well known relation exists between orthogonal projectors and closed subspaces.

Proposition 2.21. Let H be a Hilbert space with set of orthogonal projectors L (H). Thefollowing facts are valid.

(a) If P ∈ L (H), then P (H) is a closed subspace.

(b) If P ∈ L (H), then Q := I − P ∈ L (H) and Q(H) = P (H)⊥.

(c) The direct orthogonal decomposition H = P (H) ⊕ Q(H) is valid, so that, in particular,z ∈ H is uniquely decomposed as z = x + y with x ∈ P (H), y ∈ Q(H). It holds x = P (z)and y = Q(z).

(d) If H0 ⊂ H is a closed subspace, there exists exactly one P ∈ L (H) such that it projectsonto H0, i.e. P (H) = H0.

56

Proof. (a) It is clear that P (H) is a subspace. It is also closed because, if x = limn→+∞ Pxn,then x = Px. Indeed, Px = P limn→+∞ P (xn) = limn→+∞ PPxn = limn→+∞ Pxn = x since Pis continuous.(b) We have (I−P )∗ = I∗−P ∗ = I−P and (I−P )(I−P ) = I−2P+PP = I−2P+P = I−P ,so Q := I−P ∈ L (H). We pass to prove that Q(H) = P (H)⊥. First of all, observe that y ∈ Q(H)and x ∈ P (H) yield 〈y|x〉 = 〈(I−P )y|Px〉 = 〈y|(I−P )Px〉 = 〈y|(P−PP )x〉 = 〈y|(P−P )x〉 = 0.Therefore Q(H) ⊂ P (H)⊥. To conclude, we have to prove that Q(H) ⊃ P (H)⊥. If y ∈ P (H)⊥

we have 〈Py|u〉 = 〈y|Pu〉 = 0 for u ∈ H and therefore Py = 0. As a consequence, if we definez = y+x with x ∈ P (H), we obtain Qz = (I−P )y+(I−P )x = x+y−Py−Px = z−Py−Px =z − 0− x = y. In other words, if y ∈ P (H)⊥, then y ∈ Q(H), proving Q(H) ⊃ P (H)⊥.(d) and (c). Consider a closed subspace H0. It is a Hilbert space in its own right since it includesthe limits of its Cauchy sequences (which converge in H since H is Hilbert). Therefore H0 admitsa Hilbert basis N . It is easy to prove that if N ′ is a Hilbert basis of H⊥0 , then N ∪N ′ is a Hilbertbasis of H and the orthogonal direct decomposition holds H = H0 ⊕ H⊥0 as stated in (2.3) withM = H0 so that spanM = H0. Consider the operator Px :=

∑z∈N 〈z|x〉z for x ∈ H. Using the

Hilbert decomposition u =∑z∈N∪N ′〈z|u〉z, one immediately proves that ||P || ≤ 1, PP = P ,

〈Px|y〉 = 〈x|Py〉 and thus P = P ∗, so that P ∈ L (H). Finally P (H) = H0 since N is a Hilbertbasis of H0.Let us demonstrate that the orthogonal projector P such that P (H) = H0 is uniquely determinedby H0. The proof also establishes (c). Since P (H) ∩ Q(H) = 0, because the subspaces aremutually orthogonal, and I = P + Q, we conclude that z ∈ H can be uniquely decomposedas z = x + y with x ∈ P (H) and y ∈ Q(H) and x = Pz, y = Qz. This fact proves thatthe orthogonal projector P with P (H) = H0 is unique: if P ′(H) = H0, we would have thatQ′ := I − P ′ projects onto H⊥0 and z ∈ H is uniquely decomposed as z = x + y with x ∈ H0,y ∈ H⊥0 where x = Pz = P ′z and y = Qz = Q′z. Hence P ′z = Pz for all z ∈ H.

If P ∈ L (H), then P and Q := I − P project onto mutually orthogonal subspaces andPQ = QP = 0. This fact is general according to the next elementary result.

Proposition 2.22. Let H be a Hilbert space. P,Q ∈ L (H) project onto orthogonal subspacesif and only if PQ = 0. In this case QP = 0 is also valid.

Proof. If P (H) ⊥ Q(H) then for every x, y ∈ H we have 0 = 〈Px|Qy〉 = 〈x|PQy〉. ThereforePQ = 0. Taking the adjoint of both sides we have also QP = 0. If conversely PQ = 0, from theidentity above we have 〈Px|Qy〉 = 0 for every x, y ∈ H so that P (H) ⊥ Q(H).

Let us pass to state and prove some further properties of orthogonal projectors related with anatural ordering relation which will play a crucial role in the next chapter.

Notation 2.5. Referring to Proposition 2.21, if P,Q ∈ L (H), we write P ≥ Q if and only ifP (H) ⊃ Q(H).

Proposition 2.23. If H is a Hilbert space and P,Q ∈ L (H). The following facts are valid.

57

(a) P ≥ Q is equivalent to PQ = Q. In this case QP = Q is also valid.

(b) P ≥ Q is equivalent to 〈x|Px〉 ≥ 〈x|Qx〉 for every x ∈ H.

Proof. (a) If P (H) ⊃ Q(H) then there is a Hilbert basis of P (H) NP = NQ ∪ N ′Q where NQ

ia a Hilbert basis of Q(H) and N ′Q of Q(H)⊥P , the notion of orthogonal being referred to theHilbert space P (H). From Q =

∑z∈NQ〈z|·〉z and P = Q +

∑z∈N ′Q〈z|·〉z we have PQ = Q.

The converse implication is obvious. Assume PQ = Q. If x ∈ Q(H) then Qx = x. ThereforePx = PQx = Qx = x so that x ∈ P (H), so that Q(H) ⊂ P (H) as wanted. Finally, taking theadjoint of both sides of PQ = Q, we have QP = Q since P and Q are selfadjoint.(b) Assume P ≥ Q, i.e. Q(H) ⊂ P (H). So, if x ∈ H, the vector Px ∈ P (H) can be decomposedinto y + z where y := QPx ∈ Q(H) and z ∈ P (H) is orthogonal to y. Therefore ||Px||2 =||QPx||2 + ||z||2. From (a), ||Px||2 = ||Qx||2 + ||z||2 which implies ||Px||2 ≥ ||Qx||2, namely〈x|Px〉 ≥ 〈x|Qx〉 for every x ∈ H. Conversely, if 〈x|Px〉 ≥ 〈x|Qx〉 for every x ∈ H, then||Px||2 ≥ ||Qx||2 for every x ∈ H, so that Px = 0 implies Qx = 0 for every x ∈ H. In otherwords P (H)⊥ ⊂ Q(H)⊥. Applying ⊥ again, it arises P (H) ⊃ Q(H).

Proposition 2.24. If H is a Hilbert space and Pnn∈N ∈ L (H) is a sequence such thateither (i) Pn ≤ Pn+1 for all n ∈ N or (ii) Pn ≥ Pn+1 for all n ∈ N, then Pnx → Px, for everyx ∈ H and some P ∈ L (H), as n→ +∞.

Proof. Assume Pn ≤ Pn+1 for all n ∈ N. For x ∈ H, Pnxn∈N is Cauchy. Indeed, for n > mand using (a) of Proposition 2.23 together with selfadjointness and idempotence of orthogonalprojectors, ||Pnx− Pmx||2 equals

〈x|(Pn − Pm)(Pn − Pm)x〉 = 〈x|(Pn − Pm − Pm + Pm)x〉 = ||Pnx||2 − ||Pmx||2 .

Since the sequence of numbers ||Pnx||2 = 〈x|Pnx〉 is non-decreasing and bounded by ||x||2, itconverges to some real and hence it is of Cauchy type. This implies that Pnxn∈N is Cauchyas well. The map P : H 3 x 7→ limn→+∞ Pnx ∈ H is linear by construction. Furthermore〈Px|y〉 = 〈x|Py〉 for every x, y ∈ H by continuity of the scalar product, so that P = P ∗. Finally,for every x, y ∈ H we also have 〈Px|Py〉 = limn→+∞〈Pnx|Pny〉 = limn→+∞〈x|Pny〉 = 〈x|Py〉,so that PP = P and therefore P ∈ L (H). The other case as an identical proof barring trivialchanges.

2.4.2 Projector-valued measures (PVM)

We can now state one of the most important definitions in spectral theory.

Definition 2.12. Let H be a Hilbert space and Σ(X) a σ-algebra over X. A projector-valued measure (PVM) on X, is a map P : Σ(X) 3 E 7→ PE ∈ L (H) such that

(i) PX = I,

(ii) PEPF = PE∩F ,

58

(iii) If N ⊂ N and Ekk∈N ⊂ Σ(X) satisfies Ej ∩ Ek = ∅ for k 6= j, then∑j∈N

PEjx = P∪j∈NEjx for every x ∈ H.

If N is infinite, the sum on the left hand side of (iii) is computed referring to the topology of H.We say that P is concentrated on S ∈ σ(X) if PE = PE∩S for every E ∈ Σ(X).

Remark 2.14.(a) (i) and (iii) with N = 1, 2 imply that P∅ = 0 using E1 = X and E2 = ∅. Next (ii)

entails that PEPF = 0 if E ∩ F = ∅ from Proposition 2.22. In particular, the vectors PEjx inthe sum on the left hand side of (iii) are mutually orthogonal. Therefore a series (we treat thecase N = N) ∑

j∈NPEjx , (2.36)

where Ej ∩ Ek = ∅ for k 6= j, always converges. Alternatively, the convergence of the saidseries immediately arises from Proposition 2.24, since the operators

∑nj=0 PEj are orthogonal

projectors and∑nj=0 PEj ≤

∑n+1j=0 PEj . (The series (2.36) can be re-ordered arbitrarily without

changing the final sum because, from Bessel inequality (2.1) we have∑j∈N||PEjx||2 ≤

∑j∈N

∑u∈Mj

|〈u|x〉|2 < +∞

where Mj ⊂ PEj (H) is a Hilbert basis of PEj (H). Now we can apply Lemma 2.1 proving that(2.36) converges and can be re-ordered arbitrarily.) It is however a useful exercise to explicitlyprove that the series converges. For a given ε > 0, taking advantage of the continuity of thescalar product and of the fact that PEjx ⊥ PEkx if j 6= k, we have for m > n∣∣∣∣∣∣∣∣∣∣∣∣m∑j=0

PEjx−n−1∑j=0

PEjx

∣∣∣∣∣∣∣∣∣∣∣∣2

=

∣∣∣∣∣∣∣∣∣∣∣∣j=m∑j=n

PEjx

∣∣∣∣∣∣∣∣∣∣∣∣2

=

∞j=m∑j=n

PEjx

∣∣∣∣∣∣k=m∑k=n

PEkx

∫=

j=m∑j=n

∞PEjx

∣∣∣∣∣∣k=m∑k=n

PEkx

∫=

j=m∑j=n

∞x

∣∣∣∣∣∣PEjk=m∑k=n

PEkx

∫=

j=m∑j=n

∞x

∣∣∣∣∣∣k=m∑k=n

PEjPEkx

∫=

j=m∑j=n

∞x

∣∣∣∣∣∣k=m∑k=n

δjkPEkx

∫=

j=m∑j=n

⟨x∣∣∣PEjx⟩ =

j=m∑j=n

⟨x∣∣∣PEjPEjx⟩ =

j=m∑j=n

⟨PEjx

∣∣∣PEjx⟩ =j=m∑j=n

||PEjx||2 < ε

if n,m > Nε and thus the series (2.36) converges as their truncated sums form a Cauchy sequence.In summary, (iii) can be viewed as a condition on the value of the sum of the series and not anassumption about the convergence of the series.

59

(b) If x, y ∈ H, Σ(X) 3 E 7→ 〈x|PEy〉 =: µ(P )xy (E) is a complex measure whose (finite) total

variation [Rud91] will be denoted by |µ(P )xy |. All that arises from the definition of PVM, in

particular continuity of the scalar product implying σ-additivity: If the sets En ⊂ Σ(X), withn ∈ N ⊂ N are pairwise disjoint (En ∩ Em = ∅ for n 6= m),

µ(P )xy (∪n∈NEn) =

⟨x∣∣P∪n∈NEny ⟩ =

∞x

∣∣∣∣∣∣∑n∈N PEny∫

=∑n∈N〈x |PEny 〉 =

∑n∈N

µ(P )xy (En) .

From the definition of µxy, we immediately have three important facts.

(i) µ(P )xy (X) = 〈x|y〉.

(ii) µ(P )xx is always positive and finite and µ

(P )xx (X) = ||x||2.

(iii) Consider a simple function [Rud91] s =∑nk=1 skχEk , where sk ∈ C and the sets Ek ∈ Σ(X),

with k = 1, . . . , n are pairwise disjoint. If h is the Radon-Nikodym derivative of µxy withrespect to its total variation |µxy| (see, e.g., [Mor18]), we have

∫Xsdµxy =

∫Xshd|µxy| =

n∑k=1

sk

∫Ek

hd|µxy| =n∑k=1

skµxy(Ek) =

∞x

∣∣∣∣∣∣n∑k=1

skPEky

∫=

≠x

∣∣∣∣∫Xs(λ)dP (λ) y

∑(2.37)

where we have defined ∫Xs(λ)dP (λ) :=

n∑k=1

skPEk .

All the machinery of spectral theory and measurable functional calculus relies upon (2.37) ex-tended from simple functions s to general measurable functions f .

Example 2.4.(1) The simplest example of PVM is related to a Hilbert basis N in a Hilbert space H. We candefine Σ(N) as the class of all subsets of N itself. Next, for E ∈ Σ(N) and z ∈ H we define

PEz :=∑x∈E〈x|z〉x

and P∅ := 0. It is easy to prove that the class of all PE defined this way forms a PVM on N .(This definition can be also given if H is non-separable and N is uncountable, since for everyy ∈ H only an at most countable subset of elements x ∈ E satisfy 〈x|y〉 6= 0). Observe that

60

PNx =∑u∈N 〈u|x〉u = x for every x ∈ H, so that PN = I as required.

In particular µ(P )xy (E) = 〈x|PEy〉 =

∑z∈E〈x|z〉〈z|y〉 and µ

(P )xx (E) =

∑z∈E |〈x|z〉|2.

(2) A more complicated version of (1) consists of a PVM constructed out of a Hilbert directsum

⊕j∈J Hj of a family of non-trivial pairwise orthogonal closed subspaces Hjj∈J of a Hilbert

space H, supposing that furthermore⊕

j∈J Hj = H. Again defining Σ(J) as the family of subsetsof J , for E ∈ Σ(J) and z ∈ H we define P∅ = 0 and

PEz :=∑j∈E

Qjz ,

where Qj is the orthogonal projector onto Hj . It is easy to prove that the class of PEs definedthis way form a PVM on N. Again, from the fact that

⊕j∈J Hj = H, it arises

∑j∈J Qjx = x for

every x ∈ H so that PJ = I as requested.

In particular µ(P )xy (E) = 〈x|PEy〉 =

∑j∈E〈x|Qjy〉 and µ

(P )xx (E) =

∑j∈E ||Qjx||2.

The reader can simply prove that, with the said definition∫Jf(j)dµxx(j) =

∑j∈J

f(j)||Qjx||2 (2.38)

if f is µxx integrable. This identity is trivially valid on simple functions and easily extends togeneric functions using dominated convergence theorem.

(3) A PVM of completely different flavour can be constructed in L2(Rn, dnx) as follows. Toevery E ∈ B(Rn), the Borel σ-algebra, associate the orthonormal projector PE such that, if χEis the characteristic function of E – χE(x) = 0 if x 6∈ E and χE(x) = 1 if x ∈ E –

(PEψ)(x) := χE(x)ψ(x) ∀ψ ∈ L2(Rn, dnx) .

Moreover P∅ := 0. It is easy to prove that the collection of the PE is a PVM. In particular

µ(P )fg (E) = 〈f |PEg〉 =

∫E f(x)g(x)dnx and µ

(P )ff (E) =

∫E |f(x)|2dnx.

The reader can easily check that, with the said definition,∫Rnf(x)dµgg(x) =

∫Rnf(x)|g(x)|2dnx (2.39)

if f is µgg integrable. This identity is trivially true on simple functions and easily extends togeneric measurable functions using dominated convergence theorem.

We have the following pivotal result [Rud91, Mor18, Schm12] that extends (2.37) form simplefunctions to measurable functions of a suitable class.

Theorem 2.6. Let H be a Hilbert space, P : Σ(X) → L (H) a PVM, and f : X → C ameasurable function. Define

∆f :=

ßx ∈ H

∣∣∣∣ ∫X|f(λ)|2µ(P )

xx (λ) < +∞™.

61

The following facts hold.

(a) ∆f is a dense subspace of H and there is a unique operator, we denote by∫Xf(λ)dP (λ) : ∆f → H , (2.40)

such that ≠x

∣∣∣∣∫Xf(λ)dP (λ)y

∑=

∫Xf(λ)dµ(P )

xy (λ) ∀x ∈ H , ∀y ∈ ∆f . (2.41)

(b) The operator in (2.40) is closed and normal.

(c) The adjoint of the operator in (2.40) satisfiesÅ∫Xf(λ) dP (λ)

ã∗=

∫Xf(λ) dP (λ) . (2.42)

(d) The operator in (2.40) satisfies∣∣∣∣∣∣∣∣∫Xf(λ) dP (λ)x

∣∣∣∣∣∣∣∣2 =

∫X|f(λ)|2dµ(P )

xx (λ) ∀x ∈ ∆f . (2.43)

Proof. (I. Existence and uniqueness.) We start by proving that, if ∆f is subspace of H, thenthere is a unique operator denoted by

∫X f(λ)dP (λ) satisfying (2.41). The proof of this fact

relies upon a preliminary lemma we state and immediately prove.

Lemma 2.2. If f : X → C is measurable, the following inequality holds∫X|f(λ)| d|µ(P )

xy |(λ) ≤ ||x|| ∫

X|f(λ)|2dµ(P )

yy (λ) ∀y ∈ ∆f , ∀x ∈ H . (2.44)

Proof. We henceforth write µxy in place of µ(P )xy for the sake of shortness. The idea is to initially

establish the inequality for simple functions and then to extend it to general functions. Let x ∈ Hand y ∈ ∆f . Let s : X → C be a simple function, h : X → C the Radon-Nikodym derivative ofµxy with respect to |µxy| so that |h(x)| = 1 and µxy(E) =

∫E hd|µxy|. We have for an increasing

sequence of simple functions zn such that zn → h−1 pointwise, with |zn| ≤ |h−1| = 1, due to thedominate convergence theorem,

∫X|s|d|µxy| =

∫X|s|h−1dµxy = lim

n→+∞

∫X|s|zndµxy = lim

n→+∞

∞x

∣∣∣∣∣∣Nn∑k=1

zn,kPEn,k y

∫.

62

In the last step we have made use of (iii)(b) in remark 2.14 for the simple function

|s|zn =Nn∑k=1

zn,kχEn,k

where we have supposed that, for fixed n, the sets En,k are pairwise disjoint. Cauchy Schwartzinequality immediately yields∫

X|s|d|µxy| ≤ ||x|| lim

n→+∞

∣∣∣∣∣∣∣∣∣∣∣∣Nn∑k=1

zn,kPEn,ky

∣∣∣∣∣∣∣∣∣∣∣∣ = ||x|| lim

n→+∞

∫X|szn|2dµyy ,

where we have used, in computing the norm, P ∗En,kPEn,k′ = PEn,kPEn,k′ = δkk′PEn,k since En,k ∩En,k′ = ∅ for k 6= k′. Next observe that, as |szn|2 → |sh−1|2 = |s|2, dominate convergencetheorem leads to ∫

X|s|d|µxy| ≤ ||x||

∫X|s|2dµyy .

Finally, replace s above for a sequence of simple functions sn → f ∈ L2(X, dµyy) point wise,with |sn| ≤ |sn+1| ≤ |f |. Monotone convergence theorem and dominate convergence theorem,respectively applied to the left and right-hand side of the found inequality, produce inequality(2.44).

To go on with the main proof, we notice that inequality (2.44) also proves that f ∈ L2(X, dµ(P )yy )

implies f ∈ L1(X, d|µ(P )xy |) for x ∈ H, so that the right-hand side of (2.41) makes sense. Since

from the general measure theory∣∣∣∣∫Xf(λ) dµ(P )

xy (λ)

∣∣∣∣ ≤ ∫X|f(λ)| d|µ(P )

xy |(λ) ,

(2.44) implies that H 3 x 7→∫X f(λ) dµ

(P )xy (λ) is continuous at x = 0. This map is also anti-

linear if f is a simple function as follows from the definition of µxy and anti linearity of thescalar product in the left entry. Anti linearity extends to measurable functions f using the usualapproximation procedure of measurable functions by means of simple functions. We concludethat, for y ∈ ∆f , the map

H 3 x 7→∫Xf(λ) dµ

(P )xy (λ)

is linear and continuous. Riesz’ lemma proves that there exists a unique vector, indicated by∫X f(λ)dP (λ)y, satisfying ∫

Xf(λ) dµ

(P )xy (λ) =

≠∫Xf(λ)dP (λ)y

∣∣∣∣x∑ .Taking the complex conjugation of both sised we obtains (2.41). As we have assumed ∆f is asubspace, the map

∆f 3 y 7→∫Xf(λ) dµ(P )

xy (λ)

63

is linear when f is a simple function as immediately follows from the definition of µ(P )xy taking

advantage of the linearity of the scalar product in the right entry. Again, linearity extends tomeasurable functions f using the usual approximation procedure of measurable functions bysimple functions. As a consequence of (2.41)

∆f 3 y 7→∫Xf(λ)dP (λ)y

is linear as well. Uniqueness of this operator is immediate consequence of the uniqueness state-ment in Riesz’ lemma.(II. ∆f is a dense subspace.) We pass to prove that ∆f is a subspace. Notice that it includes 0so it is not empty. Moreover, directly from the definition of ∆f , it is clear that if x ∈ ∆f , then

ax ∈ ∆f for every a ∈ C, because µ(P )ax,ax(E) = |a|2µ(P )

xx (E) independently from E and so∫X|f |2dµ(P )

ax,ax = |a|2∫X|f |2dµ(P )

x,x < +∞ .

Next suppose that x, y ∈ ∆f . We therefore have ||PE(x+y)||2 ≤ (||PEx||+||PEy||)2 ≤ 2||PEx||2+

2||PEy||2. As a consequence µ(P )x+y,x+y(E) = ||PE(x+ y)||2 ≤ 2µ

(P )xx (E) + 2µ

(P )yy (E) so that∫

X|f |2dµ(P )

x+y,x+y ≤ 2

∫X|f |2dµ(P )

xx +

∫X|f |2dµ(P )

yy < +∞ ,

hence x + y ∈ ∆f . Let us pass to prove that ∆f is dense. Consider the countable partition ofX made of the measurable sets Fn := λ ∈ X | n ≤ |f(λ)|2 < n + 1, for n = 0, 1, 2, . . .. Fromσ-additivity of P , if z ∈ H, then z = PXz =

∑+∞n=0 PFnz. Therefore the finite span of the union

of closed subspaces Hn := PFn(H) is dense in H. If we prove that Hn ⊂ ∆f for every n, since ∆f

is a subspace, we immediately infer that it is dense. Let us prove it. If x ∈ Hn, then x = PFnx

and therefore µ(P )xx (E) = 〈PFnx|PEPFnx〉 = 〈x|PE∩Fnx〉 = µ

(P )xx (E ∩ Fn). As a consequence∫

X|f |2dµ(P )

xx =

∫Fn

|f |2dµ(P )xx ≤

∫Fn

(n+ 1)dµ(P )xx ≤ (n+ 1)||x||2 < +∞

and therefore x ∈ ∆f as wanted.(III. Proof of Eq.(2.43).) For x ∈ ∆f and taking advantage of (2.41), we obtain∣∣∣∣∣∣∣∣∫

XfdPx

∣∣∣∣∣∣∣∣2 =

≠∫XfdPx

∣∣∣∣∫XfdPx

∑=

∫Xfdν (2.45)

where

ν(E) = µ(P )∫XfdPx,x

(E) =

≠∫XfdPx

∣∣∣∣PEx∑ =

∫Xfdµ

(P )PEx,x

.

Since µ(P )PEx,x

(F ) = 〈PEx|PFx〉 = 〈x|PE∩Fx〉, we have

ν(E) =

∫Efdµ(P )

xx .

64

Using the definition of integral (of complex measure), it immediately arises that, if s is a simplefunction ∫

Xsdν =

∫Xs · fdµ(P )

xx .

Therefore, a standard argument based of dominated convergence theorem, using a sequence ofsimple functions sn point wise tending to f with |sn| ≤ |f |, establishes that, as |f |2 is µxxintegrable, ∫

Xfdν =

∫X|f |2dµ(P )

xx .

Inserting this result in (2.45) we obtain (2.43) as wanted.(IV. Proof of Eq.(2.42) and the fact that

∫X fdP is closed.) Since the adjoint is always closed,

Eq.(2.42),∫X fdP = (

∫X fdP )∗, would imply that

∫X fdP is closed. Let us prove Eq.(2.42).

It is easy to see that, from (2.41),∫X fdP ⊂ (

∫X fdP )∗ since, if x, y ∈ ∆f , noticing that

µ(P )yx (E) = µ

(P )xy (E)≠

y

∣∣∣∣∫XfdPx

∑=

∫Xfdµ(P )

yx =

∫Xfdµ

(P )xy =

≠x

∣∣∣∣∫XfdPy

∑=

≠∫XfdPy

∣∣∣x∑ . (2.46)

So, we only have to prove that∫X fdP ⊃ (

∫X fdP )∗. This is equivalent to establish that if

y ∈ D((∫X fdP )∗), then y ∈ ∆f = ∆f . Let us prove it. We need another lemma.

Lemma 2.3. With the main hypotheses,(i) for every E ∈ Σ(X), it holds that

∫X χEdP = PE,

(ii) for every E ∈ Σ(X) we have the identity∫X fdPPE =

∫X f · χEdP ,

(iii) if f is bounded over E ∈ Σ(X), (∫X f · χEdP )∗ =

∫X f · χEdP .

Proof. (i) is true since, by direct inspection, 〈x|PEy〉 = µxy(E) =∫E 1dµ

(P )xy so that (2.41) holds

true and it uniquely determines∫X χEdP .

Concerning (ii), the domain of∫X fdPPE is made of the elements x ∈ H such that PEx ∈ ∆f ,

that is∫X |f |2dµ

(P )PEx,PEx

< +∞. Since µ(P )PEx,PEx

(F ) = 〈PEx|PFPEx〉 = 〈x|PE∩Fx〉 = µ(P )xx (E ∩

F ), the said condition can be re-phrased to∫X χE · |f |2dµ

(P )xx < +∞, namely,

∫X |χE ·f |2dµ

(P )xx <

+∞. Therefore∫X fdPPE and

∫X χE · fdP have the same domain. If x ∈ H and y ∈ ∆χ·f ,

〈x|∫X fdPPEy〉 =

∫X fdµ

(P )x,PEy

=∫X fdµ

(P )PEx,PEy

=∫E fdµ

(P )x,y =

∫E f · χEdµ

(P )x,y which implies∫

X fdPPE =∫X f · χEdP again from (2.41).

(iii) is true because ∆f ·χE = H and∫X f ·χEdP ∈ B(H) from (2.43), so that (2.46) for f replaced

by f · χE proves that∫X f · χEdP =

∫X f · χEdP is the adjoint of

∫X f · χEdP .

To conclude, we prove that (i), (ii), and (iii) entail that if y ∈ D((∫X fdP )∗) then y ∈ ∆f . We

start by observing that, defining En := λ ∈ X | |f(λ)| < n, form (i)-(iii) we have

PEn

Å∫XfdP

ã∗= P ∗En

Å∫XfdP

ã∗⊂Å∫

XfdPPEn

ã∗=

Å∫Xf · χEndP

ã∗=

∫Xf · χEndP .

65

Hence, if y ∈ D((∫X fdP )∗), we can infer that∫

Xf · χEndPy = PEn

Å∫XfdP

ã∗y ,

so ∣∣∣∣∣∣∣∣∫Xf · χEndPy

∣∣∣∣∣∣∣∣2 =

∣∣∣∣∣∣∣∣PEn Å∫XfdP

ã∗y

∣∣∣∣∣∣∣∣2 ≤ ∣∣∣∣∣∣∣∣Å∫XfdP

ã∗y

∣∣∣∣∣∣∣∣2 .Taking advantage of (2.43), ∫

X|f · χEn |2dµ(P )

yy ≤∣∣∣∣∣∣∣∣Å∫

XfdP

ã∗y

∣∣∣∣∣∣∣∣2Since |f · χEn |2 ≤ |f · χEn+1 |2 → |f |2 for n→ +∞, monotone convergence theorem implies that∫

X|f |2dµ(P )

yy ≤∣∣∣∣∣∣∣∣Å∫

XfdP

ã∗y

∣∣∣∣∣∣∣∣2 < +∞

namely, y ∈ ∆f as wanted.

(V. Proof that∫X fdP is normal.) With the same argument as that in the previous lemma for

establishing (ii), we have that PE∫X fdPx =

∫X χE ·fdPx if x ∈ ∆f . Next consider the domain

of∫X fdP

∫X fdP . It is made of the vectors x ∈ ∆f such that∫

X|f |2dµ(P )∫

XfdPx,

∫XfdPx

< +∞ . (2.47)

Let us write this condition into another simpler way. First observe that

µ(P )∫XfdPx,

∫XfdPx

(E) =

≠∫XfdPx

∣∣∣∣PE ∫XfdPx

∑=

≠PE

∫XfdPx

∣∣∣∣PE ∫XfdPx

∑=

≠∫XχE · fdPx

∣∣∣∣∫XχE · fdPx

∑=

∫E|f |2dµ(P )

xx .

Starting from simple functions and extending the result to measurable functions, it is thereforeeasy to prove that ∫

Xgdµ

(P )∫XfdPx,

∫XfdPx

=

∫X|f |2gdµ(P )

xx .

In summary, (2.47) states that

D

Å∫XfdP

∫XfdP

ã= ∆|f |2 .

As a consequence of the first statement of this theorem for f replaced by |f |2, that domain isdense and D(

∫X fdP

∫X fdP ) = D(

∫X fdP

∫X fdP ). To terminate the demonstration, consider

x ∈ D(∫X fdP

∫X fdP ) = D(

∫X fdP

∫X fdP ). It holds that≠

x

∣∣∣∣∫XfdP

∫XfdPx

∑=

≠∫XfdPx

∣∣∣∣∫XfdPx

∑=

∫X|f |2dµ(P )

xx =

≠∫XfdPx

∣∣∣∣∫XfdPx

∑66

=

≠x

∣∣∣∣∫XfdP

∫XfdPx

∑.

In other words ≠x

∣∣∣∣Å∫XfdP

∫XfdP −

∫XfdP

∫XfdP

ãx

∑= 0 .

With the standard procedure based on the polarization identity, we finally also obtain≠y

∣∣∣∣Å∫XfdP

∫XfdP −

∫XfdP

∫XfdP

ãx

∑= 0 ,

for every x, y ∈ D(∫X fdP

∫X fdP ) = D(

∫X fdP

∫X fdP ). Since this domain is dense, we

conclude that∫X fdP

∫X fdP −

∫X fdP

∫X fdP = 0 as wanted.

The proved theorem has some technically important consequences listed in the following corol-lary and in the subsequent proposition.

Corollary 2.4. With the hypotheses of Theorem 2.6, the following results hold.

(a) If f : X → C satisfies f(x) ≥ 0 for all x ∈ X then≠x

∣∣∣∣∫XfdPx

∑≥ 0 ∀x ∈ ∆f .

(b) If T is an operator in H with D(T ) = ∆f such that

〈x |Tx〉 =

∫Xf(λ) dµ(P )

xx (λ) ∀x ∈ ∆f , (2.48)

then

T =

∫Xf(λ)dP (λ) .

Proof. (a) The proof is evident from (2.41) with y = x, noticing that µ(P )xx is positive.

(b) From the definition of µxy we easily have (everywhere omitting (P ) for semplicity)

4µxy(E) = µx+y,x+y(E)− µx−y,x−y(E)− iµx+iy,x+iy(E) + iµx−iy,x−iy(E) .

This identity, taking advantage of the definition of integral, implies that for a simple function

4

∫Xsdµxy =

∫Xsdµx+y,x+y −

∫Xsdµx−y,x−y − i

∫Xsdµx+iy.x+iy + i

∫Xsdµx−iy,x−iy

if x, y ∈ ∆s. With the standard procedure of approximation of measurable functions f withsimple functions and taking advantage of dominated convergence theorem, the identity extendsto

4

∫Xfdµxy =

∫Xfdµx+y,x+y −

∫Xfdµx−y,x−y − i

∫Xfdµx+iy.x+iy + i

∫Xfdµx−iy,x−iy

67

for x, y ∈ ∆f . Similarly, from the elementary properties of the scalar product, when x, y ∈ D(T ),

4〈x|Ty〉 = 〈x+ y|T (x+ y)〉 − 〈x− y|T (x− y)〉 − i〈x+ iy|T (x+ iy)〉+ i〈x− iy|T (x− iy)〉 .

Collecting everything, it is now obvious that (2.48) implies

〈x |Ty 〉 =

∫Xf(λ)µ(P )

xy (λ) ∀x, y ∈ ∆f ,

so that ≠x

∣∣∣∣ÅT − ∫Xf(λ)dP (λ)

ãy

∑= 0 ∀x, y ∈ ∆f

Since x varies in a dense set ∆f , we have that Ty−∫X f(λ)dP (λ)y = 0 for every y ∈ ∆f , which

is the thesis.

Example 2.5.(1) Referring to the PVM in (2) of example 2.4, exploiting (b) in Corollary 2.4 and (2.38), wehave that ∫

Jf(λ)dP (λ)z =

∑n∈J

f(j)Qjz

for every f : J → C (which is necessarily measurable with our definition of Σ(J)). Correspond-ingly, the domain of

∫J f(λ)dP (λ) results to be

∆f :=

z ∈ H

∣∣∣∣∣∣ ∑j∈J |f(j)|2||Qjz||2 < +∞

.

In fact, for every z ∈ ∆f and according to (b) in Corollary 2.4, form (2.39), we have≠z

∣∣∣∣∫Jf(j)dP (j)z

∑=∑j∈J

f(j)||Qjz||2 =

∫Rf(j)dµzz .

(2) Referring to the PVM in (3) of example 2.4 with for the sake of semplicity, exploiting (b)in Corollary 2.4 and (2.39), we have thatÅ∫

Rnf(λ)dP (λ)ψ

ã(x) = f(x)ψ(x) , x ∈ Rn .

Correspondingly, the domain of∫Rn f(λ)dP (λ) results to be

∆f :=

ßψ ∈ L2(Rn, dnx)

∣∣∣∣ ∫Rn|f(x)|2|ψ(x)|2dnx < +∞

™.

In fact, for every ψ ∈ ∆f and according to (b) in Corollary 2.4, form (2.39), we obtain≠ψ

∣∣∣∣∫Rnf(λ)dP (λ)ψ

∑=

∫Rnf(x)|ψ(x)|2dnx =

∫Rnfdµψψ .

68

2.4.3 Integral of bounded functions

We now state and prove a proposition concerning the most important properties of∫X fdP when

f : X → C is bounded or, more weakly, P -essentially bounded. Some of these fact have beenalready exploited in the proof of Theorem 2.6, however they result to be so useful in the practicethat they deserve a separate statement.If µ is a σ-additive positive measure over a σ-algebra Σ(X),

||f ||(µ)∞ := inf r ≥ 0 | µ(x ∈ X | |f(x)| > r) = 0 .

As a consequence, since the integral sees only non-zero measure sets in Σ(X), for instance,∫X|f |dµ ≤ ||f ||(µ)

∞

∫X

1dµ .

The same definition can be extended to PVM:

||f ||(P )∞ := inf r ≥ 0 | P (x ∈ X | |f(x)| > r) = 0

and f is said to be P -essentially bounded if ||f ||(P )∞ < +∞.

Notice that if PE = 0, then µ(P )xy (E) = 0 for E ∈ Σ(X). Therefore, if f is P -essentially bounded,

it is also µ(P )xx -essentially bounded for every x ∈ ∆f . In particular, since zero measure sets with

respect to P have evidently zero measure also for µ(P )xx ,

0 ≤ ||f ||(µ(P )xx )∞ ≤ ||f ||(P )

∞ ≤ ||f ||∞ ≤ +∞ . (2.49)

A seminorm p : X → R over a complex vector space X by definition satisfies p(x) ≥ 0,p(ax) = |a|p(x) and p(x+ y) ≤ p(x) + p(y) for all x, y ∈ X and a ∈ C.

It is easy to prove that || ||(P )∞ is a seminorm over vector space of P -essentially bounded measur-

able complex-valued functions over X. Moreover, |f | ≤ |g| point wise implies ||f ||(P )∞ ≤ ||g||(P )

∞

and ||f ·g||(P )∞ ≤ ||f ||(P )

∞ ||g||(P )∞ where f ·g is the point wise product (f ·g)(x) = f(x)g(x) for x ∈ X.

Proposition 2.25. Consider a PVM P : Σ(X)→ L (H), the following facts are true.

(a) It hold that ∫Xf(λ) dP (λ) ∈ B(H)

if and only if f is P -essentially bounded. In this case,∣∣∣∣∣∣∣∣∫Xf(λ) dP (λ)

∣∣∣∣∣∣∣∣ ≤ ||f ||(P )∞ ≤ ||f ||∞ . (2.50)

(b) It holds that ∫XχE dP = PE , if E ∈ Σ(X). (2.51)

69

In particular, ∫X

1 dP = I . (2.52)

For a simple function s =∑nk=1 skχEk , where sk ∈ C and Ek ∈ Σ(X), k = 1, . . . , n,∫X

n∑k=1

skχEkdP =n∑k=1

skPEk . (2.53)

(c) Let f, fn : X → R be measurable functions such that ||f ||(P )∞ , ||fn||(P )

∞ ≤ K < +∞ for someK ∈ R and every n ∈ N. If fn → f pointwise as n→ +∞, then∫

XfndPx→

∫XfdPx for n→ +∞ and for every x ∈ H . (2.54)

(d) If f, g : X → C are P -essentially bounded and a, b ∈ C, then∫X

(af + bg) dP = a

∫XfdP + b

∫XgdP , (2.55)

∫XfdP

∫XgdP =

∫Xf · g dP . (2.56)

Proof. (a) Assume that f is P -essentially bounded. Since µxx(X) = ||x||2 < +∞ for everyx ∈ H,∫

X|f(λ)|2dµ(P )

xx (λ) ≤ (||f ||(µ(P )xx )∞ )2

∫X

1dµ(P )xx ≤ (||f ||(P )

∞ )2∫X

1dµ(P )xx = ||x||2 (|f ||(P )

∞ )2 ,

so that ∆f = H. Next, dividing by ||x||2 and taking the sup over the elements x 6= 0, (2.43)implies (2.50). If, contrarily, f is not P -essentially bounded, then for every n ∈ N, there isEn ∈ Σ(X) with PEn 6= 0 and |f(λ)| ≥ n if λ ∈ En. Pick out xn ∈ PEn(H) with ||xn|| = 1 forevery n ∈ N. If xn 6∈ ∆f for some n, then

∫X fdP 6∈ B(H) because the domain of the operator

is smaller than the entire H and the proof ends. If xn ∈ ∆f for every n ∈ N, from (d) in

Theorem 2.6, it holds that ||∫X fdPxn||2 =

∫X |f |2dµ

(P )xnxn =

∫En|f |2dµ(P )

xnxn , where we have used

the fact that µ(P )xnxn(F ) = 〈xn|PFxn〉 = 〈PEnxn|PFPEnxn〉 = 〈xn|PF∩Enxn〉 = µ

(P )xnxn(F ∩ En).

Therefore ||∫X fdPxn||2 ≥

∫Enn2dµ

(P )xnxn = n2

∫En

1dµxnxn = n2∫X 1dµ

(P )xnxn = n2||xn||2 = n2.

Hence ||∫X fdP || cannot be finite and

∫X fdP 6∈ B(H).

(b) By direct inspection

〈y |PEx〉 = µ(P )yx (E) =

∫E

1dµ(P )yx (λ) =

∫XχE(λ)dµ(P )

yx (λ) ∀x, y ∈ ∆χE = H .

70

This proves (2.51), which also implies (2.52) for E = X since PX = I. The proof of (2.53)

is a trivial extension of this argument using linearity of the integral with respect to µ(P )yx and

linearity of the scalar product.(c) Under the said hypotheses,∣∣∣∣∣∣∣∣Å∫

XfdP −

∫XfndP

ãx

∣∣∣∣∣∣∣∣2 =

∣∣∣∣∣∣∣∣∫Xf − fn dPx

∣∣∣∣∣∣∣∣2 =

∫X|f − fn|2dµ(P )

xx ,

where we have used (2.55) to achieve the first identity, whose proof is independent from thepresent one. Since |f − fn|2 ≤ 4K2 almost everywhere with respect to P and hence also to

µ(P )xx and

∫|K2|dµ(P )

xx = ||x||2K2 < +∞, here dominated convergence theorem implies that∫X |f − fn|2dµ

(P )xx → 0 as n→ +∞ proving our assert.

(d) (i) First observe that, under the said hypotheses, we have ∆af+bg,∆f ,∆g = H because

f, g, af + bg are P -essentially bounded (||af + bg||(P )∞ ≤ |a|||f ||(P )

∞ + |b|||g||(P )∞ )), so both sides of

(c)(i) are everywhere defined. Next, from standard properties of the integral, it holds for everyx ∈ H ∫

Xaf + bg dµ(P )

yx = a

∫Xfdµ(P )

yx + b

∫Xgdµ(P )

yx .

Using (2.41) it implies≠y

∣∣∣∣∫Xaf + bg dPx

∑= a

≠y

∣∣∣∣∫Xf dPx

∑+ b

≠y

∣∣∣∣∫Xg dPx

∑=

≠y

∣∣∣∣Åa ∫XfdP + b

∫Xg dP

ãx

∑.

The proof concludes since x, y ∈ H are arbitrary.Let us pass to prove (2.56). First consider a pair of simple functions s =

∑nk=1 skχEk and

t =∑mh=1 thχFh . The pointwise product s · t is, in turn, a simple function. Indeed,

s·t =n∑k=1

skχEk

m∑h=1

thχFh =∑k,h

skthχEkχFh =∑

(k,h)∈In×Im

skthχEk∩Fh =∑

(k,h)∈In×Im

(s·t)(k,h)PG(k,h),

where Il := 1, 2, . . . , l and G(k,h) := Ek ∩ Fh. Exploiting (2.53), we immediately find∫XsdP

∫XtdP =

n∑k=1

skPEk

m∑h=1

thPFh =∑h,k

skthPEkPFh

=∑

(k,h)∈In×Im

skthPEk∩Fh =∑

(k,h)∈In×Im

(s · t)(k,h)PG(k,h)=

∫Xs · tdP .

We have established the thesis for the case of simple functions f, g. Referring to generic P -essentially bounded functions f, g, consider two sequences of simple functions sn → f andtn → g point wise, such that |sn| ≤ |sn+1| ≤ |f | and |tn| ≤ |tn+1| ≤ |g| for all n ∈ N. Evidently

sn · tn → f · g and |sn · tn| ≤ |sn+1 · tn+1| ≤ |f · g| and also ||sn||(P )∞ ≤ ||f ||(P )

∞ , ||tn||(P )∞ ≤ ||g||(P )

∞ ,

||sn · tn||(P )∞ ≤ ||f · g||(P )

∞ ≤ ||f ||(P )∞ ||g||(P )

∞ . We can apply (c) obtaining, for every pair x, y ∈ H≠∫XsndPx

∣∣∣∣∫XtndPy

∑=

≠x

∣∣∣∣∫XsndP

∫XtndPy

∑=

≠x

∣∣∣∣∫Xsn · tndPy

∑→≠x

∣∣∣∣∫Xf · gdPy

∑71

for n→ +∞. On the other hand, taking advantage of (c) again and exploiting the fact that thescalar product is continuous, we also have for n→ +∞,≠∫

XsndPx

∣∣∣∣∫XtndPy

∑→≠∫

XfdPx

∣∣∣∣∫XgdPy

∑.

Summarizing, ≠∫XfdPx

∣∣∣∣∫XgdPy

∑=

≠x

∣∣∣∣∫Xf · gdPy

∑,

which, from (2.42) and using the fact that∫X fdP has domain given by the whole H, implies≠

x

∣∣∣∣∫XfdP

∫XgdPy

∑=

≠x

∣∣∣∣∫Xf · g dPy

∑.

Since x, y ∈ H are arbitrary, (2.56) holds.

Remark 2.15.(a) We stress that if f : X → C is measurable and P -essentially bounded, redefining it

as 0 over the preimage of the numbers z ∈ C with |z| > ||f ||(P )∞ we produce a measurable

function f ′ ∈Mb(X) such that∫X f′dP =

∫X fdP . So, concerning the integration of measurable

functions with respect to a PVM, bounded functions bring the same information as P -essentiallybounded functions.

(b) The first inequality in (a) Proposition 2.25 is an equality actually [Rud91, Mor18],∣∣∣∣∣∣∣∣∫Xf(λ) dP (λ)

∣∣∣∣∣∣∣∣ = ||f ||(P )∞ . (2.57)

See the solution of Exercise 2.9 for a proof of this identity.(c) Consider a set X equipped with a σ-algebra Σ(X). The set

Mb(X) := f : X → C | f is measurable and ||f ||∞ < +∞

is a commutative unital C∗-algebra. Here the norm, making Mb(X) a complete vector space, is|| · ||∞ the involution is the standard complex conjugation of functions f∗(x) = f(x) for x ∈ X,the algebra product is the commutative pointwise product of functions (f ·g)(x) = f(x)g(x) andthe complex vector space structure is the standard one: (af + bg)(x) := af(x) + bg(x) if x ∈ X,a, b ∈ C, and f, g ∈Mb(X). The unit elements is the constant function 11(x) = 1 if x ∈ X. TheC∗-property ||f∗ · f ||2 = ||f ||2 is nothing but |||f |2||∞ = ||f ||2∞.Suppose that a PVM P : Σ(X)→ L (H) is also given. The map

πP : Mb(X) 3 f 7→∫XfdP ∈ B(H)

preserves the structure of ∗-algebra and the unit. Therefore is a ∗-algebra representation. It isalso continuous and norm decreasing because of (2.50). This representation is not injective nor

72

isometric in general, however it enjoys another topological property, unrelated with the conti-nuity with respect to the natural norms of Mb(X) and B(H). It immediately arises from (2.43)

using the fact that µ(P )xx (X) < +∞.

Proposition 2.26. With the said definitions, if Mb(X) 3 fn → f point wise for n → +∞and there is a constant K ≥ 0 with |fn| ≤ K, then πP (fn)x→ πP (f)x for every x ∈ H.

(c) If X is a topological space and Σ(X) is the Borel σ-algebra B(X), the content of (c) holdsagain if replacing Mb(X) with the commutative unital C∗-algebra of complex-valued boundedcontinuous functions

Cb(X) := f : X → C | f is continuous and ||f ||∞ < +∞

An important result of C∗-algebras theory establishes that

Theorem 2.7. [Commutative Gelfand-Naimark’s theorem]A commutative unital C∗-algebra is isometrically ∗-isomorphic to a unital C∗-algebra Cb(X)where X is a compact and Hausdorff topological space.

That is the celebrated commutative Gelfand-Naimark theorem [Mor18].

2.4.4 Integral of generally unbounded functions

To conclude, we state a proposition concerning the most important and general properties ofthe integral of a measurable function with respect to a PVM in the generally unbounded case.

Proposition 2.27. Let P : Σ(X)→ H be a PVM, f, g : X → C measurable functions and letaf , f · g, and f + g respectively denote the point wise product with scalars and functions and thepoint wise sum of functions. The following facts hold.

(a) For a ∈ Ca

∫XfdP =

∫XafdP .

(b) It holds that D(∫X fdP +

∫X gdP ) = ∆f ∩∆g and∫XfdP +

∫XgdP ⊂

∫X

(f + g)dP ,

where the symbol “ ⊂′′ can be replaced by “ =′′ if and only if ∆f+g = ∆f ∩∆g.

(c) It holds that D(∫X fdP

∫X gdP ) = ∆f ·g ∩∆g and∫

XfdP

∫XgdP ⊂

∫X

(f · g)dP

73

where the symbol “ ⊂′′ can be replaced by “ =′′ if and only if ∆f ·g ⊂ ∆g.

(d) It holds that D((∫X fdP )∗

∫X fdP

)= D

(∫X fdP (

∫X fdP )∗

)= ∆|f |2 andÅ∫

XfdP

ã∗ ∫XfdP =

∫X|f |2dP =

∫XfdP

Å∫XfdP

ã∗.

(e) If U : H→ H′ is a surjective linear (or anti linear) isometry, Σ(X) 3 E 7→ P ′E := UPEU−1

is a PVM over H′ and

U

Å∫XfdP

ãU−1 =

∫XfdP ′ .

In particular, D(∫X fdP

′) = UD(∫X fdP ) = U(∆f ).

(f) If φ : X → X ′ is measurable with respect to corresponding σ-algebra Σ(X) and Σ′(X ′) andf : X ′ → C is measurable, then

(i) Σ′(X ′) 3 E′ 7→ P ′(E′) := P (φ−1(E′)) is a PVM on X ′.

(ii) it holds that ∫X′f dP ′ =

∫Xf φ dP .

Furthermore∆′f = ∆fφ ,

where ∆′f is the domain of∫X′ f dP

′.

Proof. Items (a),(e), and (f) are proved straightforwardly by checking the relevant definitions.(d) is a trivial consequence of (c) and (b)-(c) in Theorem 2.6. (b) can be proved with thesame argument used to prove the first identity in (d) Proposition 2.25 working in ∆f ∩ ∆g,the identity D(

∫X fdP +

∫X gdP ) = ∆f ∩ ∆g is the very definition of domain of a sum of

operators A+B. With this identity the last statement is obvious. Similarly, (c) can be provedwith the same argument used to prove the second identity in (d) Proposition 2.25 working inD(∫X fdP

∫X gdP ) and taking into account the identity D(

∫X fdP

∫X gdP ) = ∆f ·g ∩∆g which

can be established as follows. D(∫X fdP

∫X gdP ) is made of the vectors x ∈ H such that both

x ∈ ∆g and ∫X|f |2dµ(P )∫

XgdPx,

∫XgdPx

< +∞ .

However, applying the definition of µ(P )zz , it is easy to prove that∫

X|f |2dµ(P )∫

XgdPx,

∫XgdPx

=

∫X|f |2|g|2dµ(P )

xx ,

so that D(∫X fdP

∫X gdP ) = ∆f ·g ∩∆g. With this identity the last statement is obvious.

74

Remark 2.16. It is moreover possible to prove [Mor18] that, if P : Σ(X) → H is a PVMand f, g : X → C are measurable functions, then∫

XfdP

∫XgdP =

∫X

(f · g)dP ,

and ∫XfdP +

∫XgdP =

∫X

(f + g)dP ,

the bar denoting the closure.

Exercise 2.9. Prove (2.57) when f : X → C is measurable and P -essentially bounded.

Solution. We already know that ||∫X fdP || ≤ ||f ||

(P )∞ so that, in particular, if ||f ||(P )

∞ = 0

the thesis is obvious. Assume ||f ||(P )∞ > 0. Using the same argument as in the proof of (a)

Proposition 2.25, for n > 0 there is En ∈ Σ(X) such that PE 6= 0 and |f(λ)| ≥ ||f ||(P )∞ −1/n > 0

if λ ∈ En and n is sufficiently large. Choosing xn ∈ PEn(H) with ||xn|| = 1, we have that∣∣∣∣∣∣∣∣∫XfdPxn

∣∣∣∣∣∣∣∣2 =

∫X|f |2dµ(P )

xnxn ≥Ä||f ||(P )

∞ − 1/nä2 ∫

En

1dµ(P )xnxn =

Ä||f ||(P )

∞ − 1/nä2

,

that is

||f ||(P )∞ ≤

∣∣∣∣∣∣∣∣∫XfdPxn

∣∣∣∣∣∣∣∣+ 1/n .

Since we already know that ||∫X fdPxn|| ≤ ||f ||

(P )∞ (notice that ||xn|| = 1), this prove that there

is a sequence of unit vectors xn such that ||∫X fdPxn|| → ||f ||

(P )∞ for n → +∞, demonstrating

the assert.

Exercise 2.10. Suppose that fn → f pointiwise as n → +∞ where fn : X → C aremeasurable and |fn| ≤ |f |, prove that∫

XfndPx→

∫XfdPx if n→ +∞

for every x ∈ ∆f .

Solution. Evidently ∆fn ⊂ ∆f , so x ∈ ∆fn if x ∈ ∆f . Next, taking advantage of (b)

Proposition 2.27 and (2.43), we obtain ||∫X fndPx −

∫X f dPx||2 =

∫X |f − fn|2dµ

(P )xx → 0 as

n→ +∞ by direct application of dominated convergence theorem. 2

2.5 Spectral Decomposition of Selfadjoint Operators

We are in a position to state the fundamental result of the spectral theory of selfadjoint operators,which extends the expansion (1.4) to an integral formula valid also in the infinite dimensional

75

case, and where the set of eigenvalues is replaced by the full spectrum of the selfadjoint oper-ator. After this we shall focus on some consequences of the theorem relevant in quantum physics.

Notation 2.6. From now on B(T ) denotes the Borel σ-algebra on the topological space T .

Definition 2.13. Giving a PVM P : B(X)→ L (H) over the Borel σ-algebra of a topolog-ical space X, the support supp(P ) of P is the complement in X of the union of all open setsO ⊂ X with PO = 0.

Remark 2.17. If X is second countable, P is necessarily concentrated on supp(P ), i.e.,

PE = PE∩supp(P ) if E ⊂ X.

In fact, D := X \ supp(P ) is the union of a number of open sets O with PO = 0. Since the

topology is second countable, we can extract a countable subcovering. By subadditivity of µ(P )xx

it holds that µ(P )xx (D) = 0 for every x ∈ H. This can be rephrased as ||PDx|| = 0 for every x ∈ H.

So PD = 0. If E ∈ B(X), we therefore have PE = PE∩supp(P ) + PE∩D = PE∩supp(P ).

2.5.1 The spectral theorem for selfadjoint generally unbounded operators

To state the theorem, we preventively notice that (2.42) implies that∫f(λ)dP (λ) is selfadjoint

if f is real: The idea of the theorem is to prove that every selfadjoint operator can be writtenthis way for a specific f and with respect to a PVM on R associated with the operator itself.

Theorem 2.8. [Spectral Decomposition Theorem for Selfadjoint Operators]Let A be a selfadjoint operator in the complex Hilbert space H.

(a) There is a unique PVM P (A) : B(R) → L (H), called the spectral measure of A, suchthat

A =

∫RλdP (A)(λ) .

In particular D(A) = ∆ı, where ı : R 3 λ 7→ λ.

(b) It resultssupp(P (A)) = σ(A)

so that, as the standard topology of R is second-countable, P (A) is concentrated on σ(A):

P (A)(E) = P (A)(E ∩ σ(A)) , ∀E ∈ B(R) . (2.58)

(c) λ ∈ σp(A) if and only if P (A)(λ) 6= 0, in particular it happens if λ is an isolated point

of σ(A). Finally P(A)λ is the orthogonal projector onto the eigenspace of λ ∈ σp(A).

(d) λ ∈ σc(A) if and only if P (A)(λ) = 0 but P (A)(E) 6= 0 if E 3 λ is an open set of R.

76

Proof. (a) The existence part of the proof is involved and we postpone it to Appendix 2.8:Theorem 2.11 for the bounded case and Theorem 2.13 for the unbonded case (see also [Rud91,Mor18, Schm12]). Let us pass to the uniqueness issue. Suppose there are two PVMs P1 and P2

over B(R) satisfying

A =

∫RλdPk(λ) k = 1, 2 .

Consider the bounded normal operators

Tk :=

∫R

1

r − idPk(r) .

As we shall see below, Tk = Ri(A) the resolvent operator of A for λ = i. So, these operators areactually identical and we shall write simply T .Taking advantage of (f) Proposition 2.27, we can define new PVMs over the image Γ′ ⊂ C ofthe continuous injective map φ : R 3 r 7→ 1

r−i ∈ Γ (which turns out to be a homeomorphismonto its image equipped with the topology induced by C), and we also assume Σ(Γ′) := B(Γ)so that φ : R→ Γ′ is measurable.

Q′k(E) := Pk(φ−1(E)) , E ∈ B(Γ′) , k = 1, 2 .

With this choices,

T =

∫Γ′zdQ′k(z, z) , k = 1, 2 .

In Cartesian coordinates,

Γ =

®x+ iy ∈ C \ 0

∣∣∣∣∣ x2 +

Åy − 1

2

ã2

=1

4

´and this punctured circle – centred on i/2 with radius 1/2 – is drawn in an anti-clockwisedirection starting from 0, formally corresponding to r = −∞, and reaching 0 again for r = +∞.It is finally convenient to consider the complete circle given by the compact set Γ := Γ′ = Γ∪0,again assuming Σ(Γ) = B(Γ) and to extend the PVMs into a trivial way

Qk(F ) := Q′k(F \ 0) , F ∈ B(Γ) , k = 1, 2 .

The reader can easily prove that this extension defines in fact well-behaved PVMs over B(Γ).This way, the added point satisfies Qk(0) = 0, even if it belongs to the supports of the measures(defined as we did for P (A)). For this reason we also have

T =

∫ΓzdQk(z, z) , k = 1, 2 .

It is also convenient to write down the adjoint of T ,

T ∗ =

∫ΓzdQk(z, z) , k = 1, 2 .

77

Notice that these operators are bounded and therefore we can apply (d) of Proposition 2.25,obtaining in particulare that, if p is a complex polynomial of z, z,

p(T, T ∗) =

∫Γp(z, z)dQk(z, z) ,

where the polynomial in the left-hand side is defined thinking of the product of operators as theircomposition. We also have, for x, y ∈ H,∫

Γp(z, z)dµ(Q1)

xy = 〈x|p(T, T ∗)y〉 =

∫Γp(z, z)dµ(Q2)

xy . (2.59)

Since Γ is Hausdorff and compact and the algebra of complex polynomials in z and z (i) includesthe constant polynomial 1, (ii) is closed under complex conjugation and (iii) separates the pointsof C and thus also of Γ (i.e. if γ, γ′ ∈ Γ and γ 6= γ′, there is a polynomial p with p(γ) 6= p(γ′)),Stone-Weierstass theorem implies that these polynomials are || · ||∞-dense in the Banach spaceC(Γ) of continuous complex-valued functions defined over Γ. Using a continuity argumentimmediately arising from (2.50), approximating continuous functions over Γ in terms of saidpolynomials, the found identity (2.59) implies∫

Γf(z, z)dµ(Q1)

xx =

∫Γf(z, z)dµ(Q2)

xx for every continuous function f ∈ C(Γ).

Since in the locally compact Hausdorff space Γ open sets are countable union of compact sets

with finite µ(Q2)xx -measure, these Borel measures are regular [Rud86]. Hence, uniqueness part of

Riesz’ theorem for positive Borel measures [Rud86] implies that µ(Q1)xx (E) = µ

(Q2)xx (E) for every

E ∈ B(Γ). In particular, for every E ∈ B(Γ) and every x ∈ H,

〈x|(Q1(E)−Q2(E))x〉 =

∫ΓχEdµ

(Q1)xx −

∫ΓχEdµ

(Q2)xx = 0 ,

proving that Q1(E) = Q2(E) for every E ∈ B(Γ). Passing to the initial PVMs, noticingthat φ : R → Γ′ is bijectve and bi-continuous, so that φ−1 : Γ′ → R is measurable and thusφ(F ) ∈ B(Γ′) if F ∈ B(R),

P1(F ) = Q′1(φ(F )) = Q1(φ(F )) = Q2(φ(F )) = Q′2(φ(F )) = P2(F ) , F ∈ B(R) .

We have established that P (A) is uniquely determined by A.(b) If λ 6∈ supp(P (A)), the map C 3 r 7→ 1

r−λ = g(r) is P -essentially bounded, so∫R

1r−λdP (r) ∈

B(H) and ∆g = H. According (c) Proposition 2.27,

(A− λI)

∫R

1

r − λdP (r) =

∫R

r − λr − λ

dP (A)(r) =

∫R

1dP (A)(r) = I

and ∫R

1

r − λdP (r)(A− λI)x =

∫R

r − λr − λ

dP (A)(r)x =

∫R

1dP (A)x = x if x ∈ D(A) .

78

We conclude that∫R

1r−λdP (r) = Rλ(A) and λ 6∈ σ(A). Suppose conversely that λ 6∈ σ(A) and

so Rλ(A) := (A− λI)−1 exists in B(H). Therefore, for x ∈ D(A), we have x = Rλ(A)(A− λI)xand ||x|| ≤ ||Rλ(A)|| ||(A − λ)x|| and so ||(A − λ)x||2 ≥ ||x||2/||Rλ(A)||2. According to (2.43),taking ||x|| = 1, ∫

R|r − λ|2dµ(P (A))

xx (r) ≥ 1

||Rλ(A)||2> 0 . (2.60)

If λ ∈ supp(P (A)), we would have P(A)(λ−1/n,λ+1/n) 6= 0 and we would be consequently able to

pick out a sequence xn ∈ P(A)(λ−1/n,λ+1/n)(H) with ||xn|| = 1, finding

∫R |r − λ|2dµ(P (A))

xx (r) ≤4||xn||/n2 = 4/n2 → 0 as n → +∞ that is impossible due to (2.60). So λ 6∈ supp(P (A)). Thisconcludes the proof of (b).

(c) If P(A)λ 6= 0, let 0 6= x ∈ P (A)

λ (H). We have, form (2.51) and (c) Proposition 2.27,

Ax = AP(A)λx =

∫RrdP (A)(r)

∫Rχλ(r)dP (r)x =

∫Rrχλ(r)dP

(A)x

=

∫Rλχλ(r)dP

(A)x = λP(A)λx = λx .

So λ ∈ σp(A). If conversely λ ∈ σp(A), we have Ax = λx for some eigenvector x ∈ D(A) with||x|| = 1, so that (A− iI)x = (1− i)x and (A− iI)−1x = (λ− i)−1x. Similarly, (A+ iI)−1x =(λ+ i)−1x. Exploiting the same argument we used in proving uniqueness of P (A), writing Q inplace of Q1 = Q2, the found identities can be rephrased as

Tx =

∫ΓzdQ(z, z)x =

1

λ− ix and T ∗x =

∫ΓzdQ(z, z)x =

1

λ+ ix .

Considering polynomial compositions of the operators T and T ∗ these relations can be extended,for instance∫

Γ(az + bzz)dQ(z, z)x = aT ∗ + bTTx = a

1

λ− ix+ b

1

λ+ iTx =

ña

1

λ− i+ b

Å1

λ+ i

ã2ôx ,

and so on. In complete generality, defining t := 1λ−i , we have∫

Γp(z, z)dQ(z, z)x = p(T, T ∗)x = p(t, t)x

for every polynomial p in the variables z and z. As before, we can extend this identity to con-tinuous functions f : Γ→ C exploiting Stone-Weierstrass theorem and uniformly approximatinga continuous functions f = f(z, z) over the compact Γ by means of a sequence of polynomialspn = pn(z, z) restricted thereon. As ||f−pn Γ ||∞ → 0 for n→ +∞, (2.50) implies in particular

pn(t, t)x =

∫Γpn(z, z)dQ(z, z)x→

∫Γf(z, z)dQ(z, z)x if n→ +∞ ,

79

so that, since pn(t, t)→ f(t, t), we finally obtain∫Γf(z, z)dQ(z, z)x = f(t, t)x . (2.61)

Eventually, it is easy to construct a sequence of continuous functions over Γ such that fn → χtpoinwise on Γ as n → +∞ and |fn(z, z)| < K < +∞ for some K > 0 and every (z, z) ∈ Γ. (c)and (b) in Proposition 2.25 imply from (2.61),

Qtx =

∫Γχt(z, z)dQ(z, z)x = lim

n→+∞

∫Γfn(z, z)dQ(z, z)x = lim

n→+∞fn(t, t)x = χt(t, t)x = x .

Since t ∈ Γ′ by construction, Qt = Q′t = P(A)φ−1(t) = P

(A)λ . We have found that P

(A)λx = x.

Since x 6= 0, we also have P(A)λ 6= 0 concluding the proof.

It is clear that if λ ∈ σ(A) = supp(P (A)) is an isolated point so that there is an open set

O 3 λ such that O \ λ stays in R \ supp(P (A)), then P(A)λ 6= 0. Otherwise we would have

P(A)O = 0 by additivity for some open set O 3 λ forbidding λ ∈ supp(P (A)). Let us prove the

last statement in (c): P(A)λ (H) = Hλ, where Hλ is the eigenspace of λ ∈ σp(A). We established

that if P(A)λ 6= 0 (and it happens iff λ ∈ σp(A)), x ∈ P

(A)λ (H) satisfies Ax = λx. Therefore

P(A)λ (H) ⊂ Hλ. We have also proved that x ∈ Hλ implies P

(A)λx = x, that is Hλ ⊂ P

(A)λ (H). In

summary, P(A)λ (H) = Hλ.

(d) If λ ∈ σc(A), due to (c), it must be P(A)λ = 0 otherwise λ ∈ σp(A) which is disjoint from

σc(A). On the other hand, since λ ∈ supp(A), for every open set O including λ, it must be

P(A)O 6= 0. Suppose that P

(A)O 6= 0 for every open set O including λ. This fact entails that

λ ∈ supp(P (A)) = σ(A) and the further requirement P(A)λ = 0 yields λ ∈ σc(A) in view of

(c).

Remark 2.18.(a) If P is a PVM on R and f : R→ C is measurable, we can always write∫

Rf(λ)dP (λ) = f(A)

where we have introduced the selfadjoint operator A obtained as

A =

∫Rı(λ)dP (λ) , (2.62)

due to (2.42) and where ı : R 3 λ → λ. Evidently P (A) = P due to the uniqueness part of thespectral theorem. This fact leads to the conclusion that, in a Hilbert space H, all the PVM overB(R) are one-to-one associated to all selfadjoint operators in H.

80

(b) Theorem 2.8 is a particular case of a more general theorem (see [Rud91, Mor18] andespecially [Schm12]) valid when A is a (densely defined closed) normal operator. The generalstatement is identical, it is sufficient to replace everywhere R for C. A particular case is thatof A unitary. A statement and a proof of the spectral theorem for normal operators in B(H)appears as Theorem 2.12 in Appendix 2.8

Notation 2.7. In view of the said theorem, and (b) in particular, if f : σ(A)→ C is measur-able with respect to the σ-algebra obtained by restricting the elements of B(R) to σ(A), whichcoincides with B(σ(A)) when equipping σ(A) with the relative topology, we will use indifferentlythe notations

f(A) :=

∫σ(A)

f(λ)dP (A)(λ) :=

∫Rg(λ)dP (A)(λ) =: g(A) . (2.63)

where g : R→ C is the extension of f to the zero function outside σ(A) or any other measurablefunction which coincides with f on supp(P (A)) = σ(A). Obviously g(A) = g′(A) if g, g′ : R→ Ccoincide in supp(P (A)) = σ(A).

Example 2.6.(1) Let us focus on the m-axis position operator Xm in L2(Rn, dnx) introduced in (1) of example2.2. We know that σ(Xm) = σc(Xm) = R from example 2.3. We are interested in the PVMP (Xm) of Xm defined on R = σ(Xm). Let us fix m = 1 the other cases are analogous. The PVMassociated to X1 is

(P(X1)E ψ)(x) = χE×Rn−1(x)ψ(x) ψ ∈ L2(Rn, dnx) , (2.64)

where E ∈ B(R) is here identified with a subset of the first factor of R × Rn−1 = Rn. In-deed, indicating by Pψ the right-hand side of (2.64), one easily verifies that ∆x1 = D(X1) andapproximating the function Rn 3 x 7→ x1 ∈ R with simple functions2∫

Rnx1|ψ(x)|2dnx =

∫Rx1µ

(P )ψ,ψ(x1) =

∫Rλµ

(P )ψ,ψ(λ) ∀ψ ∈ D(X1) = ∆x1

where µ(P )ψ,ψ(E) = 〈ψ|PEψ〉 =

∫E×Rn−1 |ψ(x)|2dnx. Since the left-hand side is nothing but

〈ψ|X1ψ〉, (b) Corollary 2.4 proves that (2.64) holds true.

(2) Considering the m-axis momentum operator Pm in L2(Rn, dnx) introduced in (2) of example2.2, taking (2.22) into account where F (and thus F ∗) is unitary, in view of (i) in Proposition2.32 we immediately have that the PVM of Pm is

Q(Pm)E := F ∗P

(Km)E F .

2More generally∫R

∫Rn−1 g(x1)|ψ(x)|2dxdn−1x =

∫R g(x1)dµ

(P )ψ,ψ(x1) is evidently valid for simple functions and

then it extends to generic measurable functions when both sides make sense in view of, for instance, Lebesgue’sdominate convergence theorem for positive measures.

81

Above Km is the operator Xm represented in L2(Rn, dnk) as in (1) of example 2.2.

(3) With a similar argument it easily arises that the PVM of the operator H = H0 of theharmonic oscillator in Example (2.2) (4) is, for E ∈ B(R),

PE =∑

λ∈E∩~ω(N+1/2)

〈ψλ|·〉ψλ

where

H =∑

λ∈~ω(N+1/2)

λ〈ψλ|·〉ψλ =∑n∈N

~ω(n+ 1/2)〈ψn|·〉ψn . (2.65)

with domain

D(H) =

ψ ∈ L2(R, dx)

∣∣∣∣∣ +∞∑n=0

(n+ 1/2)2|〈ψn|ψ〉|2 < +∞.

Indeed, using the fact that ψnn∈Nis a Hilbert basis of L2(R, dx), the right-hand side of (2.65)is selfadjoint as it is the integral of the (real) function ı : R 3 λ 7→ λ ∈ R of the said PVM(notice that D(H) = ∆ı) and therefore right-hand side of (2.65) is a selfadjoint extension of H0

in Example (2.2) (4) which is essentially selfadjoint so that H = H0 as wanted. Let us provethat

σ(H) = σp(H) = ~ω(n+ 1/2) | n = 0, 1, . . . .

Evidently σ(H) includes the closed set of eigenvalues ~ω(n + 1/2). It cannot include any fur-ther points λ different from the numbers ~ω(n + 1/2). Indeed, suppose that such a further

element λ belongs to σp(H) so that P(H)λ 6= 0. If x ∈ P

(H)λ (H), we would have 〈x|ψn〉 =

〈P (H)λ x|P

(H)~ω(n+1/2)ψn〉 = 〈x|P (H)

λ∩~ω(n+1/2)ψn〉 = 〈x|P (H)∅ ψn〉 = 0. Therefore x = 0 because

it is orthogonal to a Hilbert basis, and P(H)λ = 0 contrarily to the hypothesis. So assume the

only other possibility λ ∈ σc(H). Since ~ω(n + 1/2) | n = 0, 1, . . . is closed and λ does notbelong to that set, it cannot be an accumulation point. We can therefore find δ > 0 such that(λ−δ, λ+δ)∩~ω(n+1/2) |n = 0, 1, . . . = ∅. With the same argument as before we can prove

that, if x ∈ P (H)(λ−δ,λ+δ)(H) then x = 0 and thus P

(H)(λ−δ,λ+δ) = 0 contrarily to (d) in Theorem 2.8.

We conclude that σ(H) = σp(H) = ~ω(n+ 1/2) | n = 0, 1, . . ..

(4) We stress that it is in general false that if a selfadjoint operator A admits a Hilbert basis ofeigenvectors then its spectrum is made of eigenvalues only. Since σ(A) is closed, if σp(A) is notclosed, points of the continuous spectrum may exist as accumulation points of σp(A).Using the Hilbert basis ψnn∈N of the previous example, consider the selfadjoint boundedoperator

A =∑

λ∈Q∩[0,1]

λ〈ψnλ |·〉ψnλ : L2(R, dx)→ L2(R, dx)

82

where Q ∩ [0, 1] 3 q 7→ nq ∈ N a bijection. As a matter of fact, we can re-phrase the definitionof A as

A =

∫RλdP (λ)

where, for every E ∈ B(R),PE =

∑λ∈E∩Q∩[0,1]

λ〈ψnλ |·〉ψnλ .

The operator A is evidently bounded and it is easy to prove that ||A|| = 1. The domain of Ais therefore the whole L2(R, dx) = ∆ı. Evidently, with the same argument as in the previousexample, Q ∩ [0, 1] = σp(A) because ψnn∈N is a Hilbert basis of L2(R, dx). As σp(A) ⊂ σ(A)that is closed, we have Q ∩ [0, 1] = [0, 1] = σp(A) ⊂ σ(A). It is easy to prove from (2.66) thatσ(A) ⊂ [0, 1] because ||A|| = 1. We conclude that σ(A) = [0, 1] and that elements [0, 1]\Q mustbe part of σc(A).

(5) More complicated cases exist. Considering an operator of Schrodinger form (∆ is the Laplaceoperator in Rn)

H :=1

2m

n∑k=1

P 2k + U(x) = − 1

2m∆ + U(x)

where Pk is the momentum operator in L2(Rk, dkx) associated to the k-th coordinate, m > 0 is aconstant and U is a real valued function on R used as multiplicative operator. If U = U1+U2 withU1 ∈ L2(Rk, dkx) and U2 ∈ L∞(Rk, dkx) real valued where k = 1, 2, 3, and D(H) = C∞(R),H turns out to be (trivially) symmetric but also essentially selfadjoint [ReSi80, Mor18] as aconsequence of a well known result (Kato-Rellich’s theorem). The unique selfadjoint extensionH = (H∗)∗ of H physically represents the Hamiltonian operator of a quantum particle livingalong R with a potential energy described by U . In this case, generally speaking, σ(H) has both

point and continuous part.∫σp(H) λdP

(H)(λ) has a form like this∫σp(H)

λdP (H)(λ) =∑

λ∈σp(H)

λP(H)λ

where Pλ is the orthogonal projector onto the eigenspace of H with eigenvalue λ. Conversely,∫σc(H) λdP

(H)(λ) has an expression much more complicated and, under a unitary transform,

is similar to the integral decomposition of X, but including many spaces L2 in general. If

Hp := P(H)

σp(H)(H) is the closed subspace spanned by all eigenspaces of H and Hc := P

(H)

σc(H)(H), we

have the orthogonal decomposition H = Hc ⊕ Hp. The operator Hp :=∫σp(H) λdP

(H)(λ) leavesinvariant a subspace of Hp

D(Hp) :=

ψ ∈ Hp

∣∣∣∣∣∣∣∑

E∈σp(H)

E2||P (H)E ψ||2 < +∞

83

and Hc :=∫σc(H) λdP

(H)(λ) leaves fixed a subspace of Hs

D(Hc) :=

®ψ ∈ Hp

∣∣∣∣∣∫σc(H)

E2dµP(H)

ψ,ψ (E) < +∞´.

In this sense,H = Hc ⊕Hp .

A possible situation (not the only) is that Hc is isomorphic to a direct sum ⊕Nn=1L2(σc(H), dE)

and here,Hc : (ψ1, . . . , ψN ) 7→ (ı · ψ1, . . . , ı · ψN )

acts as a the multiplicative operator in terms of identity function in each slot:

(ı · ψk)(E) := Eψk(E) .

Definition 2.14. Selfadjoint operators admitting a Hilbert basis of eigenvectors are said tohave a pure point spectrum.

Remark 2.19. It does not automatically mean that σp(A) = σ(A) as illustrated in theexample (4) above. However having pure point spectrum implies that σc(A) cannot have internalpoints (this is forbidden by (d) in Theorem 2.8).

2.5.2 Some technically relevant consequences of the spectral theorem

The spectral theorem has many implications towards several directions, we mention here justa few of them which have a relevant impact on the quantum theory. The first fact concernspositivity of a selfadjoint operator.

Proposition 2.28. If A is a selfdjoint operator in the Hilbert space H, it holds that A ≥ 0 –that is 〈x|Ax〉 ≥ 0 for every x ∈ D(A) – if and only if σ(A) ⊂ [0,+∞).

Proof. Suppose that σ(A) ⊂ [0,+∞). If x ∈ D(A) we have 〈x|Ax〉 =∫σ(A) λdµx,x ≥ 0 in view of

(2.41) (where we write µ in place of µ(P (A))) the spectral decomposition theorem, since µx,x is apositive measure ad σ(A) ∈ [0,+∞). To conclude, we prove that A ≥ 0 is false if σ(A) includesnegative elements. To this end assume that, conversely, σ(A) 3 λ0 < 0. Using (c) and (d) of

Theorem 2.8, one finds an interval [a, b] ⊂ σ(A) with [a, b] ⊂ (−∞, 0) and P(A)[a,b] 6= 0 (possibly

a = b = λ0). If x ∈ P (A)[a,b](H) with x 6= 0, it holds that µxx(E) = 〈x|PEx〉 = 〈x|P ∗[a,b]PExP[a,b]〉 =

〈x|P[a,b]PEP[a,b]x〉 = 〈x|P[a,b]∩Ex〉 = 0 if [a, b] ∩ E = ∅. Therefore, 〈x|Ax〉 =∫σ(A) λdµx,x =∫

[a,b] λdµx,x ≤∫[a,b] bµx,x < b||x||2 < 0.

84

Another remarkable result concerns a bound on the extension of the spectrum. This result canbe proved also for normal operators independently from the spectral theorem (it can be used toprove the spectral theorem, actually) following a much more elementary route as we shall do inProposition 2.40.

Proposition 2.29. A selfadjoint operator is bounded (and its domain coincide to the wholeH) if and only if σ(A) is bounded. In this case

||A|| = sup|λ| | λ ∈ σ(A) .

Proof. From Proposition 2.18, we have that if A ∈ B(H) then ||A|| ≥ sup|λ| | λ ∈ σ(A). Ifconversely σ(A) is bounded and thus compact (since it is closed), restricting the integrationspace to X = σ(A), the continuous function ı : σ(A) 3 λ → λ is bounded and (2.43) impliesthat A =

∫σ(A) ıdP

(A) is bounded and the inequality holds

||Ax||2 =

∫σ(A)|λ|2dµ(P (A))

xx (λ) ≤ (sup|λ| | λ ∈ σ(A))2∫σ(A)

1dµ(P (A))

xx (λ)

= (sup|λ| | λ ∈ σ(A))2 ||x||2 .

Hence ||A|| ≤ sup|λ| | λ ∈ σ(A), so that

||A|| = sup|λ| | λ ∈ σ(A) . (2.66)

In this case it also holds that D(A) = ∆ı = H.

Remark 2.20. Proposition 2.29 explains the reason why observables A in QM are veryoften represented by unbounded selfadjoint operators. σ(A) is the set of values of the observableA. When, as it frequently happens, that observable is allowed to take arbitrarily large values(think of X or P ), it cannot be represented by a bounded selfadjoint operator just because itsspectrum is not bounded.

Another elementary but technically important result is the following concerning the covariance ofa selfadjoint operator and its PVM under unitary (or isometric surjective linear transformations).

Proposition 2.30. If A : D(A) → H is a selfadjoint operator in the Hilbert space H andU : H → H′ is an isometric surjective linear (or anti linear) map, then UAU−1, with domainD(UAU−1) = U(D(A)) is selfadjoint as well (Proposition 2.9 and the subsequent remark) and

P(UAU−1)E = UP

(A)E U−1 for every E ∈ B(R).

85

Proof. If x ∈ D(A),∫Rı dµ(P (A))

xx = 〈x|Ax〉 = 〈Ux|UAU−1Ux〉 =

∫Rı dµ

(P (UAU−1))Ux,Ux =

∫Rı dµ(U−1P (UAU−1)U)

x,x .

In the last passage, we used

µ(P (UAU−1))Ux,Ux (E) = 〈Ux|P (UAU−1)

E Ux〉 = 〈x|U−1P(UAU−1)E Ux〉 = µ(U−1P (UAU−1)U)

x,x (E) .

Applying (b) Corollary 2.4, we conclude that

A =

∫Rı d U−1P (UAU−1)U .

Uniqueness of the PVM of A implies

P(A)E = U−1P

(UAU−1)E U , if E ∈ B(R),

that is the thesis we wanted to prove.

The notion (2.63) of a function of a selfadjoint operator is just an extension of in the analogousnotion (1.7) introduced for the finite dimensional case and thus may be used in QM applications.In the finite-dimensional case, the set of eigenvalues of f(A) is just the image of f of the set ofeigenvalues of A: σ(f(A)) = f(σ(A)). What happens in the infinite dimensional case?If f : R → C is Borel measurable (we can equivalently use a function f : σ(A) → C Borelmeasurable with respect to B(σ(A))) and A : D(A)→ H is selfadjoint, it is quite evident that

f(σp(A)) ⊂ σp(f(A)) . (2.67)

For if λ ∈ σp(A), then there is x = P(A)λx 6= 0, for the spectral theorem. Therefore,∫

RfdP (A)x =

∫RfdP (A)P

(A)λx =

∫RfdP (A)

∫RχλdP

(A)x =

∫Rf · χλdP (A)x

=

∫Rf(λ)χλdP

(A)x = f(λ)

∫RχλdP

(A)x = f(λ)x ,

hence f(λ) ∈ σp(f(A)). In the infinite-dimensional case, there are simple counter-examplesproving that the converse inclusion f(σp(A)) ⊃ σp(f(A)) may be false. The most elementary

is χE(A) = P(A)E . This operator is an orthogonal projector and thus it has only point spec-

trum given by a non-empty subset of 0, 1, even for the cases where σ(A) = σc(A) so thatχE(σp(A)) = ∅.To go on, a new notion is relevant.

Definition 2.15. Let P : B(X) → L (H) be a PVM over a topological space X. Iff : X → C is measurable, its P -essential rank is

essrank(f) := z ∈ C | Pf−1(O) 6= 0 if O is open and O 3 z .

86

Notice that the definition makes sense because f−1(O) ∈ B(X) since f is Borel measurableand O is open and hence belongs to B(C). An almost immediate consequence of the definitionfollows.

Proposition 2.31. Let P : B(X) → L (H) be a PVM over a topological space X. Iff : X → C is measurable, then

σ

Å∫XfdP

ã= essrank(f) .

Proof. If z 6∈ essrank(f), then there is an open set B 3 z in C with Pf−1(B) = 0. If Br(z) is anopen ball of radius r centred on z with Br(z) ⊂ B, by additivity it must be Pf−1(Br(z)) = 0 (and

Pf−1(B\Br(z)) = 0). The map X 3 λ 7→ g(λ) := 1f(λ)−z is therefore P -essentially bounded with

||g||(P )∞ ≤ 1/r since Pλ∈X | |g(λ)|>1/r = 0. Hence

∫X

1f(λ)−zdP (λ) ∈ B(H) from (a) Proposition

2.25. In addition, from (c) Proposition 2.27 and again (a) Proposition 2.25,∫X

1

f(λ)− zdP (λ)

∫X

(f(λ)− z)dP (λ)x =

∫X

f(λ)− zf(λ)− z

dP (λ)x = x if x ∈ D(∫x fdP )

so that z ∈ ρ(∫X fdP

), namely z 6∈ σ

(∫X fdP

).

If z ∈ essrank(f), then Pf−1(O) 6= 0 for every open set O containig z. It therefore happens forevery ball B1/n(z) of radius 1/n, n = 1, 2, . . ., centered in z. (In particular f−1(B1/2(z)) 6= ∅otherwise Pf−1(B1/2(z)) = 0.) Let us prove that, if R := (

∫X(f − zI)dP )−1 exists, it cannot be

bounded and thus z ∈ σc(∫X fdP

). Indeed, ||x|| = ||R

∫X(f − zI)dPx|| would imply, taking

||x|| = 1,

||R||2 ≥ 1

||∫X(f − zI)dPx||2

=1∫

X |f − zI|2dµ(P )xx

≥ 1

supf(λ)∈B1/n(z) |f(λ)− z|2∫X 1dµ

(P )xx

= n2 ,

which is not bounded as n = 1, 2, . . .. If R := (∫X(f −zI)dP )−1 does not exist, z ∈ σp

(∫X fdP

).

Since the residual spectrum is empty as∫X(f − zI)dP is normal, we have established that

z ∈ essrank(f) implies z ∈ σ(∫X fdP

)concluding the proof.

Remark 2.21. A finer argument [Rud91, Mor18] proves that z ∈ essrank(f) belongs toσp(∫X fdP

)if and only if Pf−1(z) 6= 0.

The relevant corollary of Proposition 2.31 and of the spectral theorem is the following one.

Corollary 2.5. Let A be a selfadjoint operator in the Hilbert space H and f : σ(A) → C acontinuous map. Then

σ(f(A)) = f(σ(A)) , (2.68)

87

where the bar denotes the closure and it is not necessary if A is bounded.

Proof. In view of Proposition 2.31 and Theorem 2.8, we have just to prove that essrank(f) =

f(supp(P (A))). If z = f(r) for some r ∈ supp(P (A)) and O 3 z is open, then f−1(O) is open sincef is continuous and includes r. Hence Pf−1(O) 6= 0 for the very definition of support. This proves

that essrank(f) ⊂ f(supp(P (A))). Since essrank(f) is closed by definition (its complement is

open), we also have essrank(f) = essrank(f) ⊂ f(supp(P (A))). To conclude, suppose z ∈f(supp(P (A))). If O 3 z is open, it must have a nonempty intersection with f(supp(P (A)).Hence f−1(O) is open, non empty and f−1(O) ∩ supp(P (A)) 6= ∅. From the definition of

support, P(A)f−1(O) 6= 0. By definition z ∈ essrank(f). We have established that essrank(f) ⊃

f(supp(P (A))) concluding the proof. Regarding the last statement we observe that, if A isbounded, σ(A) is compact in view of (b) Proposition 2.19. Since f is continuous, f(σ(A)) iscompact as well and also closed because C is Hausdorff so that f(σ(A)) = f(σ(A)).

Remark 2.22. It is fundamental to stress that, in QM, (2.68) permits us to adopt thestandard operational approach to intepret the observable f(A). It is the observable whose setof possible values is (the closure of) the set of reals f(a) where a is a possible value of A.

2.5.3 Joint spectral measure

The last spectral tool we introduce is the notion of joint spectral measure (see, e.g., [ReSi80,Mor18]). Everything is stated in the following theorem, whose proofs are usually quite long andtechnical. We present an original proof in Appendix 2.8 which naturally follows our presentationof the spectral machinery.

Theorem 2.9. [Joint spectral measure]Let A := A1, A2, . . . , An be a set of selfadjoint operators in the Hilbert space H. Suppose thatthe spectral measures of those operators pairwise commute:

P(Ak)Ek

P(Ah)Eh

= P(Ah)Eh

P(Ak)Ek

∀k, h ∈ 1, . . . , n ,∀Ek, Eh ∈ B(R) .

There is a unique PVM , P (A), on Rn such that

P(A)E1×···×En = P

(A1)E1· · ·P (An)

En, ∀E1, . . . , En ∈ B(R) .

For every f : R→ C measurable, it holds that∫Rnf(xk)dP

(A)(x) = f(Ak) , k = 1, . . . , n (2.69)

where x = (x1, . . . , xk, . . . , xn) and f(Ak) :=∫R f(λ)dP (Ak).

Finally, B ∈ B(H) commutes with P (A) if and only if it commutes with all P (Ak), k = 1, 2, . . . , n.

88

Proof. See Appendix 2.8.

Definition 2.16. Referring to Theorem 2.9, the PVM P (A) is called the joint spectralmeasure of A1, A2, . . . , An and its support supp(P (A)), i.e. the complement in Rn to the largest

open set O with P(A)O = 0, is called the joint spectrum of A1, A2, . . . , An.

Example 2.7. The simplest example is provided by considering the n position operators Xm

in L2(Rn, dnx). It should be clear that the n spectral measures commute because P(Xk)E , for

E ∈ B(R), is the multiplicative operator for χR×···×R×E×R×···×R the factor E staying in the k-thposition among the n Cartesian factors. In this case the joint spectrum of the n operators Xm

coincides with Rn itself.A completely analogous discussion holds for the n momentum operators Pk, since they arerelated to the position ones by means of the unitary Fourier-Plancherel operator as already seenseveral times. Again the joint spectrum of the n operators Pm coincides with Rn itself.

2.5.4 Measurable functional calculus

The following proposition states some useful properties of f(A), where A is selfadjoint andf : R → C is Borel measurable. These properties define the so called measurable functionalcalculus. We suppose here that A = A∗, but the statements can be reformulated for normaloperators [Mor18].

Proposition 2.32. Let A be a selfadjoint operator in the complex Hilbert space H and letf, g : σ(A)→ C be measurable functions. Let af , f ·g ,and f+g respectively denote the pointwiseproducts with scalars and functions and the pointwise sum of functions. The following facts hold.

(a) If f(λ) = pn(λ) :=∑nk=0 akλ

k with an 6= 0, then

pn(A) =n∑k=0

akAk with D(pn(A)) = ∆pn = D(An),

where the right-hand side is defined in its standard domain and A0 := I, A1 := A, A2 :=AA, and so on.

(b) If f = χE the characteristic function of E ∈ B(σ(A)), then

f(A) = P (A)(E) .

(c) The bar denoting the complex conjugation,

f(A)∗ = f(A) .

(d) For a ∈ C,af(A) = (af)(A) .

89

(e) D(f(A) + g(A)) = ∆f ∩∆g and

f(A) + g(A) ⊂ (f + g)(A) .

The symbol “ ⊂′′ can be replaced by “ =′′ if and only if ∆f+g = ∆f ∩∆g.

(f) D(f(A)g(A)) = ∆f ·g ∩∆g and

f(A)g(A) ⊂ (f · g)(A) .

The symbol “ ⊂′′ can be replaced by “ =′′ if and only if ∆f ·g ⊂ ∆g.

(g) It holds that D(f(A)∗f(A)) = ∆|f |2 and

f(A)∗f(A) = |f |2(A) .

(h) if f ≥ 0 then〈x|f(A)x〉 ≥ 0 for x ∈ ∆f .

(i) If x ∈ ∆f ,

||f(A)x||2 =

∫σ(A)|f(λ)|2dµ(P (A))

xx (λ) :

In particular, if f is bounded or P (A)-essentially bounded on σ(A), f(A) ∈ B(H) and

||f(A)|| ≤ ||f ||(P (A))∞ ≤ ||f ||∞ .

(j) If U : H→ H′ is and isometric surjective linear (or antilinear) map, then

Uf(A)U−1 = f(UAU−1)

and, in particular, D(f(UAU−1)) = UD(f(A)) = U(∆f ).

(k) If φ : R → R is measurable, then B(R) 3 E 7→ P ′(E) := P (A)(φ−1(E)) is a PVM on R.Introducing the selfadjoint operator

A′ =

∫Rλ′dP ′(λ′)

such that P (A′) = P ′, we haveA′ = φ(A)

andf(A′) = (f φ)(A) and ∆′f = ∆fφ

for every f : R→ C is measurable.

90

Proof. Everything but (a), (b), (c) and (i) are trivial re-formulations of corresponding statementsin Proposition 2.27. As a matter of fact, (b), (c), (h) and (i) are nothing but (2.51), (2.42),(a) Corollary 2.4 and (2.43) respectively. (a) is easy to prove. Let us initially focus on thecase pn(λ) = λn. First observe that A =

∫σ(A) λdP

(A)λ = p1(A). Let us prove that the thesis

is true for n if it is true for n − 1. In fact, An = AAn−1 =∫R λdP

(A)(λ)∫R λ

n−1dP (A)(λ) =∫R λ

ndP (A)(λ) = pn(A). In the penultimate identity we used (c) Proposition 2.27, observing

that the condition ∆f ·g ⊂ ∆g is satisfied for f = ı and g = ın−1 because the measure µ(P )xx is

finite and thus∫R |λ|2ndµxx(λ) < +∞ implies

∫R |λ|2(n−1)dµxx(λ) < +∞.

Let us pass to polynomials. For every polynomial pm(λ) =∑mk=0 akλ

k of order m (i.e. am 6= 0)define pm(A) :=

∑mk=0 akA

k. For m = 0 it is clear that p1(A) =∫a0dP

(A)(λ) = a0I. Letus extend the result inductively. Suppose that pn−1(A) =

∫σ(A) pn−1(λ)dP (A)(λ). Form (b) in

Proposition 2.27, if an 6= 0, it arises anAn + pn−1(A) =

∫R anλ

n + pn−1(λ)dP (A)(λ). This isbecause the condition ∆f+g = ∆f ∩∆g in (b) in Proposition 2.27 is satisfied for f = an ı

n and

g = pn−1 since ∆an ın+pn−1 = ∆ın again from finiteness of µ(P )xx . Putting all together, we have

that∑nk=0 akA

k =∫σ(A) p(λ)dλ for every polynomial p(λ) =

∑nk=0 akλ

k of order n. It is obvious

that D(pn(A)) = D(An) (if an 6= 0) from the definition of standard domain.

Exercise 2.11. Prove that if A : D(A) → H is a selfadjoint operator in the Hilbert spaceH, then

R 3 t 7→ Ut := eitA

is a one-parameter group of unitary operators, i.e. every Ut is unitary, U0 = I, UtUs = Ut+s,U−t = U∗t = (Ut)

−1 for every t, s ∈ R.

Solution. Ut =∫R e

itλdP (A)(λ) is an element of B(H) because the function in the integralis bounded due to (i) in Proposition 2.32. Then the thesis immediately arises form (b), (c) and(f) in Proposition 2.32, noticing that ei0 = 1, eitλeisλ = ei(t+s)λ and eitλ = e−itλ. 2

We have a pair of important technical facts about the one-parameter group of unitary operatorspresented in the previous exercise.

Proposition 2.33. If A : D(A) → H is a selfadjoint operator in the Hilbert space H, thenthe one-parameter group of unitary operators

R 3 t 7→ Ut := eitA

is strongly continuous, i.e. Utx→ Usx if t→ s for every fixed x ∈ H. Furthermore

Ut(D(A)) = D(A) and UtA = AUt for t ∈ R.

91

Proof. Since Uu is isometric, ||Utx − Usx|| = ||Us(Ut−sx − x)|| = ||Ut−sx − x||. Thereforecontinuity at any s ∈ R is equivalent to continuity at 0. Next (i) in Proposition 2.32 entails

||Utx− x||2 =

∫R|eitλ − 1|2dµ(P (A))

xx → 0 for t→ 0,

where we have used dominated convergence theorem noticing that µ(P (A))xx is finite and |eitλ−1|2 ≤

4. Regarding the second statement, observe that

UtP(A)E =

∫ReitλdP (A)

∫RχEdP

(A) =

∫RχEe

itλdP (A) =

∫RχEdP

(A)∫ReitλdP (A) = P

(A)E Ut ,

(i), (b) and (f) in Proposition 2.32. As a consequence, µ(P (A))Utx,Utx

(E) = ||P (A)E Utx||2 = ||UtP (A)

E x||2 =

||P (A)E x||2 = µ

(P (A))xx (E). Therefore

∫R |λ|2dµ

(P (A))xx =

∫R |λ|2dµ

(P (A))Utx,Utx

that means Ut(D(A)) =

D(A). Now (f) in Proposition 2.32 proves UtA =∫R e

itλλdP (A) = AUt writing these operatorsin terms of integrals and observing that the condition on the domains necessary and sufficientto write = in place of ⊂ is satisfied with either order of the factors.

Proposition 2.34. If A : D(A) → H is a selfadjoint operator in the Hilbert space H andx ∈ D(A), then

−i ddt

∣∣∣∣t=s

eitAx = eisAAx = AeisAx .

Proof. Let us start with s = 0. Notice that if x ∈ D(A), (i) in Proposition 2.32 yields∣∣∣∣∣∣∣∣1h(eihAx− x)− iAx∣∣∣∣∣∣∣∣2 =

∫R

∣∣∣∣1h(eihr − 1)− ir∣∣∣∣2 dµ(P (A))

xx (r) . (2.70)

The integrand tends to 0 pointwise as h→ 0. On the other hand, Lagrange theorem applied tothe real and imaginary part of h 7→ eihr, for [−|H|, |H|] 3 h 7→ 1

h(eihr − 1) = coshr−1h + i sinhr

hsays that, for some h0, h

′0 ∈ [−|H|, |H|],∣∣∣∣1h(eihr − 1)− ir

∣∣∣∣2 =∣∣−r sin(h0r) + ir cos(h′0r)− ir

∣∣2 =∣∣− sin(h0r) + i cos(h′0r)− i

∣∣2 r2 ≤ 9r2.

The map R 3 r 7→ r2 is µ(P (A))xx -integrable since x ∈ D(A) = ∆ı2 . Finally, dominated convergence

theorem proves that the limit of the left-hand side of (2.70) vanishes for h→ 0 establishing thethesis for s = 0. The case s 6= 0 can be proved observing that∣∣∣∣∣∣∣∣1h(ei(s+h)Ax− eisAx)− ieisAAx

∣∣∣∣∣∣∣∣2 =

∣∣∣∣∣∣∣∣eisA ï1

h(eihAx− x)− iAx

ò∣∣∣∣∣∣∣∣2 =

∣∣∣∣∣∣∣∣1h(eihAx− x)− iAx∣∣∣∣∣∣∣∣2 ,

and eventually applying the result of the previous proposition.

92

Exercise 2.12. Prove that if A ∈ B(H) is selfadjoint in the Hilbert space H, then

eitA =+∞∑n=0

(it)n

n!An

for every t ∈ R, where the convergence of the series is in the operator norm topology.

Solution. From (i) Proposition 2.32, using the fact that eitA −∑Nn=0

(it)n

n! An is bounded,∣∣∣∣∣∣

∣∣∣∣∣∣eitA −N∑n=0

(it)n

n!An

∣∣∣∣∣∣∣∣∣∣∣∣ =

∣∣∣∣∣∣∣∣∣∣∣∣∫σ(A)

eitr −N∑n=0

(it)n

n!rn dP (A)

∣∣∣∣∣∣∣∣∣∣∣∣ ≤ sup

r∈σ(A)

∣∣∣∣∣∣eitr −N∑n=0

(itr)n

n!

∣∣∣∣∣∣ .For t ∈ R fixed, the limt as N → +∞ of the right-most term above vanishes proving the thesis.This is because the power series

ez =+∞∑n=0

zn

n!

uniformly converges in every closed disk centred at the origin with finite radius, since the con-vergence radius of the series is +∞. So the convergence is uniform in any compact set of C, initσ(A) in particular which is compact as A is bounded and Proposition 2.29 is valid. 2.

2.6 Elementary Quantum Formalism: A rigorous approach

Coming back to the discussion in the introduction, let us show how practically the physicalhypotheses on quantum systems (1)-(3) have to be mathematically interpreted (again reversingthe order of (2) and (3) for our convenience) in the general case of infinite dimensional Hilbertspaces.

2.6.1 Elementary formalism for the infinite dimensional case

Our general assumptions on the mathematical description of quantum systems are the followingones.

1. A quantum mechanical system S is always associated to Hilbert space H, finite or infinitedimensional;

2. observables are pictured in terms of (generally unbounded) self-adjoint operators A in H,

3. states are of equivalence classes of unit vectors ψ ∈ H

[ψ] = eiαψ | α ∈ R

(the equivalence realtion being ψ ∼ ψ′ iff ψ = eiaψ′ for some a ∈ R).

93

Let us show how the mathematical assumptions (1)-(3) permit us to set the physical propertiesof quantum systems (1)-(3) of Section 1.1.2 into mathematically nice form in the general caseof an infinite dimensional Hilbert space H.

(1) Randomness: The Borel subset E ⊂ σ(A), represents the outcomes of measurementprocedures of the observable associated with the selfadjoint operator A. (In case of continuousspectrum the outcome of a measurement is at least an interval in view of the experimentalerrors.) Given a state represented by the unit vector ψ ∈ H, the probability to obtain E ⊂ σ(A)as an outcome when measuring A is

µ(P (A))ψ,ψ (E) := ||P (A)

E ψ||2 , (2.71)

where we have used the PVM P (A) of the operator A.Going along with this interpretation, the expectation value, 〈A〉ψ, of A when the state isrepresented by the unit vector ψ ∈ H, turns out to be

〈A〉ψ :=

∫σ(A)

λ dµ(P (A))ψ,ψ (λ) . (2.72)

This identity makes sense provided ı : σ(A) 3 λ → λ belongs to L1(σ(A), µ(P (A))ψ,ψ ) (which is

equivalent to say that ψ ∈ ∆|ı|1/2 and, in turn, that ψ ∈ D(|A|1/2)), otherwise the expectationvalue is not defined.Since

L2(σ(A), µ(P (A))ψ,ψ ) ⊂ L1(σ(A), µ

(P (A))ψ,ψ )

because µ(P (A))ψ,ψ is finite, we have the popular identity arising from (2.41),

〈A〉ψ = 〈ψ|Aψ〉 if ψ ∈ D(A) . (2.73)

The associated standard deviation, ∆Aψ, results to be

∆Aψ :=

∫σ(A)

(λ− 〈A〉ψ)2 dµ(P (A))ψ,ψ (λ) . (2.74)

This definition makes sense provided ı ∈ L2(σ(A), µ(P (A))ψ,ψ ) (which is equivalent to say that

ψ ∈ ∆ı and, in turn, that ψ ∈ D(A)).As before, the functional calculus permits us to write the other popular identity

∆Aψ =»〈ψ|A2ψ〉 − 〈ψ|Aψ〉2 if ψ ∈ D(A2) ⊂ D(A) . (2.75)

We stress that Heisenberg inequalities, as established in (1) Exercise 1.1, are now completelyjustified as the reader can easily check.

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(3) Collapse of the state: If the Borel set E ⊂ σ(A) is the outcome of the (idealized)measurement of A, when the state is represented by the unit vector ψ ∈ H, the new stateimmediately after the measurement is represented by the unit vector

ψ′ :=P

(A)E ψ

||P (A)E ψ||

. (2.76)

Remark 2.23. Obviously this formula does not make sense if µ(P (A))ψ,ψ (E) = 0 as expected.

Moreover the arbitrary phase affecting ψ does not give rise to troubles due to the linearity of

P(A)E .

(2) Compatible and Incompatible Observables: Two observables A, B are compatible– i.e. they can be simultaneously measured – if and only if their spectral measures commutewhich means

P(A)E P

(B)F = P

(B)F P

(A)E , E ∈ B(σ(A)) , F ∈ B(σ(B)) . (2.77)

In this case||P (A)

E P(B)F ψ||2 = ||P (B)

F P(A)E ψ||2 = ||P (A,B)

E×F ψ||2

where P (A,B) is the joint spectral measure of A and B, has the natural interpretation of theprobability to obtain the outcomes E and F for a simultaneous measurement of A and B. Ifinstead A and B are incompatible it may happen that

||P (A)E P

(B)F ψ||2 6= ||P (B)

F P(A)E ψ||2 .

Sticking to the case of A and B incompatible, exploiting (2.76),

||P (A)E P

(B)F ψ||2 =

∣∣∣∣∣∣∣∣∣∣∣∣P (A)E

P(B)F ψ

||P (B)F ψ||

∣∣∣∣∣∣∣∣∣∣∣∣2

||P (B)F ψ||2 (2.78)

has the natural meaning of the probability of obtaining first F and next E in a subsequent mea-surement of B and A.

Remark 2.24. It is worth stressing that the notion of probability we are using here cannot bea classical notion because of the presence of incompatible observables. The theory of conditionalprobability cannot follows the standard rules. The probability Pψ(EA|FB), that (in a statedefined by a unit vector ψ) a certain observable A takes the value EA when the observable Bhas the value FB, cannot be computed by the standard procedure

Pψ(EA|FB) =Pψ(EA AND FB)

Pψ(FB)

95

if A and B are incompatible, just because, in general, nothing exists which can be interpreted

as the event “EA AND FB” if P(A)E and P

(B)F do not commute! The correct formula is

Pψ(EA|FB) =〈ψ|P (B)

F P(A)E P

(B)F ψ〉

||P (B)F ψ||2

which leads to well known different properties with respect to the classical theory, the so calledcombination of “probability amplitudes” in particular. As a matter of fact, up to now we do nothave a clear notion of (quantum) probability. This issue will be clarified in the next chapter.

2.6.2 Commuting spectral measures

The reason to pass from operators to their spectral measures in defining compatible observablesis that, if A ad B are selfadjoint and defined on different domains, AB = BA does not makesense in general. Moreover it is possible to find counterexamples (due to Nelson) where commu-tativity of A and B on common dense invariant subspaces does not implies that their spectralmeasures commute. However, from general results again due to Nelson, one has the followingnice result we will prove later (see Exercise 5.2).

Proposition 2.35. If selfadjoint operators, A and B, in a Hilbert space H commute on acommon dense invariant domain D where A2 + B2 is essentially selfadjoint, then the spectralmeasures of A and B commute.

In addition to the afore-mentioned direct result by Nelson, there are several technical facts pro-viding necessary and sufficient conditions for commutativity of the spectral measures of pairs ofselfadjoint operator. The most elementary and perhaps useful is the following one.

Proposition 2.36. Let A, B be selfadjoint operators in the complex Hilbert space H. Thefollowing facts are equivalent,

(i) the spectral measures of A and B commute (i.e. (2.77) holds),

(ii) eitAeisB = eisBeitA for every s, t ∈ R,

(iii) eitAP(B)E = P

(B)E eitA for every t ∈ R and E ∈ B(R),

(iv) eitAB ⊂ BeitA for all t ∈ R, that is equivalent to eitAB = BeitA for all t ∈ R.

Holding the above statements, we also have eitA(D(B)) = D(B) for all t ∈ R.

Proof. Evidently (i) implies (ii) since∫R sdP

(A)∫R tdP

(B) =∫R tdP

(B)∫R sdP

(A) if s and t arecomplex simple functions due to (2.53), and the result extends to the exponentials by exploit-ing (c) Proposition (2.25) with suitable sequences of bounded simple functions tending to the

96

exponential functions. Let us prove that (ii) entails (iii). From (ii) and for x, y ∈ H, we have〈x|e−itAeisBeitAy〉 = 〈x|eisBy〉, that can be re-phrased as∫

Reisrdµ

(P (B))Utx,Uty

(r) =

∫Reisrdµ(P (B))

xy (r) ,

where Ut := eitA. If f ∈ S (R), since both µ(P (B))xy and µ

(P (B))Utx,Uty

are complex measure (so thattheir absolute variations are finite measures) we have∫

R|f(s)|

∫R|eisr| d|µ(P (B))

Utx,Uty|(r)ds < +∞ ,

∫R|f(s)|

∫R|eisr| d|µ(P (B))

x,y |(r)ds < +∞ .

So, the very definition of integral with respect to a complex measure and Fubini and Tonellitheorems imply that∫

R

Å∫Rf(s)eisrds

ãdµ

(P (B))Utx,Uty

(r) =

∫R

Å∫Rf(s)eisreisrds

ãdµ(P (B))

xy (r) .

Since the Fourier transform is a bijection from S (R) onto S (R), the provious identity can bere-written ∫

Rg(r)dµ

(P (B))Utx,Uty

(r) =

∫Rg(r)dµ(P (B))

xy (r) , (2.79)

for every g ∈ S (R). Using Stone-Weierstrass theorem and a further smoothing procedure, itis possible to prove that if f is a complex continuous compactly supported function over R,there is a sequence of smooth functions fn with compact support included in a large compact[−2a, 2a] (obtained out of approximating polynomials by truncating them outside of [−2a, 2a]and smoothing there all them), with supp(f) ∈ [−a, a], such that ||f − fn||∞ → 0 for n→ +∞.Since the measures in (2.79) are finite, this fact immediately implies that (2.79) holds true alsoif g is continuous and compactly supported. Both Borel measures are regular because [Rud86],since they are finite, open sets are countable union of compact with finite measure. Riesz’

theorem for positive (regular) Borel measures [Rud86] implies that µ(P (B))xy (E) = µ

(P (B))Utx,Uty

(E) for

every Borel set E ∈ B(R). In other words 〈x|(U∗t P(B)E Ut−P (B)

E )y〉 = 0 for every x, y ∈ H which,

in turn, means UtP(B)E = P

(B)E Ut, namely, (iii). To prove that (iii) implies commutativity of the

measures P (A) and P (B), we proceed as above starting by observing that for x, y ∈ H we have

〈x|eitAP (B)E y〉 = 〈x|P (B)

E eitAy〉. The same argument previously exploited leads to µ(A)

P(B)E x,y

(F ) =

µ(A)

x,P(B)E y

(F ), namely 〈x|P (B)E P

(A)F y〉 = 〈x|P (A)

F P(B)E y〉 for all x, y ∈ H and E,F ∈ B(R). This is

equivalent to (i).Finally, assuming eitAB ⊂ BeitA for all t ∈ R, applying e−itA on the right of both sides andusing the fact that t is arbitray, proves BeitA ⊂ eitAB for all t ∈ R so that eitAB = BeitA t ∈ R.This identity is equivalent to eitABe−itA = B. In turn, the found identity is the same as saying

that (iii) holds, eitAP(B)E e−itA = P

(B)E for all t ∈ R and E ∈ B(R), in view of Proposition 2.30.

The last statement immediately arises from the second assert in (iv), using the fact that eitA isbijective.

97

With similar arguments the following proposition regarding a special case of A ∈ B(H) can beproved straightforwardly.

Proposition 2.37. Let A, B be selfadjoint operators in the complex Hilbert space H. IfA ∈ B(H) the following facts are equivalent,

(i) the spectral measures of A and B commute (i.e. (2.77) holds),

(ii) AB ⊂ BA (where = replaces ⊂ if also B ∈ B(H)) ,

(iii) Af(B) ⊂ f(B)A, if f : σ(B)→ R is Borel measurable ,

(iv) P(B)E A = AP

(B)E if E ∈ B(σ(B)) .

Proof. (i) implies (iv) just using the definition of integral with respect to a PVM integratig thefunction ı with respect to P (A). Integrating again f with respect to P (B) we obtain (iii) from

(iv). To do it observe that µ(P (B))Ax,Ax(E) ≤ ||A||2µ(P (B))

x,x (E) (since P (B) and A commute) so thatAx ∈ D(f(B)) if x ∈ D(f(B)). The special case f = ı produces (ii) from (iii). Finally (ii)implies AnB ⊂ BAn and also, using Exercise 2.12 and the fact that our B is closed becauseselfadjoint, we have eitAB ⊂ BeitA for every t ∈ R. Proposition 2.36 finally proves that (i) isvalid closing the proof.

Another useful result toward the converse direction is the following.

Proposition 2.38. Let A, B be selfadjoint operators in the complex Hilbert space H suchthat their spectral measures commute. The following facts hold.

(a) ABx = BAx if x ∈ D(AB) ∩D(BA) .

(b) 〈Ax|By〉 = 〈Bx|Ay〉 if x, y ∈ D(A) ∩D(B).

Proof. (a) Take y ∈ D(B) and x ∈ D(AB). Since eitBeisA = eisAeitB, we have

〈e−itBy|eisAx〉 = 〈y|eisAeitBx〉

computing the t-derivative at t = 0 according to Proposition 2.34 and using continuity of eisA,we have

〈By|eisAx〉 = 〈y|eisABx〉 .

From the definition of adjoint, we have that eisAx ∈ D(B∗) = D(B) and eisABx = B∗eisAx =BeisAx. Assuming x ∈ D(BA) and exploiting Proposition 2.34 once more, we can finally cal-culate the s-derivative at s = 0 of eisABx = BeisAx, using the fact that B is closed, obtainingABx = BAx.(b) It is sufficient to compute the relevant derivatives of 〈e−itBy|eisAx〉 = 〈e−isAy|eitBx〉 exploit-ing Proposition 2.34.

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2.6.3 A first look to time evolution of quantum states

As we already said, studying a quantum system in an inertial frame and assuming temporal ho-mogeneity, time evolution of states is described in terms of a strongly-continuous one-parameter

group of unitary operators of the form Ut := e−it~H , with t ∈ R, and where the selfadjoint opera-

tor H is called the Hamiltonian operator of the quantum system (depending on the referenceframe). The observable H has the physical meaning of the energy of the quantum system in theconsidered reference frame.If a quantum state at time t = 0 is represented by the unit vector ψ ∈ H, where H is the Hilbertspace of the system, the evoluted state ψt at a generic instant of time t is therefore

ψt = Utψ . (2.80)

We do not discuss here the motivations of this description of time evolution, we just make someimportant observations.

Remark 2.25.(a) If we represent the state ψ by another vector ψ′ := eiαψ at t = 0, the evoluted state is

coherently represented by ψ′t = Utψ′ = eiαUtψ in view of linearity of Ut, so that the description

of time evolution is phase-independent as expected: it preserves equivalence classes

[ψ] = eiαψ | α ∈ R

of unit vectors, i.e. states. As a consequence, we can unambiguously define an action of timeevolution on states: Ut[ψ] := [Utψ].

(b) Since Ut is isometric, the unit normalization of ψt is preserved by time evolution and

this is in agreement with the intepretation of the measures µ(P (A))ψt,ψt

which must always satisfy

µ(P (A))ψt,ψt

(R) = 1 as they are probability measures.

According to Propositions 2.33 and 2.34, if ψ ∈ D(H), form (2.80) we have

d

dtψt =

d

dte−i

t~Hψ = −i1

~He−i

t~Hψ = −i1

~Hψt .

We have this way found the celebrated Schrodinger equation:

i~dψtdt

= Hψt . (2.81)

It is worth stressing that the correct topology to interpret the derivative is that of the Hilbertspace. In other words, Schrodinger equation is not a standard PDE in the case of H = L2(R3, d3x)that is the most elementary case in standard quantum mechanics where, for some real functionV : Rn → R,

H := H0 and H0 = − ~2

2m∆ + V .

99

Above, H0 is defined on a suitable dense linear domain D(H0) ⊂ H of smooth functions whereit is essentially selfadjoint. Nevertheless, it is possible to prove that, under suitable hypotheses,jointly regular solutions ψ : R× R3 → C of the PDE transcription of (2.81),

i~∂ψ(t, x)

∂t+

~2

2m∆xψ(t, x)− V (x)ψ(t, x) = 0

define proper solutions of (2.81).A very particular class of physically interesting solutions are the so called stationary statesof a given Hamiltonian operator H. They are defined when σp(H) 6= ∅. If E ∈ σp(H) andψE ∈ D(H) is a corresponding eigenstate so that HψE = EψE , its time evolution is trivial

e−it~HψE = e−i

t~EψE .

The quantum state [ψE ] associated to ψE is a stationary state with energy E. Notice that, this

state is fixed under time evolution, since states are (normalized) vectors up to phases and e−it~E

is such.For a non-relativistic spinless particle described in H = L2(R3, d3x), where the position operatorsalong the Cartesian axes of the used inertial reference frame is represented by the multiplicativeoperator Xj as discussed in the Example 2.2, a stationary state ψE ∈ L2(R3, dx) has constantprobability density |ψEt(x)|2 = |ψE(x)|2 of finding the particle at x ∈ R3. When discussing forexample the electron in the Hydrogen atom (with mass m and electrical charge e and assumingthe proton at the origin of coordinates generating the Coulomb force as a geometric point ofthe matter), the stationary states of the electron with the energy levels corresponding to thespectrum of the Coulomb Hamiltonian H0

H0 := − ~2

2m∆− e2

||x||: S (R3)→ L2(R3, d3x) ,

define the orbitals of the atom.

Remark 2.26. Roughly speaking, stationary states are stable states of the matter andall relatively stable structures of physical objects are described in terms of stationary quantumstates of the corresponding Hamiltonian operator of the system. These states may cease to bestable when the Hamiltonian changes because of interactions with some external quantum sys-tem. For instance, the stationary states of the electron of the Hydrogen atom are stationary assoon as the system is kept isolated. If interacting with other systems (especially photons), thesestates becomes non-stationary because they are not represented by eigenvectors of the completeHamiltonian operator of the overall system. Even for an isolated Hydrogen atoms the protonshould be treated quantistically, and the complete system is made of a pair of quantum particlesdescribed in a overall Hilbert space L2(R3

e ×R3p, d

3xe⊗ d3xp). Usually the motion of the protonis neglected and it is classically treated. This is because its mass is around 2000 times that ofthe electron and in many applications where one is essentially interested in the motion of the

100

electron, it can be considered as a fixed classical particle.

Example 2.8. Let us consider the case of a free spinless particle of mass m > 0. Referring toorthonormal Cartesian coordinates of an inertial reference frame, its Hilbert space is L2(R3, d3x).This explicit representation of the Hilbert space of a non-relativistic particle where the positionoperators are multiplicative is called position representation. The Hamiltonian operator His the unique selfadjoint extension of the essentially selfadjoint operator

H0 :=1

2m

3∑k=1

P 2k : S (R3)→ L2(R3, d3x) .

It is evident that it includes only the kinetic part of the energy. In this sense the particle is free.To go on, it is more easy to represent the Hilbert space as a L2 space where the momentumoperators are described by multiplicative operators. As we know from the content of (2) Example2.2 (use Eq.(2.23) in particular), this realisation of the Hilbert space is connected to the positionrepresentation by means of a Hilbert-space isomorphism given by the Fourier-Plancherel operator

F : L2(R3, d3x) 3 ψ 7→ ψ ∈ L2(R3, d3k) .

Observe that this Hilbert space isomorphism reduces to the standard integral Fourier trans-form on S (R3) and bijectively transform this subspace into itself (changing the varible of thefunctions from x to k). The representation L2(R3, d3k) of the Hilbert space, where the momen-tum operators are multiplicative, is popularly named momentum representation. Here, theHamiltonian operator H = H0 is correspondingly represented by the selfadjoint operator

H ′ := FHF−1 .

Since it is the square of the momentum operator up to a constant factor (2m)−1, we immediatelyhave that it must act as Ä

H ′ψä

(k) =k2

2mψ(k) (2.82)

where k2 :=∑3j=1 k

2j , and

D(H ′) :=ψ ∈ L2(R3, d3k)

∣∣∣ k2ψ ∈ L2(R3, d3k).

The spectrum of H is continuous and it is not difficult to prove that σ(H) = σc(H) = [0,+∞) asa by-product of (2.82), as is expected form physical considerations, since the energy is completelykinetic.Time evolution has a direct representation here:Ä

e−itH′ψä

(k) := e−itk2

2m ψ(k) . (2.83)

101

Notice that the right-hand side belongs to S (R3) at every t if it does at t = 0.Time evolution has a corresponding representation in the space L2(R3, d3x) obtained throughthe action of the Fourier-Plancherel isomorphism

e−itH = F−1e−itH′F .

If ψ ∈ S (R3), we can take advantage of the standard integral Fourier transform,

ψ(k) =1

(2π)3/2

∫R3e−ikxψ(x)d3x and ψ(x) =

1

(2π)3/2

∫R3eikxψ(k)d3k . (2.84)

Composing these transformations with (2.83), it immediately arisesÄe−itHψ

ä(x) =

1

(2π)3/2

∫R3ei(kx−

k2t2m

)ψ(k)d3k for ψ ∈ S (R3) .

In particular, notice that the time evolution leaves fixed the space S (R3).

2.6.4 A first look to (continuous) symmetries and conserved quantities

As we shall discuss better later, physical operations acting on a given quantum system changingits states are pictured in terms of either unitary or anti unitary transformations U : H→ H andthese operations are called (quantum) symmetries.Symmetries U transform vectors ψ 7→ ψU := Uψ preserving their norms (since U is isometricby hypotheses) and the transformation is phase independent (eiαψ is transformed into eiαψU ),therefore symmetries acts on states unambiguously and we can define an action on states viewedas equivalence classes: U [ψ] := [Uψ].A particular subclass of symmetries are provided by continuous symmetries. These arestrongly-continuous one-parameter groups of unitary operators eisAs∈R generated by someselfadjoint operator A : D(A) → H interpreted as an observable somehow related with thecontinuous symmetry. A is called the generator of the continuous symmetry.When a continuous symmetry commutes with the time evolution, i.e. (always assuming ~ = 1)

eisBe−itH = e−itHeisB for all t, s ∈ R , (2.85)

the symmetry is said to be a dynamical symmetry. This feature has a fundamental conse-quence. The generator B becomes a constant of motion, in the sense that all of the statisticalproperties of outcomes of measurements of B over a given state represented by ψ ∈ H turn outto be independent from the time evolution of ψ. If E ∈ B(R), applying Proposition 2.36, theprobability that the outcome of a measurement of B at time t belongs to E is

µP(B)

Utψ,Utψ(E) = ||P (B)E Utψt||2 = ||UtP (B)

E ψ||2 = ||P (B)E ψ||2 = µP

(B)

ψ,ψ (E) ,

which coincides to the probability of obtaining E at time t = 0 measuring B. The crucial pas-

sage above is P(B)E Ut = UtP

(B)E that is consequence of (2.85) through Proposition 2.36 for A = H.

Remark 2.27. If B is a constant of motion as defined above, the expectation value of Band its standard deviation are constant in time in particular, just applying the definition of of

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expectation value and standard deviation.These results, though immediately arising form the definition of expectation value and standarddeviation, are popularly obtained by physicists using identities (2.73) and (2.74) directly (whenthe requirements on domains are fulfilled), again using Proposition 2.36:

〈B〉ψt = 〈Utψ|BUtψ〉 = 〈ψ|U∗t BUtψ〉 = 〈ψ|BU∗t Utψ〉 = 〈ψ|Bψ〉 = 〈B〉ψ ,

and

∆Bψt = 〈Utψ|B2Utψ〉 − 〈B〉2ψt = 〈ψ|U∗t B2Utψ〉 − 〈B〉2ψ = 〈ψ|B2U∗t Utψ〉 − 〈B〉2ψ = ∆Bψ .

Example 2.9. Consider the momentum operator Pj along the j-th axis in R3. We want tofocus on the strongly continuous one-parameter group of unitary operators Va := e−iaPj witha ∈ R. It is convenient to deal with the momentum representation. As we know, here Pj is

nothing but the multiplicative operatorÄP ′jψä

(k) = kjψ(k) for every vector of the Hilbert space

ψ ∈ L2(R3, d3k). As in Example 2.8, we adopt the notation A′ := FAF−1 to write down themomentum representation A′ of operators given by A in position representation. It is easy toprove that Ä

V ′aψä

(k) = e−ikjaψ for every ψ ∈ L2(R3, d3k) .

Taking advantage of (2.84), if ψ ∈ S (R3), so that ψ ∈ S (R3) and vice versa, we immediatelyfind

(Vaψ) (x) =1

(2π)3/2

∫R3eikxe−ikjaψ(k)d3k =

1

(2π)3/2

∫R3eikx−kjaψ(k)d3k = ψ(x− aej) .

In other words, Va pushes wavefunctions of S (R3) along the coordinate unit vector ej of alength a. Note that S (R3) is dense in L2(R3, d3x) and Va is continuous. Moreover, if S (R3) 3ψn → ψ ∈ L2(R3, d3x) as n → +∞, then S (R3) 3 ψn(· − aej) → ψ(· − aej) ∈ L2(R3, d3x) asn→ +∞ in view of the translational invariance of Lebesgue measure d3x. Summing up,

e−iaPjψ = e−iaPj limn→+∞

ψn = limn→+∞

e−iaPjψn = limn→+∞

ψn(· − aej) = ψ(· − aej) .

In other words, Äe−iaPjψ

ä(x) = ψ(x− aej) for every ψ ∈ L2(R3, d3x) . (2.86)

In the language of physicists, the momentum along the j-th direction is the generator of physicalspace translations of the quantum system along the j-th direction.This is not the whole story if we also assume that the Hamiltonian of the particle is the free onegiven by (2.82) in momentum representation. In this juncture time evolution is represented by(2.83) again in momentum representation. It is therefore evident that

e−itHe−iaPj = e−iaPje−itH for every t, a ∈ R.

103

We conclude that, with the said free Hamiltonian, the momentum operator along the j-th direc-tion is a constant of motion. Therefore the statistical features of the measurements of Pj areinvariant along the temporal evolution of the state of the system.

2.7 Some Basic Operator Topologies

There are at least 7 to 9 relevant topologies [KaRi97, BrRo02] in Quantum Theory which enterthe game discussing sequences of operators. Here, we limit ourselves to quickly illustrate a fewof the most important ones [Mor18]. We assume that H is a Hilbert space though some of theillustrated examples may be extended to more general context with some re-adaptation.

(a) The strongest topology is the uniform operator topology in B(H): It is the Hausdorfftopology induced by the operator norm || || defined in (2.8).As a consequence of the definition of this topology, a sequence of elements An ∈ B(H) is said touniformly converge to A ∈ B(H) when ||An −A|| → 0 for n→ +∞.We already know that B(H) is a Banach algebra with respect to that norm and also a unitalC∗-algebra.

(b) If L(D;H), where D ⊂ H is a subspace, denotes the complex vector space of the operatorsA : D → H, the strong operator topology on L(D;H) is the Hausdorff topology induced bythe seminorms px with x ∈ D and px(A) := ||Ax|| if A ∈ L(D;H). By definition of topologyinduced by a family of seminorms, the open sets of this topology are the empty set and setsmade of unions of arbitrary many intersections of an arbitrary large but finite number n of open

balls B(x1,...,xn)r1,...,rn (A0) associated to the seminorms pxi , with different xi ∈ D, arbitrary finite radii

ri > 0, and a common centre A0 ∈ L(D;H) arbitrarily fixed:

B(x1,...,xn)r1,...,rn (A0) := A ∈ L(D;H) | pxi(A−A0) ≤ ri , i = 1, . . . , n .

As a consequence of the definition of this topology, a sequence of elements An ∈ L(D;H) is saidto strongly converge to A ∈ L(D;H) when ||(An −A)x|| → 0 for n→ +∞ for every x ∈ D.It should be evident that, if we restrict ourselves to work in B(H), the uniform operator topologyis stronger than the strong operator topology.

(c) The weak operator topology on L(D;H) is the Hausdorff topology induced by theseminorms px,y with x ∈ H, y ∈ D and px,y(A) := |〈x|Ay〉| if A ∈ L(D;H). In other words,the open sets of this topology are the empty set and sets which are unions of arbitrary many

intersections of an arbitrary large but finite number n of open balls B(x1,y1,...,xn,yn)r1,...,rn (A0) associated

to the seminorms pxi,yi , with different xi ∈ H, yi ∈ D arbitrary finite radii ri > 0, and a commoncentre A0 ∈ L(D;H) arbitrarily fixed

B(x1,y1,...,xn,yn)r1,...,rn (A0) := A ∈ L(D;H) | pxi,yi(A−A0) ≤ ri , i = 1, . . . , n .

As a consequence of the definition of this topology, a sequence of elements An ∈ L(D;H) is said

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to weakly converge to A ∈ L(D;H) when |〈x|(An − A)y〉|| → 0 for n → +∞ for every x ∈ Hand y ∈ D. It should be evident that the strong operator topology is stronger than the weakoperator topology.

We introduce another pair of topologies which depend on the space B1(H) of trace-class opera-tors we will discuss later.

(d) The ultrastrong topology (also known as σ-strong topology) on B(H) is the Haus-dorff topology induced as above by the family seminorms pT with T ∈ B1(H) with T ≥ 0 and

pT (A) :=»tr(TA∗A) if A ∈ B(H). In spite of the name, it is weaker than the uniform operator

topology.

(e) The ultraweak topology (also known as σ-weak topology) on B(H) is the Haus-dorff topology induced as above by the family of seminorms qT with T ∈ B1(H) defined as

qT (A) := |»tr(TA)| if A ∈ B(H). It is stronger than the weak operator topology.

Another relevant topology concerns the topological dual space of B(H).

(f) As a normed space B(H) induces an important weak topology on its topological dual

B(H)∗ := f : B(H)→ C | f linear and continuous .

The ∗-weak topology on B(H)∗ is that induced as above by the family of seminorms pAA∈B(H)

defined as pA(f) := |f(A)| for every f ∈ B(H)′. This definition is general, it is valid replacingB(H) for a normed space B and B(H)∗ for its topological dual B∗. Hahn-Banach theorem provesthat the ∗-weak topology is Hausdorff because the functionals in B′ separate the elements of B.Notice that B′ is also a normed Banach space with respect to the standard operator norm

||f || = sup06=A∈B

|f(A)|||A||B

.

This normed topology is stroger than the ∗-weak one. The relevance of the ∗-weak topologyis in particular due to the so called Banach-Alaoglu theorem: the closed unit ball of B(H)∗ iscompact with respect to the ∗-weak topology.

Example 2.10.(1) If f : R→ C is Borel measurable, and A a selfadjoint operator in H, consider the sets

Rn := r ∈ R | |f(r)| < n for n ∈ N .

It is clear that χRnf → f pointwise as n → +∞ and that |χRnf |2 ≤ |f |2. As a consequence,restricting to ∆f the operators appearing on the left hand side,∫

σ(A)χRnfdP

(A)

∣∣∣∣∣∆f

→ f(A) strongly, for n→ +∞,

105

as an immediate consequence of Lebesgue’s dominate convergence theorem and the first part of(i) in Proposition 2.32. (See also exercise 2.9.)

(2) If in the previous example f is bounded on σ(A), and fn → f uniformly on σ(A) (or

P -essentially uniformly ||f − fn||(P(A))

∞ → 0 for n→ +∞), then

fn(A)→ f(A) uniformly, as n→ +∞,

again for the second part of (i) in Proposition 2.32.

Exercise 2.13. Prove that a selfadjoint operator A in the Hlbert H admits a dense set ofanalytic vectors in its domain.

Solution. Consider the class of functions fn = χ[−n,n] where n ∈ N. As in (1) Example 2.10,

we have ψn := fn(A)ψ =∫

[−n,n] 1dP (A)ψ →∫R 1dP (A)ψ = P

(A)R ψ = ψ for n → +∞. Therefore

the set D := ψn | ψ ∈ H , n ∈ N is dense in H. The elements of D are analytic vectors

for A as we go to prove. Clearly ψn ∈ D(Ak) since µ(P (A))ψn,ψn

(E) = µ(P (A))ψ,ψ (E ∩ [−n, n]) as im-

mediate consequence of the definition of the measure µ(P (A))x,y , therefore

∫R |λk|2dµ

(P (A))ψn,ψn

(λ) =∫[−n,n] |λ|2kdµ

(P (A))ψ,ψ (λ) ≤

∫[−n,n] |n|2kdµ

(P (A))ψ,ψ (λ) ≤ |n|2k

∫R dµ

(P (A))ψ,ψ (λ) = |n|2k||ψ||2 < +∞.

Similarly ||Akψn||2 = 〈Akψn|Akψn〉 = 〈ψn|A2kψn〉 =∫R λ

2kdµ(P (A))ψn,ψn

(λ) ≤ |n|2k||ψ||2. We

conclude that∑+∞k=0

(it)k

k! ||Akψn|| conveges for every t ∈ C as it is dominated by the series∑+∞

k=0|t|kk! |n|

2k||ψ||2 = e|t| |n|2 ||ψ||2.

2.8 Appendix. On the spectral theorem and joint spectral mea-sures again

This appendix is devoted to prove in particular the existence of a PVM P (A) : B(R) → L (H)for a selfadjoint operator A : D(A) → H in a Hilbert space H according to Theorem 2.8. Theremaining statements of that theorem have been already established. As an intermediate resultwe also demonstrate the spectral theorem for normal operators in B(H). Finally, we also furnisha proof of the Theorem 2.9 regarding the notion of joint spectral measure.

2.8.1 Continuous functional calculus for bounded selfadjoint operators andC∗-algebras

Let us start by establishing some general properties of spectral theory for bounded operatorsand unital C∗-algebras.

Proposition 2.39. Let A ∈ B(H) for a Hilbert space H and let p : C → C a complexpolynomial of fixed degree n = 0, 1, . . ., then

σ(p(A)) = p(σ(A)) , (2.87)

106

where p(A) is interpreted as in (a) Proposition 2.32. Furthermore

σ(A∗) = λ | λ ∈ σ(A) ,

The results are valid replacing A ∈ B(H) for a ∈ A, where A is any unital C∗-algebra.

Proof. We explicitly take advantage of Proposition 2.16: for A ∈ B(H), we have that λ ∈ σ(A)if and only if A− λI is a bijection from H to H.First of all, we decompose the polynomials according to the fundamental theorem of alge-bra, p(z) = c(z − λ1)n1 · · · (z − λk)nr , where λ1, . . . , λr are its complex roots with multiplicityn1, . . . , nr > 0 such that

∑l nk = n and c 6= 0. As a consequence, a corresponding decom-

position holds for p(A) = c(A − λ1I)n1 · · · (A − λkI)nk . Define µ := p(λ). The polynomialC 3 z 7→ p′(z) := p(z)− µ vanishes at z = λ so that its decomposition includes a factor (z − λ)and thus p(A)− µI has a factor (A− λI) in the corresponding decomposition. If λ ∈ σ(A), theoperator (A − λI) is not bijective and therefore p′(A) := p(A) − µI (decomposed into its ownfactors (A − λ′kI)n

′k) cannot be a bijection from H to H. Indeed, if (A − λI) is not injective,

moving it on the right-most place of the product decomposition of p′(A) (all factors pairwisecommute), p′(A) cannot be injective. If (A − λI) is not surjective, moving it on the left-mostplace of the product decomposition of p′(A), p′(A) cannot be surjective. We have so far estab-lished that λ ∈ σ(A) implies µ = p(λ) ∈ σ(p(A)), i.e. p(σ(A)) ⊂ σ(p(A)). Let us prove the

converse inclusion. Suppose µ ∈ σ(p(A)). We know that p(z)− µ = c(z − α1)n′1 · · · (z − αk′)n

′r′ .

If all αk′ belonged to ρ(A), the operator p(A) : H → H would be bijective with left and right

inverse c−1(A− α1I)−n′1 · · · (A− αk′I)−n

′r′ that is impossible. So at least one of the αk′ belongs

to σ(A) and so p(αk′)− µ = 0. In other words, µ ∈ p(σ(A)) proving that σ(p(A)) ⊂ p(σ(A)).The second statement is quite obvious observing that if T ∈ B(H) then T ∗ is bijective if andonly if T is (Exercise 2.5). In this case (T ∗)−1 = (T−1)∗. Applying this fact to A−λI, we provethe thesis.With the obvious changes, this proof is valid also replacing B(H) for any unital C∗-algebraA.

We pass now to an important consequence stated in the next proposition, which is also valid forany unital C∗-algebra in place of B(H) with the same proof. The first assert extends Proposition2.29 and proves that it is actually independent from the spectral theorem.

Proposition 2.40. If A ∈ B(H) is normal (A∗A = AA∗) then

sup|λ| | λ ∈ σ(A) = ||A|| . (2.88)

If A = A∗ and p : R→ C is a polynomial, then

||p(A)|| = ||pσ(A) ||∞ . (2.89)

The results are valid also replacing A for a ∈ A where A is any unital C∗-algebra.

107

Proof. Let us prove (2.88). We need a preliminary quite interesting lemma.

Lemma 2.4. [Gelfand’s formula for the spectral radius]If A ∈ B(H) for a Hilbert space H, then

sup|λ| | λ ∈ σ(A) = limn→+∞

||An||1/n . (2.90)

The formula is valid also replacing A for a ∈ A, where A is any unital C∗-algebra.

Proof. Define rA := sup|λ| |λ ∈ σ(A). If |λ| > rA, then Rλ(A) exists by definition of resolvent.ρ(A) 3 λ 7→ Rλ(A) is holomorphic (interpreted as a Banach-space valued function) ad admitsTaylor expansion, defining ζ = 1/λ,

Rλ(A) = −+∞∑n=0

ζn+1Tn

which converges for |ζ| < 1/||A|| at least, as we already established (Proposition 2.18). It isknown from a celebrated theorem by Hadamard (whose proof is immediately generalizable toBanach-space valued holomorphic functions) that the convergence radius is defined by the firstsingularity which necessarily belongs to σ(A). The series −∑+∞

n=0 ζn+1Tn therefore converges

for |ζ| < 1/rA. The convergence radius R of the series must therefore satisfy R ≥ 1/rA, namely,1/R ≤ rA. formula of convergence radius R of holomorphic functions implies

1/R = lim supn||Tn||1/n ≤ rA .

On the other hand σ(An) = µn | µ ∈ σ(A) due to (2.87), so that, exploiting Proposition 2.18once more,

rnA = rAn ≤ ||An|| .

Hence rA ≤ lim infn ||An||1/n. In summary rA ≤ lim infn ||An||1/n ≤ lim supn ||An||1/n = rAwhich implies the thesis.

If A = A∗, we have ||A2|| = ||A∗A|| = ||A||2 and, similarly, ||(A2)2|| = ||A2||2 = ||A||4,||(A4)2|| = ||A4||2 = ||A||8 and so on. In general ||A2n || = ||A||2n . Applying (2.90), we find

sup|λ| | λ ∈ σ(A) = limn→+∞

||An||1/n = limn→+∞

||A2n ||1/2n = limn→+∞

||A||2n/2n = ||A|| .

Now consider A ∈ B(H), it holds that ||An|| = ||(An)∗An||1/2 = ||(A∗)nAn||1/2. If A is normal,all operators commute and the identity can be re-arranged to ||An|| = ||(A∗A)n||1/2. Since A∗Ais selfadjoint, we can implement the result above:

sup|λ| | λ ∈ σ(A) = limn→+∞

||An||1/n = limn→+∞

||(A∗A)n||1/(2n) =

Ålim

n→+∞||(A∗A)n||1/n

ã1/2

108

= ||A∗A||1/2 = ||A|| .

Let us pass to prove (2.89). Since A is selfadjoint, p(A) is normal. Therefore

||p(A)|| = sup|λ| | λ ∈ σ(p(A)) = sup|λ| | λ ∈ p(σ(A)) = ||pσ (A)||∞ ,

where we have exploited (2.87) in the last passage.

The utmost consequence of these propositions is the following theorem establishing existenceand continuity of the so-called continuous functional calculus for bounded selfadjoint operators.Actually, the same result holds true as it stands for unital C∗-algebras.

Theorem 2.10. Let A ∈ B(H) be a selfadjoint operator in the Hilbert space H. There is aunique unital ∗-algebra representation (Definition 2.5), called continuous functional calculus,

Ψ : C(σ(A)) 3 f → f(A) ∈ B(H)

that is continuous with respect to || · ||∞ in the domain and the operator norm in the codomainand such that

Ψ(ı) = A .

(Where ı : σ(A) 3 x 7→ x ∈ R.) It turns out that

(a) Ψ is isometric and hence injective,

(b) B ∈ B(H) commutes with every f(A) for f ∈ C(σ(A)) if B commutes with A.

The results are valid also replacing B(H) for any unital C∗-algebra A and A for every selfadjointelement a ∈ A.

Proof. If f ∈ C(σ(A)), there is a sequence of complex polynomials such that pn → f uniformlyon σ(A) as n → +∞ in view of Stone-Weierstrass theorem. Define f(A) := limn→+∞ pn(A).Due to (2.89), the sequence pn(A) is Cauchy. Hence there is a limit element in B(H) becausethis space is complete (Theorem 2.2). It is evident that the limit point does not depend on thesequence as a different sequence would satisfy ||p′n(A) − pn(A)|| = ||p′nσ(A) −pnσ(A) ||∞ → 0.With this definition, the map f 7→ f(A) is evidently isometric. Next observe that, restricting topolynomials, f 7→ f(A) is linear, preserves the product, and f 7→ f(A)∗, so that these featuresare preserved by the limit procedure when f ∈ C(σ(A)) is generic. By construction f(1) = Iand f(ı) = A. If B commutes with A, it commutes with all polynomials p(A). Hence

Bf(A) = B limn→+∞

pn(A) = limn→+∞

Bpn(A) limn→+∞

pn(A)B = f(A)B .

To conclude, we prove that a continuous unital ∗-representation Φ : C(σ(A))→ B(H) coincideswith Ψ if Φ(ı) = A. In fact, Ψ(ı) = Φ(ı) = A and Ψ(1) = Φ(1) = I, therefore Ψ(p) = Φ(p) for

109

every polynomial. By continuity, if pn → f as n→ +∞ in the norm || · ||∞ over σ(A), we haveΨ(f) = Φ(f).All presented arguments are valid if replacing B(H) with a general unital C∗-algebra A and Afor a = a∗ ∈ A.

2.8.2 Existence of the spectral measure for bounded selfadjoint operators

A fundamental consequence of Theorem 2.10 is the following proposition in the direction of thespectral theorem. As already introduced, Mb(σ(A)) is the unital C∗-algebra of the complexbounded Borel-measurable functions over σ(A), the relavant norm being || · ||∞ once again. Westress that, to formulate this proposition, the Hilbert-space structure is necessary and thereforeno straightforward generalizations also valid for for abstract C∗-algebras exist.

Proposition 2.41. Let A ∈ B(H) be a bounded selfadjoint operator in the Hilbert space H.There is a norm-decreasing (hence continuous) unital ∗-algebra representation (Definition 2.5),

Ψ′ : Mb(σ(A))→ B(H)

such thatΨ′(ı) = A ,

The representation also satisfies:

(a) Ψ′C(σ(A))= Ψ,

(b) B ∈ B(H) commutes with Ψ′(f) for every f ∈Mb(σ(A)) if B commutes with A,

(c) Mb(σ(A)) 3 fn → f point wise as n → +∞ and |fn| ≤ K for some K ∈ [0,+∞) and alln entails

Ψ′(fn)x→ Ψ′(f)x for every x ∈ H.

Proof. Taking x, y ∈ H, the linear map C(σ(A)) 3 f 7→ Fx,y(f) := 〈x|Ψ(A)y〉 satisfies |Fx,y(f)| ≤||x|| ||y|| ||f ||∞. Riesz’s theorem for complex measures [Rud91] implies that there exists a uniquecomplex regular Borel measure µxy : B(σ(A))→ C such that

〈x|Ψ(f)y〉 =

∫σ(A)

fdµxy ∀f ∈ C(σ(A)) , (2.91)

and also ||Fxy|| = |µxy|(σ(A)) ≤ ||x|| ||y||. Actually, all complex Borel measures over B(σ(A))are regular since the open sets of σ(A) are union of countably many compact sets [Rud91]. Inview of their uniqueness, the said Borel measure satisfy µxy(E) = µyx(E) as these complexmeasures produce the same result when integrating continuous functions in view of the propertyΨ(f) = Ψ(f)∗ and the standard properties of the scalar product. Using Riesz’ Lemma, it is easyto prove that, if f ∈Mb(σ(A)), there exists a unique operator Ψ′(f) ∈ B(H) such that

〈x|Ψ′(f)y〉 =

∫σ(A)

fdµxy ∀x, y ∈ H , (2.92)

110

and |〈x|Ψ′(f)y〉| ≤ ||f ||∞|µxy|(σ(A)) ≤ ||f ||∞||x|| ||y||, so that ||Ψ′(f)|| ≤ ||f ||∞. By construc-tion Ψ′(1) = I and Ψ′(ı) = A, furthermore Mb(σ(A)) 3 f 7→ Ψ′(f) is linear and thereforecoincides with Ψ on polynomials. Continuity implies that it coincides with Ψ on C(σ(A)) prov-ing (a). Ψ′ satisfies Ψ′(f)∗ = Ψ′(f) as a consequence of (2.92), the fact that the scalar productis Hermitian, and µxy(E) = µyx(E). To conclude the proof of the first statement it is sufficientto prove that Ψ′(f)Ψ′(g) = Ψ′(f · g). First consider f, g ∈ C(σ(A)). Since Ψ(f · g) = Ψ(f)Ψ(g)and Ψ′ extends Ψ:∫

σ(A)f · gdµx,y = 〈x|Ψ′(f · g)y〉 = 〈x|Ψ′(f)Ψ′(g)y〉 =

∫σ(A)

fdµx,Ψ′(g)y .

Riesz’ theorem implies that µx,Ψ′(g)y coincides to the complex regular Borel measure λ such that

λ(E) =

∫σ(A)

gdµxy .

Therefore∫σ(A)

f · gdµxy =

∫σ(A)

fdλ =

∫σ(A)

fdµx,Ψ′(g)y if f ∈Mb(σ(A)) and g ∈ C(σ(A)).

As a consequence∫σ(A)

f · gdµxy =

∫σ(A)

fdµx,Ψ′(g)y = 〈x|Ψ′(f)Ψ′(g)y〉 = 〈Ψ′(f)∗x|Ψ′(g)y〉 =

∫σ(A)

gdµΨ′(f)∗x,y

for x, y ∈ H, f ∈Mb(σ(A)), g ∈ C(σ(A)). With a similar reasoning we obtain that∫σ(A)

f · gdµxy =

∫σ(A)

gdµΨ′(f)∗x,y

must hold true also if g ∈Mb(σ(A)). Summing up, for x, y ∈ H, f, g ∈Mb(σ(A)), we have

〈x|Ψ′(f · g)y〉 =

∫σ(A)

f · gµxy =

∫σ(A)

gµΨ′(f)∗x,y = 〈Ψ′(f)∗x|Ψ′(g)y〉 = 〈x|Ψ′(f)Ψ′(g)y〉

so that Ψ′(f · g) = Ψ′(f)Ψ′(g) as wanted.The proof of (b) is analogous: If B ∈ B(H) commutes with A, it also commutes with everypolynomial p(A) and with every Ψ′(f) with f ∈ C(σ(A)) by continuity. Therefore, for everyf ∈ C(σ(A)).∫

σ(A)fdµx,By = 〈x|Ψ′(f)By〉 = 〈x|BΨ′(f)y〉 = 〈B∗x|Ψ′(f)y〉 =

∫σ(A)

fdµB∗x,y .

Riesz’ theorem implies that µx,By = µBx,y. The definition of Ψ′ immediately entails that〈x|Ψ′(f)By〉 = 〈B∗x|Ψ′(f)y〉 = 〈x|BΨ′(f)y〉 for every f ∈ Mb(σ(A)). That is the thesis, since

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x, y ∈ H are arbitrary.Let us prove (c). Exploiting the fact that Ψ′ is a ∗-representation we immediately have

||Ψ′(fn)x−Ψ′(f)x||2 = ||Ψ′(f − fn)x||2 = 〈Ψ′(f − fn)x|Ψ′(f − fn)x〉 = 〈x|Ψ′(|f − fn|2)x〉 .

Taking (2.92) into account,

||Ψ′(fn)x−Ψ′(f)x||2 =

∫σ(A)|f − fn|2dµxy → 0

as n→ +∞ in view of dominated convergence theorem since |µxy| is finite.

We are in a position to prove the existence part of the Spectral Theorem (Theorem 2.8) forbounded selfadjoint operators.

Theorem 2.11. If A ∈ B(H) is selfadjoint over the Hilbert space H, there exists a PVMP (A) : B(R)→ L (H) such that

A :=

∫Rı dP (A) .

More generally, if Ψ′ : Mb(σ(A))→ B(H) is defined as in Proposition 2.41, it holds that

Ψ′(f) =

∫σ(A)

f dP (A)

for every f ∈Mb(σ(A)).

Proof. Referring to Proposition 2.41, the wanted PVM is nothing but P(A)E := Ψ′(χE∩σ(A)) for

every E ∈ B(R) also assuming P(A)∅ := 0. Indeed, suppose that the said P (A) is a PVM.

If s =∑Nj=1 sjχEj is a simple function, linearity of Ψ′ immediately proves that Ψ′(s) =∑N

j=1 sjΨ′(χEj ) =

∫R s dP

(A). Now consider a sequence of simple functions sn such that|sn| ≤ |sn+1| ≤ |ı| over the compact σ(A), vanishing outside σ(A), and pointwise converg-ing to ı on σ(A). From (a) and (c) of Proposition 2.41 and (c) of Proposition 2.25 observingthat the PVM is concentrated on σ(A) per construction, it holds that∫

Rı dP (A)x =

∫σ(A)

ı dP (A)x = limn→+∞

∫σ(A)

sn dP(A)x = lim

n→+∞Ψ′(sn) = Ψ′(ı)x = Ax .

Since x ∈ H is arbitrary, we gets A =∫R ıdP

(A) as wanted. The same procedure, using a sequenceof simple functions sn such that |sn| ≤ |sn+1| ≤ |f | pointwise converging to f ∈ Mb(σ(A))produces the second identity in the thesis.

To end the proof, it is sufficient to prove that P(A)E := Ψ′(χE∩σ(A)) with E ∈ B(R) (and

obviously P(A)∅ := 0) defines a PVM. P

(A)R = I, P

(A)E P

(A)F = P

(A)E∩F , (P

(A)E )∗ = P

(A)E (in particular

P(A)E ∈ L (H)) are immediate consequences of the fact that Ψ′ is a ∗-representation, trivial

112

properties of characteristic functions χE , and Ψ′(1) = Ψ′(χσ(A)) = I. Finally, σ-additivityimmediately arises from (c) of Proposition 2.41 since, referring to a countable class of Ek ∈ B(R),

N∑k=1

χEk∩σ(A) → χσ(A)∩∪Nk=1

Ekpointwise as n→ +∞

if Ek ∩ Eh = ∅ for k 6= k and all functions are bounded by the constant 1.

2.8.3 Spectral theorem for normal operators in B(H)

The functional calculus developed in the previous section permits us to state and prove thespectral theorem for normal operators in B(H). So, in particular, the theorem embodies thecase of selfadjoint operators in B(H) and unitary operators.

Theorem 2.12. [Spectral Decomposition Theorem for Normal Operators in B(H)]Let T ∈ B(H) be a normal operator in the complex Hilbert space H.

(a) There is a unique PVM P (A) : B(C) → L (H), called the spectral measure of T , suchthat

T =

∫CzdP (T )(z, z) .

In particular D(T ) = ∆ı, where ı : C 3 z 7→ z.

(b) It resultssupp(P (T )) = σ(T )

so that, as the standard topology of C is second-countable, P (T ) is concentrated on σ(T ):

P (T )(E) = P (T )(E ∩ σ(T )) , ∀E ∈ B(C) . (2.93)

(c) z ∈ σp(T ) if and only if P (T )(z) 6= 0, in particular it happens if z is an isolated point of

σ(T ). Finally P(T )z is the orthogonal projector onto the eigenspace of z ∈ σp(A).

(d) z ∈ σc(T ) if and only if P (T )(λ) = 0 but P (T )(E) 6= 0 if E 3 λ is an open set of C.

Proof. (a) Let us prove that a PVM over C with T =∫C zdP

(T )(z) exists. Decompose T = A+iBwhere A = 1

2(T +T ∗) and A = 12i(T −T

∗) are selfadjont, belong to B(H) and commute becauseT and T ∗ do by hypothesis. Notice that, as a consequence of (b) Proposition 2.41, the spectral

measure P (A) of A, that exists for Theorem 2.11 and satisfies P(A)E = Ψ′A(χE), commutes with B.

With the same argument, the spectral measure P (B) of B commutes with the spectral measureof A.Next consider the compact K = [−||A||, ||A||] × [−||B||, ||B||] ⊂ R2 ≡ C and the class of stepfunctions thereon. A step function is a simple function of the form

s(x, y) =N∑i=1

M∑j=1

sijχIi(x)χJj (y) , z = x+ iy ∈ K (2.94)

113

where I1 := [−||A||, a2], J1 := [−||B||, b2] but Ii := (ai, ai+1], Jj := (bj , bj+1] for i, j > 1,aN+1 = ||A||, bn+1 = ||B||, for fixed numbers sij ∈ C. Exploiting the fact that the decompositionof s ∈ S(K) as in (2.94) is not unique, since every such decomposition can be refined addingpoints ai of bj , it is easy to prove that the class of step functions over K, denoted by S(K),is closed with respect to linear combinations and products. Since S(K) evidently includes theconstant function 1 and is closed with respect to complex conjugation, S(K) is a sub ∗-algebrawith unit of Mb(K). Let us define Φ0 : S(K)→ B(H), referring to (2.94), by means of

Φ0(s) :=N∑i=1

M∑j=1

sijP(A)Ii

P(B)Jj

=N∑i=1

M∑j=1

sijP(B)Jj

P(A)Ii

. (2.95)

The definition is well-posed independently from the fact that s admits several expansions as in(2.94). By direct inspection, one sees that Φ0 is a homomorphism of unital ∗-algebras and alsothat

||Φ0(s)ψ||2 =N∑i=1

M∑j=1

|sij |2||P (A)Ii

P(B)Jj

ψ||2 ≤ supi,j|sij |2

N∑i=1

M∑j=1

||P (A)Ii

P(B)Jj

ψ||2 = supij|sij |2||ψ||2 ,

using the fact that the sets Ii × Jj are pairwise disjoint and that∑i,j P

(A)Ii

P(B)Jj

= I because∪i.jIi × Jj = K . As a consequence

||Φ0(s)|| ≤ ||s||∞ if s ∈ S(K).

Since S(K) is dense in C(K) with respect to the norm || · ||∞ (because a continuous functionon a compact set is uniformly continuous thereon), with the same proof as for Theorem 2.10,the continuous unital ∗-homomorphism Φ0 generates a norm decreasing unital ∗-homomorphismΦ : C(K) → B(H). Notice however that here Φ is not an extension of Φ0, since the domainof the former includes continuous functions only, whereas the domain of the latter includesdiscontinuous functions. By definition Φ(1) = I and also, defining ı1 : K 3 (x, y) 7→ x andı2 : K 3 (x, y) 7→ y, we have

Φ(ı1) = A and Φ(ı2) = B .

Indeed, if sn : [−||A||, ||A||] × [−||B||, ||B||] → R is a sequence of step functions, constant inthe variable y ∈ [−||B||, ||B||] and uniformly converging to the map ı1, by direct application of(2.95) we have, with obvious notation,

Φ0(sn) =

∫RsndP

(A) → Φ(ı1) =

∫Rı1dP

(A) = A , in the uniform topology as n→ +∞

where we exploited (2.50). The case of ı2 is identical.As the last step, going on as in the proof of Proposition 2.41, we can extend Φ to a unital∗-algebra homomorphism Φ′ : Mb(K)→ B(H) completely defined by the requirement

〈ψ|Φ′(f)φ〉 =

∫Kfdνψ,φ ψ, φ ∈ H , f ∈Mb(K)

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where νψ.φ : B(K) → C is the unique complex regular Borel measure satisfying the aboveidentity when f ∈ C(K). With a proof essentially identical to that of Proposition 2.41, wehave that the homomorphism of unital ∗-algebras Φ′ : Mb(K) → B(H) is norm-decreasing(||Φ′(f)|| ≤ ||f ||∞), satisfies

Φ′(ı1) = A and Φ′(ı2) = B , (2.96)

and finally

Φ′(fn)ψ → Φ′(f)ψ for every ψ ∈ H, (2.97)

if Mb(K) 3 fn → f pointwise as n→ +∞ and |fn| ≤M for some M ∈ [0,+∞) and all n.The last convergence property in particular, with the same proof as for Theorem 2.11, implies

P(T )E := Φ′(χE∩K) (with P

(T )∅ := 0) defines a PVM on C ≡ R2 when E varies in B(C) and that,

according to (2.96)∫Cı1dP

(T ) = Φ′(ı1) = A ,

∫Cı2dP

(T ) = Φ′(ı2) = B . (2.98)

Since T = A+ iB and T ∗ = A− iB, these relations can be rephrased to∫CzdP (T )(z, z) = T ,

∫CzdP (T )(z, z) = T ∗ . (2.99)

Let us pass to the uniqueness issue. First of all observe that, if T =∫C zdP (z, z) then

P must have bounded support otherwise, for every n ∈ N, we can find En ∈ B(C) out-side the disk of radius n centred on the origin of C with PEn 6= 0. So we can pick out

xn ∈ PEn(H) with ||xn|| = 1. As a consequence ||Txn||2 ≥ |n|2∫C 1dµ

(P )xnxn = |n|2 → +∞

as n → +∞ and this is not possible since ||T || < +∞. We conclude that there is a suffi-ciantly large compact K := [a, b] × [c, d] ⊂ R2 ≡ C (we can always assume to be larger that[−||A||, ||A||] × [−||B||, ||B||]) such that supp(P ) ⊂ K and we can reduce ourselves to discusstherein. If

∫K zdP (z, z) = T =

∫K zdP

(T )(z, z), taking the adjoint we also have∫K zdP (z, z) =

T ∗ =∫K zdP

(T )(z, z). Using standard properties of bounded PVMs, we immediately have that∫K p(z, z)dP (z, z) =

∫K p(z, z)dP

(T )(z, z) for every polynomial p defined on K. Since poly-nomials are || · ||∞-dense in C(K) (Stone-Weierstrass theorem), identity (2.50) entails that∫K f(z, z)dP (z, z) =

∫K f(z, z)dP (T )(z, z) for every f ∈ C(K). As a consequence, applying

Riesz theorem for positive Borel measures to∫Kfdµ

(P )ψψ =

≠ψ

∣∣∣∣∫KfdP ψ

∑=

≠ψ

∣∣∣∣∫KfdP (T ) ψ

∑=

∫Kfdµ

(P (T ))ψψ ∀f ∈ C(K) .

We conclude that µ(P (T ))ψψ (E) = µ

(P )ψψ (E) for every E ∈ B(K). Since the supports of the two

measures stay in K, the found identity can be re-stated as µ(P (T ))ψψ (E) = µ

(P )ψψ (E) for every

E ∈ B(C), i.e. 〈ψ|(P (T )E − PE)ψ〉 = 0 for every ψ ∈ H. This result immediately leads to the

thesis, P(T )E = PE for every E ∈ B(C).

The proofs of (b), (c) and (d) are identical to those of the corresponding statements of Theorem2.8, with trivial changes (R replaced by C and λ for z).

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2.8.4 Existence of the spectral measure for unbounded selfadjoint operators

At the end of this long detour, we are eventually prompt to establish the existence part of theSpectral Theorem (Theorem 2.8) for generally unbounded selfadjoint operators.

Theorem 2.13. If A is a (generally unbounded) selfadjoint operator over the Hilbert spaceH, there exists a PVM P (A) : B(R)→ L (H) such that

A :=

∫Rı dP (A) .

Proof. First of all observe that, if A is normal, then its resolvent satisfies Rλ(A)∗ = Rλ(A).Indeed, we know that λ ∈ ρ(A) iff λ ∈ ρ(A∗) in view of (c) Proposition 2.20. In this caseRλ(A − iλI) = ID(A) implies (A − iλI)∗R∗λ = I∗D(A)= I, namely (A + iλI)R∗λ = I. Since we

also have (A+ iλI)Rλ = I and the inverse is unique, it must be Rλ(A)∗ = Rλ(A). This resultsholds true in particular when A = A∗.Next, assuming A = A∗, consider the operator

U := I − 2iR−i(A)

Taking advantage of the resolvent identity (2.31) and Rλ(A)∗ = Rλ(A), one immediately provesthat

UU∗ = U∗U = I ,

so U is unitary and consequently its spectrum is a closed subset of in the unit circle T = z ∈C | |z| = 1 with the topology induced by C for Proposition 2.20 and it admits a spectraldecomposition

U =

∫σ(U)

zdP (U)(z, z)

according to Theorem 2.12. We want to prove that the initial selfadjoint operator A coincideswith the selfadjoint operator (the integrand is real since z = 1/z as z ∈ T)

A′ :=

∫σ(U)

i1 + z

1− zdP (U)(z, z) . (2.100)

In fact, since R−i(A) = i2(U − I) and taking (c) Proposition 2.27 into account,

(A′ + iI)R−i(A) =

∫σ(U)

ïi1 + z

1− z+ i

òdP (U)(z, z)

∫σ(U)

i

2(z − 1)dP (U)(z, z)

=

∫σ(U)

ïi1 + z

1− z+ i

òi

2(z − 1)dP (U)(z, z) =

∫σ(U)

1dP (U)(z, z) = I .

We conclude that A′ + iI is defined on a domain that includes Ran(R−iA) = D(A) and thereincoincides with the unique left inverse of R−i(A). In other words A′+ iI is an extension of A+ iI.

116

So that A′ ⊃ A. Since A′ and A are selfadjoint, it must be A′ = A due to (b) Propostion 2.8.To conclude, we prove that (2.100) can be re-written as a spectral decomposition over R. Themap

φ : T 3 z 7→ i1 + z

1− z∈ R ∪ ∞

is a homeomorphism (equipping R ∪ ∞ with its natural topology). So that

A = A′ :=

∫Ti1 + z

1− zdP (U)(z, z) =

∫R∪∞

rdP (r) ,

where we have defined the PVM PE = P(T )φ−1(E) for E ∈ B(R∪+∞) according to (f) Proposition

2.27. However ∞ is reached by φ only for z = 1 and P(U)1 = 0. This is because, if it were

P(U)1 6= 0, we would have Ux = x for some x ∈ P (U)

1 (H)\0. Since U := I−2iR−i(A), it would

imply R−i(A)x = 0 which is impossible because R−i(A) is invertible since A is selfadjoint andthus −i ∈ ρ(A). We can therefore re-write the found identity as

A =

∫T\1

i1 + z

1− zdP (U)(z, z) =

∫RrdP (r) .

It is easy to check that the restriction P ′ of P to B(R) is still a PVM over R and the integralabove can be intepreted as

A =

∫RrdP ′(r) .

The proof is over defining P (A) := P ′.

2.8.5 Existence of joint spectral measures

We provide here a proof of Theorem 2.9. The way we follows diffears from that appearing in[Mor18] since our presentation of spectral technology is different. In particular the presenteddemonstration does not require that the Hilbert space is separable.

Theorem 2.9. [Joint spectral measure]Let A := A1, A2, . . . , An be a set of selfadjoint operators in the Hilbert space H. Suppose thatthe spectral measures of those operators pairwise commute:

P(Ak)Ek

P(Ah)Eh

= P(Ah)Eh

P(Ak)Ek

∀k, h ∈ 1, . . . , n ,∀Ek, Eh ∈ B(R) .

There is a unique PVM , P (A), on Rn such that

P(A)E1×···×En = P

(A1)E1· · ·P (An)

En, ∀E1, . . . , En ∈ B(R) . (2.101)

For every f : R→ C measurable, it holds that∫Rnf(xk)dP

(A)(x) = f(Ak) , k = 1, . . . , n (2.102)

117

where x = (x1, . . . , xk, . . . , xn) and f(Ak) :=∫R f(λ)dP (Ak).

Finally, B ∈ B(H) commutes with P (A) if and only if it commutes with all P (Ak), k = 1, 2, . . . , n.

Proof. (Existence) We start by assuming that Ak ∈ B(H) for k = 1, . . . , n. In this case we canrepeat all the initial part of the proof of Theorem 2.12, replacing the two commuting selfadjointoperators in A,B ∈ B(H) for n pairwise commuting selfadjoint operators Ak ∈ B(H). This waywe prove that, if K := [−a, a]n ⊂ Rn is sufficiently large compact set such that K ⊃ ×nk=1σ(Ak),then a map Φ′ : Mb(K) → B(H) exists with the following features. It is a norm-decreasing∗-homomorphism of unital ∗-algebras and furthermore,

Φ′(ık) = Ak for k = 1, . . . , n (2.103)

where ık : Rn 3 (x1, . . . , xn) 7→ xk ∈ R, and finally

Φ′(fn)ψ → Φ′(f)ψ for every ψ ∈ H, (2.104)

if Mb(K) 3 fn → f point wise as n→ +∞ and |fn| ≤M for some M ∈ [0,+∞) and all n.The last convergence property in particular, with the same proof as for Theorem 2.11, implies

P(A)E := Φ′(χE∩K) (2.105)

(with P(A)∅ := 0) defines a PVM on Rn when E varies in B(Rn) and that, according to (2.103)∫

RnıkdP

(A) = Φ′(ık) = Ak , k = 1, . . . , n . (2.106)

Now observe that, if varying E ∈ B(R), the family of orthogonal projectors PE := P(A)E×Rn−1

defines a PVM over R. Taking a sequence of simple functions sn over K, constant in the variablex2, . . . , xn and such that sn → ı1 pointwise with |sn| ≤ |ı1| (which is bounded over K), in viewof (2.104) and (c) Proposition 2.25, we can rephrase (2.106) for k = 1 as∫

Rı dP = A1 . (2.107)

The uniqueness property of the spectral measure of A1 (Theorem 2.8) implies that

P(A)E×Rn−1 = PE = P

(A1)E ∀E ∈ B(R) .

With the same argument, we have

P(A)

Rk−1×E×Rn−k = P(Ak)E , E ∈ B(R) , k = 1, 2, . . . , n .

This identity, from (2.105) and using the fact that Φ′ preserves the products, implies that (2.101)is true:

P(A)E1×···×En = Φ′(χE1×Rn−1 · · ·χRn−1×En) = Φ′(χE1×Rn−1) · · ·Φ′(χRn−1×En)

118

= P(A)E1×Rn−1 · · ·P (A)

Rn−1×En = P(A1)E1· · ·P (An)

En.

Let us pass to the case of generally unbounded selfadjoint operators Ak reducing to the case ofbounded operators. To this end, define the associated class of operators B := B1, . . . , Bn, forevery k = 1, 2, . . . , n,

Bk :=

∫R

xk»1 + x2

k

dP (Ak)(xk) .

It is clear that B∗k = Bk ∈ B(H) due to (c) Theorem 2.6 and (a) Proposition 2.25. Moreover,according to Corollary 2.5, σ(Bk) ⊂ [−1, 1], but ±1 6∈ σp(Bk). In fact, if it were ±1 ∈ σp(Bk)and ψ± ∈ H were a corresponding eigenvector, then (Bk ± I)ψ± = 0, so that

0 = ||(Bk ± I)2ψ±||2 =

∫R

Ñxk»

1 + x2k

± 1

é2

dµ(Pk)ψ±ψ±

.

Since the positive measure µ(Pk)ψ±ψ±

does not vanish (ψ± 6= 0 because it is an eigenvector), theintegrand would be zero almost everywhere. This is not possible becauseÑ

xk»1 + x2

k

± 1

é2

> 0 for every xk ∈ R.

Let us now focus on the map

φ : R 3 x 7→ x√1 + x2

∈ [−1, 1] ,

where R = R∪±∞ is equipped with its standard compactification topology and the topologyon [−1, 1] is the one induced by R. Notice also that φ(R) = (−1, 1) whereas φ(±∞) = ±1. It iseasy to see that φ is an homeomorphism. As a consequence φ and φ−1 are Borel measurable.In view of these properties of φ, it is preferable to extend the spectral measures P (Ak) to new

PVMs ‹P (Ak) defined on the Borel algebra B(R), by simply declaring that ‹P (Ak)+∞ = ‹P (Ak)

−∞ = 0

and ‹P (Ak)E = P

(Ak)E when E ∩ +∞ = E ∩ −∞ = ∅ for E ∈ B(R). This way, we can safely

write that

Bk :=

∫R

xk»1 + x2

k

d‹P (Ak)(xk) .

With the said extension, taking advantage of (f) Proposition 2.27, we can infer that

Bk =

∫[−1,1]

ykdP(Bk)(yk) ,

where

P (Bk)(F ) = ‹P (Ak)(φ−1(F )) for F ∈ B([−1, 1]) . (2.108)

119

We could extend P (Bk) on the whole B(R) by trivially defining P(Bk)1 (F ) := P

(Bk)1 (F ∩ [−1, 1])

for F ∈ B(R), however we stick to the original notation for the sake of simplicity intepretingthe relevant PVM as their extensions where necessary.Observe that the spectral measures P (Bk) pairwise commute due to (2.108) and the fact thatthe PVMs ‹P (Ak) do (the added points ±∞ are harmless). We can therefore apply the previousproof, constructing a PVM P (B) over B(Rn), with support included in [−1, 1]n, which satisfies

P(B)F1×···×Fn = P

(B1)F1· · ·P (Bn)

Fnif Fk ∈ B(R) for k = 1, . . . , n. (2.109)

Coming back to the unbounded operators Ak, we first introduce the homeomorphism

Φ : Rn 3 (x1, . . . , xn) 7→ (φ(x1), . . . , φ(xn)) ∈ [−1, 1]n

and next define the PVM over Rn as permitted by (f) Proposition 2.27 (notice that Φ = (Φ−1)−1

and Φ−1 is Borel measurable since Φ is an homeomorphism)

PE := P(B)Φ(E) E ∈ B(Rn) .

With this definition, (2.109) entails

PE1×···×En = ‹P (A1)E1· · · ‹P (An)

En, ∀E1, . . . , En ∈ B(R) . (2.110)

To conclude the proof of existence, it is enough getting rid of annoying points ±∞. To this endconsider the boundary of Rn. It is made of the union of 2n sets of the form

F(k)± := Rk−1 × ±∞ × Rn−k .

Every such set has zero P -measure. In fact, for instance, exploiting (2.110),

PF

(1)+

= ‹P (A1)+∞ · · · ‹P (An)

R = 0

because ‹P (A1)+∞ = P

(B1)+1 = 0 since +1 6∈ σp(B1) as established above and (c)-(d) Theorem 2.8

hold true. Hence the boundary of Rn has zero measure for P . This means that, restricting to

the interior Rn of Rn, the map P(A)E := PE with E ∈ B(Rn), still defines a PVM, in particular

P(A)Rn = I. By construction, P (A) satisfies (2.102) since (2.109) is valid, concluding the existence

part of the proof.(Uniqueness) Let us come to the uniqueness property. We have the following known result ofgeneral measure theory (Corollary 1.6.3 [Coh80]).

Lemma 2.5. Let Σ(X) be a σ-algebra over X and P ⊂ Σ(X) such that

(i) P is closed with respect to finite intersections;

(ii) the σ-algebra generated by P is Σ(X) itself;

120

(iii) there is an increasing sequence Cmm∈N ⊂ P such that ∪m∈N = X.

If µ and ν are positive σ-additive measures over Σ(X) such that µ(Cm) = ν(Cm) < +∞ forevery m ∈ N, then µ = ν.

Coming back to our case, define Σ(X) := B(Rn) and let P be the class of sets E1 × · · · × Enfor Ek ∈ B(R). It it known that (as R is a separable metric space) the σ-algebra generatedby P is just B(Rn). We can now define Cm = (−r, r)m with m ∈ N. Finally fix x ∈ H anddefine µ(F ) := 〈x|PFx〉 and ν(F ) := 〈x|P ′Fx〉 for F ∈ B(Rn), where both P and P ′ satisfy(2.101) in place of P (A). Notice that these measures are finite as µ(F ) = ν(F ) = ||x||2 bydefinition of PVM and satisfy µ(Cn) = ν(Cn) < +∞ in view of (2.101). Lemma 2.5 provesthat 〈x|PFx〉 = 〈x|P ′Fx〉 so that 〈x|(PF − P ′F )x〉 = 0. Arbitrariness of x ∈ H and the standardargument related to polarization identity implies that PF = P ′F for every F ∈ B(R).

(Identity (2.102)) The proof of (2.102) is easy. Consider the case k = 1 for instance. Thereis a sequence of simple functions sm over R pointi wise converging to the measurable function

f : R → C as m → +∞ and such that |sm| ≤ |sm+1| ≤ |f |. If ψ ∈ ∆(A1)f , according to (d)

Theorem 2.6 and dominated convergence theorem, we have

f(A1) =

∫Rf(x1)dP (A1)ψ = lim

m→+∞

∫RsmdP

(A1)ψ

= limm→+∞

∫Rns′mdP

(A)ψ =

∫Rnf(x1)dP (A)ψ , (2.111)

where, if sm(x1) :=∑Nr=1 crχEr we have defined s′m(x1, . . . , xn) :=

∑Nr=1 crχEr×Rn−1(x1, . . . , xn)

(so that s′m is constant in xn, . . . , xn and coincides to sm in the remaining variable) and thepenultimate identity in (2.111) is valid thanks to (2.101). The same argument, using monotone

convegence, and the identity∫R |sm|2dµ

(P (A1))ψψ =

∫R |s′m|2dµ

(P (A))ψψ proves also that ψ ∈ ∆

(A)f with

obvious notation, so that∫Rn f(x1)dP (A)ψ is well defined.

(Last statement) If B ∈ B(H) commutes with P (A) it evidently commutes with every P (Ak),k = 1, 2, . . . , n due to (2.101) just taking all Ek = R but one. Suppose conversely that U ∈B(H) unitary commutes with every P (Ak). The PVM defined by the projectors UP

(A)E U−1, for

E ∈ B(Rn), therefore coincides with P (A) when E = E1× · · · ×En with Ek ∈ B(R). In view of

the already established uniqueness property, we immediately have UP(A)E U−1 = P

(A)E for every

E ∈ B(Rn). In other words

UP(A)E = P

(A)E U for every for E ∈ B(Rn).

To pass form U unitary to a generic B ∈ B(H), it sufficies applying Lemma 3.2 (whose proofrelies only upon the spectral theorem of selfadjoint operators), decomposing B into a complexlinear combination of unitary operators B = aU + bU ′, and exploiting linearity of compositionof operators in the identity above.

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Chapter 3

Fundamental Quantum Structures inHilbert Space

The question we want to answer now is the following one:Is there anything more fundamental behind the phenomenological facts (1), (2), and (3) dis-cussed in the first chapter and their formalization presented in Sect. 2.6?An appealing attempt to answer that question and justify the formalism based on the spec-tral theory is due to von Neumann [Neu32] (and subsequently extended by Birkhoff and vonNeumann). This chapter is devoted to quickly review an elementary part of those ideas, addinghowever several more modern results (see also [Var07, Mor18] for a similar approach and [Red98]for an extensive technical account on quantum lattice theory and applications).

3.1 Lattices in CM and QM

3.1.1 A different viewpoint on classical mechanics

Let us start by analyzing Classical Mechanics (CM). Consider a classical Hamiltonian systemdescribed in symplectic manifold (Γ, ω), where ω =

∑nk=1 dq

k ∧ dpk in any system of localsymplectic coordinates q1, . . . , qn, p1, . . . , pn. The state of the system at time t is a point s ∈ Γ,in local coordinates s ≡ (q1, . . . , qn, p1, . . . , pn), whose evolution R 3 t 7→ s(t) is a solution of theHamiltonian equation of motion. Always in local symplectic coordinates, they read

dqk

dt=

∂H(t, q, p)

∂pk, k = 1, . . . , n (3.1)

dpkdt

= −∂H(t, q, p)

∂qk, k = 1, . . . , n , (3.2)

H being the Hamiltonian function of the system, depending on the reference frame. Everyphysical elementary property, E, that the system may possess at a certain time t, i.e. whichcan be true or false at that time, can be identified with a subset E ⊂ Γ. The property is true

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if s ∈ E and it is not if s 6∈ E. From this point of view, the standard set theory operations∩, ∪, ⊂, ¬ (where ¬E := Γ \ E from now on is the complement operation) have a logicalinterpretation:

(i) E ∩ F corresponds to the property “E AND F”,

(ii) E ∪ F corresponds to the property “E OR F”,

(iii) ¬E corresponds to the property “NOT E”,

(iv) E ⊂ F means “E IMPLIES F”.

In this context,

(v) Γ is the property which is always true

(vi) ∅ is the property which is always false.

This identification is possible because, as is well known, the logical operations have the samealgebraic structure of the set-theory operations.As soon as we admit the possibility to construct statements including countably infinite numberof disjunctions or conjunctions, we can enlarge our interpretation towards the abstract measuretheory, interpreting the states as probability Dirac measures supported on a single point. To thisend, we initially restrict the class of possible elementary properties to the Borel σ-algebra of Γ,B(Γ). For various reasons this class of sets seems to be sufficiently large to describe physics (inparticular B(Γ) includes the preimages of measurable sets under continuous functions). A stateat time t, s ∈ Γ, can be viewed as a Dirac measure, δs, supported on s itself. If E ∈ B(Γ),δs(E) = 0 if s 6∈ E or δs(E) = 1 if s ∈ E.If we do not have a perfect knowledge of the system, as for instance it happens in statisticalmechanics, the state at time t, µ, is a proper probability measure on B(Γ) which now, is allowedto attain all values of [0, 1]. If E ∈ B(Γ) is an elementary property of the physical system, µ(E)denotes the probability that the property E is true for the system at time t.

Remark 3.1. The evolution equation of µ, in statistical mechanics is given by the well-known Liouville’s equation associate with the Hamiltonian flow. In that case µ is proportionalto the natural symplectic volume measure of Γ, Ω = ω ∧ · · · ∧ ω (n-times, where 2n = dim(Γ)).In fact we have µ = ρΩ, where the non-negative function ρ is the so-called Liouville densitysatisfying the famous Liouville’s equation. In symplectic local coordinates that equation reads

∂ρ(t, q, p)

∂t+

n∑k=1

Å∂ρ

∂qk∂H

∂pk− ∂ρ

∂pk

∂H

∂qk

ã= 0 .

We shall not deal any further with this equation in this paper.

More complicated classical quantities of the system can be described by Borel measurable func-tions f : Γ → R. Measurability is a good requirement as it permits one to perform physical

123

operations like computing, for instance, the expectation value (at a given time) when the stateis µ:

〈f〉µ =

∫Γf dµ .

Also elementary properties can be pictured by measurable functions, in fact they are one-to-oneidentified with all the Borel measurable functions g : Γ → 0, 1. The Borel set Eg associatedto g is g−1(1) and in fact g = χEg .A generic physical quantity, a measurable function f : Γ→ R, is completely determined by the

class of Borel sets (elementary properties) E(f)B := f−1(B) where B ∈ B(R). The meaning of

E(f)B is

E(f)B = “the value of f belongs to B” (3.3)

It is possible to prove [Mor18] that the map B(R) 3 B 7→ E(f)B permits one to reconstruct the

function f . The sets E(f)B := f−1(B) form a σ-algebra as well and the class of sets E

(f)B satisfies

the following elementary properties when B ranges in B(R).

(Fi) E(f)R = Γ,

(Fii) E(f)B ∩ E(f)

C = E(f)B∩C ,

(Fiii) If N ⊂ N and Bkk∈N ⊂ B(R) satisfies Bj ∩Bk = ∅ if k 6= j, then

∪j∈NE(f)Bj

= E(f)∪j∈NBj .

These conditions just say that B(R) 3 B 7→ E(f)B ∈ B(Γ) is a homomorpism of σ-algebras.

Notice in particular that, keeping (Fi) and (Fiii), requirement (Fii) can be equivalently replaced

by E(f)R\E = Γ \ E(f)

E as the reader immediately proves.We observe that our model of classical elementary properties can be also viewed as anothermathematical structure, when referring to the notion of lattice we go to introduce.

3.1.2 The notion of lattice

We remind the reader that in a partially ordered set (X,≥), if Y ⊂ X, the symbol supY denotes,if it exists, the least element x of X such that x ≥ y for every y ∈ Y . Similarly, the symbolinf Y denotes, if it exists, the greatest element x of X such that y ≥ x for every y ∈ Y .

Definition 3.1. A partially ordered set (X,≥) is a lattice when, for any a, b ∈ X,

(a) supa, b exists in X, denoted a ∨ b (sometimes called ‘join’);

(b) infa, b exists in X, written a ∧ b (sometimes ‘meet’).

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(The partially ordered set is not required to be totally ordered.)

Remark 3.2.(a) In our considered concrete cases where X = B(R) or X = B(Γ) and ≥ is nothing but

⊃ and thus ∨ means ∪ and ∧ has the meaning of ∩.(b) In the general case ∨ and ∧ turn out to be separately associative, therefore it make sense

to write a1∨· · ·∨an and a1∧· · ·∧an in a lattice. Moreover they are also separately commutativeso

a1 ∨ · · · ∨ an = aπ(1) ∨ · · · ∨ aπ(n) and a1 ∧ · · · ∧ an = aπ(1) ∧ · · · ∧ aπ(n)

for every permutation π : 1, . . . , n → 1, . . . , n.

Definition 3.2. A lattice (X,≥) is said to be:

(a) distributive if ∨ and ∧ distribute over one another: for any a, b, c ∈ X,

a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) , a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) ;

(b) bounded if it admits a minimum 0 and a maximum 1 (sometimes called ‘bottom’ and‘top’);

(c) orthocomplemented if bounded and equipped with a mapping X 3 a 7→ ¬a, where ¬ais the orthogonal complement of a, such that:

(i) a ∨ ¬a = 1 for any a ∈ X,

(ii) a ∧ ¬a = 0 for any a ∈ X,

(iii) ¬(¬a) = a for any a ∈ X,

(iv) a ≥ b implies ¬b ≥ ¬a for any a, b ∈ X;

(d) complete (σ-complete), if every (resp. every countable) set ajj∈J ⊂ X admits infimum∨j∈Jaj and supremum ∧j∈Jaj .

A lattice with properties (a), (b) and (c) is called a Boolean algebra. A Boolean algebrasatisfying (d) with J = N is a Boolean σ-algebra.A sublattice is a subset X0 of a lattice X preserving the lattice structure with respect tothe restriction of the lattice structure of X in the following precise sense: the infimum andthe supremum of any pair of elements of X must exist and coincide with the correspondinginfimum and supremum in X. Referring to sub bounded and orthocomplemented lattices, asimilar requirement must hold for the top, the bottom and the orthocomplement of an elementof the substructure: they have to coincide with the corresponding objects in the whole lattice.

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It is easy to prove the so called De Morgan’s law for an orthocomplemented lattice [Red98,Mor18] just applying the relevant definitions.

Proposition 3.1. If (X,≥,0,1,¬) is an orthocomplemented lattice and A ⊂ X is finite then,with an obvious notation,

¬ ∨a∈A a = ∧a∈A¬a and ¬ ∧a∈A a = ∨a∈A¬a .

If A is infinite, both sides of each of the two identities above either simultaneously exist or si-multaneously do not exist. If they exist the identity is valid as well.

Definition 3.3. If X, Y are lattices, a map h : X → Y is a (lattice) homomorphismwhen

h(a ∨X b) = h(a) ∨Y h(b) , h(a ∧X b) = h(a) ∧Y h(b) , a, b ∈ X

(with the obvious notations.) If X and Y are bounded, a homomorphism h is further requiredto satisfy

h(0X) = 0Y , h(1X) = 1Y .

If X and Y are orthocomplemented, a homomorphism h also must satisfy

h(¬Xa) = ¬Y h(x) .

If X, Y are complete (σ-complete), h it is further required to satisfy (with J = N)

h(∨j∈Jaj) = ∨j∈Jh(aj) , h(∧j∈Jaj) = ∧j∈Jh(aj) if ajj∈J ⊂ X .

In all cases (bounded, orthocomplemented, (σ-)complete lattices, Boolean (σ-)algebras) if h isbijective it is called isomorphism of the relative structures.

It is clear that, just because it is a concrete σ-algebra, the lattice of the elementary propertiesof a classical system is a lattice which is distributive, bounded (here 0 = ∅ and 1 = Γ), ortho-complemented (the orthocomplement being the complement with respect to Γ) and σ-complete.

Moreover, as the reader can easily prove, the above map, B(R) 3 B 7→ E(f)B ∈ B(Γ), is also a

homomorphism of Boolean σ-algebras.

Remark 3.3. Given an abstract Boolean σ-algebra X, does there exist a concrete σ-algebraof sets that is isomorphic to the previous one? In this respect, the following general result holds,known as Loomis-Sikorski theorem [Sik48]. This guarantees that every Boolean σ-algebra isisomorphic to a quotient Boolean σ-algebra Σ/N, where Σ is a concrete σ-algebra of sets overa measurable space and N ⊂ Σ is closed under countable unions; moreover, ∅ ∈ N and for anyA ∈ Σ with A ⊂ N ∈ N, then A ∈ N. The equivalence relation is A ∼ B iff A∪B \ (A∩B) ∈ N,for any A,B ∈ Σ. It is easy to see the coset space Σ/N inherits the structure of Boolean σ-algebra from Σ with respect to the (well-defined) partial order relation [A] ≥ [B] if A ⊃ B,

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A,B ∈ Σ.If dealing with the simpler case of an abstract Boolean algebra, the celebrated Stone’s rep-resentation theorem [Sto36] proves that it is always isomorphic to a concrete algebra of sets.

3.2 The non-Boolean Logic of QM

It is evident that the classical-like picture illustrated in Sect.3.1 is untenable if referring toquantum systems. The deep reason is that there are pairs of elementary properties E,F ofquantum systems which are incompatible. Here, an elementary property is an observable which,if measured by means of a corresponding experimental apparatus, can only attain two values:0 if it is false or 1 if it is true. For instance, E = “the component Sx of the electron is ~/2”and F = “the component Sy of the electron is ~/2”. There is no physical instrument capableto establish if E AND F is true or false. We conclude that some of elementary observablesof quantum systems cannot be logically combined by the standard operation of the logic. Themodel of Borel σ-algebra seems not to be appropriate for quantum systems. However one couldtry to use some form of lattice structure different form the classical one.

3.2.1 The lattice of quantum elementary observables

The fundamental ideas by von Neumann were the following pair.

(N1) Given a quantum system, there is a complex separable Hilbert space H such that theelementary observables – the ones which only assume values in 0, 1 – are one-to-onerepresented by all the elements of L (H), the orthogonal projectors in B(H).

(N2) Two elementary observables P , Q are compatible if and only if they commute as projectors.

Remark 3.4.(a) As we shall see later, (N1) has to be changed for those quantum systems which admit

superselection rules. For the moment we stick to the above version of (N1).(b) The technical requirement of separability will play a crucial role in several places.

Let us analyse the reasons for von Neumann’s postulates. First of all we observe that L (H) is infact a lattice if one remembers the relation between orthogonal projectors and closed subspacesstated in Proposition 2.21 and equipping the set of closed subspaces with the natural orderingrelation given by set-theoretic inclusion relation.Referring to Notation 2.5, if P,Q ∈ L (H), we write P ≥ Q if and only if P (H) ⊃ Q(H). Aspreannounced, it turns out that (L (H),≥) is a lattice and, in particular, it enjoys the followingproperties.

Proposition 3.2. Let H be a complex (not necessarily separable) Hilbert space. For everyP ∈ L (H), define ¬P := I − P (the orthogonal projector onto P (H)⊥ according to Proposition

127

2.21). With this definition, (L (H),≥, 0, I,¬) turns out to be a bounded, orthocomplemented,complete (so also σ-complete) lattice which is not distributive if dim(H) ≥ 2.More precisely,

(i) P ∨Q is the orthogonal projector onto P (H) +Q(H).The analogue holds for a set Pjj∈J ⊂ L (H), namely ∨j∈JPj is the orthogonal projectoronto spanPj(H)j∈J .

(ii) P ∧Q is the orthogonal projector on P (H) ∩Q(H).The analogue holds for a set Pjj∈J ⊂ L (H), namely ∧j∈JPj is the orthogonal projectoronto ∩j∈JPj(H).

(iii) The bottom and the top are respectively 0 and I.

(iv) Referring to (i) and (ii), it turns out that, if J = N,

∨n∈N Pn = s- limk→+∞

∨n≤kPn and ∧n∈N Pn = s- limk→+∞

∧n≤kPn (3.4)

where ”s-” indicates that the limits are computed with respect to the strong operator topol-ogy.

Proof. The fact that L (H) is a lattice is evident if directly interpreting it as a partially orderedset of closed subspaces. It is clear that supP (H), Q(H) = P (H) +Q(H) if P,Q ∈ L (H),since supP (H), Q(H) includes both P (H) and Q(H) and every closed subspace including thesesubspaces must also contain P (H) +Q(H) by linearity and definition of closure. It is clear thatinfP (H), Q(H) = P (H) ∩ Q(H) if P,Q ∈ L (H), since the closed subspace P (H) ∩ Q(H) isincluded in both P (H) and Q(H) and every closed subspaces that is part of both both P (H) andQ(H) must be contained in these subspaces must be contain in the closed subspace P (H)∩Q(H).A trivial extension of the same arguments proves (i) and (ii). It is evident that L (H) is boundedwith the said top and bottom. The fact that ¬P := I − P (that is the orthogonal projecotronto P (H)⊥ as established in (b) Proposition 2.21) is an orthocomplement can be immediatelyproved by direct inspection using properties of ⊥ presented in Sect.2.1.2 and in Proposition 2.21.Failure of distributivity for dim(H) ≥ 2 immediately arises form the analog for H = C2 we goto prove. Let e1, e2 be the standard basis of C2 and define the subspaces H1 := spane1,H2 := spane2, H3 := spane1 + e2. Finally P1, P2, P3 respectively denote the orthogonalprojectors onto these spaces. By direct inspection one sees that P1 ∧ (P2 ∨ P3) = P1 ∧ I = P1

and (P1 ∧P2)∨ (P1 ∧P3) = 0∨ 0 = 0, so that P1 ∧ (P2 ∨P3) 6= (P1 ∧P2)∨ (P1 ∧P3). To end theproof, let us prove (3.4). Consider the former limit. P := s- limk→+∞ ∨n≤kPn exists in L (H) inview of Proposition 2.24 since ∨n≤kPn projects onto larger and larger subspaces as n increases.We want to prove that the limit P coincides to the projector onto spanPj(H)j∈J denoted by∨n∈NPn in (i). It is clear that ∨n≤kPn ≤ P by definition of P as it holds that

〈x| ∨n≤k Pnx〉 ≤ supk∈N〈x| ∨n≤k Pnx〉 = lim

k→+∞〈x| ∨n≤k Pnx〉 = 〈x|Px〉 ,

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so P (H) includes all subspaces ∨n≤kPn and also each single Pn(H). So P (H) includes theirfinite span, by linearity, and also the closure of the span, because P (H) is closed. HenceP (H) ⊃ spanPn(H)n∈N. On the other hand, if x ∈ P (H), then x = limk→+∞ ∨n≤kPnx ∈spanPn(H)n∈N, hence P (H) ⊂ spanPn(H)n∈N. We conclude that P (H) = spanPn(H)n∈N.For (i), this is the same as saying P = ∨n∈NPn. The proof of the second formula in (3.4) isidentical barring trivial changes.

3.2.2 Part of CM is hidden in QM

To go on, the crucial observation is that, nevertheless (L (H),≥, 0, I,¬) includes lots of Booleanσ-algebras, and precisely the maximal sets of pairwise compatible projectors. These σ-algebrasin the quantum context could be interpreted as made of classical observables at least concerningmutual relations.

Proposition 3.3. Let H be a complex separable Hilbert space and consider the lattice oforthogonal projectors (L (H),≥, 0, I,¬).Assume that L0 ⊂ L (H) is a maximal subset of pairwise commuting elements (i.e. if Q ∈ L (H)commutes with every P ∈ L0 then Q ∈ L0). Then L0 contains 0, I is ¬-closed and, if equippedwith the restriction of the lattice structure of (L (H),≥, 0, I,¬), turns out to be a Boolean σ-algebra (in particular the supremum and the infimum of countable sequences of elements in L0

computed in L0 coincide with the corresponding infimum and supremum computed in the wholeL (H)).Finally, if P,Q ∈ L0,

(i) P ∨Q = P +Q− PQ ,

(ii) P ∧Q = PQ.

Proof. L0 includes both 0 and I because L0 is maximally commutative and is ¬ closed: ¬P =I − P commutes with every element of L0 if P ∈ L0 so that ¬P ∈ L0 due to the maximalitycondition. Taking advantage of associativity of ∨ and ∧, and using (iv) in proposition 3.2, thesup and the inf of a sequence of projectors Pnn∈N ⊂ L0 commute with the elements of L0

since every element ∨n≤kPn and ∧n≤kPn does by direct application of (i) and (ii). Maximalityimplies that these limit projectors belong to L0. Finally (i) and (ii) prove by direct inspectionthat ∨ and ∧ are mutually distributive. Let us prove (ii) and (i) to conclude. If PQ = QP ,PQ is an orthogonal projector and PQ(H) = QP (H) ⊂ P (H) ∩ Q(H). On the other hand, ifx ∈ P (H)∩Q(H) then Px = x and x = Qx so that PQx = x and thus P (H)∩Q(H) ⊂ PQ(H) and

(ii) holds. To prove (i) observe that P (H) +Q(H)⊥

= (P (H) + Q(H))⊥. By linearity, (P (H) +

Q(H))⊥ = P (H)⊥ ∩Q(H)⊥. Therefore P (H) +Q(H) = (P (H) +Q(H)⊥

)⊥ = (P (H)⊥ ∩Q(H)⊥)⊥.Using (ii), and the fact that I−R is the orthogonal projector onto R(H)⊥, this can be rephrasedas P ∨Q = I − (I − P )(I −Q) = I − (I − P −Q+ PQ) = P +Q− PQ.

Remark 3.5.

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(a) Every set of pairwise commuting orthogonal projectors can be completed to a maximalset as an elementary application of Zorn’s lemma. However, since the commutativity propertyis not transitive, there are many possible maximal subsets of pairwise commuting elements inL (H) with non-empty intersection.

(b) As a consequence of the stated proposition, the symbols ∨, ∧ and ¬ have the sameproperties in L0 as the corresponding symbols of classical logic OR, AND and NOT . MoreoverP ≥ Q can be interpreted as “Q IMPLIES P”.

(c) There were and are many attempts to interpret ∨ and ∧ as connectives of a new non-distributive logic when dealing with the whole L (H): a quantum logic. The first noticeableproposal was due to Birkhoff and von Neumann [BivN36]. Nowadays there are lots of quantumlogics [BeCa81, Red98, EGL09] all regarded with suspicion by physicists. Indeed, the mostdifficult issue is the physical operational interpretation of these connectives taking into accountthe fact that they put together incompatible propositions, which cannot be measured simultane-ously. An interesting interpretative attempt, due to Jauch, relies upon a result by von Neumann.

Proposition 3.4. In a Hilbert space H, the identity holds

(P ∧Q)x = limn→+∞

(PQ)nx (3.5)

for every P,Q ∈ L (H) and x ∈ H.

Proof. See Appendix 3.5 and also [Red98, Mor18] for alternate proofs.

Notice that the result holds in particular if P and Q do not commute, so they are incompatibleelementary observables. The right-hand side of the identity above can be interpreted as theconsecutive and alternated measurement of an infinite sequence of elementary observables Pand Q. As

||(P ∧Q)x||2 = limn→+∞

||(PQ)nx||2 for every P,Q ∈ L (H) and x ∈ H,

the probability that P ∧Q is true for a state represented by the unit vector x ∈ H is the proba-bility that the infinite sequence of consecutive alternated measurements of P and Q produce istrue at each step.

Exercise 3.1. Prove that, if P,Q ∈ L (H), then P +Q ∈ L (H) if and only if P and Q projectonto orthogonal subspaces.

Solution. If P and Q project onto orthogonal subspaces then PQ = QP = 0 (Proposition2.22), so that L (H) 3 P ∨Q = P +Q−PQ = P +Q due to Proposition 3.3. Suppose converselythat P +Q ∈ L (H). Therefore (P +Q)2 = P +Q. In other words, P 2 +Q2 +PQ+QP = P +Q,namely P + Q + PQ + QP = P + Q so that we end up with PQ = −QP . Applying P on theright, we obtain PQP = −QP and applying P on the left we produce PQP = −PQP . HencePQP = 0. From PQP = −QP , we also have QP = 0 and also PQ = 0 if taking the adjoint.Proposition 2.22 implies that P and Q project onto orthogonal subspaces. 2

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3.2.3 A reason why observables are selfadjoint operators

We are in a position to clarify why, in this context, observables are PVMs over B(R) andtherefore they are also selfadjoint operators in view of the spectral integration and disintegrationprocedure, since PVMs over B(R) are one-to-one with selfadjoint operators. Exactly as in CM,an observable A can be viewed as collection of elementary YES-NO observables PEE∈B(R)

labelled on the Borel sets E of R. Exactly as for classical quantities, (3.3) we can say that themeaning of PE is

PE = “the value of the observable belongs to E” . (3.6)

Assuming, as is obvious, that all those elementary observables are pairwise compatible, we cancomplete PEE∈B(R) to a maximal set of compatible elementary observables L0 and we candeal therein forgetting Quantum Theory. We therefore expect that they also satisfy the sameproperties (Fi)-(Fiii) as for classical quantities. Notice that (Fi)-(Fiii) immediately translateinto

(i)’ PR = I,

(ii)’ PE ∧ PF = PE∩F ,

(iii)’ If N ⊂ N and Ekk∈N ⊂ B(R) satisfies Ej ∩ Ek = ∅ for k 6= j, then

∨j∈NPEjx = P∪j∈NEjx for every x ∈ H.

Next, taking Proposition 3.3 into account, these properties become

(i) PR = I,

(ii) PEPF = PE∩F ,

(iii) If N ⊂ N and Ekk∈N ⊂ B(R) satisfies Ej ∩ Ek = ∅ for k 6= j, then∑j∈N

PEjx = P∪j∈NEjx for every x ∈ H.

(The presence of x is due to the fact that the convergence of the series if N is infinite isin the strong operator topology as declared in the last statement of Proposition 3.2.)

In other words we have just found Definition 2.12, specialized to PVM on R: Observables inQM (viewed as collections of elementary propositions labelled over the Borel sets of R) arePVM over R. We also know that all PVM over R are one-to-one associated to all selfadjointoperators in view of the results presented in the previous chapter. Indeed, integrating thefunction ı : R 3 r 7→ r ∈ R with respect to P we have the normal operator

AP =

∫Rr dP (r)

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according to Theorem 2.6. This operator is selfadjoint because the integrand function is real-valued ((c) Theorem 2.6). Finally, Theorem 2.8 proves that P is the unique PVM associatedto the operator AP and the support of P is σ(AP ). The operator AP encapsulates all informa-tion of the PVM PEE∈B(R), i.e. of the associated observable A as a collection of elementarypropositions labelled over the Borel sets of R.We conclude that, adopting von Neumann’s framework, in QM observables are naturally de-scribed by selfadjoint operators, whose spectra coincide with the set of values attained by theobservables.

3.3 Recovering the Hilbert space structure: The “coordinatiza-tion” problem

A reasonable question to ask is whether there are better reasons for choosing to describe quan-tum systems via a lattice of orthogonal projectors, other than the kill-off argument “it works”.To tackle the problem we start by listing special properties of the lattice of orthogonal pro-jectors, whose proof is elementary. The notion of orthomodularity shows up below. It is aweaker version of distributivity of ∨ with respect to ∧ that we know to be untenable in L (H).Another notion is that of atom. (See [Red98] for a concise discussion on these properties ad alist of alternative and equivalent reformulations of orthomodularity condition.)

Definition 3.4. If (L ,≥,0,1) is a bounded lattice, a ∈ L \ 0 is called atom, if p ≤ aimplies p = 0 or p = a.

The following theorem collects all relevant properties of the special lattice L (H), simultaneouslydefining them. These definitions may actually apply to a generic orthocomplemented lattice.

Theorem 3.1. In the bounded, orthocomplemented, σ-complete lattice L (H) of Propositions3.2 and 3.3, the orthogonal projectors onto one-dimensional spaces are the only atoms of L (H).Moreover L (H) satisfies these additional properties:

(i) separability (for H separable): if Paa∈A ⊂ L (H) \ 0 satisfies Pi ≤ ¬Pj, i 6= j, thenA is at most countable;

(ii1) atomicity: for any P ∈ L (H) \ 0 there exists an atom A with A ≤ P ;

(ii2) atomisticity: for every P ∈ L (H) \ 0, then P = ∨A ≤ P |A is an atom of L (H);

(iii) orthomodularity: P ≤ Q implies Q = P ∨ ((¬P ) ∧Q);

(iv) covering property: if A,P ∈ L (H), with A an atom, satisfy A ∧ P = 0, then(1) P ≤ A ∨ P with P 6= A ∨ P , and(2) P ≤ Q ≤ A ∨ P implies Q = P or Q = A ∨ P ;

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(v) irreducibility: only 0 and I commute with every element of L (H).

Proof. Everything has an immediate elementary proof. The only pair of properties which arenot completely trivial are orthomodularity and irreducibility. The former immediately arisesform the observation that P ≤ Q is equivalent to PQ = QP = Q (Proposition 2.22) so that,in particular P and Q commute. Embedding them in a maximal set of pairwise commutingprojectors, we can use Proposition 3.3:

P∨((¬P )∧Q) = P∨((I−P )Q) = P∨(Q−P ) = P+(Q−P )−P (Q−P ) = P+Q−P−P+P = Q.

Irreducibility can easily be proved observing that if P ∈ L (H) commutes with all projectorsalong one-dimensional subspaces, Px = λxx for every x ∈ H. Thus P (x+ y) = λx+y(x+ y) butalso Px+Py = λxx+λyy and thus (λx−λx+y)x = (λx+y−λy)y, which entails λx = λy if x ⊥ y.If N ⊂ H is a Hilbert basis, Pz =

∑x∈N 〈x|z〉λx = λz for some fixed λ ∈ C. Since P = P ∗ = PP ,

we conclude that either λ = 0 or λ = 1, i.e. either P = 0 or P = I, as wanted.

Actually, each of the listed properties admits a physical operational interpretation (e.g. see[BeCa81]). So, based on the experimental evidence of quantum systems, we could try to prove,in the absence of any Hilbert space, that elementary propositions with experimental outcome in0, 1 form a poset. More precisely, we could attempt to find a bounded, orthocomplementedσ-complete lattice that verifies conditions (i)–(v) above, and then try to prove this lattice isdescribed by the orthogonal projectors of a Hilbert space. This is known as the coordinatizationproblem [BeCa81], problem that can be traced back to von Neumann’s first works on the subject.The partial order relation of elementary propositions can be defined in various ways. But it willalways correspond to the logical implication, in some way or another. Starting from [Mac63] anumber of approaches (either of essentially physical nature, or of formal character) have beendeveloped to this end: in particular, those making use of the notion of (quantum) state, whichwe will see in a short while for the concrete case of propositions represented by orthogonalprojectors. The object of the theory is now [Mac63] the pair (O,S ), where O is the class ofobservables and S the one of states. The elementary propositions form a subclass L of O

equipped with a natural poset structure (L ,≥) (also satisfying a weaker version of some of theconditions (i)–(v)). A state s ∈ S , in particular, defines the probability ms(P ) that P is truefor every P ∈ L [Mac63]. As a matter of fact, if P,Q ∈ L , P ≥ Q means by definition thatthe probability ms(P ) ≥ ms(Q) for every state s ∈ S . More difficult is to justify that the posetthus obtained is a lattice, i.e. that it admits a greatest lower bound P ∨ Q and a least upperbound P ∧Q for every P,Q. There are several proposals, very different in nature, to introducethis lattice structure (see [BeCa81] and [EGL09] for a general treatise) and make the physicalmeaning explicit in terms of measurement outcome. See Aerts in [EGL09] for an abstract butoperational viewpoint and [BeCa81, §21.1] for a summary on several possible ways to introducethe lattice structure on the partially ordered sets.If we accept the lattice structure on elementary propositions of a quantum system, then wemay define the operation of orthocomplementation by the familiar logical/physical negation.An apparent problem is the abstract definition of the notion of compatible propositions, since

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this notion makes explicit use of the structure of L (H) as set of operators. Actually also thisnotion is general and can be defined for generic orthocomplemented lattices.

Definition 3.5. Let (L ,≥,0,1,¬) be an orthocomplemented lattice and consider two elementsa, b ∈ L .

(a) They are said to be orthogonal written a ⊥ b, if ¬a ≥ b (or equivalently ¬b ≥ a).

(b) They are said to be commuting, if a = c1 ∨ c3 and b = c2 ∨ c3 with ci ⊥ cj if i 6= j.

Remark 3.6.

(a) These notions of orthogonality and compatibility make sense beacuse, a posteriori, theyturn out to be the usual ones when propositions are interpreted via projectors.

Proposition 3.5. If H Let H a Hilbert space and think of L (H) as an orthocomplementedlattice. Two elements P,Q ∈ L (H)

(i) are orthogonal in the sense of Definition 3.5 if and only if they project onto mutuallyorthogonal subspaces, which it is equivalent to saying PQ = QP = 0;

(ii) commute in accordance with Definition 3.5 if and only if PQ = QP .

Proof. (i) It is evident since ¬P ≥ Q is equivalent to Q(H) ⊂ P (H)⊥, in turn, this is the sameas PQ = QP = 0 for Proposition 2.22.(ii) Assume that P = P1 ∨P3 and Q = P1 ∨P2 where PiPj = 0 if i 6= j so that, in particular, Piand Pj commute. Therefore, embedding the Pjs in to a maximal set of commuting projectors L0,in view of Proposition 3.3 we have P = P1+P2−P1P2 = P1+P2 and Q = P1+P3−P1P3 = P1+P3

and also PQ = QP since Pi and Pj commute. If conversely, PQ = QP , the said decompositionarises for P3 := PQ, P1 := P (I −Q), P2 := Q(I − P ).

(b) It is not difficult to prove that [BeCa81, Mor18] that, in an orthocomplemented latticeL , a pair of elements p, q commutes if and only if the intersection of all sub orthocomplementedlattices including both p and q (which is a sub orthocomplemented lattice in its own right) isBoolean.

Now, fully-fledged with an orthocomplemented lattice and the notion of compatible proposi-tions, we can attach a physical meaning (an interpretation backed by experimental evidence) tothe requests that the lattice be orthocomplemented, complete, atomistic, irreducible and that ithave the covering property [BeCa81]. Under these hypotheses and assuming there exist at least4 pairwise-orthogonal atoms, Piron ([Pir64, JaPi69],[BeCa81, §21], Aerts in [EGL09]) used pro-jective geometry techniques to show that the lattice of quantum propositions can be canonicallyidentified with the closed (in a generalised sense) subsets of a generalised Hilbert space of sorts.In the latter:

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(a) the field is replaced by a division ring (usually not commutative) equipped with an invo-lution, and

(b) there exists a certain non-singular Hermitian form associated with the involution.

It has been conjectured by many people (see [BeCa81]) that if the lattice is also orthomodularand separable, the division ring can only be picked among R,C or H (quaternion algebra).More recently first Soler [Sol95] and next Holland [Hol95] and Aerts–van Steirteghem [AeSt00]have found sufficient hypotheses, in terms of the existence of infinite orthogonal systems, forthis to happen. Under these hypotheses, if the ring is R or C, we obtain precisely the latticeof orthogonal projectors of the separable Hilbert space. In the case of H, one gets a similargeneralised structure (see, e.g., [GMP13, GMP17]).In all these arguments the assumption of irreducibility is not really crucial: if property (v) fails,the lattice can be split into irreducible sublattices [Jau78, BeCa81]. Physically-speaking thissituation is natural in the presence of superselection rules, of which more later.An evident issue arises here: why aren’t quantum systems described in real or quanternionicHilbert spaces known to physicists?This is a long standing problem which received a recent solution at least for the physical de-scription of elementary relativistic systems [MoOp17a, MoOp17b]. It seems that the complexstructure is just a sort of accident imposed by relativistic symmetry.

Remark 3.7. It is worth stressing that the covering property in Theorem 3.1 is a crucialproperty. Indeed there are other lattice structures relevant in physics verifying all the remainingproperties in the afore-mentioned theorem. Remarkably the family of the so-called causallyclosed sets in a general spacetime satisfies all the said properties but the covering law (see, e.g.[Cas02]). This obstruction prevents one from endowing a spacetime with a natural (generalized)Hilbert space structure, while it suggests some ideas towards a formulation of quantum gravity.

3.4 Quantum states as probability measures and Gleason’s The-orem

As commented in Remark 2.24, the probabilistic interpretation of quantum states is not welldefined because there is no a true probability measure in view of the fact that there are in-compatible observables. The idea is to re-define the notion of probability in the bounded,orthocomplemented, σ-complete lattice like L (H) instead of on a σ-algebra. The study of thesegeneralized measures is the final goal of this section.

3.4.1 Probability measures over L (H)

Exactly as in CM, where the generic states are probability measures on Boolean lattice B(Γ) ofthe elementary properties of the system (Sect.3.1), we can think of states of a quantum system

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as σ-additive probability measures over the non-Boolean lattice of the elementary observablesL (H). A state is therefore a map ρ : L (H) 3 P 7→ ρ(P ) ∈ [0, 1] that satisfies ρ(I) = 1 and aσ-additive requirement

ρ (∨n∈NPn) =∑n∈N

ρ(Pn) ,

where the sequence Pnn∈N ⊂ L (H) are made of simultaneously compatible (PiPj = PjPi) andindependent (Pi∧Pj = 0 if i 6= j) elementary propositions. In orther words, since Pi∧Pj = PiPjwhen the projectors commute, the said condition reads PiPj = PjPi = 0 for i 6= j. Making useof associativity of ∨ and (i) Proposition 3.3, we have

∨n≤kPn =k∑

n=0

Pn .

Next, exploiting (iv) Proposition 3.2, we can write down the projector ∨n∈NPn into a moreeffective way:

∨n∈NPn = s- limk→+∞

∨n≤kPn = s- limk→+∞

k∑n=0

Pn = s-∑n∈N

Pn .

(As usual “s-” denotes the limit in the strong operator topology.) The σ-additivity requirementcan be re-phrased as

ρ

(s-∑n∈N

Pn

)=∑n∈N

ρ(Pn) (3.7)

where the sequence Pnn∈N ⊂ L (H) satisfies PnPm = 0 for n 6= m. Notice that simple addi-tivity is included just assuming that Pn = 0 for all n excluding a finite subset of N.

Remark 3.8.(a) The class Pnn∈N can always be embedded into a maximal set of commuting elementary

observables L0 that has the structure of a Boolean σ-algebra. A quantum state ρ restricted to L0

is a standard Kolmogorov probability measure. Its quantum nature relies on the peculiarity thatit acts also on projectors which are not contined in a common Boolean, namely, incompatibleelementary observables. σ-algebra.

(b) That introduced is the most general notion of quantum state. The issue remain openof the existence of sharp states associating either 0 or 1 and not intermediate values to everyelmentary proposition, as the non-probabilistic states in the phase space do. If they exist, theymust be a special case of these probability measures. We shall see that actually sharp statesdo not exist in quantum theories, in the Hilbert space formulation, differently from classicaltheories. In this sense quantum theory is intrinsically probabilistic.

We face now two fundamental questions.

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(1) Do quantum states pictured as above exist?

The answer is positive: if ψ ∈ H and ||ψ|| = 1, the map ρψ : L (H) 3 P 7→ 〈ψ|Pψ〉 ∈ [0, 1]satisfies the said requirement as the reader immediately proves: ρψ(I) = 〈ψ|ψ〉 = 1 and (3.7) isvalid simply because the scalar product is continuous. It is also worth stressing that, as expectedform elementary formulations, ρψ depends on ψ up to a phase, since ρλψ = ρψ if a ∈ C with|a| = 1.

(2) Are unit vectors up to phases the unique quantum states?

The answer is negative and quite articulate. The rest of this section is mainly devoted toproperly answer this question. To do it, we need to focus on a particular class of operatorscalled trace-class operators because they play a central role in a celebrated characterizationof the afore-mentioned measures due to Gleason. To define trace-class operators we need twoingredients, the polar decomposition theorem and the class of compact operators.

3.4.2 Polar decomposition

Complex numbers z 6= 0 can be decomposed into the product of a positive number, the absolutevalue |z|, and the phase u, with |u| = 1, z = u|z|. Bounded (actually closed) operators A 6= 0can be analogously decomposed into the composition of their absolute value |A|, that is positive,and a “partial isometry” U with ||U || = 1, A = U |A|. To explain how this decomposition workwe needs a preliminary result.

Proposition 3.6. Let H be a Hilbert space and A : H → H a positive operator: 〈x|Ax〉 ≥ 0for every x ∈ H. There exists a unique positive operator B : H → H such that A = B2. Thisoperator is bounded and commutes with every operator in B(H) commuting with A.That operator is called the square root of A and is denoted by

√A.

Proof. We remind the reader that a positive operator T : H → H is necessarily in B(H) andselfadjoint in view of (3) Exercise 2.7. As A ∈ B(H) is selfadjoint, A =

∫σ(A) λdP

(A)(λ) for The-

orem 2.8. Moreover σ(A) ∈ [0,+∞) as proved in Proposition 2.28. So B :=∫σ(A)

√λdP (A)(λ)

is selfadjoint positive (using the same proof as for Proposition 2.28) and

BB =

∫σ(A)

√λdP (A)(λ)

∫σ(A)

√λdP (A)(λ) =

∫σ(A)

λdP (A)(λ) = A ,

for (d) Proposition 2.25 as all operators are in B(H). If B′ ∈ B(H) is positive and B′B′ = A wehave

∫[0+∞) r

2dP (B′)(r) = A =∫

[0,+∞) r2dP (B)(r), that is

∫[0+∞) sdQ

′(s) = A =∫

[0,+∞) sdQ(s)

where we have defined QE := P(B′)φ−1(E) and QE := P

(B)φ−1(E) and the homeomorphism φ : [0,+∞) 3

r 7→ r2 ∈ [0,+∞) according to (f) Proposition 2.27. Uniqueness of the spectral measure of aselfadjoint operator (extending Q and Q′ over B(R) in the simplest way, i.e. Q1E := QE∩[0,+∞))

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implies that Q = Q′ = P (A) so that P(B)E = Qφ(E) = Q′φ(E) = P

(B′)E . Hence B = B′.

To conclude, observe that if D∗ = D ∈ B(H) commutes with A, then D commutes with An andhence with eitA for every t ∈ R as a consequence of Exercise 2.12. Proposition 2.36 entails thatD commutes with the spectral measure of A and thus with every operator s(A) =

∫σ(A) sdP

(A),where s is a simple function. Approximating essentially bounded functions f with simple func-tions according to (c) Proposition 2.25, we extend the result to operators f(A). In particular Dcommutes with

√A (that is bounded over σ(A) since it is compact). If D ∈ B(H) is not selfad-

joint, the previous argument holds true for the selfadjoint operators 12(D+D∗) and 1

2i(D−D∗).

Hence, it holds for their sum D.

Definition 3.6. If A ∈ B(H) for a Hilbert space H, the absolute value of A is the operator|A| :=

√A∗A.

We are ready for the polar decomposition theorem, an extensive discussion of that subject, alsoapplied to closed unbounded operators, appears in [Mor18].

Theorem 3.2. [Polar decomposition]Let A ∈ B(H) for a Hilbert space H. There is a unique pair P ∈ B(H), U ∈ B(H) such that

(a) A = UP (called polar decomposition of A),

(b) P is positive,

(c) U vanishes on Ker(A) and is isometric on Ran(P ).

It turns out that P = |A| and that Ker(U) = Ker(A) = Ker(P ).

Proof. Let us start by observing that A and |A| have the same kernel, since we have |||A|x||2 =〈|A|x||A|x〉 = 〈x||A|2x〉 = 〈x|A∗Ax〉 = 〈Ax|Ax〉 = ||Ax||2 Hence, on Ker(A)⊥ = Ker(|A|)⊥ =Ran(|A|∗) = Ran(|A|) they are injective. So define U : Ran(|A|) → H by means of Uy :=A|A|−1y if y ∈ Ran(|A|). With this definition, we have A = U |A| no matter how we ex-tend U outside Ran(|A|). Now notice that ||Ax||2 = ||U |A|x||2 = |||A|x||2 as establishedabove. This identity proves that U is isometric on Ran(|A|) and, with the standard argumentbased on polarization identity, we also have that 〈Uu|Uv〉 = 〈u|v〉 provided u, v ∈ Ran(|A|).The operator U is in particular continuous and can be extended over Ran(|A|) by continu-ity remaining isometric thereon. Since H = Ker(A) ⊕ Ker(A)⊥ = Ker(A) ⊕ Ker(|A|)⊥ =Ker(A) ⊕ Ran(|A|∗) = Ker(A) ⊕ Ran(|A|), if we define U = 0 over Ker(A), we have con-structed an operator U ∈ B(H) such that, together with P := |A|, all requirements (a),(b)and (c) are valid and also Ker(U) = Ker(A) = Ker(P ). In particular Ker(U) cannotinclude non-vanishing vectors orthogonal to Ker(A), i.e. in Ran(|A|), since U is isometricthereon. Suppose conversely that there exist U ′, P ′ ∈ B(H) satisfying (a),(b) and (c). FromA = U ′P ′, we have A∗ = P ′∗U ′∗ = P ′ and thus A∗A = P ′U ′∗U ′P ′ = P ′P ′ = P ′2 (wherewe have used the fact that, since U ′ is isometric over Ran(P ′), for every x, y ∈ H, we have〈x|P ′P ′y〉 = 〈P ′x|P ′y〉 = 〈U ′P ′x|U ′P ′y〉 = 〈x|P ′U ′∗U ′P ′y〉, so that P ′P ′ = P ′U ′∗U ′P ′). Since

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P ′ is positive, we have P ′ =√A∗A = |A| by uniqueness of the square root. As A is injec-

tive on Ran(|A|), the identity A = U ′|A| implies Uy := A|A|−1y = Uy if y ∈ Ran(|A|). Asbefore, since U ′ is bounded, U ′ = U over Ran(|A|) by continuity. Finally U = U ′ also on

Ran(|A|)⊥ = Ker(|A|) since both vanish by hypothesis there. Summing up, U = U ′.

Remark 3.9. Observe that if A 6= 0, U cannot vanish. Since ||Ux|| ≤ ||x|| by constructionand ||Ux|| = ||x|| on a non-trivial subspace (Ran(P ) 6= 0 if A 6= 0), we conclude that ||U || = 1.

Another related technically useful notion is that of partial isometry.

Definition 3.7. If H is an Hilbert space, an operator U ∈ B(H) such that it is an isome-try over K1 := Ker(U)⊥ is called partial isometry with initial space K1 and final spaceK2 := Ran(U).

Evidently U of the polar decomposition A = UP is a partial isometry with inital space Ker(A)⊥.

Exercise 3.2. Prove that if U ∈ B(H) is a partial isometry with initial space K1 and finalspace K2, then K2 is closed.

Solution. First of all, if y ∈ Ran(U) = K2, there is a sequence of vectors xn ∈ Hwith Uxn → y. Decomposing xn = x′n + x′′n with respect to the standard decompositionKer(U)⊥ ⊕Ker(U), we can omit the part x′′n ∈ Ker(U) since Ux′′n = 0, and we are allowed toassume Ux′n → y. Since U acts isometrically on x′n, and the sequence of Ux′ns is Cauchy, thesequence of the x′ns must be Cauchy as well. By continuity of U , y = U(limn→+∞ x

′n) ∈ Ran(U).

Therefore Ran(U) = Ran(U), namely K2 is closed. 2

Exercise 3.3. Prove that U ∈ B(H) is a partial isometry with initial space K1 if and only ifU∗U is the orthogonal projector onto K1

Solution. If U is a partial isometry with initial space K1 = Ker(U)⊥, then 〈Ux|Uy〉 = 〈x|y〉for x, y ∈ K1. However, since H = K1 ⊕ Ker(U), we can extend by linearity this identity to〈Ux|Uy〉 = 〈x|y〉 for x ∈ K1 and y ∈ H. This is equivalent to 〈U∗Ux|y〉 = 〈x|y〉 for x ∈ K1 andy ∈ H, namely U∗Ux = x if x ∈ K1. On the other hand, U∗Ux = 0 if x ∈ Ker(U) = K⊥1 . Inother words, U∗U : K1⊕K⊥1 3 x+y 7→ x+0 ∈ K1⊕K⊥1 , so that it coincides with the orthogonalprojector onto K1. If, conversely U ∈ B(H) is such that U∗U is the orthogonal projector ontothe closed subspace K1, we have that 〈Ux|Uy〉 = 〈U∗Ux|y〉 = 〈x|y〉 for x, y ∈ K1, so that U is anisometry thereon. Furthermore Ux = 0 is equivalent to ||Ux||2 = 0, that is 〈x|U∗Ux〉 = 0. SinceU∗U is idempotent and selfadjoint, this is equivalent to 〈U∗Ux|U∗Ux〉 = 0, namely ||U∗Ux|| = 0.We have proved that Ker(U) = U∗U(H)⊥ = K⊥1 . In other words, K1 = Ker(U)⊥. In summary,

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U is a partial isometry with initial space K1. 2

Exercise 3.4. Prove that if U ∈ B(H) is a partial isometry with initial space K1 and finalspace K2, then U∗ is a partial isometry with initial space K2 and final space K1. Consequently,UU∗ is the orthogonal projector onto K2

Solution. From the previous exercise, it holds that U∗(Ux) = x if x ∈ K1, so that Ux ∈ K2.Since ||Ux|| = ||x||, we have obtained that U∗ is isometric over K2 = Ran(U) = Ker(U∗)⊥. Wefurthermore have Ker(U∗) = Ran(U) = K1. The last statement immediately follows form theprevious exercise notincing that (U∗)∗ = U . 2

3.4.3 The two-sided ∗-ideal of compact operators

We give here the definition of compact operator in a Hilbert space. However the definition ismuch more general and can be given for operators A ∈ B(X,Y ) with X,Y normed spaces,preserving many properties of these types of bounded operators (see, e.g., [Mor18]).

Definition 3.8. Let H be a Hilbert space. An operator A ∈ B(H) is said to be compact ifAxnn∈N admits a converging subsequence if xnn∈N ⊂ H is bounded. The class of compactoperators over H is indicated by B∞(H).

Example 3.1.(1) As an example, every operator A ∈ B(H) such that Ran(A) is a finite-dimensional subspaceof H is necessarily compact. In fact, let us identify Ran(A) with Cn for n given by the (finite)dimension of Ran(A) by fixing a Hilbert basis of Ran(A) (that coincides with Ran(A), sinceall finite dimensional subspaces are closed, the proof being elementary). If xnn∈N ⊂ H isbounded, i.e. ||xn|| ≤ C for all n ∈ N and some (finite) constant C > 0, then ||Axn|| ≤ ||A||Cfor n ∈ N. The vectors Axn are therefore included in the closed ball in Cn of radius ||A||Cand centred on the origin which is necessarily compact. Hence Axnn∈N admits a convergingsubsequence. An example of such type of compact operator is a finite linear combination ofoperators Ax,y : H 3 z 7→ 〈x|z〉y, for x, y ∈ H fixed.

(2) If A ∈ B(H) and P ∈ L (H) is an orthogonal projector onto a finite dimensional subspace,then AP ∈ B∞(H). In fact, if e1, . . . , en is an orthonormal basis of P (H), we have

Ran(AP ) =

n∑j=1

cjfj

∣∣∣∣∣∣ cj ∈ C , j = 1, . . . , n

,

where fj := Aej for j = 1, . . . , n. Therefore Ran(AP ) has dimension ≤ n and AP is compactdue to (1).

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We summarize below the most important properties of compact operators in Hilbert spaces.

Theorem 3.3. Let H be a Hilbert space and focus on the set of compact operators B∞(H).A ∈ B(H) is compact if and only if |A| is compact.Furthermore B∞(H) is:

(a) a linear subspace of B(H);

(b) a two-sided ∗-ideal of B(H), i.e,

(i) AB,BA ∈ B∞(H) if B ∈ B(H) and A ∈ B∞(H),

(ii) A∗ ∈ B∞(H) if A ∈ B∞(H).

(c) a C∗-algebra (without unit if H is not finite dimensional) with respect to the structureinduced by B(H). In particular B∞(H) is a closed subspace of B(H).

Proof. The first statement immediately arises from the definition of compact operator and iden-tity |||A|x||2 = 〈|A|x||A|x〉 = 〈x||A|2x〉 = 〈x|A∗Ax〉 = 〈Ax|Ax〉 = ||Ax||2, that implies that|A|xnn∈N is Cauchy if and only if Axnn∈N is Cauchy.(a) Fix a, b ∈ C, A,B ∈ B∞(H), and a bounded sequence xnn∈N. Extract a subsequencexnkk∈N such that Axnk → y as k → +∞. xnkk∈N is bounded, so that there is a subsequencexnkhh∈N such that Bxnkh → z as h → +∞. By construction (aA + bB)xnkh → ay + bz forh→ +∞. Hence, aA+ bB is compact.(b) The fact that AB and BA are compact if A ∈ B∞(H) and B ∈ B(H) are immediate con-sequences of the fact that B is bounded and the definition of compact operator. The fact thatA∗ is compact if A is compact now immediately follows from the first statement and the polardecomposition (Theorem 3.2). In fact, from A = U |A| we have A∗ = |A|U∗, since |A| is compactand U∗ ∈ B(H), A∗ is compact as well.(c) Let us prove that B∞(H) is a Banach space with respect to the operator norm sincethe remaining requirements for defining a C∗-algebra are valid because of (a) and (b). LetB(H) 3 A = limi→+∞Ai with Ai ∈ B∞(H). Take a bounded sequence xnn∈N in H: ||xn|| ≤ Cfor any n. We want to prove the existence of a convergent subsequence of Axn. Using ahopefully-clear notation, we build recursively a family of subsequences:

xn ⊃ x(1)n ⊃ x(2)

n ⊃ · · · (3.8)

such that, for any i = 1, 2, . . ., x(i+1)n is a subsequence of x(i)

n with Ai+1x(i+1)n convergent.

This is always possible, because any x(i)n is bounded by C, being a subsequence of xn, and

Ai+1 is compact by assumption. We claim that Ax(i)i is the subsequence of Axn that will

converge. From the triangle inequality

||Ax(i)i −Ax

(k)k || ≤ ||Ax

(i)i −Anx

(i)i ||+ ||Anx

(i)i −Anx

(k)k ||+ ||Anx

(k)k −Ax

(k)k || .

With this estimate,

||Ax(i)i −Ax

(k)k || ≤ ||A−An||(||x

(i)i ||+ ||x

(k)k ||) + ||Anx(i)

i −Anx(k)k ||

141

≤ 2C||A−An||+ ||Anx(i)i −Anx

(k)k || .

Given ε > 0, if n is large enough then 2C||A − An|| ≤ ε/2, since An → A. Fix n and take

r ≥ n. Then An(x(r)p )p is a subsequence of the converging sequence An(x

(n)p )p. Consider

the sequence An(x(p)p )p, for p ≥ n: it picks up the “diagonal” terms of all those subsequences,

each of which is a subsequence of the preceding one by (3.8); moreover, it is still a subsequence

of the convergent sequence An(x(n)p )p, so it, too, converges (to the same limit). We conclude

that if i, k ≥ n are large enough, then ||Anx(i)i − Anx

(k)k || ≤ ε/2. Hence if i, k are large enough

then ||Ax(i)i − Ax

(k)k || ≤ ε/2 + ε/2 = ε. This finishes the proof, for we have produced a Cauchy

subsequence in the Banach space H, which must converge in the space.To end the proof of (c), we notice that, evidently, I cannot be compact if H is infinite dimensional,since every orthonormal sequence unn∈N cannot admit a converging subsequence because||un − um||2 = 2 for n 6= m.

To conclude this essential summary of properties of compact operators in Hilbert spaces, westate and prove the version of spectral theorem for selfadjoint compact operators due to Hilbertand Schmidt. (An alternate proof of this classical theorem can be found in [Mor18].)

Theorem 3.4. [Hilbert-Schmidt decomposition]Let H be a Hilbert space and consider T ∗ = T ∈ B(H) a compact operator with T 6= 0.The following facts hold.

(a) σ(T ) \ 0 = σp(T ) \ 0, so that, if 0 ∈ σ(T ), either 0 ∈ σp(T ) or 0 is the unique elementof σc(T ).

(b) σ(T ) is finite or both infinite and countable. In the latter case 0 is unique accumulationpoint of σp(T ).

(c) There exists λ ∈ σp(T ) with ||T || = |λ|.

(d) If λ ∈ σp(T ) \ 0, its eigenspace has dimension dλ < +∞.

(e) The spectral decomposition holds (the order of the sum is irrelevant)

Tx =∑n∈N

λn〈un|x〉un ∀x ∈ H , (3.9)

and for a finite (N ( N) or countable (N = N) Hilbert basis of eigenvectors unn∈N ofRan(T ), where λn ∈ σp(T ) is the eigenvalue of un.

(f) If N = N and the ordering of the uns is such that |λn| ≥ |λn+1|, then

T =+∞∑n=0

λn〈un| 〉un , (3.10)

in the uniform operator topology.

142

Proof. (a) Take λ ∈ σc(T ) \ 0 assuming that it exists. Due to Proposition 2.13, for everynatural n > 0 there is xn ∈ H with ||xn|| = 1 and ||Txn − λxn|| < 2

n . In particular, if P (T ) isthe PVM of T , we can always fix xn in the closed subspace P[λ−1/n,λ+1/n](H). This subspace isnot trivial because of (d) Theorem 2.8, as it includes the non-trivial subspace P(λ−1/n,λ+1/n)(H).Consequently,

||xn − xm|| = |λ|−1 ||λxn − λxm|| ≤ |λ|−1 ||λxn − Txn − λxm + Txm||+ |λ|−1 ||Txn − Txm|| .

Hence

||xn − xm|| ≤4

|λ|n+

1

|λ|||Txn − Txm|| ,

so that

||Txn − Txm|| ≥ |λ|||xn − xm|| −4

n. (3.11)

Moreover, since λ ∈ σc(T ) and invoking (c) Proposition 2.25, we have for m→ +∞

P(T )[λ−1/m,λ+1/m]xn =

∫Rχ[λ−1/m,λ+1/m]dP

(T )xn →∫RχλdP

(T )xn = P(T )λxn = 0xn = 0 .

This fact has the implication that, if n is fixed, then 〈xn|xm〉 → 0 if m → +∞, because

〈xn|xm〉 = 〈xn|P (T )[λ−1/m,λ+1/m]xm〉 = 〈P (T )

[λ−1/m,λ+1/m]xn|xm〉 → 0. Hence, we also have that

||xn − xm||2 = 2 − 2Re〈xn|xm〉 → 2 as m → +∞. In summary, looking at (3.11), if n issufficiently large such that

4

n<|λ|√

2

4,

we can always take m sufficiently large so that ||xn − xm|| ≥√

22 , obtaining

||Txn − Txm|| ≥ |λ|√

2

2− |λ|

√2

4= |λ|

√2

4.

Even if the sequence xnn∈N is bounded (because ||xn|| = 1 for every n ∈ N), its image

Txnn∈N cannot include Cauchy subsequences since ||Txn − Txm|| ≥ |λ|√

24 if n and m are

sufficiently large. This is impossible because T is compact. The only possibility is λ = 0concluding the proof of (a).(b) Suppose that for some sequence of elements σp(T ) 3 λn → a 6= 0 as n → +∞. Considereigenvectors xn with Txn = λnxn for ||xn|| = 1. Since xn ⊥ xm if n 6= m ((d) Proposition 2.20)and λn → a as n→ +∞, we have

||Txn − Txm||2 = ||λnxn − λmxm||2 = |λn|2 + |λm|2 ≥ 2|a|2 − ε

for n,m > Nε. If |a| > 0, taking ε = |a|2, we conclude that the sequence Txnn∈N cannot admitCauchy subsequences, since ||Txn − Txm||2 ≥ |a|2 > 0 for sufficiently large n,m as it instead

143

should, since T is compact and xnn∈N is bounded. In summary, the accumulation pointa 6= 0 does not exist. Now remember that σ(T ), and hence σp(T ), are included in [−||T ||, ||T ||](Proposition 2.29). In every compact set [−||T ||,−1/n] ∪ [1/n, ||T ||] for n ∈ N with 1/n < ||T ||,the elements of σp(T ) must be finitely many (possibly no one), otherwise an accumulation pointmust exist in that set and this is forbidden since the set does not include 0. We have found thatσp(T ) is either finite or countably infinte and, in this case, 0 is the only accumulation point.(c) Since sup|λ| | λ ∈ σ(T ) = sup|λ| | λ ∈ σp(T ) = ||T || (Proposition 2.29), and ||T || 6= 0cannot be an accumulation point of σp(T ), there must be λ ∈ σp(T ) with |λ| = ||T ||.(d) If λ ∈ σp(T ) 6= 0, define Hλ as the corresponding eigenspace of T and let xjj∈J be aHilbert basis of Hλ. As a consequence ||Txj − Txk||2 = |λ|2||xj − xk||2 = |λ|22 if j 6= k. SoTxjj∈J cannot admit a Cauchy subsequence when J is not finite in spite of xjj∈J beingbounded and T compact. We conclude that J is finite, namely dim(Hλ) < +∞.(e) We assume N = N since the finite case is trivial. Consider a class of sets En ⊂ σp(T )with n ∈ N such that every set En is finite, En+1 ⊃ En and ∪n∈NEn = σp(T ). Notice thatσp(T ) = σ(T ), possibly up to the point 0 ∈ σc(T ) that however does not play any role in the

following because P(T )λ = 0 if λ ∈ σc(T ) as we know for Theorem 2.8. The sequence of functions

χEnı point wise tends to χσp(T )ı and is bounded by the constant ||T || since σ(T ) ⊂ [−||T ||, ||T ||].Applying (c) Proposition 2.25, we have since P (T ) is concentrated on the eigenvalues,

Tx =

∫Rı dP (T )x = lim

n→+∞

∫RχEnı dP

(T )x = limn→+∞

∑λ∈En

λP(T )λx =

∑λ∈σp(T )

λP(T )λx ,

where in the final formula the procedure we adopt to enumerate the eigenvalues does notmatter because the class of sets En is arbitrarily chosen. If we fix an orthornormal basis

Nλ = u(λ)j j=1,...,dλ in every eigenspace Pλ(H) with λ 6= 0 (if λ = 0 is an eigenvalue it does

not give contribution to the total sum defining Tx), so that P(T )λ =

∑dλj=1〈u

(λ)j | 〉u

(λ)j , we can

re-arrange the formula as

Tx =∑

λ∈σp(T )

dλ∑j=1

λ〈u(λ)j |x〉u

(λ)j .

According to Lemma 2.1, since the vectors in the sum are pairwise orthogonal, the sum canbe re-arranged arbitrarily and written into the form where the pairwise orthogonal un are thevectors in the union of bases ∪λ∈σp(T )\0Nλ,

Tx =∑n∈N

λn〈un|x〉un ∀x ∈ H ,

with Tun = λnun. Observe that, from the formula above, the set of orthonormal vectors unspans the whole range of T and also its closure, so that they form a Hilbert basis of Ran(T ).The proof of (e) is over.(f) Suppose again that N = N, otherwise everything becomes trivial. In this case 0 must bethe unique limit point of the λns in view of (b). Assuming to order the eigenvectors so that

144

|λn+1| ≤ |λn|, consider the operators TN :=∑N−1n=0 λn〈un| 〉un. It holds that

||(T − TN )x||2 =

∣∣∣∣∣∣∣∣∣∣ +∞∑n=N

λn〈un|x〉un

∣∣∣∣∣∣∣∣∣∣2

=+∞∑n=N

|λn|2|〈un|x〉|2 ≤ |λN |2+∞∑n=N

|〈un|x〉|2 ≤ |λN |2||x||2 ,

Hence, dividing for ||x|| and taking the sup over the vectors x with ||x|| 6= 0,

||T − TN || ≤ |λN | → 0 if N → +∞.

We have proved that (3.10) is valid in the uniform operator topology completing the proof.

Example 3.2. Let us come back to the Hamiltonian operator H of the harmonic oscillatordiscussed in (3) Example 2.6. It turns out that H−1 ∈ B∞(H). Since 0 6∈ σ(H), it must beH−1 = R0(H) (the resolvent operator for λ = 0), hence H−1 ∈ B(H). Moreover applyingCorollary 2.5,

σ(H−1) = 0 ∪®

1

~ω(n+ 1/2)

∣∣∣∣∣n = 0, 1, 2, . . .

´,

where the points 1~ω(n+1/2) are in the point spectrum as they are isolated points (Theorem 2.8).

Using the same proof as for proving (f) of Theorem 3.3, we have that

H−1 = limN→+∞

N∑n=0

1

~ω(n+ 1/2)〈ψn|·〉ψn

where the ψns are the eigenvectors of H, according to (3) Example 2.6, and the limit is in theuniform operator topology. Since the operators after the limit symbol are of finite rank andthus compact, applying (c) Theorem 3.3, we have that also H−1 is compact. The same resultactually holds true for H−α with α > 0.

3.4.4 Trace-class operators

Let us finally introduce an important family of compact operators called trace class. As a matterof fact, these operators A : H→ H are those which admit well-defined trace

tr(A) =∑u∈N〈u|Au〉 ,

where N ⊂ H is every Hilbert basis and tr(A) does not depend on the choice of the Hilbertbasis. This notion of trace is evidently the direct generalization of the analogous notion in finitedimensional vectors spaces. This family of compact operators will plays a decisive role in theissue concerning the characterisation of the whole class of quantum states.The traditional procedure to introduce them (see, e.g., [Mor18]) passes through the Hilbert-Schmidt class or even trace-class operators are presented as a specific case of Schatten-classoperators. However, since these further types of operators are not of great relevance in our

145

concise presentation, we follow here a much more direct route.We start with a definition which becomes illuminating if we think of the trace as an integrationprocedure: we should deal with absolutely integrable functions to make effective the notion ofintegral. The same happens for the trace.

Definition 3.9. If H is a Hilbert space, B1(H) ⊂ B(H) denotes the set of trace class ornuclear operators, i.e. the operators T ∈ B(H) satisfying∑

z∈M〈z||T |z〉 < +∞ (3.12)

for some Hilbert basis M ⊂ H.

A technical proposition is in order after an important remark concerning different alternate andequivalent defintions of B1(H).

Remark 3.10. A weaker version of condition (b) below, namely,∑u∈N|〈u|Tu〉| < +∞ for every Hilbert basis N ,

is equivalent to T ∈ B1(H) in complex Hilbert spaces [Mor18] (but not in real Hilbert spaces).This condition is sometimes adopted as alternate and equivalent definition of B1(H) in complexHilbert spaces.

Proposition 3.7. Let H be a complex Hilbert space, the following facts are valid for everyT ∈ B1(H).

(a) For every Hilbert basis N ⊂ H,

||T ||1 :=∑u∈N〈u||T |u〉 < +∞

and ||T ||1 does not depend on N .

(b) For every Hilbert basis N ⊂ H,∑u∈N|〈u|Tu〉| ≤ ||T ||1 < +∞ .

(c) T , |T | and»|T | belong to B∞(H).

Proof. (a) From Definition 3.9,

+∞ >∑z∈M〈z||T |z〉 =

∑z∈M

⟨»|T |z

∣∣∣»|T | z⟩ =∑z∈M

∣∣∣∣∣∣»|T |z∣∣∣∣∣∣2 =∑z∈M

∑u∈N

∣∣∣⟨u ∣∣∣»|T | z⟩∣∣∣2146

=∑z∈M

∑u∈N

∣∣∣⟨»|T |u∣∣∣ z⟩∣∣∣2 =∑u∈N

∑z∈M

∣∣∣⟨»|T |u∣∣∣ z⟩∣∣∣2 =∑u∈N

∣∣∣∣∣∣»|T |u∣∣∣∣∣∣2 =∑u∈N〈u||T |u〉 .

The crucial passage is swapping the summations∑z∈M

∑u∈N →

∑u∈N

∑z∈M . This interchange

is allowed by intepreting the sum procedure as a product integration of a pair of countingmeasures over a product space N ×M and using Tonelli and Fubini theorems. Observe thatonly a countable set of terms |〈u|

»|T |z〉|2 of the Cartesian product N ×M do not vanish, so

that the spaces are σ-finite and their product can be defined.(b) Making use of the polar decomposition of T (Theorem 3.2), we have∑

u∈N|〈u|Tu〉| =

∑u∈N|〈u|U |T |u〉| =

∑u∈N

∣∣∣⟨u ∣∣∣U»|T |»|T |u⟩∣∣∣ =∑u∈N

∣∣∣⟨»|T |U∗u ∣∣∣»|T |u⟩∣∣∣≤∑u∈N

∣∣∣∣∣∣»|T |U∗u∣∣∣∣∣∣ ∣∣∣∣∣∣»|T |u∣∣∣∣∣∣ ≤ √∑u∈N

∣∣∣∣∣∣»|T |U∗u∣∣∣∣∣∣2√∑u∈N

∣∣∣∣∣∣»|T |u∣∣∣∣∣∣2 ≤ C»||T ||1 , (3.13)

where

C :=

√∑u∈N

∣∣∣∣∣∣»|T |U∗u∣∣∣∣∣∣2 = ∑u∈N〈u|U |T |U∗u〉 .

Let us study the value of C, proving that it is finite. We start by noticing that U |T |U∗ is positiveand thus coincides with |U |T |U∗|. On the other hand, U |T |U∗ ∈ B1(H) since it satisfies (3.12)for a Hilbert basis M we go to construct. First observe that U∗ is a partial isometry accordingto Exercise 3.4, so that it is an isometry over a closed subspace K = Ker(U∗)⊥. If L is a Hilbertbasis of K, the vectors U∗v for v ∈ L are an orthonormal system in Ran(U∗) and this systemcan always be completed to a Hilbert basis M of H. In summary,

+∞ > ||T ||1 =∑z∈M〈z||T |z〉 =

∑v∈L〈U∗v||T |U∗v〉+

∑v∈M\L

〈z||T |z〉 ≥∑v∈L〈v|U |T |U∗|v〉

=∑v∈N ′〈v|U |T |U∗|v〉 = ||U |T |U∗||1 ,

where, in the last line, we have completed the basis L of K with a Hilbert basis L′ of K⊥ =Ker(U∗), obtaining a Hilbert basis N ′ = L ∪ L′ of H, so that 〈v|U |T |U∗v〉 = 〈U∗v||T |U∗v〉 =0 when v ∈ L′. Since we have this way established that U |T |U∗ ∈ B1(H), the value of||U |T |U∗||1 ≤ ||T ||1 must be independent from the used basis and we can conclude that

C = ∑u∈N〈u|U |T |U∗|u〉 ≤

»||T ||1 .

Inserting in (3.13), we finish the proof of (b),∑u∈N|〈u|Tu〉| ≤

»||T ||1

»||T ||1 = ||T ||1 < +∞ .

147

(c) Consider a Hilbert basis M ⊂ H. If T ∈ B1(H), we have

||T1|| =∑u∈M

∣∣∣∣∣∣»|T |u∣∣∣∣∣∣2 < +∞ .

As a consequence, the elements u ∈ M such that∣∣∣∣∣∣»|T |u∣∣∣∣∣∣ 6= 0 form a finite or countably

infinite subset unn∈N . We henceforth assume N = N, the finite case being trivial. Consider

the compact operator»|T |PN (see (2) Example 3.1), where PN =

∑N−1n=0 〈un| 〉un. We have

∣∣∣∣∣∣(»|T | −»|T |PN)x∣∣∣∣∣∣ =

∣∣∣∣∣∣∣∣∣∣ +∞∑n=N

〈un|x〉»|T |un

∣∣∣∣∣∣∣∣∣∣ ≤ +∞∑

n=N

|〈un|x〉|∣∣∣∣∣∣»|T |un∣∣∣∣∣∣

≤

Ã+∞∑n=N

|〈un|x〉|2Ã

+∞∑n=N

∣∣∣∣∣∣»|T |un∣∣∣∣∣∣2 ≤ ||x||Ã +∞∑n=N

∣∣∣∣∣∣»|T |un∣∣∣∣∣∣2 .Hence ∣∣∣∣∣∣»|T | −»|T |PN ∣∣∣∣∣∣ ≤ Ã +∞∑

n=N

∣∣∣∣∣∣»|T |un∣∣∣∣∣∣2 .The right-hand side vanishes as N → +∞ because the series

∑+∞n=1

∣∣∣∣∣∣»|T |un∣∣∣∣∣∣2 converges to

||T ||1 < +∞. Since»|T |PN ∈ B∞(H) and this space is closed in the uniform topology, it being

a C∗-algebra in B(H) (Theorem 3.3), we have»|T | ∈ B∞(H). Since B∞(H) is a two-sided ideal

(Theorem 3.3 again) we have both that |T | =»|T |»|T | ∈ B∞(H), and T = U |T | ∈ B∞(H),

where we have used the polar decomposition of T , so that U ∈ B(H).

The general properties of B∞(H) are listed in the next proposition whose proof appears in Ap-pendix 3.5 but the fact that B1(H) is Banach with respect to || ||1.

Proposition 3.8. Let H a Hilbert space, B1(H) satisfies the following properties.

(a) B1(H) is a subspace of B(H) which is moreover a two-sided ∗-ideal, namely

(i) AT, TA ∈ B1(H) if T ∈ B1(H) and A ∈ B(H),

(ii) T ∗ ∈ B1(H) if and only if T ∈ B1(H).

(b) || ||1 is a norm on B1(H) making it a Banach space and satisfying

(i) ||TA||1 ≤ ||A|| ||T ||1 and ||AT ||1 ≤ ||A|| ||T ||1 if T ∈ B1(H) and A ∈ B(H),

(ii) ||T ||1 = ||T ∗||1 if T ∈ B1(H).

We are now in a position to introduce the central mathematical tool of this section, i.e. the no-tion of trace of a trace-class operator, listing and proving its main properties with direct interest

148

in quantum physics.

Proposition 3.9. Let H be a Hilbert space and focus on the space of operators B1(H). IfN ⊂ H is a Hilbert basis, the map

B1(H) 3 T 7→ tr(T ) :=∑u∈N〈u|Tu〉 , (3.14)

is well defined, the sum can be re-ordered arbitrarily and more strongly it does not depend onthe choice of N .The complex number tr(T ) is called the trace of T and satisfies the following further properties.

(a) tr(aA+ bB) = a tr(A) + b tr(B) for every a, b ∈ C and A,B ∈ B1(H).

(b) tr(A∗) = tr(A) for every A ∈ B1(H).

(c) tr(AB) = tr(BA) if A ∈ B1(H) and B ∈ B(H).

(d) For every A ∈ B1(H),

(i) |tr(A)| ≤ tr(|A|) = ||A||1,

(ii) ||A|| ≤ tr(|A|) = ||A||1.

(e) If A∗ = A ∈ B1(H) thentr(A) =

∑λ∈σp(A)

dλλ

where dλ is the dimension of the λ-eigenspace and we assume +∞ · 0 = 0.

(f) If U ∈ B(H) is a bijective operator (in particular unitary), then tr(UAU−1) = tr(A) forevery A ∈ B1(H).

(g) If A ≥ 0 and A ∈ B1(H), then tr(A) ≥ 0.

Proof. First of all we notice that∑u∈N 〈u|Tu〉 absolutely converges due to (b) Proposition 3.7,

so that it can be re-ordered arbitrarily. Let us prove that the sum is even independent fromthe basis N . Since T = A + iB with A = 1

2(T + T ∗) and B = 12i(T − T

∗), where A and B areselfadjoint and belong to B1(H) because of (a) Proposition 3.8, it would be enough demonstratingthe assert for the case T = T ∗, simply exploiting linearity of the trace ((a) below whose proofdoes not depend on the present argument). If T ∗ = T ∈ B(H), we can decompose it as

T = T+ − T− where T+ :=∫

[0,+∞) ıdP(T ) = TP

(T )[0,+∞) and T− := −

∫(−∞,0) ıdP

(T ) = −TP (T )(−∞,0).

Since T ∈ B1(H), also T± ∈ B1(H) due to (a) Proposition 3.8. Since T± ≥ 0, exploitingagain linearity of the trace, to complete the proof it is sufficient establishing it in the caseT ∗ = T ∈ B1(H) with T ≥ 0. In this case however T = |T | and therefore (a) Proposition 3.7proves that tr(T ) =

∑u∈N 〈u|Tu〉 =

∑u∈N 〈u||T |u〉 does not depend on N , concluding the proof.

(a) and (b). Observing that aA+ bB, A∗ ∈ B1(H) if A,B ∈ B1(H) due to (a) Proposition 3.9,

149

the proofs of statements (a) and (b) immediately arise from elementary properties of the scalarproduct and the scalar product of operators in B(H), using the fact that 〈u|(aA + bB)u〉 =a〈u|Au〉+ b〈u|Au〉 and 〈u|Au〉 = 〈u|A∗u〉.(c) It is sufficient to prove the statement with A∗ = A ∈ B1(H) and B ∈ B(H), since we canalways decompose a generic A ∈ B1(H) into a linear combination of a pair of selfadjoint trace-class operators 1

2(A + A∗) and 12i(A − A

∗) taking advantage of (a) Proposition 3.8 and finallyexploiting linearity of the trace function. So let us stick to A∗ = A ∈ B1(H) and B ∈ B(H). Weknow that AB,BA ∈ B1(H) (a) Proposition 3.8. Moreover, we compute the traces with respectto a Hilbert basis obtained by completing the Hilbert basis of Ran(A) made of eigenvectors ofA according to (e) Theorem 3.4 noticing that A ∈ B∞(H) from (c) Proposition 3.7. Notice thatthe added elements to the initial basis do not give contribution to the trace as they belong toKer(A) = Ker(A∗), so we can ignore them in the sums below.

tr(AB) =∑n∈N〈un|ABun〉 =

∑n∈N〈Aun|Bun〉 =

∑n∈N

λn〈un|Bun〉 =∑n∈N

λn〈un|Bun〉 ,

where we have used σ(A) ⊂ R since A = A∗. Similarly

tr(BA) =∑n∈N〈un|BAun〉 =

∑n∈N〈un|Bun〉λn =

∑n∈N

λn〈un|Bun〉 = tr(AB) .

(d) First of all take advantage of the polar decomposition A = U |A|. Here |A| is compact dueto (c) Proposition 3.7. Since |A| is selfadjoint it being positive (so it is selfadjoint in view of (3)in exercise 2.7)), there is a Hilbert basis N of eigenvectors of |A| obtained by completing thatin (e) Theorem 3.4. We have

|tr(A)| =

∣∣∣∣∣∣∑u∈N〈u|U |A|u〉∣∣∣∣∣∣ =

∣∣∣∣∣∣∑u∈N〈u|Uu〉λu∣∣∣∣∣∣ ≤ ∑

u∈N|λu| |〈u|Uu〉| .

Next observe that |λu| = λu because |A| ≥ 0 and |〈u|Uu〉| ≤ ||u|| ||Uu|| ≤ 1||Uu|| ≤ ||u|| = 1(||U || ≤ 1 since it is a partial isometry). Hence

|tr(A)| ≤∑u∈N

λu =∑u∈N〈u||A|u〉 = tr|A| = ||A||1 .

The second statement is obvious. Since A ∈ B∞(H) ((c) Proposition 3.7), there is λ ∈ σp(A)such that |λ| = ||A|| because of (c) Theorem 3.4. On the other hand form (e) whose proof isindependent from this argument, ||A||1 ≥ |λ| = ||A||.(e) Since A∗ = A ∈ B∞(H), there is a Hilbert basis of eigenvectors of A obtained by completingthat in (e) Theorem 3.4. Computing the trace using this basis, taking (d) Theorem 3.4 intoaccount, we immediately have the thesis.(f) Exploiting (c), we immediately have tr(UAU−1) = tr((UA)U−1) = tr(U−1UA) = tr(A).(g) The proof is evident form the definition of trace.

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Remark 3.11. It is easy to prove that (c) can be generalized to

tr(T1 · · ·Tn) = tr(Tπ(1) · · ·Tπ(n))

if at least one of the Tk belongs to B1(H), the remaining ones are in B(H), and

π : 1, . . . , n → 1, . . . , n

is a cyclic permutation. The elementary proof arises by decomposing π into a product of one-step cyclic permutations and finally using (c) recursively, re-defining A and B appearing in (c)at every action of the said elementary cyclic permutations. The found identity is known as thecyclic property of the trace.

Example 3.3. Consider the Hamiltonian operator H of the harmonic oscillator discussedin (3) Example 2.6. It holds that H−2 ∈ B1(H). The proof is easy: since 0 6∈ σ(H), it mustbe H−2 = R0(H2) (the resolvent operator for λ = 0), hence H−2 ∈ B(H). Moreover H−2 ≥ 0because its spectrum is positive

σ(H−2) =

®1

~2ω2(n+ 1/2)2

∣∣∣∣∣n = 0, 1, 2, . . .

´.

Finally, computing ||H−2||1 using the Hilbert basis of eigenvectors of H, we have

||H−2||1 =+∞∑n=0

1

~2ω2(n+ 1/2)2< +∞ .

The same result actually holds true for H−α with α > 1.

3.4.5 The mathematical notion of quantum state and Gleason’s theorem

We have constructed all the mathematical machinery to pursue the description of quantumstates in terms of probability measures of L (H) as discussed in Sect.3.4.1. According to thediscussion in the said section, we can give the following general definition.

Definition 3.10. Let H be a Hilbert space. A quantum probability measure in H is amap ρ : L (H)→ [0, 1] such that the following requirements are satisfied.

(1) ρ(I) = 1 .

(2) If Pnn∈N ⊂ L (H) satisfies PkPh = 0 when h 6= k for h, k ∈ N, then

ρ

(s-∑n∈N

Pn

)=∑n∈N

ρ(Pn) . (3.15)

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The convex set of quantum probability measures in H will be denoted by M (H).

The last statement refers to the evident fact that λρ1 + (1 − λ)ρ2 ∈ M (H) if λ ∈ [0, 1] andρ1, ρ2 ∈M(H). This result extends trivially to a finite convex combination

ρ =n∑k=1

pkρk ,

where pk ∈ [0, 1] and∑nk=1 pk = 1, which defines an element of M (H) if all ρk ∈M (H).

Remark 3.12. We stress that in these notes the term quantum state corresponds to themathematical notion of quantum probability measure. We prefer to explicitly use the latter inmathematical statements because the former is therefore ambiguously used in the physical dis-cussion, where quantum states are confounded with quantum state operators we will introduceshortly. This confusion is usually harmless, but becomes significant when dealing with superse-lection rules we shall discuss later.

As already observed in Sect.3.4.1, unit vectors ψ ∈ H, up to phases, define quantum probabilitymeasures by ρψ(P ) := 〈ψ|Pψ〉 for every P ∈ L (H). This is not the only case, since finite convexcombination of quantum probability measures are quantum probability measures as well as justsaid. Suppose in particular that 〈ψk|ψh〉 = δhk and consider the finite convex combination

ρ =n∑k=1

pkρψk ,

where pk ∈ [0, 1] and∑nk=1 pk = 1. By direct inspection, completing the finite orthonormal

system ψkk=1,...,n to a full Hilbert basis of H, one quickly proves that, defining

T =n∑k=1

pk〈ψk| 〉ψk , (3.16)

ρ(P ) can be computed asρ(P ) = tr(TP ) , P ∈ L (H) ,

In particular, it turns out that T is in B1(H), satisfies T ≥ 0 (so it is selfadjoint due to (3) inexercise 2.7) and tr(T ) = 1. As a matter of fact, (3.16) is just the spectral decomposition of T ,whose spectrum is pkk=1,...,n. This result is general.

Proposition 3.10. Let H be a Hilbert space and let define the convex subset of B1(H) of thequantum states or quantum state operators

S (H) := T ∈ B1(H) | T ≥ 0 , tr(T ) = 1 .

If T ∈ S (H), the mapρT : L (H) 3 P 7→ tr(TP ) = tr(PT )

152

is well defined and ρT ∈M(H).

Proof. Observe that tr(TP ) = tr(PT ) is valid in view of (c) Proposition 3.9. The trace-classoperator T is positive, hence self-adjoint so that their eigenvalues λ belong to [0,+∞). Further-more, according to (e) Proposition 3.9, 1 = tr(T ) =

∑λ∈σp(A) dλλ and thus λ ∈ [0, 1]. Exploiting

in particular (e) Proposition 3.4, since T ∈ B∞(H) for Proposition 3.7,

tr(TP ) =∑n∈M〈un|TPun〉 =

∑n∈M

λn〈un|Pun〉 ≤∑n∈M

λn||un||||Pun|| ≤∑n∈M

λn = 1 ,

where M ⊂ N and unn∈M is a Hilbert basis of Ran(T ) which can be completed to a Hilbertbasis of Ker(T ) = Ran(T )⊥, however these added vectors do not give contribution to thecomputed traces as the reader immediately proves. On the other hand, since T ≥ 0,

0 ≤∑n∈M〈Pun|TPun〉 = tr(PTP ) = tr(TPP ) = tr(TP )

and, trivially, tr(IT ) = tr(T ) = 1. Let us prove that the map L (H) 3 P 7→ tr(PT ) is σ-additive to conclude that it fulfils Definition 3.10. If Pnn∈N ⊂ L (H) satisfies PnPm = 0 forn 6= 0, taking advantage of a Hilbert basis of H completing the Hilbert basis of Ran(T ) made ofeigenvectors of T as said in (e) Proposition 3.4,

tr

(T s-

∑n∈N

Pn

)=∑l∈M

∞ul

∣∣∣∣∣∣T ∑n∈NPnul∫

=∑l∈M

∑n∈N〈ul |TPnul 〉 =

∑l∈M

∑n∈N

λl 〈ul |Pnul 〉 .

In other words, since 〈ul|Pnul〉 = 〈ul|PnPnul〉 = 〈Pnul|Pnul〉,

tr

(T s-

∑n∈N

Pn

)=∑l∈N

∑n∈N

λn||Pnul||2 .

Applying Fubini and Tonelli theorems, since λn||Pnul||2 ≥ 0, the sums can be interchanged,obtaining

tr

(T s-

∑n∈N

Pn

)=∑n∈N

∑l∈N

λn||Pnul||2 =∑n∈N

tr(TPn) ,

proving σ-additivity.

The very remarkable fact is that these operators exhaust S (H) if H is separable with dimension6= 2, as established by Gleason in 1957 in a celebrated theorem [Gle57] we restate re-adaptingit to these lectures (see [Dvu92] for a general treatise on the subject).

Theorem 3.5. [Gleason’s Theorem]Let H be a Hilbert space of finite dimension 6= 2, or infinite-dimensional and separable. The set

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of quantum probability measures ρ ∈M (H) is one-to-one with the set of quantum state operatorsT ∈ S (H). The bijection is such that

tr(TP ) = ρ(P ) for every P ∈ L (H),

The said bijection preserves the convex structures of the two sets. Finally, quantum probabilitymeasures separate the elements of L (H) because quantum-state operator do it.

Comments on the proof. The very hard part of Gleason’s theorem is just existence and wewill not try to address it here (see [Dvu92]). The remaining parts are quite easy. Preservationof the complex structures is evident by linearity of the trace. T associated to ρ is unique forthe following elementary reason. Any other T ′ of trace class such that ρ(P ) = tr(T ′P ) for anyP ∈ L (H) must also satisfy 〈x|(T − T ′)x〉 = 0 for any x ∈ H. If x = 0 this is clear, whileif x 6= 0 we may complete the vector x/||x|| to a basis, in which tr((T − T ′)Px) = 0 reads||x||−2〈x|(T − T ′)x〉 = 0, where Px is the projector onto spanx. By (3) in exercise 2.7, weobtain T −T ′ = 01. The fact that quantum-state operators separe the elements of L (H) is quiteobvious since, if tr(TP ) = tr(TP ′) for all T ∈ S (H), we have in particular 〈x|Px〉 = 〈x|P ′x〉,where we have chosen T = 〈x|·〉x for every x ∈ H with ||x|| = 1. As before, it implies thatP = P ′. 2

Remark 3.13.(a) Imposing dimH 6= 2 is mandatory, since a well known counterexample exists. Identifying

H to C2, one-dimensional projectors Pn are one-to-one with unit vectors n = (n1, n2, n3)t ∈ R3

by means of Pn = 12

ÄI +

∑3j=1 njσj

ä, where σj are the standard Pauli matrices. Observe

that we have Pn ⊥ Pn′ if and only if n = −n′. If m ∈ R3 is a fixed unit vector, the mapρ(Pn) := 1

2

Ä1 +

∑3j=1(njmj)

3ä

uniquely extends to a quantum probability measure over L (C2)by additivity, as the reader immediately proves. However, there is no T as in Gleason’s the-orem such that ρ(TPn) = ρ(Pn) for every one-dimensional orthogonal projector Pn. Thisis because, imposing this identity leads to

∑3j=1 njTj =

∑3j=1 n

3jm

3j for a fixed unit vector

m := (m1,m2,m3)t and all unit vectors n. It is easy to prove that this is impossible for everychoice of the constants Tj = tr(Tσj).

(b) Particles with spin 1/2, like electrons, admit a Hilbert space – in which the observablespin is defined – of dimension 2. The same occurs to the Hilbert space in which the polarisa-tion of light is described (cf. helicity of photons). When these systems are described in full,however, for instance including degrees of freedom relative to position or momentum, they arerepresentable on a separable Hilbert space of infinite dimension.

Gleason’s characterization of quantum states has an important consequence. It proves that thereare no sharp states in QM, i.e. probability measures assigning 1 to some elementary observables

1In a real Hilbert space 〈x|Ax〉 = 0 for all x does not imply A = 0. Think of real anti symmetric matricesin Rn equipped with the standard scalar product. Gleason’s theorem is valid in real and quaternionic Hilbertspaces, in the former case uniqueness is valid requiring explicitly that T = T ∗.

154

and 0 to the remaining ones, differently to what happens in CM. In a sense, QM is intrinsicallyprobabilistic since it does not admit sharp measures as instead happens in CM.

Theorem 3.6. Let H be a Hilbert space of finite dimension > 2, or infinite-dimensional andseparable. There is no quantum probability measure ρ : L (H)→ [0, 1], in the sense of Def. 3.10,such that ρ(L (H)) = 0, 1.

Proof. Define S := x ∈ H | ||x|| = 1 endowed with the topology induced by H, and letT ∈ B1(H) be the representative of ρ using Gleason’s theorem. The map

fρ : S 3 x 7→ 〈x|Tx〉 = ρ(〈x| 〉x) ∈ C

is continuous because T is bounded. We have fρ(S) ⊂ 0, 1, where 0, 1 is equipped withthe topology induced by C. Since S is connected (because it is connected by continuous curvesas the reader can prove easily) its image must be connected, too. So either fρ(S) = 0 orfρ(S) = 1. In the first case T = 0 which is impossible because tr(T ) = 1, in the second casetr(T ) > 2 which is similarly impossible.

This negative result, which can be made stronger in the formulation of the so-called Kochen-Specker theorem [SEP], produces no-go theorems in some attempts to explain QM in terms ofCM based on the introduction of so-called hidden variables. We will prove an alternative for-mulation of the established result in Theorem 3.7 below.

Remark 3.14.(a) In view of Proposition 3.10 and Theorem 3.5, when dealing with Hilbert spaces with

physical meaning, we assume that H has finite dimension or is separable, we automaticallyidentify M (H) with S (H). We simply disregard the quantum measures in H with dimension 2which are not represented by elements of S (H), especially taking (b) Remark 3.13 into account.

(b) Since most of the next propositions are valid for the elements of S (H) also if H doesnot respect the said requirements, for mathematical convenience we stick to a general con-text, dealing with the class of quantum-state operators S (H) without restrictions on H. Westress that the corresponding set of measures is just a non-exhaustive part of the space of thequantum measures when H does not satisfy Gleason’s hypotheses. When H is not separable,the elements of S (H) correspond to the so-called completely additive measures which satisfiesa stronger requirement than σ-additivity and define a proper subset of M (H) [Lan17, Mor18].

We are in a position to state some definitions of interest for physicists, especially the distinc-tion between pure and mixed states, so we proceed to analyse the structure of the space of thequantum-state operators. To this end, we remind the reader that, if C is a convex set in a vectorspace, e ∈ C is called extreme if it cannot be written as e = λx + (1 − λ)y, with λ ∈ (0, 1),x, y ∈ C \ e.We have the following simple result.

Proposition 3.11. Let H be a Hilbert space, the following facts hold.

155

(a) The extreme points of the convex set S (H) are those of the form: ρψ := 〈ψ| 〉ψ for everyvector ψ ∈ H with ||ψ|| = 1. (This sets up a bijection between extreme state operators andelements of the complex projective space PH.)When the hypotheses of Gleason’s theorem are satisfied, the extreme points of S (H) areone-to-one with the extreme points of M (H).

(b) Any quantum state operator T ∈ S (H) is a linear combination of extreme quantum-stateoperators, including infinite combinations in the strong operator topology. In particularthere is always a decomposition

T =∑u∈M

pu〈u| 〉u ,

where M is an T -eigenvector Hilbert basis of H, pu ∈ [0, 1] for any u ∈M , and∑u∈M

pu = 1 .

Proof. We start by proving (b). The expansion in statement is a trivial consequence of (e)Theorem 3.4, since trace-class operators are compact because of (c) Proposition 3.7. Nextobserve that T is positive, hence selfadjoint, so that their eigenvalues pu belong to [0,+∞). Mis obtained by completing the Hilbert basis of Ran(T ) by adding a Hilbert space of Ker(T ).Furthermore, according to (e) Proposition 3.9, 1 = tr(T ) =

∑u∈M pu and and also pu ∈ [0, 1].

(a) Consider T ∈ B1(H) and refer to the expansion used in the proof of (b), T =∑u∈N pu〈u| 〉u.

If T is not an one-dimensional orthogonal projector there are at least two different u, say u1

and u2 with pu1 > 0 and 1 − pu1 ≥ pu2 > 0. As a consequence, T decomposes into the convexexpansion T = pu1T1 + (1− pu1)T2 for

T1 = 〈u1| 〉u1 and T2 :=∑u6=u1

pu1− pu1

〈u| 〉u .

Notice that (i) T1 6= T2, (ii) T1, T2 6= 0, (iii) T1, T2 ∈ B1(H) by construction, (iv) they areselfadjoint, (v) T1, T2 ≥ 0 and (vi) tr(T1) = tr(T2) = 1, so T1 and T2 belong to S (H). Weconclude that T cannot be extremal. To complete the proof, let us prove that P = 〈ψ| 〉ψ, with||ψ|| = 1, does not admit non-trivial convex decompositions. Suppose that

P = λT1 + (1− λ)T2 for λ ∈ (0, 1) and T1, T2 ∈ B1(H).

We want to prove that T1 = T2 = P . As a consequence of the hypothesis, if P⊥ = I − P ,

0 = P⊥P = λP⊥T1 + (1− λ)P⊥T2 ,

so that

0 = λtr(P⊥T1) + (1− λ)tr(P⊥T2) = λtr(P⊥T1P⊥) + (1− λ)tr(P⊥T2P

⊥) .

156

Since λ, (1−λ) > 0 and both P⊥TjP⊥ ≥ 0 for j = 1, 2, it must be tr(P⊥T1P

⊥) = tr(P⊥T2P⊥) =

0. Since Tj ≥ 0, if N is a Hilbert basis of P⊥(H) = P (H)⊥, the said conditions can be rephrasedas∑u∈N ||

√Tju||2 = 0, so that TjP

⊥ = 0 and, taking the adjoint, P⊥Tj = 0 because Tj = T ∗j .

Decomposing Tj = PTjP + P⊥TjP + PTjP⊥ + P⊥TjP

⊥, we conclude that

Tj = PTjP = tj〈ψ| 〉ψ

for some tj ∈ C and j = 1, 2. The condition tr(Tj) = 1 fixes tj = 1.

Exercise 3.5. Consider T ∈ S (H). Prove that

(i) T 2 ≤ T (i.e. 〈x|T 2x〉 ≤ 〈x|Tx〉 for all x ∈ H);

(ii) T is extreme if and only if T 2 = T .

Solution. From decomposition of T along the Hilbert basis of eigenvectors of T ∈ S (H), wehave T 2 =

∑u∈N p

2u〈u| 〉u. Since pu ∈ [0, 1], it follows that 0 ≤ p2

u ≤ pu so that 〈x|T 2x〉 ≤ 〈x|Tx〉for all x ∈ H. Since tr(T 2) =

∑u∈N p

2u, if T 2 = T is valid so that

∑u∈N p

2u − pu = 0, and

p2u−p ≤ 0, we conclude that pu = p2

u for all u, so that pu = 0 or pu = 1. Since∑u∈N pn = 1, this

is possible only if all pu vanishes but one which takes the value 1. In other words T = 〈u| 〉u.Conversely, if T = 〈u| 〉u, evidently T 2 = T . 2.

Exercise 3.6. Prove that the quantum probability measure ρ : L (H) → [0, 1] associatedto T ∈ S (H) according to Proposition 3.10 satisfies the so-called Jauch-Piron property: ifρ(P ) = ρ(Q) = 0 is true for P,Q ∈ L (H), then ρ(P ∨Q) = 0.

Solution. tr(TP ) = 0 can be re-written as∑u∈N ||

√TPu||2 = 0 for every Hilbert basis

N ⊂ H. Fixing N completing a Hilbert basis of P (H), the written identity entails√Tx = 0 if x

belong to that basis and also for x ∈ P (H) in view of the continuity of√T . As a consequence

Tx =√T√Tx = 0 for x ∈ P (H). The same result is true replacing P for Q. Every vector in

P ∨ Q(H) is the limit of linear combinations of vectors in P (H) and Q(H). Hence Tx = 0 ifx ∈ P ∨ Q(H) by linearity and continuity of T . Computing tr(TP ∨ Q) using a Hilbert basiswhich completes a Hilbert basis of P ∨ Q(H) by adding a Hilbert basis of (P ∨ Q(H))⊥, weimmediately find tr(TP ∨Q) = 0, namely ρ(P ∨Q) = 0. 2

3.4.6 Physical interpretation

The stated proposition allows us to introduce some notions and terminology relevant in physics.

(a) First of all, extreme elements in S (H) are usually said to describe pure states by physi-cists. We shall denote their set by Sp(H).

(b) Non-extreme quantum state operators are called statistical operators or also densitymatrices. They are said to describe mixed states, mixtures or non-pure states.

157

(c) Ifψ =

∑i∈I

aiφi ,

with I finite or countable (and the series converges in the topology of H in the second case),where the vectors φi ∈ H are all non-null and 0 6= ai ∈ C, physicists say that the stateoperator 〈ψ| 〉ψ is called an coherent superposition of the state operators 〈φi| 〉φi/||φi||2.

(d) The possibility of creating pure states by non-trivial combinations of vectors associatedto other pure states is called, in the jargon of QM, superposition principle of (pure)states

(e) There is however another type of superposition of states. If T ∈ S (H) satisfies:

T =∑i∈I

piTi

with I finite, Ti ∈ S (H), 0 6= pi ∈ [0, 1] for any i ∈ I, and∑i pi = 1, the state operator T

is said to describe an incoherent superposition of the states described by the operatorsTi (possibly pure).

(f) If ψ, φ ∈ H satisfy ||ψ|| = ||φ|| = 1 the following terminology is very popular: Thecomplex number 〈ψ|φ〉 is the transition amplitude or probability amplitude of thestate operator 〈φ| 〉φ on the state operator 〈ψ| 〉ψ, moreover the non-negative real number|〈ψ|φ〉|2 is the transition probability of the state operator 〈φ| 〉φ on the state operator〈ψ| 〉ψ.

We make some comments about these notions. Consider the extreme state operator Tψ ∈ Sp(H),written Tψ = 〈ψ| 〉ψ for some ψ ∈ H with ||ψ|| = 1. What we want to emphasise is that thisextreme state operator is also an orthogonal projector Pψ := 〈ψ| 〉ψ, so it must correspondto an elementary observable of the system (an atom using the terminology of Theorem 3.1).The naıve and natural interpretation2 of that observable is this: “the system’s state is the purestate given by the vector ψ”. We can therefore interpret the square modulus of the transitionamplitude 〈φ|ψ〉 as follows. If ||φ|| = ||ψ|| = 1, as the definition of transition amplitude imposes,tr(TψPφ) = |〈φ|ψ〉|2, where Tψ := 〈ψ| 〉ψ and Pφ = 〈φ| 〉φ. Using (4) we conclude:|〈φ|ψ〉|2 is the probability that the state, given (at time t) by the vector ψ, following a measure-ment (at time t) on the system becomes determined by φ.Notice |〈φ|ψ〉|2 = |〈ψ|φ〉|2, so the probability transition of the state determined by ψ on thestate determined by φ coincides with the analogous probability where the vectors are swapped.This fact is, a priori, highly non-evident in physics.

2We cannot but notice how this interpretation muddles the semantic and syntactic levels. Although this couldbe problematic in a formulation within formal logic, the use physicists make of the interpretation eschews theissue.

158

3.4.7 Post measurement states: Meaning of Luders-von Neumann’s postulate

Since we have introduced a new notion of state, the axiom concerning the collapse of the state(Sect.2.6) must be improved in order to encompass all state operators of S (H). The standardformulation of QM assumes the following axiom (introduced by von Neumann and generalised byLuders) about what occurs to the physical system, in a state described by the operator T ∈ S (H)at time t, when subjected to the measurement of an elementary observable P ∈ L (H), if thelatter is true (so in particular tr(TP ) > 0, prior to the measurement). We are referring to non-destructive testing, also known as indirect measurement or first-kind measurement, where thephysical system examined (typically a particle) is not absorbed/annihilated by the instrument.They are idealised versions of the actual processes used in labs, and only in part they can bemodelled in such a way.

Collapse of the state: General formulation. If the quantum system is in the state describedby T ∈ S (H) at time t and proposition P ∈ L (H) is true after a measurement at time t, thesystem’s state immediately afterwards is described by

TP :=PTP

tr(TP ). (3.17)

In particular, if T is pure and determined by the unit vector ψ, the state immediately aftermeasurement is still pure, and determined by:

ψP =Pψ

||Pψ||.

(Obviously, in either case TP and ψP define states. In the former, in fact, TP is positive of traceclass, with unit trace, while in the latter ||ψP || = 1.)

The stated postulate has an important characterization. Suppose that the initial state wasdescribed by T ∈ S (H), we measured P ∈ L (H) and we want to know the probability to measureQ ∈ L (H). This is a problem of conditional probability. In general, if Q is not compatible withP , i.e. if P and Q do not commute, the rules to handle conditional probability are different fromthe classical ones as physicists known very well. However, if we deal with compatible elementaryobservables, we expect that the quantum rules and the classical ones coincide, immerging theseobservables in a maximal set of commuting elementary observables as we already did elsewhere.In particular, let us assume Q ≤ P . In this case P ∧Q = PQ = QP = Q (Proposition 2.23), sothe classical rule is expected to hold with an obvious meaning of the symbols,

PT (Q|P ) =PT (P ∧Q)

PT (P )=

PT (Q)

PT (P ).

This requirement, if assumed, completely characterizes the post-measurement state and impliesthat Luders-von Neumann’s postulate holds, as established in the following proposition.

Proposition 3.12. Let T ∈ S (H) be a quantum state operator for a Hilbert space H andsuppose that, for P ∈ L (H), it holds tr(TP ) > 0. There exists exactly one other quantum state

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operator T ′ ∈ S (H) such that

tr(T ′Q) =tr(TQ)

tr(TP )for every Q ∈ L (H) with Q ≤ P . (3.18)

It holds that

T ′ =PTP

tr(TP ).

Proof. One immediately proves that T ′ satisfies the said condition. Let us prove the conversestatement. If x ∈ H0 := P (H) has unit norm, consider the orthogonal projector Qx := 〈x| 〉x.Since Q ≤ P , condition (3.18) specialises to tr(T ′Qx) = tr(TP )−1tr(TQx). Computing thetraces completing x to a basis of H, we have 〈x|T ′x〉 − tr(TP )−1〈x|Tx〉 = 0 and, since x = Px,it can be re-arranged to 〈x|T ′x〉 − tr(TP )−1〈Px|TPx〉 = 0, so that

〈x|(T ′ − tr(TP )−1PTP )x〉 = 0 for every x ∈ H0 . (3.19)

Now observe that condition (3.18) for Q = P leads to tr(T ′P ) = 1. Taking also advantage of thecyclic property of the trace and PP = P , we have tr(T ′P ) = tr(PT ′P ) = 1. On the other hand,using the decomposition T ′ = PTP +P⊥TP⊥+P⊥TP +PTP⊥, (whereP⊥ := I −P ), the nor-malization condition tr(T ′) = 1 implies 1 = tr(PT ′P ) + tr(P⊥T ′P⊥). Comparing the obtainedresults, we conclude that tr(P⊥T ′P⊥) = 0, namely tr(P⊥

√T√TP⊥) =

∑u∈N ||

√Tu||2 = 0,

where N is a Hilbert basis of P⊥(H). We have found that T ′P⊥ = 0 and also, taking the adjointP⊥T ′ = 0. Coming back to the decomposition T ′ = PT ′P + P⊥T ′P⊥ + P⊥T ′P = PT ′P⊥, werealize that T ′ = PT ′P . In view of the analogous form PTP

tr(TP ) , we can restrict our analysis to

the Hilbert space H0 := P (H), since both operators vanish on the orthogonal of H0 and theirimages are included in H0 viewed as a Hilbert space in its own right. To this regard, Proposition2.2 implies that (3.19) is therefore equivalent to (T ′− tr(TP )−1PTP )z = 0 when z ∈ H0. Since,as we said, both operators vanish on the orthogonal of H0, we have that T ′y = tr(TP )−1PTPyfor every y ∈ H proving our assert.

Conditional probability is an articulated part of quantum logic (quantum conditional and quan-tum conditional probability) with profound differences between the classical analogs and openissues, see [Red98] for a technical account.

Remark 3.15.(a) Measuring a property of a physical quantity goes through the interaction between the

system and an instrument (supposed to be macroscopic and obeying the laws of classical physics).Quantum Mechanics, in its standard formulation, does not establish what a measuring instru-ment is, it only says they exist; nor is it capable of describing the interaction of instrumentand quantum system set out in the von Neumann Luders’ postulate discussed above. Severalviewpoints and conjectures exist on how to complete the physical description of the measuringprocess; these are called, in the slang of QM, collapse, or reduction, of the state or of

160

the wavefunction also described in terms of decoherence (see [BLPY16, Lan17] for completediscussions and references).

(b) Measuring instruments are commonly employed to prepare a system in a certain purestate. Theoretically-speaking the preparation of a pure state is carried out like this. A finitecollection of compatible propositions P1, . . . , Pn is chosen so that the projection subspace ofP1 ∧ · · · ∧ Pn = P1 · · ·Pn is one-dimensional. In other words P1 · · ·Pn = 〈ψ| 〉ψ for some vectorwith ||ψ|| = 1. The existence of such propositions is seen in practically all quantum systemsused in experiments. (From a theoretical point of view these are atomic propositions) Thenpropositions Pi are simultaneously measured on several identical copies of the physical systemof concern (e.g., electrons), whose initial states, though, are unknown. If for one system themeasurements of all propositions are successful, the post-measurement state is determined bythe vector ψ, and the system was prepared in that particular pure state.Normally each projector Pi belongs to the PVM P (A) of an observable Ai whose spectrum is

made of isolated points (thus a pure point spectrum according to Definition 2.14) and Pi = P(A)λi

with λi ∈ σp(Ai). We will come back to this issue in Section 4.2.2.(c) Let us finally explain how to practically obtain non-pure states from pure ones. Consider

q1 identical copies of system S prepared in the pure state associated to ψ1, q2 copies of S preparedin the pure state associated to ψ2 and so on, up to ψn. If we mix these states each one willbe in the non-pure state: T =

∑ni=1 pi〈ψi| 〉ψi , where pi := qi/

∑ni=1 qi. In general, 〈ψi|ψj〉 is

not zero if i 6= j, so the above expression for T is not the decomposition with respect to aneigenvector basis for T . This procedure may seem to suggest the existence of two different typesof probability, one intrinsic and due to the quantum nature of state associated to ψi, the otherepistemic, and encoded in the probability pi. But this is not true: once a non-pure state has beencreated, as above, there is no way, within QM, to distinguish the states forming the mixture.For example, the same state operator T could have been obtained mixing other pure states thanthose determined by the ψi. In particular, one could have used those in the decomposition of Tinto a basis of its eigenvectors. For physics, no kind of measurement would distinguish the twomixtures.

3.4.8 General interplay of quantum observables and quantum states

Dealing with mixed states, definitions (2.72) and (2.74) for, respectively the expectation value〈A〉ψ and the standard deviation ∆Aψ of an observable A referred to the pure state defined by〈ψ| 〉ψ with ||ψ|| = 1 are no longer valid. Extended natural definitions can be stated referringto the probability measure associated to both the mixed state defined by T ∈ S (H) and theobservable A, more precisely its PVM P (A). In practice, we can define

µ(A)T : B(σ(A)) 3 E 7→ tr(P

(A)E T ) ∈ [0, 1] (3.20)

with the meaning ofthe probability to obtain E after a measurement of A in the quantum state defined by T ∈

S (H).

161

In particular, if T is pure, so that T = ψ〈ψ|·〉 for some unit vector ψ ∈ H, we find again theprobabilty already seen in (2.71),

µ(A)T (E) = ||P (A)

E ψ||2 = µ(A)ψ,ψ(E) .

The proof is trivial, just complete ψ as a Hilbert basis of H and compute the trace along thatbasis.Adopting the definition of µ

(A)T introduced in (3.20),

(a) the expectation value of A with respect to the state described by T is defined as

〈A〉T :=

∫σ(A)

λ dµ(A)T (λ) ,

provided the function σ(A) 3 λ→ λ ∈ R is L1(σ(A), µ(A)T );

(b) the standard deviation is defined as

∆AT :=

∫σ(A)

(λ− 〈A〉T )2 dµ(A)T (λ) =

∫σ(A)

λ2 dµ(A)T (λ)− 〈A〉2T ,

provided σ(A) 3 λ → λ ∈ R is L2(σ(A), µ(A)T ). (Notice L2(σ(A), µ

(A)T ) ⊂ L1(σ(A), µ

(A)T )

since the measure is finite.)

The next proposition establishes that the usual formal results handled by physicists (see formu-las in (b)-(d) below) are valid3. Referring to domain issues in (b) and (c) below we observe thatwe have D(A2) = ∆ı2 ⊂ ∆ı = D(A) = D(|A|).

Proposition 3.13. Let H be a Hilbert space, T ∈ S (H) a quantum state operator andA : D(A)→ H, densely defined, an observable (i.e. A = A∗). The following facts hold.

(1) µ(A)T as in (3.20) is a well-defined probability measure over B(σ(A)).

(2) If Ran(T ) ⊂ D(A) and |A|T ∈ B1(H) (always valid if A ∈ B(H)), then

(a) 〈A〉T is defined,

(b) 〈A〉T = tr(AT ).

(3) If Ran(T ) ⊂ D(A2) and |A|T,A2T ∈ B1(H) (always valid if A ∈ B(H)), then

(a) ∆AT is defined,

(b) ∆AT =»tr(A2T )− (tr(AT ))2.

3Weaker necessary and sufficient conditions assuring that these formulas are valid can be found in [Mor18],referring to the Hilbert-Schmidt class of the operators we do not consider here.

162

(4) Assume that T = ψ〈ψ| 〉 with ||ψ|| = 1

(a) If ψ ∈ D(A) then the hypotheses in 2 are valid and 〈A〉T = 〈ψ|Aψ〉,(b) If ψ ∈ D(A2) then the hypotheses in 3 are valid and ∆AT =

»〈ψ|A2ψ〉 − 〈ψ|Aψ〉2.

Proof. (1) Taking the definition PVM into account, the proof is a trivial re-adaptation of theproof of Proposition 3.10.(2)(a) Let us assume Ran(T ) ⊂ D(A) and |A|T ∈ B1(H) that are automatically true if A ∈B(H). As already stressed, D(|A|) = D(A) so Ran(T ) ⊂ D(A) = D(|A|) is true and bothAT , |A|T are well defined with the said hypotheses. Next, polar decomposition theorem for(generally unbounded) selfadjoint operators A = U |A| (immediately obtained from the spectraldecomposition in the three considered cases with |A| and U := sign(A) ∈ B(H) spectrallydefined) implies AT = U |A|T ∈ B1(H), because U ∈ B(H) and B1(H) is two-sided ideal. Now,referring to the Borel σ-algebra over σ(A) ⊂ R, we can construct a sequence of real simplefunctions

sn =∑in∈In

c(n)inχE

(n)in

: σ(A)→ R with c(n)in∈ R, and In finite

which satisfies0 ≤ |sn| ≤ |sn+1| ≤ |ı| , sn → ı pointwise for n→ +∞, (3.21)

where ı : σ(A) 3 λ 7→ λ ∈ R. By direct application of the given definitions, if

An :=

∫σ(A)

sndP(A) =

∑in∈In

c(n)inP

(A)

E(n)in

∈ B(H) ,

exploiting (c) Proposition 2.25, monotone and Lebesgue dominated convergence theorems, wehave both

〈ψ|Anψ〉 → 〈ψ|Aψ〉 , 〈ψ||An|ψ〉 → 〈ψ||A|ψ〉 ∀ψ ∈ D(A) as n→ +∞ (3.22)

and also|〈ψ|Anψ〉| ≤ 〈ψ||An|ψ〉 ≤ 〈ψ||A|ψ〉 . (3.23)

On the other hand, if M is a Hilbert basis of H obtained by completing a Hilbert basis N ofKer(T )⊥ made of eigenvectors of T according to (e) Theorem 3.4 and taking advantage of thecyclic property of the trace, we have both

tr(AnT ) = tr

Ñ∑in∈In

c(n)inP

(A)

E(n)in

T

é=∑in∈In

c(n)intr(P

(A)

E(n)in

T ) =∑in∈In

c(n)intr(TP

(A)

E(n)in

) =

=∑in∈In

c(n)inµT (E

(n)in

) =

∫σ(A)

sndµ(A)T

(3.24)

and similarly

tr(|An|T ) =

∫σ(A)|sn| dµ(A)

T . (3.25)

163

Looking at the identity (3.25), by monotone convergence theorem, for n→ +∞,

tr(|An|T ) =

∫σ(A)|sn|(λ) dµ

(A)T (λ)→

∫σ(A)|λ| dµ(A)

T (λ) ,

and simultaneously we have

tr(|An|T ) =∑u∈N

s(u)〈u||An|u〉 →∑u∈N

s(u)〈u|Au〉 = tr(|A|T ) ,

where s(u) ≥ 0 are the eigenvalues of T , again by monotone convergence theorem and (3.22).Putting all together, we find

tr(|A|T ) =

∫σ(A)|λ| dµ(A)

T (λ) .

We have in particular established that the integral in the right-hand side is finite (because theleft-hand side exists by hypothesis) and thus 〈A〉T is well defined.(2)(b) Let us look at the identity in (3.24). From dominated convergence theorem taking (3.21)into account, we obtain for n→∞

tr(AnT ) =

∫σ(A)

sn(λ) dµ(A)T →

∫σ(A)

λ dµ(A)T .

On the other hand,

tr(AnT ) =∑u∈N〈u|Anu〉s(u)→

∑u∈N〈u|Au〉s(u) = tr(AT ) ,

where we have once again applied dominated convergence theorem as is permitted by (3.23).Putting all together we get

tr(AT ) =

∫σ(A)

λ dµ(A)T (λ) =: 〈A〉T ,

concluding the proof of 2.(b).(3) The proof is strictly analogous to that of (2) also noticing that the hypotheses of (3) implies

those of (2) and that L2(σ(A), µ(A)T ) ⊂ L1(σ(A), µ

(A)T ) because µ

(A)T is finite.

(4) The thesis consists of trivial subcases of (2) and (3) in particular completing ψ to a Hilbertbasis of H to be used to explicitly compute the various traces.

Example 3.4. Let us consider a quantum spinless particle of mass m > 0, living on the realline, whose Hamiltonian operator

H = s-+∞∑n=0

~ω(n+ 1/2)〈ψn| 〉ψn

164

is that of an harmonic oscillator (see (3) Example 2.6). In this case H = L2(R, dx). If the systemis in contact with a heat bath at the (absolute) temperature (kBβ)−1 > 0 (kB being Boltzmann’sconstant), its state is mixed and is described by the statistical operator

Tβ = Z−1β e−βH ,

where expanding the trace along the Hilbert basis of eigenvectors ψn of H,

Zβ = tr(e−βH) =+∞∑n=0

e−β~ω(n+1/2) =e−β~ω/2

1− e−β~ω. (3.26)

is the so-called canonical partition function. In other words

Tβ = s-+∞∑n=0

e−β~ω(n+1/2)

Zβ〈ψn| 〉ψn .

It is easy to check that Tβ ∈ S (H). Furthermore the elements of Ran(Tβ) have the form

+∞∑n=0

e−β~ω(n+1/2)cnψn with∑+∞n=0 |cn|2 < +∞.

It is therefore evident that Ran(Tβ) ⊂ D(Hm) for m = 1, 2, . . . and that |H|Tβ = HβTβ andH2Tβ ∈ B1(H). For instance

HmTβ = s-+∞∑n=0

(~ω)me−β~ω(n+1/2)(n+ 1/2)m

Zβ〈ψn| 〉ψn ,

so that HmTβ ∈ B1(H) with

||HmTβ|| = supn∈N

(~ω)2e−β~ω(n+1/2)(n+ 1/2)m

Zβ=

(~ω)me−β~ω/2

2mZβ,

||HmTβ||1 =+∞∑n=0

(~ω)me−β~ω(n+1/2)(n+ 1/2)m

Zβ< +∞ .

Therefore we can apply Proposition 3.13. For instance

〈H〉Tβ = tr(HTβ) =~ωZβ

+∞∑n=0

e−β~ω(n+1/2)(n+ 1/2) = − 1

Zβ

d

dβZβ = − d

dβlnZβ ,

where in the penultimate passage we have passed a β-derivative under the symbol of summationof Zβ defined as in (3.26), as it is allowed by standard elementary theorems of calculus, sincethe series converges and that of derivatives uniformly does.

165

3.4.9 Non existence of dispersion-free quantum probability measures

With the hypotheses of Gleason’s theorem, quantum state operators and quantum probabilitymeasures are one-to-one, so that the notion of expectation value and standard deviation of anobservable can be ascribed to quantum probability measures ρ ∈M (H), in particular 〈A〉ρ and∆Aρ can be defined when A is bounded (simply replacing ρ for the corresponding state T andusing the already known definitions).A quantum probability measure ρ ∈ M (H) is called dispersion free if ∆Aρ = 0 for ev-ery bounded observable A. An important consequence of Gleason’s theorem (however alreadyknown to von Neumann in 1932 and physically equivalent to Theorem 3.6) is the fact that thereare no dispersion free quantum probability measures as soon as one assumes the physically quitemild hypotheses of Gleason’s theorem.

Theorem 3.7. Let H be a Hilbert space with finite dimension dim(H) > 2 or infinite dimen-sional and separable. There are no quantum quantum probability measures ρ ∈M (H) satisfying∆Aρ = 0 for every bounded observable A.

Proof. Suppose that such a ρ ∈ M (H) exists and let T ∈ S (H) be the associated quantumstate operator. If A = P ∈ L (H), it must hold that 0 = (∆PT )2 = tr(TPP ) − tr(TP )2 =tr(TP ) − tr(TP )2. As a consequence, either tr(TP ) = 0 or tr(TP ) = 1 for every P ∈ L (H).This is impossible for Theorem 3.6.

3.5 Appendix. On L (H) and trace-class operators again.

3.5.1 Proof of some technical results about L (H).

We prove here Proposition 3.4. We will take advantage of the machinery of spectral theoryproducing an original proof. More elementary proofs appear in [Red98] and [Mor18], based ontechnical propositions we did not discuss in these lectures.

Proposition 3.4. In a Hilbert space H the identity holds

(P ∧Q)x = limn→+∞

(PQ)nx (3.27)

for every P,Q ∈ L (H) and x ∈ H.

Proof. Fix x ∈ H and (uniquely) decompose it as

x = x0 + y , where x0 ∈ (P ∧Q)(H) = P (H) ∩Q(H) and y ∈ (P (H) ∩Q(H))⊥. (3.28)

Consider the sequence of operators

A1 := P, An := QP, A3 := PQP, A4 := QPQP, · · ·

166

We want to prove that

Any → 0 . (3.29)

This concludes the proof because Anx0 = x0 since x0 ∈ P (H) and x0 ∈ Q(H), so that Px0 =Qx0 = x0 and Anx → x0 + 0 = x0; finally, the sequence (PQ)nxn∈N is a subsequence ofAnxn∈N and thus it converges to the same limit x0, proving (3.27).To prove (3.29), observe that the sequence of operators applied to y, Anyn∈N, satisfies

||An+1y|| ≤ ||Any|| ,

since either An+1 = PAn or An+1 = QAn and ||P ||, ||Q|| ≤ 1. The non-increasing sequence||Any||n∈N must therefore admit a limit in view of elementary results of calculus. If we founda subsequence of Anyn∈N converging to 0 we would prove that also ||Any|| → 0 as n → +∞which, in turn, would entail (3.29). The following lemma concludes the proof.

Lemma 3.1. The subsequence A2n+1yn∈N tends to 0 as n→ +∞.

Proof. Consider the subsequence of operators A2n+1n∈N. Remembering that PP = P , wehave

A3 = PQP =: B, A5 = PQPQP = (PQP )2 = B2, A7 = PQPQPQP = (PQP )3 = B3 , · · ·

· · · , A2n+1 = Bn, · · ·

Notice that(1) B∗ = (PQP )∗ = P ∗Q∗P ∗ = PQP = B ∈ B(H),(2) ||B|| ≤ ||P ||||Q||||P || ≤ 1,(3) σ(B) ⊂ [−||B||, ||B||] (Proposition 2.29),(4) σ(B) ∈ [0,+∞) (Proposition 2.28) as 〈z|Bz〉 = 〈Pz|QPz〉 = 〈Pz|QQPz〉 = ||QPz||2 ≥ 0.

Collecting together these results, we have from the spectral theory that

Bnz =

∫[0,1]

λndP (B)(λ)z if z ∈ H .

Since λn → χ1(λ) pointwise for λ ∈ [0, 1] if n → +∞, exploiting (c) Proposition 2.25, weconclude that

Bnz → Ez := P(B)1 z as n→ +∞ and z ∈ H. (3.30)

With the same argument we can also prove that

Cnz → Fz := P(C)1 z as n→ +∞ and z ∈ H, (3.31)

where we have defined the other sequence of operators (which is not a subsequence of Ann∈N)

C := QPQ, C2 = (QPQ)2 = QPQPQ, B3 = (QPQ)3 = QPQPQPQ , · · · .

167

We now prove that the identity of orthogonal projectors holds E = F . To this end, notice that

(PQP )n(QPQ)m(PQP )lz = (PQP )n+m+l+1z ,

which implies EFE = E. (To prove it, take first the limit for m → +∞ using continuity of(PQP )n, next compute the limit for l → +∞ using continuity of (PQP )nF and eventuallycompute the limit for n→ +∞.) Swapping the role of P and Q we also have FEF = F . FromEFE = E we obtain

0 = 〈z|(E−EFE−EFE+EFE)z〉 = 〈z|(E2−EFE−EFE+EFE)z〉 = 〈z|(E−EF )(E−FE)z〉

= 〈z|(E − FE)∗(E − FE)z〉 = ||(E − FE)z||2 for z ∈ H.

Hence E = FE. Starting from FEF = F , with the same argument, we find F = EF . Puttingtogether the found results, we find F = E as wanted, since F = F ∗ = (EF )∗ = FE = E.To go on, observe that, by construction of E and F , it holds that PE = E and QF = F , so that

E(H) = F (H) ⊂ P (H) ∩Q(H) .

If we apply the result to the sequence Bny in (3.30) with y in (3.28), we obtain

A2n+1y = Bny → Ey ∈ P (H) ∩Q(H) . (3.32)

However we also have that

A2n+1y = Bny → Ey ∈ (P (H) ∩Q(H))⊥ (3.33)

because (P (H) ∩ Q(H))⊥ is closed and every A2n+1y belongs to (P (H) ∩ Q(H))⊥ since, if s ∈P (H) ∩ Q(H) then 〈s|A2n+1y〉 = 〈s|(QP · · ·QP )y〉 = 〈(PQ · · ·PQ)s|y〉 = 〈s|y〉 = 0 becausey ∈ (P (H) ∩Q(H))⊥ by (3.28).The only possibility permitted by (3.32) and (3.33) is A2n+1y → 0.

As said above, the validity of the lemma ends the proof.

Remark 3.16. The proof actually proves that more strongly

Px, QPx, PQPx, QPQPx, PQPQPQPx, · · · → (P ∧Q)x ∀x ∈ H .

We also have

Qx, PQx, QPQx, PQPQx, QPQPQPQx, · · · → (P ∧Q)x ∀x ∈ H ,

since P ∧Q = Q ∧ P .

168

3.5.2 Proof of some technical results about trace-class operators.

We prove here Proposition 3.8. A preliminary lemma is useful not only in this context.

Lemma 3.2. If H is a Hilbert space and B ∈ B(H), then B is a linear combination of unitaryoperators.

Proof. As we know, B can be decomposed as a complex linear combination of selfadjoint op-erators B = 1

2(B + B∗) + i 12i(B − B∗), so it is sufficient to prove the thesis for selfadjoint

operators. Consider A∗ = A ∈ B(H), if ||A|| = 0 the thesis is trivial, so we assume ||A|| > 0.In this case, A′ := 1

||A||A satisfies ||A|| ≤ 1 so that σ(A′) ⊂ [−1, 1] for Proposition 2.29 and

A′± := A′± i√I −A′2 ∈ B(H) are well defined via spectral theory integrating the corresponding

functions over σ(A′). It is easy to prove that A′± are unitary exploiting Theorem 2.6 and Propo-sition 2.25 and finding A′±

∗A′± = A′±A′±∗ = I. By construction, A′ = 1

2A+ + 12A− concluding

the proof.

We state Proposition 3.8 for reader’s convenience and we prove it (closely following a corre-sponding proof in [ReSi80] for part of (a)).

Proposition 3.8. Let H a Hilbert space, B1(H) satisfies the following properties.

(a) B1(H) is a subspace of B(H) which is moreover a two-sided ∗-ideal, namely

(i) AT, TA ∈ B1(H) if T ∈ B1(H) and A ∈ B(H),

(ii) T ∗ ∈ B1(H) if and only if T ∈ B1(H).

(b) || ||1 is a norm on B1(H) making it a Banach space and satisfying

(i) ||TA||1 ≤ ||A|| ||T ||1 and ||AT ||1 ≤ ||A|| ||T ||1 if T ∈ B1(H) and A ∈ B(H),

(ii) ||T ||1 = ||T ∗||1 if T ∈ B1(H).

Proof. (a) First of all, observe that |aA| = |a||A| for a ∈ C so that, to prove that B1(H) is avector space it suffices to check that A + B ∈ B1(H) for A,B ∈ B1(H). Let U , V , and W thepartial isometries arising from the polar decompositions of A+B, A, and B

A+B = U |A+B| , A = V |A| , B = W |B| .

As a consequence, if N is a Hilbert basis of H,∑u∈N〈u||A+B|u〉 =

∑u∈N〈u|U∗(A+B)u〉 ≤

∑u∈N|〈u|U∗V |A|u〉|+

∑u∈N|〈u|U∗W |B|u〉| .

However,∑u∈N|〈u|U∗V Au〉| ≤

∑u∈N||»|A|V ∗Uu||||

»|A|u|| ≤

√∑u∈N||»|A|V ∗Uu||2

√∑u∈N||»|A|u||2 .

169

The same argument is valid for B. Hence, if we can prove that∑u∈N||»|A|V ∗Uu||2 ≤ tr(|A|) , (3.34)

we can conclude that ∑u∈N〈u||A+B|u〉 ≤ tr(|A|) + tr(|B|) < +∞ ,

establishing that A+B ∈ B1(H) as wanted. To show (3.34) we need only to prove that

tr(U∗V |A|V ∗U) ≤ tr(|A|) .

Referring to a Hilbert basis N 3 u whose elements satisfy either u ∈ Ker(U) or u ∈ Ker(U)⊥,we see that

tr(U∗V |A|V ∗U) ≤ tr(V |A|V ∗) .

Iterating the procedure for tr(V |A|V ∗), using a Hilbert basis N 3 u whose elements satisfyeither u ∈ Ker(V ) or u ∈ Ker(V )⊥, we also conclude that

tr(V |A|V ∗) ≤ tr(|A|) ,

proving our assert.(a)(i) Since Lemma 3.2 is valid, exploiting the fact that B1(H) is a linear space, we have onlyto prove that UT, TU ∈ B1(H) if T ∈ B1(H) and U ∈ B(H) is unitary. Observe that |UT |2 =T ∗U∗UT = |T |2 so |UT | = |T | and thus tr(|UT |) = tr(|T |) < +∞ proving that UT ∈ B1(H).Similarly |TU |2 = U∗T ∗TU = U∗|T |2U , so that |TU | = U∗|T |U (because this operator is positiveand its square is U∗|T |2U). Therefore we have tr(|TU |) = tr(U∗|T |U) =

∑u∈N 〈Uu||T |Uu〉 =

tr(|T |) < +∞ (because Uuu∈N is a Hilbert basis ifN is since U is unitary) and so TU ∈ B1(H).(a)(ii) Let T = U |T | the polar decomposition of T . Therefore T ∗ = |T |U∗ and |T ∗|2 = TT ∗ =U |T |2U∗. Since U |T |U∗ U |T |U∗ = U |T |2U∗ because U∗U is the orthogonal projecotr ontoRan(|A|) (Theorem 3.2 and Exercise 3.3), we conclude that |T ∗| = U |T |U∗. Now (i) impliesthat T ∗ ∈ B1(H) if T ∈ B1(H). Since (T ∗)∗ = T we have also that T ∗ ∈ B1(H) entailsT ∈ B1(H).(b) If a ∈ C and A ∈ B1(H), we find

||aA||1 =∑u∈N〈u||aA|u〉 =

∑u∈N〈u||a||A|u〉 = |a|

∑u∈N〈u||A|u〉 = |a|||A||1 .

Proving (a), we have established that ||A+B||1 ≤ ||A||1 + ||B||1 for A,B ∈ B1(H), so that || ||1 :B1(H)→ C is a seminorm. On the other hand, if ||A||1 = 0 it means that

∑u∈N 〈u||A|u〉 = 0 for

every Hilbert basis N . Since every unit vector x ∈ H can be completed to a basis, this implies in

particular that ||»|A|x||2 = 〈x||A|x〉 = 0 and thus |A|x =

»|A|

2x = 0 for every x ∈ H, so that

||Ax||2 = 〈Ax|Ax〉 = |||A|2x|| = 0 for every x ∈ H meaning that A = 0. Hence || ||1 : B1(H)→ C

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is a norm. The proof of the fact that the norm makes B1(H) a Banach space can be found in[Scha60].(b)(i) It is sufficient to check that ||AT ||1 ≤ ||A||||T ||1. Indeed, assuming it, from (ii) whose proofis independent form the present one, we have ||TA||1 = ||A∗T ∗||1 ≤ ||T ∗||||A∗||1 = ||T ||||A||1.Let us prove ||AT ||1 ≤ ||A||||T ||1. Consider the polar decomposition T = U |T | and also |AT | =W |AT |, so that |AT | = W ∗(AT ) = W ∗AU |T |. Putting S = W ∗AU , we have, exploiting theusual Hilbert basis N of eigenvectors of the selfadjoint positive compact operator |T |

||AT ||1 = tr(|AT |) = tr(S|T |) =∑u∈N〈u|S|T |u〉 =

∑u∈N

λu〈u|Su〉 ≤∑u∈N|λu〈u|Su〉|

≤∑u∈N

λu|〈u|Su〉| ≤∑u∈N

λu||S|| = ||S||||T ||1 .

Since W ∗ and U are partial isometries, ||S|| ≤ ||A||, proving that ||AT ||1 ≤ ||A||||T ||1.(b)(ii) The proof of (a)(ii) established that |T ∗| = U |T |U∗. Making use of a Hilbert basisN whose elements belong either to Ker(U∗) or Ker(U∗)⊥, we immediately have ||T ∗||1 =∑u∈N 〈U∗u||T |Uu〉 = ||T ||1.

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Chapter 4

Von Neumann algebras ofObservables and SuperselectionRules

The aim of this chapter is to focus on the class of observables of a quantum system, described inthe Hilbert space H, exploiting some elementary results of the theory of von Neuman algebras.This mathematical tool will be in particular employed to formalise the notion of superselectionrule.

4.1 Introduction to von Neumann algebras

Up to now, we have tacitly supposed that all selfadjoint operators in H represent observables, allorthogonal projectors represent elementary observables, all normalized vectors represent purestates. This is not the case in physics due to the presence of the so-called superselection rulesintroduced by Wigner (and developed together with Wick and Wightman around 1952) and alsodue to the possible presence of a so-called (non-Abelian) gauge group and several others theoreti-cal and experimental facts. Within the Hilbert space approach, the appropriate tool to deal withthese notions is the mathematical structure of a von Neumann algebra. The idea of restrictingthe algebra of observables entered the formulation of Quantum Mechanics quite early, around1936, by von Neumann who tried to justify the intrinsic stochasticity of quantum systems with aphysically sound notion of quantum probability “a priori” (see [Red98] for a historical account).Von Neumann’s ideas were valid only dealing with a specific type of von Neumann algebrascalled type-II1 von Neumann algebras satisfying a stronger version of orthomodularity knownas modularity. Though nowadays von Neumann’s ideas are considered physically untenable, thetheory of von Neumann algebras has become an important part of pure mathematics [KaRi97]with many intersections with areas also different from functional analysis (non-commutativegeometry for instance). Moreover, the idea of restricting the algebra of observables survivedvon Neumann’s approach since, as said above, it received a strong physical support from the

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experimental evidence of Wigner’s idea of superselection rules, the formulation of non-Abeliangauge theories, and from Quantum Field Theory also formulated in terms of Fermionic Fields(which are not observables) [Emc72, Haa96, Ara09, Lan17].For all these reasons, we spend the initial part of this chapter, of pure mathematical flavour, todiscuss the basic properties of elegant notion of von Neumann algebra.

4.1.1 The mathematical notion of von Neumann algebra

Before we introduce the notion of von Neumann algebra, let us define first the commutant of asubset of B(H) and state an important preliminary theorem.

Definition 4.1. Consider a Hilbert space H. If M ⊂ B(H), the set of operators

M′ := T ∈ B(H) | TA−AT = 0 for any A ∈M (4.1)

is called commutant of M.

Remark 4.1. It is evident from the definition that, if M1,M2,N ⊂ B(H), then

(1) M1 ⊂M2 implies M′2 ⊂M′1

(2) N ⊂ (N′)′.

Further properties of the commutant are stated below.

Proposition 4.1. Let H a Hilbert space and M ⊂ B(H). The commutant M′ enjoys thefollowing properties.

(a) M′ is a unital C∗-algebra (C∗-subalgebra) in B(H) if M is ∗-closed (i.e. A∗ ∈ M ifA ∈M).

(b) M′ is both strongly and weakly closed.

(c) M′ = ((M′)′)′. Hence we cannot reach beyond the second commutant by iteration.

Proof. (a) I ∈ M′ in any cases. Furthermore, if A ∈ B(H) satisfies AB − BA = 0 for everyB ∈ M, then B∗A∗ − A∗B∗ = 0 for every B ∈ M. If C ∈ M, C∗ ∈ M by hypothesis andC = (C∗)∗, so CA∗ − A∗C = 0 for every C ∈ M and thus A∗ ∈ M′ if A ∈ M′. To concludethe proof of (a) it is enough to prove that M′ is closed with respect to the uniform topologyof operators. If AnB = BAn and An → A uniformly, where A,An ∈ B(H) and B ∈ M, thenA ∈M′ because

||AB −BA|| = || limn→+∞

AnB −B limn→+∞

An|| = || limn→+∞

AnB − limn→+∞

BAn|| = 0

= limn→+∞

||AnB −BAn|| = limn→+∞

0 = 0 .

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(b) Actually strong closure arises from weak closure, but we give an explicit independent proof asan exercise. An → A strongly means that Anx→ Ax for every x ∈ H. Assuming AnB−BAn = 0where A ∈ B(H), An ∈M′ and B ∈M, we have that A ∈M′ since, for every x ∈ H,

ABx−BAx = limn→+∞

An(Bx)−B limn→+∞

Anx = limn→+∞

(AnBx−BAnx) = limn→+∞

0 = 0 .

The case of the weak operator topology is treated similarly. An → A weakly means that〈y|Anx〉 → 〈y|Ax〉 for every x, y ∈ H. Assuming AnB − BAn = 0 where A ∈ B(H), An ∈ M′

and B ∈ M, we have 〈y|ABx〉 − 〈y|BAx〉 = limn→+∞〈y|An(Bx)〉 − limn→+∞〈B∗y|Anx〉 =limn→+∞〈y|(AnB −BAn)x〉 = limn→+∞ 0 = 0 , so that 〈y|(AB −BA)x〉 = 0 for every x, y ∈ Hwhich implies A ∈M′.(c) If N = M′, (2) Remark 4.1 implies M′ ⊂ ((M′)′)′. On the other hand M ⊂ (M′)′ impliesvia (1) Remark 4.1 that ((M′)′)′ ⊂M′. Summing up, M′ = ((M′)′)′.

In the sequel we shall adopt the standard convention used for von Neumann algebras and writeM′′ in place of (M′)′ etc. The next crucial classical result is due to von Neumann. It remarkablyconnects algebraic properties to topological ones.

Theorem 4.1. [von Neumann’s double commutant theorem]If H is a Hilbert space and A a unital ∗-subalgebra in B(H), the statements are equivalent.

(a) A = A′′.

(b) A is weakly closed.

(c) A is strongly closed.

Proof. (a) implies (b) because A = (A′)′ and (c) Proposition 4.1 is valid, moreover (b) implies(c) immediately since the strong operator topology is stronger than the weak operator topology.To conclude, we prove that (c) implies (a). Since A′′ = (A′)′ is strongly closed ((c) Proposition4.1), the thesis is true if we establish that A is dense in A′′ in the strong operator topology. Inother words, following the definitions presented in (b) Sect.2.7, assume that Y ∈ A′′ and theset xii∈I ⊂ H, with I finite, are fixed. Then, for every choice of εi > 0, i ∈ I, there must beX ∈ A with ||(X − Y )xi|| < εi for i ∈ I. To prove this assert, first consider the case I = 1and define x := x1. Let us focus on the closed subspace K := Xx |X ∈ A, noticing that x ∈ Kbecause I ∈ A by hypothesis, and let P ∈ L (H) be the ortogonal projector onto K. EvidentlyZ(K) ⊂ K if Z ∈ A, since products of elements in A are in A (it is an algebra) and elements ofA are continuous. Z(K) ⊂ K is the same as ZP = PZP , for every Z ∈ A. Taking the adjoint,we also have PZ = PZP for every Z ∈ A (since A is ∗-closed by hypothesis) and, comparingthe identities, we conclude that PZ = ZP for Z ∈ A. We have found that P ∈ A′ = (A′′)′ andin particular PY = Y P since Y ∈ A′′. In turn, this identity proves that Y (K) ⊂ K so that,in particular, Y x ∈ K. In other words, Y x belongs to the closure of Xx | X ∈ A. Hence||Xx− Y x|| < ε if X ∈ A is suitably chosen.The result generalizes to I ⊃ 1 finite, by defining the direct sum HI :=

⊕i∈I H equipped with

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the scalar product 〈⊕i∈Ixi| ⊕i∈I yi〉I :=∑i∈I〈xi|yi〉, making HI a Hilbert space. The set of

operators AI := XI |X ∈ A(H) ⊂ B(HI), where

XI(⊕i∈Ixi) := ⊕i∈IXxi ∀ ⊕i∈I xi ∈⊕i∈I

H , (4.2)

is a unital sub ∗-algebra of B(HI). Now, for Y ∈ A′′, define YI ∈ B(HI) according to (4.2)finding YI ∈ A′′I . With a trivial extension of the reasoning above, we prove that if ε > 0, there isXI ∈ AI with ||XI⊕i∈I xi−YI⊕i∈I xi||I < ε. Therefore ||(X−Y )xi||2 ≤

∑j∈I ||(X−Y )xj ||2 ≤ ε2

for every i ∈ I. Taking ε = minεii∈I , the thesis is proved.

At this juncture we are ready to define von Neumann algebras.

Definition 4.2. Let H be a Hilbert space. A von Neumann algebra A in B(H) is a∗-subalgebra of B(H) with unit, that satisfies any of the equivalent properties appearing in vonNeumann’s theorem 4.1. The center of A is the set A ∩ A′.

Von Neumann algebras as defined above are also known as (concrete) W ∗-algebras.

Remark 4.2.(a) The statement of Thereom 4.1 is valid also replacing the strong topology with the ultra-

strong topology and the weak topology with the ultraweak topolgy and another pair of operatortopologies (see, e.g., [BrRo02].)

(b) If M is a ∗-closed subset of B(H), since (M′)′′ = M′ ((c) Proposition 4.1), then M′ is avon Neumann algebra. In turn, M′′ = (M′)′ is also a von Neumann algebra. As an elementaryconsequence, the centre of a von Neumann algebra is a commutative von Neumann algebra.

(c) A von Neumann algebra R in B(H) is a special case of C∗-algebra with unit, or better,a C∗-subalgebra with unit of B(H). This immediately arises from (a) Proposition 4.1 becauseR = (R′)′.

(d) The intersection of a family (with arbitrary cardinality) of von Neumann algebrasRjj∈J over a Hilbert space H is a von Neumann algebra over H. (In fact, it easy to seethat

⋂j∈J Rj is a unital sub ∗-algebra of B(H). Furthermore, if

⋂j∈J Rj 3 An → A ∈ B(H)

strongly, then Rj 3 An → A strongly for every fixed j ∈ J , so that A ∈ Rj since Rj is a vonNeumann algebra. Hence A ∈ ⋂j∈J Rj . This proves that

⋂j∈J Rj is strongly closed and so it is

a von Neumann algebra.)

If M ⊂ B(H) is ∗-closed, the smallest (set-theoretically) von Neumann algebra containing M asa subset – the intersection of all von Neumann algebras including M – has a precise form. In-deed, if U is a von Neumann algebra and M ⊂ U, taking the commutant twice, we have U′ ⊂M′

and M′′ ⊂ U′′ = U, so that M′′ ⊂ U for every von Neumann algebra U ⊃M. As a consequenceM′′ coincides with the intersection of all von Neumann algebras including M. All that leads tothe following definition.

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Definition 4.3. Let H be a Hilbert space and consider a ∗-closed set M ⊂ B(H). The doublecommutant M′′ is called the von Neumann algebra generated by M.

A topological characterization of the generated von Neumann algebra appears in Exercise 4.1when M is a unital sub ∗-algebra of B(H).

Definition 4.4. A pair of unital ∗-algebras R1 ⊂ B(H1) and R2 ⊂ B(H2) over the respectiveHilbert spaces H1 and H2 are

(a) isomorphic (also said quasi equivalent) if there exists a unital ∗-algebra isomorphismφ : R1 → R2;

(b) completely isomorphic if the unital ∗-algebra isomorphism φ in (a) is also a homeomor-phism with respect to weak and strong topology;

(c) spatially isomorphic if there is an isometric surjective linear map V : H1 → H2 suchand R1 3 A 7→ V AV −1 ∈ R2 is surjective hence defining a complete isomorphism.

Actually, dealing with von Neumann algebras the first two types of isomorphism coincide inview of the following result [BrRo02] proving an even stronger property.

Proposition 4.2. A unital ∗-algebra isomorphism between two von Neumann algebras isa norm-preserving complete isomorphism. In particular isomorphic von Neumann algebras arealso isometrically ∗-isomorphic as unital C∗-algebras.

4.1.2 Unbounded selfadjoint operators affiliated to a von Neumann algebra

Dealing with unbounded selfadjoint operators is quite standard in Quantum Theory, the defini-tion of commutant and generated von Neumann algebra should be therefore extended to the caseof a set of generally unbounded selfadjoint operators (a further extension may concern closedoperators, see, e.g., [Mor18]).

Definition 4.5. If N is a set of (generally unbounded) selfadjoint operators in the Hilbertspace H, the following definitions are valid.

(a) The commutant N′ of N, is defined as the commutant in the sense of Definition 4.1 ofthe set of all the spectral measures P (A) of every A ∈ N.

(b) The von Neumann algebra N′′ generated by N is defined as (N′)′, where the externalprime is that in Definition 4.1.

If M is a von Neumann algebra over H, a selfadjoint operator A : D(A)→ H with D(A) ⊂ H issaid to be affiliated to H if its PVM P (A) belongs to M.

Remark 4.3.

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(a) When N ⊂ B(H) the commutant N′ computed as in (a) coincides with the standardcommutant as in Definition 4.1 as a consequence of of (ii) and (iv) of Proposition 2.37.

(b) If A∗ = A ∈ N, then A is automatically affiliated to N because P (A) commutes with allselfadjoint operators in B(H) commuting with A (due to Proposition 2.37) and, in particular,with every operator in B(H) commuting with A, because these operators are linear combinationsof similar selfadjoint operators. Therefore P (A) ⊂ (N′)′ = N. In this sense “to be affiliated” isa weaker form of “to belong to”.

Let us discuss how unbounded selfadjoint operators affiliated to a von Neumann algebra are limitpoints of the algebra in the strong operator topology over the domain of the operator. We havethe following elementary result.

Proposition 4.3. If A : D(A) → H is a selfadjoint operator over the Hilbert space H and Ais affiliated to the von Neumann algebra R, then A is the strong limit over D(A) of a sequenceof selfadjoint operators in R. Furthermore A ∈ R if D(A) = H.

Proof. Let us start by observing that, if A is an unbounded selfadjoint operator, for everyx ∈ D(A) we have

Ax = limn→+∞

∫[−n,n]∩σ(A)

λdP (A)(λ)x n ∈ N

as a consequence of (d) Proposition 2.6 and dominated convergence theorem. In other words, Ais the strong limit over D(A) of the sequence of operators An ∈ B(H) defined by

An :=

∫[−n,n]∩σ(A)

λdP (A)(λ) .

These operators are in B(H) according to Proposition 2.25, since the map ı : R 3 λ → λ ∈ Ris bounded over [−n, n], so that ||An|| ≤ ||ı [−n,n] ||∞. Moreover, if A is affiliated to a vonNeumann algebra R, then An ∈ R as we go to prove. First notice that An is the stronglimit, over the whole H, of integrals of simple functions sn → ı point wise over [−n, n] suchthat |sn| ≤ |ı|, using again (d) Proposition 2.6 and dominated convergence theorem. Here, the

integrals∫

[−n,n] sndP(A) are linear combinations of projectors P

(A)E ∈ R by hypothesis, so that∫

[−n,n] sndP(A) ∈ R. Hence An ∈ R as said, it being the strong limit of elements of R which is

strongly closed. If D(A) = H so that A ∈ B(H) (Theorem 2.4) is the strong limit of elements ofR everywhere in H, then A ∈ R since R is strongly closed.

Exercise 4.1.(1) If H is a Hilbert space, let A ⊂ B(H) be a unital ∗-algebra. Prove that the von Neumannalgebra generated by A satisfies

A′′ = Astrong

= Aweak

,

with the obvious meaning of the symbols regarding topological closures.

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Solution. Evidently Astrong ⊂ A

weak. Next observe that, as A′′ is a von Neumann algebra,

it is weakly closed due to Theorem 4.1. Since it includes A, we have A ⊂ Astrong ⊂ A

weak ⊂ A′′.

It is enough proving that A′′ ⊂ Astrong

to conclude. This fact has been established in the proof

of Theorem 4.1 when we proved that A is dense in A′′ in the strong topology: Astrong ⊃ A′′. 2.

(2) If M is a von Neumann algebra in the Hilbert space H and A : D(A) → H is a selfadjointoperator with D(A) ⊂ H, prove that the following facts are equivalent.

(a) A is affiliated to M.

(b) UA ⊂ AU for every unitary operator U ∈M′.

(c) UAU−1 = A for every unitary operator U ∈M′.

Solution. Assume (a) is valid and consider a sequence of simple functions sn → ı point-wise such that |sn| ≤ |ı|. With these hypotheses, If x ∈ D(A), then

∫R sndP

(A)x → Ax (using(d) Proposition 2.6, dominated convergence theorem and Theorem 2.8). On the other hand,

since UP(A)E = P

(A)E U (because U ∈M′ whereas P

(A)E ∈M), (b) immediately follows, observing

in particular that µ(P (A))xx (E) = ||P (A)

E x||2 = ||UP (A)E x||2 = ||P (A)

E Ux||2 = µ(P (A))Ux,Ux(E) since U

is unitary, so that U(D(A)) = U(∆ı) ⊂ ∆ı = D(A). Next suppose that (b) is valid so thatUA ⊂ AU for every unitary operator U ∈M. As a consequence, UAU−1 ⊂ A for every unitaryoperator U ∈ M. Since U−1 = U∗ ∈ M if U ∈ M, we also have U−1AU ⊂ A that impliesA ⊂ UAU−1. Putting all together UAU−1 ⊂ A ⊂ UAU−1, hence (c) is valid. To conclude it issufficient proving that (c) implies (a). From Proposition 2.30 we have that, if (c) holds true, P (A)

commutes with all unitary operators in M′. As a consequence of Lemma 3.2, B ∈ M′ can bedecomposed into a linear combination of unitary operators U . These operators U are obtainedas spectral functions of the selfadjoint operators B +B∗ ∈M′ and i(B −B∗) ∈M′. So the Uscan be constructed as strong limits of linear combinations of elements in the PVMs of B + B∗

and i(B − B∗). These PVM belong to M′ as we prove at the end of this proof. Since M′ is avon Neumann algebra and thus strongly closed, we conclude that U ∈ M′. Summing up P (A)

commutes with every element of M′ since every element of M′ is a linear combination of unitaryelements in M′ and P (A) commutes with these operators. We have found that P (A) ⊂M′′ = Mas wanted. We have only to demonstrate that, if B∗ = B ∈M′, then P (B) ⊂M′ as well. FromProposition 2.37, we can assert that P (B) commutes with all operators in B(H) commuting withB. In other words, P (B) ⊂ (M′)′′ = M′ as wanted. 2

(3) Let A,B ⊂ B(H) be ∗-closed, define A ∨B := (A ∪B)′′ and A ∧B := A ∩B. Prove that

(a) (A ∨B)′ = A′ ∧B′,

(b) (A ∧B)′ ⊃ A′ ∨B′,

(c) (A ∧B)′ = A′ ∨B′ if A,B are more strongly von Neumann algebras.

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Solution. Direct inspection and M′′′ = M′ prove (a) and (b). (c) follows from (a) replacingA for A′, B for B′ and using A = A′′, B = B′′, (A′ ∨B′)′′ = A′ ∨B′. 2

4.1.3 Lattices of orthogonal projectors of von Neumann algebras and factors

To conclude this elementary mathematical survey about basic properties of von Neumann alge-bras, we should say some words about the lattices of orthogonal projectors associated to them,since they play a pivotal role in the physical formalisation. The related notion of factor will bealso introduced.Consider a von Neumann algebra R on the Hilbert space H and focus attention on the intersec-tion R ∩L (H), with ∨, ∧ and ¬ being those of the lattice L (H).

(1) We see from (3.5) that, if P,Q ∈ R ∩L (H) then P ∧ Q ∈ L (H) must also belong to Rsince R is strongly closed (it being a von Neumann algebra) and (3.5) just says that P ∧Qis the strong limit of the sequence of elements (PQ)n which, in turn, belong to R since itis closed with respect to the product. Also notice that infL (H)P,Q =: P ∧ Q ∈ R, sothat infR∩L (H)P,Q exists and satisfies infR∩L (H)P,Q = infL (H)P,Q = P ∧Q.

(2) It is similarly possible to prove that P ∨ Q ∈ R ∩L (H) if P,Q ∈ R ∩L (H), concludingas before that supR∩L (H)P,Q = supL (H)P,Q = P ∨ Q. To this end we make use of(3.5) and Proposition 3.1, obtaining

P ∨Q = ¬((¬P ) ∧ (¬Q)) = I −Å

s- limn→+∞

[(I − P )(I −Q)]nã.

Since evidently 0, I ∈ L (H) ∩ R and ¬P := I − P ∈ L (H) ∩ R for P ∈ L (H) ∩ R, weconclude that R ∩L (H) is closed with respect to the supremum of pair of elements P,Qand this supremum coincides with P ∨Q as wanted.

(3) As a by-product of the results above we also have that (L (H) ∩R,≥, 0, I,¬) is a latticewhich is bounded and orthocomplemented and these structures are those induced by L (H).

(4) L (H) ∩R is σ-complete because this notion involves only the strong topology for (iv) inProposition 3.2 and R is closed with respect to that topology in view of Theorem 4.1 (itis however possible to prove that more strongly L (H) ∩R is complete [Red98, Mor18]).

(5) L (H)∩R is orthomodular and separable, assuming for the latter that H is separable, theproofs being trivial since these properties immediately arise form the analogs of L (H).

(6) Finer properties like irreducibility, atomicity, atomisticity and covering property are notguaranteed and should be studied case by case.

The above listed universally valid properties (1)-(5) permit however to restate most of the quan-tum intepretations we developed in the previous chapters, thinking of the elements of L (H)∩Ras elementary observables of a quantum system as we will do later.

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From the mathematical side, it is interesting to remark that L (H)∩R contains all informationabout R itself, since the following result holds.

Proposition 4.4. Let R be a von Neumann algebra on the Hilbert space H and define thelattice LR(H) := R ∩L (H). It holds LR(H)′′ = R .

Proof. Since LR(H) ⊂ R, we have LR(H)′ ⊃ R′ and LR(H)′′ ⊂ R′′ = R. Let us prove theother inclusion. A ∈ R can always be decomposed into a linear combination of two selfadjointoperators of R, A+A∗ and i(A−A∗). So, since R is a complex linear space in particular, we canrestrict ourselves to the case of A∗ = A ∈ R, proving that A ∈ LR(H)′′ if A ∈ R. The PVM ofA belongs to R because of (ii) and (iv) of Proposition 2.37: P (A) commutes with every boundedselfadjoint operator B which commutes with A. With the same argument as above, decomposinga generic element of B(H) into a linear combination of selfadjoint operators, we have that P (A)

commutes with every B ∈ B(H) commuting with A. So P (A) commutes, in particular, with the

elements of R′ because R 3 A. We conclude that every P(A)E ∈ R′′ = R, namely P (A) ⊂ LR(H)

if A ∈ R. Finally, as we know, there is a sequence of simple functions sn uniformly convergingto ı in a compact [−a, a] ⊃ σ(A). By construction

∫σ(A) sndP

(A) ∈ LR(H)′′ because it is a linear

combination of elements of P (A) and LR(H)′′ is a linear space. Finally∫σ(A) sndP

(A) → A for

n → +∞ uniformly, and thus strongly, as seen in (2) of example 2.10. Since LR(H)′′ is closedwith respect to the strong topology, we must have A ∈ LR(H)′′, proving that LR(H) ⊃ R aswanted.

A natural question is whether R is ∗-isomorphic to B(H1) for a suitable Hilbert space H1 (ingeneral different from the original H!), which for instance, would automatically imply that alsothe remaining properties of L (H1) were true for LR(H). In particular there would exist atomicelements in LR(H), the covering property would be satisfied and the irreducibility propertywould be valid. A necessary (but by no means sufficient) condition is that, exactly as it happensfor B(H1), there are no non-trivial elements in R∩R′, since B(H1)∩B(H1)′ = B(H1)′ = cIc∈C.

Definition 4.6. A factor is a von Neumann algebra R with trivial center:

R ∩R′ = CI

where CI := cIc∈C from now on.

The centre, the commutant and the notion of factor enter both the mathematical and the phys-ical theory in several crucial spots. First of all, they are connected to the irreducibility propertyof the lattice of the studied von Neumann algebra.

Proposition 4.5. A von Neumann algebra R over the Hilbert space H is a factor if andonly if the associated lattice LR(H) is irreducible.

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Proof. First observe that if P ∈ LR(H) commutes with every Q ∈ LR(H), then it commutesalso with the selfadjoint operators constructed out of the PVMs in R – as they are strong limitsof linear combinations of these PVMs as established in the proof of Proposition 4.4 – and moregenerally with every operator in R when decomposing them as a complex linear combinationof selfadjoint operators. So if P ∈ LR(H) commutes with every Q ∈ LR(H), it belongs to thecentre of R. If R is a factor, the only orthogonal projectors in R ∩ R′ are 0 and I (the proofis obvious) so that LR(H) is irreducible. Suppose conversely that R is not a factor, so thereis A 6= cI in R ∩ R′. Thus, at least one of A + A∗ or i(A − A∗) must be different form cIfor c ∈ C. In other words there is a non-trivial selfadjoint operator S ∈ R commuting withall operators in R. As we know from the proof of Proposition 4.4, its PVM belongs to LR(H)and commutes with all operators commuting with S, in particular its PVM commutes with allelements of LR(H). The PVM of S cannot contain only 0 and I, otherwise S would be of theform cI. Hence LR(H) includes a non-trivial projector commuting with all projectors in LR(H):it cannot be irreducible by definition.

It is possible to prove that, on separable Hilbert spaces, a von Neumann algebra is always a directsum or a direct integral of factors. Therefore factors play an important role. The classification offactors, started by von Neumann and Murray and based on the properties of elements in LR(H),is one of the key chapters in the theory of operator algebras, and has enormous consequencesin the algebraic theory of quantum fields. Type-I factors are defined by requiring that theyadmit minimal projectors (atoms). It turns out that, as unital ∗-algebra, a type-I factor R isisomorphic to B(H1) for some Hilbert space H1

1. Consequently they are atomic, atomistic anffulfil the covering property. Regarding separability, it depends on separability of H1 and it leadsto finer classification of factors of type In where n is a cardinal number: the dimension of H1.There are however factors of type II and III which do not admit atoms and are not importantin elementary QM. In particular, a type-III factor R is by definition a factor such that, ifP ∈ LR(H)\0, then P = V V ∗ for some V ∈ R with V ∗V = I. A finer analysis of type III hasbeen produced by Connes using Tomita-Takesaki modular theory (see [KaRi97, BrRo02, Tak10]and also [HaMu06] for a recent review). Type-III factors play a crucial role in the descriptionof extended (quantum) thermodynamic systems and also in algebraic relativistic quantum fieldtheory [Yng05]. Under standard hypotheses, every von Neumann algebra of observables localizedin a sufficiently regular open bounded region of Minkowski spacetime is isomorphic to the uniquehyperfinite factor of type III1. Moreover, as a consequence of the so called split property (validin particular for the free theory), every one of these factors is contained in a type-I factor that,in turn is included in another local algebra associated to a slightly larger spacetime region.Von Neumann algebras are analogously classified into different types and the classification is suchthat, in separable Hilbert spaces, a von Neumann algebra of a certain type is the direct sum orthe direct integral of factors of the same type. Generic von Neumann algebras can be uniquelydecomposed into direct sums of definite-type von Neumann algebras also if the Hilbert spaceis not separable. See [Mor18] for a brief account, [Red98] for a more extended discussion with

1This is equivalent to saying that H is isomorphic to H1 ⊗H2 and the Hilbert space isomorphism identifiesthetype-I factor R with B(H1)⊗ I2.

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several technical details and historical remarks, and [KaRi97, BrRo02] for complete treatises onthe subject. Several physical implications are discussed in [Haa96], [Ara09] especially for QFTand in [BrRo02] concerning statistical mechanics.

4.1.4 Schur’s Lemma

As an important mathematical tool we introduce an elementary yet crucial technical result. Itwill be discussed after the following general definition. The ∗-closed set M below could be a vonNeumann algebra but also, for instance, it may consists of the set Ugg∈G of the images of aunitary representation of a group G 3 g 7→ Ug (Definition 5.4). Yet, one can also include the setof unitary representatives of a unitary projective representation (Definition 5.5) of a group as weshall discuss later (re-arranging the phases in order to produce a ∗-closed set to apply Theorem4.2). Finally, M may be the image of a ∗-representation of a ∗-algebra. So the definition andstatement below encompasses quite general cases.

Definition 4.7. Let H 6= 0 be a Hilbert space and M ⊂ B(H) a set of operators.

(a) A closed subspace H0 ⊂ H is said to be invariant for M, if A(H0) ⊂ H0 for every A ∈M.

(b) M is said to be topologically irreducible, if the only invariant closed subspaces for Mare H0 = 0 and H0 = H.

Remark 4.4. “Topologically” above refers to closedness of the invariant subspaces and wehenceforth omit this specification for the sake of shortness: irreducible means topologically irre-ducible from now on.

Let us state and prove the most elementary classical version of Schur’s lemma in (complex)Hilbert spaces, using the language of von Neumann algebras.

Theorem 4.2. [Schur’s lemma]Consider a set M ⊂ B(H) over the Hilbert space H 6= 0 and assume that M is ∗-closed.The following facts are equivalent.

(a) M is irreducible.

(b) M′ = CI.

(c) M′′ = B(H).

Proof. We assume that (a) is valid and we prove (b). If A ∈M′ (so that A∗ ∈M′ as well), wecan decompose it into A = B + iB′ where B := 1

2(A + A∗) ∈ M′, B′ := 12i(A − A

∗) ∈ M′ areselfadjoint. The spectral measures of B and B′ commute with all operators commuting withB and B′ respectively, for Proposition 2.37. In turn, these PVMs commute with all operatorscommuting with A and A∗, so that the PVMs belong to M′ as well. Let P be an orthogonalprojector of P (B) or P (B′). Since PC = CP for every C ∈M, the closed subspace H0 := P (H)

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satisfies C(H0) ⊂ H0 and thus, holding (a), either H0 = 0, namely P = 0, or H0 = H, namelyP = I. Integrating the said PVMs, whose projectors are either 0 or I, we have B = bI andB′ = b′I for some b, b′ ∈ R so that A = cI for some c ∈ C. We have established (b). We nextprove that (b) implies (c). If (b) is true, M′′ = CI ′ = B(H), so that (c) is true. To conclude,we prove that (c) implies (a). If H0 is a closed subspace invariant under every operator in M,the orthogonal projector P onto H0 commutes with every A ∈ M. Indeed A(H0) ⊂ H0 impliesAP = PAP . Taking the adjoint, PA∗ = PA∗P . Since M is ∗-closed and A = (A∗)∗, we canre-write this identity like this PA = PAP . Comparing with AP = PAP , we have AP = PA.We therefore have P ∈M′ = M′′′ which means P ∈ B(H)′ when assuming (c). In particular, Pmust commute with every Q ∈ L (H). Since L (H) is irreducible (Theorem 3.1), either P = 0,namely H0 = 0, or P = I, namely H0 = H. Hence (a) is valid and the proof ends.

Corollary 4.1. Let π : G→ B(H) (π : A→ B(H)) be a unitary representation of the groupG (a ∗-representation of the ∗-algebra A) over the Hilbert space H 6= 0. If G (resp. A) isAbelian, the image of π is irreducible if and only if dim(H) = 1.

Proof. Observe that M := π(G), respectively M := π(A), is ∗-closed and every π(A) is a complexnumber due to Schur’s lemma since it commutes with M. If ψ ∈ H with ||ψ|| = 1, the closure ofthe set of finite linear combinations of all elements π(a)ψ is a closed M-invariant subspace andthus it must coincide with H if the image of π is irreducible. In other words ψ is a Hilbertbasis of H, so that dim(H) = 1. The converse implication is obvious.

4.1.5 Von Neumann algebra associated to a PVM

The last mathematical feature of von Neumann algebras we discuss concerns the interplay withPVMs. We only mention the following important technical result.

Proposition 4.6. Let P : Σ(X) → L (H) be a PVM over the measurable space (X,Σ(X))taking values on the lattice of orthogonal projectors over the Hilbert space H. If H is separable,then

PE | E ∈ Σ(X)′′ =ß∫

XfdP

∣∣∣∣ f ∈Mb(X)

™.

If H is not separable, the above statement is however valid if replacing = for ⊃.

Proof. See the Appendix 4.4.

4.2 Von Neumann algebras of observables

Let us pass to physics and we apply the presented notions and results to the formulation ofquantum physics in Hilbert spaces.

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4.2.1 The von Neumann algebra of a quantum system

Relaxing the hypothesis that all selfadjoint operators in the Hilbert space H associated to aquantum system represent observables, there are many reasons to assume that the observablesof a quantum system are represented (in the sense we are going to illustrate) by the selfadjointelements of an algebra of von Neumann, we hereafter indicated by R, called the von Neumannalgebra of observables (though only the selfadjoint elements are observables). In a sense,according to Proposition 4.4, R is the maximal set of operators we can construct out of thelattice of elementary propositions viewed as orthogonal projectors (lattice not coinciding withthe whole L (H)), using algebra operations, the adjoint operation, and the strong operatortopology (the most relevant one when dealing with spectral theory) necessary for physicallymotivating the relation PVM (elementary observables) - self adjoint operators (observables).Some physical comments are important.

(1) Including non-selfadjoint elements B ∈ R is harmless, as they can be one-to-one decom-posed into a pair of selfadjoint elements

B = B1 + iB2 =1

2(B +B∗) + i

1

2i(B −B∗) .

So these elements are nothing but complex linear combinations of bouded observables.

(2) Requiring that all the elements of R are bounded, so that unbounded observabels are ruledout, does not seem a physical problem. If A = A∗ is unbounded, the associated class ofbounded selfadjoint operators Ann∈N where

An :=

∫[−n,n]∩σ(A)

λdP (A)(λ) ,

embodies the same information as A itself. An is bounded due to Proposition 2.29 becausethe support of its spectral measures is included in [−n, n]. Physically speaking, we can saythat An is nothing but the observable A when it is measured with an instrument unableto produce outcomes larger than [−n, n]. All real measurement instruments are similarlylimited. We can safely assume that every An belongs to R. Mathematically speaking, thewhole (unbounded) observable A is recovered as the limit in the strong operator topologyover D(A):

Ax = limn→+∞

Anx if x ∈ D(A),

as established in Proposition 4.3. Finally the spectral measure of A belongs to R (A isaffiliated to R) as consequence of (2) Exercise 4.1 and the limit above.

(3) In a sense, a more precise physical picture woluld be obtained by restricting ourselves tothe only real vector space of bounded selfadjoint operators of R equipped with a naturalproduct (where A and B are bounded selfadjoint operators) called Jordan product

A B =1

2(AB +BA) .

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The arising mathematical structure, disregarding topological features, is called Jordanalgebra. Though physically appealing, it presents some mathematical drawbacks whichmake it a more cumbersome tool than ∗-algebras. In particular, the afore-mentionedproduct is not associative. In [Emc72], Jordan algebras are intensively used to describingphysical systems (see [Mor18] for some further comments).

We stress again that, within the framework of von Neumann algebras of observables, the or-thogonal projectors P ∈ R represent all elementary observables of the system. The lattice ofthese projectors, LR(H), encompass the amount of information about observables as establishedProposition 4.4. As said above LR(H) ⊂ R is bounded, orthocomplemented, σ-complete, or-thomodular and separable like the whole L (H) (assuming that H is separable) but there is noguarantee for the validity of the other properties listed in Theorem 3.1.

4.2.2 Maximal set of compatible observables and preparation of vector states

A technically important result concerning both the spectral theory and the notion of von Neu-mann algebra is the following one.

Proposition 4.7. Let A = A1, . . . , An be a finite collection of selfadjoint operators in theseparable Hilbert space H whose spectral measures commute pairwise. The von Neumann algebraA′′ generated by A satisfies

A′′ = f(A1, . . . , An) | f ∈Mb(Rn) with f(A1, . . . , An) :=∫Rn f(x1, . . . , xn)dP (A),

where P (A) is the joint spectral measure (Theorem 2.9) of the set of operators A = A1, . . . , An.

Proof. The thesis immediately arises from Proposition 4.6 assuming taking P = P (A) therein.Observe that if the Ak belong to B(H), then the von Neumann algebra they generate is thesame as that generated by their spectral measures (see (a) Remark 4.3).

The afore-mentioned result authorizes us to introduce a relevant notion. A common situationdealing with quantum systems is the existence of a maximal set of compatible observables.

Definition 4.8. Let R be a von Neumann algebra of observables in the Hilbert space H. Afinite set A = A1, . . . , An of pairwise compatible observables – that is, generally unboundedselfadjoint operators affiliated to R whose PVM mutually commute – is said to be a maximalset of compatible (or commuting) observables if every selfadjoint operator B ∈ B(H)commuting with all the PVMs of the operators in A is a function of them according to Theorem2.9:

B = f(A1, . . . , An) :=

∫Rnf(x1, . . . , xn)dP (A) ,

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for some (real-valued) function f ∈Mb(Rn).

Remark 4.5. A maximal set of compatible observables A satisfies A′ ⊂ A′′ due to Proposition4.7 whereas the converse inclusion A′′ ⊂ A′ is automatic since the PVM P (A) commutes withevery single PVM P (Ak) as it is part of P (A) itself (e.g., P

(A1)E = P

(A)E×R×···×R). So A′ = A′′. In

particular, a bounded selfadjoint operator B commuting with the PVMs of A must belong toA′ = A′′ ⊂ R′′ = R and therefore B is an observable as well.

An important physical consequence of the introduced notion is the following one. Suppose thatthe observables Ak, k = 1, . . . , n forming a maximal set of compatible observables have purepoint spectrum (Definition 2.14). In this case, it easy to check that the spectral measure overRn defined by

PE := s-∑

(a1,...,an)∈E∩×nk=1

σp(Ak)

P(A1)a1 · · ·P

(An)an , E ∈ B(Rn) (4.3)

satisfies the condition in Theorem 2.9 which defines the joint measure of A = A1, . . . , An andtherefore is that joint measure.Let Hα1,...,αn be a common eigenspace associated to eigenvalues αk ∈ σ(Ak). We argue thatdim(Hα1,...,αn) = 1.Indeed, if Hα1,...,αn included a pair of non-vanishing orthogonal vectors x1, x2, the orthogonal

projector P := 〈x1| 〉x1 would commute with every P (Ak) because PP(Ak)αk = P

(Ak)αkP = P and

PP(Ak)ak = 0 for ak 6= αk. However the selfadjoint operator P ∈ B(H) can not be a function of

A1, . . . , An as it should due according to Definition 4.8. This is because, from (4.3), the functionsof A1, . . . , An have the form

f(A1, . . . , An) = s-∑

a1∈σ1(A1),...,an∈σ1(An)

f(a1, . . . , an)P(A1)a1 · · ·P

(An)an ,

therefore f(A1, . . . , An)x = f(α1, . . . , αn)x for every x ∈ Hα1,...,αn = P(A1)α1 · · ·P

(An)αn(H) so that,

in particular, f(A1, . . . , An)x1 = f(A1, . . . , An)x2. Conversely Px1 = x1 and Px2 = 0, in spiteof xj ∈ Hα1,...,αn . We conclude that every common eigenspace Hα1,...,αn must be one dimesional.The above discussion has an important practical consequence for “preparing quantum states”as they can be prepared just measuring A1, . . . , An. After a simultaneous measurement ofA1, . . . , An, the post-measurement state is necessarily represented by a unique unit vector (upto phasis) included in the one-dimensional space Hα1,...,αn where α1, . . . , αn are the outcomes ofthe measurements. In fact, if T ∈ S (H) is the unknown initial state, according to Luders-vonNeumann postulate, after having measured α1 for A1, α2 for A2, etc., the outcoming state isalways

T ′ =P

(A1)α1 · · ·P

(An)αnTP

(A1)α1 · · ·P

(An)αn

tr(P

(A1)α1 · · ·P

(An)αnT

) = 〈ψα1,...,αn | 〉ψα1,...,αn

186

where, up to phases, ψα1,...,αn ∈ Hα1,...,αn is the unique vector with unit norm.

Another physically relevant consequence is asserted in the following proposition and the remarkbelow it.

Proposition 4.8. If a quantum physical system admits a maximal set of compatible observ-ables A, then the commutant R′ of the von Neumann algebra of observables R is Abelian becauseit coincides with the center of R.

Proof. As the spectral measure of each A ∈ A belongs to R, it must be (i) A′′ ⊂ R. SinceA′ = A′′, (i) yields A′ ⊂ R and thus, taking the commutant, (ii) A′′ ⊃ R′. Comparing (i) and(ii) we have R′ ⊂ R. In other words R′ = R′ ∩R. In particular, R′ must be Abelian becauseevery element of R′ must commute with all elements of R′ itself since R′ ⊂ R.

Remark 4.6.(a) Observe that R′ is Abelian if and only if it coincides with the center. One implication

has been proved above, the other is similarly obvious: if R′ is Abelian, then R′ ⊂ R′′ = R, soR′ = R ∩R′ once more.

(b) As soon as R′ is not Abelian, as it happens dealing with the so-called non-Abelian gaugetheories, no maximal sets of compatible observables can exist and it is impossible to preparevector states by measuring a maximal set of compatible observables with pure point spectra,since they simply do not exist.

Example 4.1.(1) In L2(R, dx), the Hamiltonian operator H of the harmonic oscillator alone is a maximalset of commuting observables with pure point spectrum. The proof is easy, according to (3)Example 2.6,

H = s-∑n∈N

~ωÅn+

1

2

ãPn

where we have defined the one-dimensional orthogonal projectors Pn := 〈ψn| 〉ψn. If B∗ = B ∈B(H) commutes with H, according to Proposition 2.37, it commutes with the spectral measureof H. So, since x =

∑n∈N Pnx for every x ∈ H and PnPm = 0 if n 6= m,

Bψ =∑n∈N

PnBψ =∑n∈N

PnPnBψ =∑n∈N

PnBPnψ .

Since Pn projects onto a one-dimensional subspace, the selfadjoint operator PnBPn takes nec-essarily the form bnPn for some bn ∈ R. We have so far obtained that

B = s-∑n∈N

bnPn ,

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which means that B = f(H), defining f : σ(H) → R as f(~ω(n + 1/2)) := bn. f must bebounded otherwise B would be unbounded contraily to our hypothesis, since∣∣∣∣∣∣

∣∣∣∣∣∣s- ∑n∈N bnPn∣∣∣∣∣∣∣∣∣∣∣∣ = sup

n∈N|bn| .

(2) Considering a quantum particle without spin and referring to the rest space R3 of an in-ertial reference frame, H = L2(R3, d3x). A maximal set of compatible observables is the setof the three position operators A1 = X1, X2, X3. The set of the three momentum operatorsA2 = P1, P2, P3 is a maximal family of commuting operators as well since the two sets ofoperators are related by means of the Fourier-Plancherel unitary transform ((2) Examples 2.2).The fact that X1, X2, X3 is a maximal set of compatible observables can be proved as follows.If A ∈ B(H) commutes with the joint spectral measure P (A1) of X1, X2, X3, representing theHilbert space as L2(R3, d3x), it turns out that A(χE) = fE for every bounded E ∈ B(R3), where

fE ∈ L2(R3, d3x) vanishes a.e. outside E. (This is because P(A1)E is the multiplicative operator

with χE , but also χE ∈ P (A1)E (L2(R3, d3x)), so that A(χE) must belong to the same subspace

P(A1)E (L2(R3, d3x)) since A commutes with P

(A1)E . Hence A(χE) is a function fE which vanishes

a.e. outside E.) Using linearity of A, one sees that, if F ∩ E 6= ∅, then fF E∩F= fEE∩F a.e..This way, a unique measurable function f (A) turns out to be defined over the entire R3 using apartition made of bounded Borel sets, such that A(χE) = f (A) ·χE . Finally, using a sequence ofsimple functions suitably converging to ψ ∈ L2(R3, d3x) and taking continuity of A into account,we get Aψ = f (A) ·ψ a.e.. Since A is bounded, it turns out that f (A) is P (A1)-essentially bounded,so that it can be re-arranged to a bounded function redefining it over a zero-measure set. Say-ing that Aψ = f (A)·ψ for every ψ ∈ L2(R3, d3x) is the same as stating that A = f (A)(X1, X2, X3).

(3) Referring to a quantum particle without spin, the full algebra of observables R must includeA1∪A2, where A1 = X1, X2, X3 and A2 = P1, P2, P3 as before. It is possible to prove that thecommutant of (A1∪A2)′′ = (A1∪A2)′ is trivial (A1∪A2)′ = CI (as it includes a unitary irreduciblerepresentation of the Weyl-Heisenberg group) so that R = R′′ ⊃ CI ′′′ = CI ′ = B(L2(R3, d3x),so that R = B(H) necessarily for the spinless non-relativistic particle. As a consequenceLR(H) = L (L2(R3, d3x)).

(4) If adding the spin space (for instance dealing with an electron “without charge”), we haveH = L2(R3, d3x) ⊗ C2. Referring to (1.11) a maximal set of compatible observables is, for in-stance, A1 = X1 ⊗ I,X2 ⊗ I,X3 ⊗ I, I ⊗ Sz, another is A2 = P1 ⊗ I, P2 ⊗ I, P3 ⊗ I, I ⊗ Sx.As before (A1 ∪ A2)′′ is the von Neumann algebra of observables of the system (changing thecomponent of the spin passing from A1 to A2 is crucial for this result). Also in this case, it turnsout that the commutant of the von Neumann algebra of observables is trivial yielding R = B(H).

(5) It is possible to construct maximal set of commuting observables with pure point spectraalso in L2(R3, d3x) ⊗ C2s+1 or in closed subspaces of it. A typical example for an electron

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(s = 1/2) is the quadruple made of the Hamiltonian of the Hydrogen atom H, the total squaredangular momentum L2, the component Lz of the angular momentum, and the component Sz ofthe spin. This quadruple is a maximal set of commuting observables with pure point spectra, ifrestricting to the closed subspace of non-positive energy.

4.3 Superselection rules

We have so far accumulated enough formalism we can successfully investigate the structure ofthe Hilbert space and the algebra of the observables when not all selfadjoint operators rep-resent observables and not all orthogonal projectors are intepreted as elementary observables.Re-adapting the approach by Wightman [Wig95] to our framework, we start by making someassumptions generally describing the so called Abelian discrete superselection rules for QM for-mulated in a separable Hilbert space, where R denotes the von Neumann algebra of observables.Next we pass to consider the case of non-Abelian superselection rules by introducing the notionof Gauge group [JaMi61, Haa96].

4.3.1 Abelian case: superselection charges and coherent sectors

We want to study the case where a finite set of pairwise compatible observables exists whichcommute with all of the observables of the system, so they belong to the centre R ∩R′ of thealgebra of observables. The most known example is perhaps represented by the electric charge.It is known that for all quantum systems carrying electrical charge, this observables commuteswith all remaining observables of the system. It is evident that, assuming this constraint, notevery selfadjoint operator of the Hilbert space can represent an observable: operators which donot commute with the electrical charge are ruled out.The general case is described as follows, where we also consider the case of co-existence ofdifferent observables commuting with R, for example the mass and the electrical charge in non-relativistic systems, and we also assume that the said set of preferred observables is exhaustive.

(a) These special central observables have pure point spectra according to Definition 2.14 (sotheir spectra essentially consist of their point spectra, in the sense that the possible ele-ments of continuous spectra are just limit points of the eigenvectors and the continuouspart of the spectrum has no internal points).

(b) The said observables exhaust the centre R ∩R′, more precisely it is generated by them.

(c) The centre coincides with the commutant R′ = R ∩R′.

The last requirement can be motivated in particular by assuming to stick to the quite frequentphysical situation where a maximal set of commuting observables exist in R, according to Propo-sition 4.8.

Definition 4.9. [Abelian discrete superselection rules]Given a quantum system described in the separable Hilbert space H with von Neumann algebra

189

of observables R, we say that Abelian (discrete) superselection rules occur if the followingrequirements are valid.

(S1) The centre of the algebra of observables coincides with the commutant R′ = R′ ∩R.

(S2) R′ ∩R includes a finite class of observables Q = Q1, . . . , Qn such that

(i) their spectra are pure point spectra,

(ii) they generate the centre: Q′′ = R′ ∩R.

(If some of the Qks are unbounded they are supposed to be affiliated to R′ ∩R.)

The Qk are called superselection charges.

Remark 4.7. A mathematically equivalent but less physically explicative way to state (S1)and (S2) consists of postulating that in the separable Hilbert space H,

(S1)’ R = Q1, . . . , Qn′,

(S2)’ Q1, . . . , Qn are selfadjoint operators with commuting PVMs and pure point spectra.

Indeed, (S1) and (S2) imply (S1)’ and (S2)’. Conversely, from (S1)’ and (S2)’ it arises thatQ1, . . . , Qn ⊂ R. Next (S1)’ implies R′ = Q1, . . . , Qn′′ ⊂ R′′ = R so that R′ ⊂ R and thus(S1) and (S2) are valid.

We have the following remarkable result, where we occasionally adopt the notation q := (q1, . . . , qn)and σ(Q) := ×nk=1σp(Qk).

Proposition 4.9. Let H be a complex separable Hilbert and suppose that the von Neumannalgebra R in H satisfies (S1) and (S2). The following facts hold.

(a) H admits the following Hilbert direct orthogonal decomposition into closed subspaces, calledsuperselection sectors or coherent sectors,

H =⊕

q∈σ(Q)

Hq where Hq := P(Q)q (H), (4.4)

and each Hq is

(i) invariant under R, i.e. A(Hq) ⊂ Hq if A ∈ R;

(ii) irreducible under R, i.e. there is no proper non-trivial subspace of Hq which is invar-ian under R.

(b) A corresponding direct decomposition occurs for R.

R =⊕

q∈σ(Q)

Rq with Rq :=AHq : Hq → Hq

∣∣∣ A ∈ R

(4.5)

190

is a von Neumann algebra in Hq considered as Hilbert space in its own right. Finally,

Rq = B(Hq)

(c) The Rqs enjoy the following properties.

(i) Each mapR 3 A 7→ AHq∈ Rq

is an unfaithful (i.e. non-injective) unital ∗-algebra representation of R (Def.2.5)which is both strongly and weakly continuous.

(ii) Representations associated with different values of q are unitarily inequivalent: thereis no isometric surjective linear map U : Hq → Hq′ such that

UAHq U−1 = AHq′ when q 6= q′.

exercise Consider a ∗-closed set M ⊂ B(H) over the Hilbert space H and suppose that AB =BA if A,B ∈ M (this is the case if M is either the image of a ∗-representation of an Abelian∗-algebra or the image of a group representation of an Abelian group). Prove that if M isirreducible, then dim(H) = 1.

Proof. As the reader can easily prove, since the charges Qk have pure point spectra and thuseach of them admits a Hilbert basis of eigenvectors, the joint spectral measure P (Q) in Rn hassupport given by the closure of ×nk=1σp(Qk) and, if E ⊂ Rn,

P(Q)E = s-

∑(q1,...,qn)∈×n

k=1σp(Qk)∩E

P(Q1)q1 · · ·P

(Qn)qn , (4.6)

where the spectral projector P(Qk)qk , according to Theorem 2.8, is nothing but the orthogonal

projector onto the eigenspace of Qk with eigenvalue qk. Notice that every P(Q)E is an observable

as it belongs to R. In fact, using Proposition 2.37, P(Q)E commutes with all bounded operators

commuting with the PVMs of the Qk which, by definition, belong to R′, so that P(Q)E ∈ (R′)′ =

R. Not only, as the Qk commute with the whole R, we also have P(Q)E ∈ R′. In summary

P(Q)E ∈ R ∩R′.

(a) Since P(Q)q P

(Q)s = 0 if q 6= s and

∑q∈σp(Q) P

(Q)q = I, H decomposes as in (4.4). Since

P(Q)q ∈ R′, the subspaces of the decomposition are invariant under the action of each element of

R because AP(Q)q = P

(Q)q A for every A ∈ R so that A(Hq) = A(P

(Q)q (Hq)) = P

(Q)q (A(Hq)) ⊂ Hq .

Let us pass to irreducibility. We start by observing that, if P ∈ R′∩R is an orthogonal projector,it must be a function of the Qks since Q′′ = R′ ∩R by hypothesis and Proposition 4.7 is valid(H is separable). Therefore

P = s-∑

(q1,...,qn)∈×nk=1

σp(Qk)∩Ef(q1, . . . , qn)P

(Q1)q1 · · ·P

(Qn)qn

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since P = PP ≥ 0 and P = P ∗, exploiting the measurable functional calculus, we easily findthat f(q) = χE(q) for some E ⊂ ×nk=1σp(Qk). In other words P is an element of the joint PVMof Q: that PVM exhausts all orthogonal projectors in R′ ∩ R. Now, if 0 6= K ⊂ Hs is aninvariant closed subspace for R, its orthogonal projector PK must commute with every A ∈ R.In fact, PKAPK = APK and taking the adjoint PKA

∗PK = PKA∗ but, since R is ∗-closed, we

can re-write this identity PKAPK = PKA for every A ∈ R. Comparing the found identities, wehave APK = PKA. Therefore PK ∈ R′ = R ∩R′ and thus PK is an element of the PVM P (Q) asestablished above. Furthermore PK ≤ P

(Q)s because K ⊂ Hs. But there are no projectors smaller

than P(Q)s in the PVM of Q. So PK = P

(Q)s and K = Hs.

(b) Rq :=AHq

∣∣∣ A ∈ R

is a von Neumann algebra on Hq considered as a Hilbert space

in its own right as it arises by direct inspection observing in particular that this is a unital∗-subalgebra of B(Hq) and is strongy closed. (Observe that Aq := P

(Q)q AP

(Q)q ∈ R and that

An|Hqψ → Bψ for all ψ ∈ Hq and some B ∈ B(Hq) is equivalent to saying that Anqφ→ B′φ forevery φ ∈ H, where B′ extends B to the zero operator on H⊥q and therefore defines an elementof B(H). Since R is a von Neumann algebra, B′ ∈ R so that B ∈ Rq.) (4.5) holds by definition.Since Hq is irreducible for R it is evidently irreducible also for Rq by construction. Schur’slemma (Theorem 4.2) implies that R′′q = B(Hq). As R′′q = Rq since we are dealing with a vonNeumann algebra, it holds that Rq = B(Hq).(c) Each map R 3 A 7→ A Hq∈ Rq is a strongly and weakly continuous representation of ∗-

algebras as follows by direct check. This representation cannot be faithful because e.g., P(Q)q ∈ R

is represented by the zero operator in Hq′ if q′ 6= q. Furthermore, if q 6= q′ –for instance q1 6= q′1–there is no isometric surjective linear map U : Hq → Hq′ such that UAHq U

−1 = AHq′ . If such

an operator existed one would have, contrarily to our hypothesis q1 6= q′1, q1IHq′ = UQ1|HqU−1 =

Q1|Hq′ = q′1IHq′ so that q1 = q′1. (If Q1 is unbounded it is enough consider the central bounded

operator Q1n =∫

[−n,n] rdP(Q1)(r) with [−n, n] 3 q1, q2.)

We have found that, in the presence of superselection charges, the Hilbert space decomposes intopairwise orthogonal subspaces which are invariant and irreducible with respect to the algebraof the observables, giving rise to inequivalent representations of the algebra itself. There areseveral superselection structures as the one pointed out in physics. The three most known areof very different nature (see Example 4.2 and Example 5.4):

• the superselection structure of the electric charge,

• the superselection structure of integer/semi integers values of the angular momentum,

• the superselection rule of the mass in non-relativistic physics, i.e. Bargmann’s superselec-tion rule.

These superselection rules take place simultaneously and can be described by correspondingpairwise compatible superselction charges so that the picture presented above is valid. Noticethat, in each superselection sector, the physical description is essentially identical to the naiveone where every selfadjoint operator is an observable (namely R = B(H)) and the superselection

192

charges appear just in terms of fixed parameters.

Example 4.2. The electric charge is the typical example of superselction charge. For instance,referring to an electron, its Hilbert space is L2(R3, d3x) ⊗ Hs ⊗ He. The space of the electriccharge is He = C2 and therein Q = eσz (see (1.12)). Many other observables could exist in He inprinciple, but the electric charge superselection rule imposes that the only possible observablesare commute with Q and are functions of σ3. The centre of the algebra of observables isI ⊗ I ⊗ f(σ3) for every function f : σ(σz) = −1, 1 → C. We have the decomposition intocoherent sectors

H = (L2(R3, d3x)⊗ Hs ⊗ H+)⊕

(L2(R3, d3x)⊗ Hs ⊗ H−) ,

where H± are respectively the eigenspaces of Q with eigenvalue ±e.

Remark 4.8. A fundamental requirement is that the superselection charges have pure pointspectra. If instead R ∩R′ includes an operator A with a continuous part in its spectrum withnon-empty interior (A may also be the strong limit on D(A) of a sequence of elements in R∩R′),the established proposition does not hold. H cannot be decomposed into a direct sum of closedsubspaces. In this case it decomposes into a direct integral finding a much more complicatedstructure whose physical meaning seems dubious.

4.3.2 Global gauge group formulation and non-Abelian case

There are quantum physical systems such that their R′ is not Abelian (think of chromodynamicswhere R′ includes a faithful representation of SU(3)) so that the centre of R does not contain thefull information about R′. A primary notion is here represented the group of unitary operatorscalled the commutant group of R (introduced in [JaMi61] and called gauge group therein),

GR := V ∈ R′ | V is unitary .

It includes all information of R and R′ because (making use of R′′ = R and Lemma 3.2)

G′R = R and G′′R = R′ . (4.7)

In the presence of Abelian superselection rules, GR is Abelian (GR ⊂ R′ = R ∩ R′). Herethe elements of R may be extracted from B(H), expoiting the former in (4.7), but referringto a subgroup of GR constructed out of a set of physically meaningful superselection chargesQ1, . . . , Qn. A ∈ R if and only if A commute with the PVMs of Q1, . . . , Qn and decomposingA = 1

2(A+A∗) + i 12i(A−A

∗) and exploiting Proposition 2.37, this is equivalent to saying

UsA = AUs , Us := eis1Q1 · · · eisnQn for s := (s1, . . . , sn) ∈ Rn. (4.8)

U : Rn 3 s 7→ Us is a strongly-continuous unitary representation of the Abelian topological groupRn taking values in GR. Looking at Remark 4.7, occurrence of Abelian discrete superselectionrules can be condensed in (a) H is separable, (b) Q1, . . . , Qn have pure point spectra, and (c)

R = U(Rn)′ . (4.9)

193

Observe that U(Rn) is considerably smaller than GR, since other choices for the charges Qk andalso for their number are possible, producing other subgroups of GR, but preserving the validityof (4.9): it is sufficient that the joint PVM of these charges is made of the same projectors Pq

onto the sectors determined by the initial charges. We can do better using separability of H: wecan construct the unique charge out of the PVMs of the n charges Qk

Q := s-∑

q∈σ(Q)

mqPq ,

for some injective map q 7→ mq ∈ Z, which must exist because the Pqs are at most countablymany, since H is separable. Now, form Remark 4.7 and Proposition 2.37, the representation Uof Rn in (4.9) can be equivalently replaced by a faithful strongly-continuous representation ofthe Abelian compact topological group U(1),

U : U(1) 3 eis 7→ eisQ ∈ GR . (4.10)

We stress that (4.10) is well posed and is a representation of U(1), and not only of R, just becauseσ(Q) ⊂ Z. The constructed Q has however no direct physical meaning in general, except perhapsin the case n = 1 with Q = e−1Q1, where Q1 is the electric charge and e the elementary electriccharge. Decompositions (4.4)-(4.5) are valid and every R-invariant and R-irreducible closedsubspace Hq is also U -invariant, since UHq is a pure phase (so that U -irreducibility fails unlessdim(Hq) = 1).

In the non-Abelian case, a similar decomposition is expected to hold for a strongly-continuousfaithful representation U : G 3 g 7→ Ug ∈ GR of some group G, called the (compact) globalgauge group, such that U ′ = G′R = R – where we adopted the notation U ′ := U(G)′ which wewill occasionally use henceforth –

H =⊕χ∈K

Hχ , R =⊕χ∈K

Rχ , Ug =⊕χ∈K

U (χ)g . (4.11)

Above, Hχ is a non-trivial closed subspace both R-invariant and U -invariant, determining cor-responding (unfaithful, strongly and weakly continuous) representations

Rχ : R 3 A 7→ AHχ : Hχ → Hχ , U(χ) : G 3 g 7→ UgHχ : Hχ → Hχ with Rχ = (U (χ))′ (4.12)

the commutant being referred to B(Hχ). The fundamental difference with the Abelian case isthat now Rχ is more weakly a factor in B(Hχ) rather than the entire B(Hχ):

Rχ ∩R′χ = CIχ = (U (χ))′ ∩ (U (χ))′′ for every χ ∈ K . (4.13)

If everything stated is valid, the orthogonal projectors Pχ onto every subspace Hχ must commutewith U and R so that they belong to the centre R∩R′ = U ′∩U ′′. Using the Pχs, superselectioncharges can be still constructed such that their joint PVM determines the generalized superse-lection sectors Hχ. We have the following general result where separability is not necessary.

Proposition 4.10. Let R be a von Neumann algebra over the Hilbert space H 6= 0.Suppose that a faithful unitary strongly-continuous representation U : G 3 g 7→ Ug ∈ GR existsof the Hausdorff compact topological group G such that U(G)′ = R. The following facts hold.

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(a) (4.11)-(4.13) are valid where K is a set of equivalence classes of irreducible strongly-continuous unitarily-equivalent representations of G,

(b) Rχ, Rχ′ are unitarily inequivalent if χ 6= χ′.

Proof. (a) Let us start by proving (4.11). If G is Hausdorff and compact, as G 3 g 7→ Ug isstrongly continuous, Peter-Weyl’s Theorem (Theorem 5.8) implies that the orthogonal Hilbertdecomposition holds H =

⊕χ∈K Hχ where each Hχ is non-trivial closed and U -invariant. Here, K

labels some equivalence classes of irreducible strongly-continuous unitarily-equivalent represen-

tations of G. In particular, the finer Hilbert orthogonal decomposition holds Hχ =⊕

λ∈Λχ H(λ)χ

where every closed subspace H(λ)χ is U -invariant, every restriction U (χλ) := U

H(λ)χ

: H(λ)χ → H

(λ)χ

is finite-dimensional and irreducible and all U (χλ) are pairwise unitarily equivalent when varyingλ ∈ Λχ for every fixed χ. By direct inspection, using irreducibility and unitary equivalence of theU (χλ)s for fixed χ, one finds (U (χ))′′ ∩ (U (χ))′ = CI, where the commutant is referred to Hχ. Onthe other hand, using the fact that the U (χ)s with different χ are unitarily inequivalent and thatU ′ = G′R = R, one sees that every Hχ is R-invariant and that the subrepresentation Rχ obtainedby restriction satisfies Rχ = (U (χ))′ where the commutant is referred to Hχ, in particular Rχ istherefore a von Neumann algebra thereon. Hence (U (χ))′ ∩ (U (χ))′′ = CI can be translated intoRχ ∩R′χ = CI and every Rχ is a factor proving (4.13). (b) Let Pχ, Pχ′ ∈ R∩R′ the orthogonalprojectors onto Hχ and Hχ′ respectively with χ 6= χ′. Rχ, Rχ′ are unitarily inequivalent because,if there were an isometric surjective V : Hχ → Hχ′ with V AHχ V

−1 = AHχ′ we would find acontradiction representing the operator A = Pχ − Pχ′ ∈ R, namely 11χ′ = −11χ′ .

For Abelian discrete superselection rules, existence of a compact global gauge group G as in thehypotheses of the theorem is guaranteed from separability of H as we established above withG = U(1). In this case, the decomposition (4.11) coincides with the decomposition (4.4)-(4.5)but, more strongly, Rχ = B(Hχ) and U (χ) is a pure phase as we know. If GR is not Abelianthe issue of the existence of such G has to be examined case by case. In all physically interstingcases, G is a compact (hence matricial) Lie group and U(G) is considerably smaller than GR.The approach to superselection rules based on the notion of a compact global gauge group ofinternal symmetries G turns out to be powerful and deep if used in addition to the request ofspacetime locality in algebraic quantum field theory in Minkowski spacetime formulated in termsof von Neumann algebras, after the remarkable results by several authors relying on the so-calledDoplicher-Haag-Roberts (DHR) analysis and Buchholz-Fredenhagen (BF) analysis of superselec-tion sectors [Haa96] describing theories with short-range interactions without topological chargesin BF’s sense. A quite complete technical review including fundamental references is [HaMu06].

4.3.3 Quantum states in the presence of Abelian superselection rules

Let us come to the problem to characterize the states when an Abelian superselection struc-ture is assumed on a complex separable Hilbert space H in accordance with (S1) and (S2). Inprinciple we can extend Definition 3.10 already given for the case of R with trivial centre. As

195

usual LR(H) indicates the lattice of orthogonal projectors in R, which we know to be boundedby 0 and I, orthocomplemented, σ-complete, orthomodular and separable, but not atomic andit does not satisfy the covering property in general. The atoms are one-dimensional projectorsexactly as pure states in the R = B(H) case, so we may expect some difference at that levelwhen R 6= B(H).

Definition 4.10. Let H be a complex separable Hilbert space. A quantum probabilitymeasure in H, for a quantum sistem with von Neumann algebra of observables R, is a mapρ : LR(H)→ [0, 1] such that the following requirement are satisfied.

(1) ρ(I) = 1 .

(2) If Qnn∈N ⊂ LR(H), for N at most countable, satisfies Qk ⊥ Qh = 0 when h, k ∈ N ,then

ρ(∨k∈NQk) =∑k∈N

ρ(Qk) . (4.14)

The set of the quantum measures will be denoted by MR(H).

Remark 4.9. Notice that ∨k∈NQk ∈ LR(H) if every Qk ∈ LR(H), if N is at most countable,because this lattice is σ-complete. Without this fact the definition above would be meaningless.

Since a von Neumann algebra R is strongly-closed, strong topology is that used to spectrallymanipulate operators, and A = A∗ is affiliated or belongs to R if and only if its PVM belongsto LR(H), the definitions of Section 3.4.8 can be restated also in the presence of Abelian super-selection rules providing a sense to notions like expectation value and standard deviation of anobservable for a given quantum state pictured as a probability measure over LR(H).The procedures presented in Section 3.4.8 to compute those statistical objects in terms of tracesmake also sense when the quantum probability measures are represented by trace-class opera-tors. This is possible also when superselection rules occurs, as we go to prove, even if the pictureis more complicated.If there is an Abelian superselection structure, we have the decompositions we re-write downinto a simpler version,

H =⊕k∈K

Hk , R =⊕k∈K

Rk , Rk = B(Hk) , k ∈ K (4.15)

where K is some finite or countable set. The lattice LR(H), as a consequence of (4.14), decom-poses as (the notation should be obvious)

LR(H) =⊕k∈K

LRk(Hk) =⊕k∈K

L (Hk) (4.16)

196

where LRk(Hk) ∩LRh(Hh) = 0 if k 6= h.In other words Q ∈ LR(H) can uniquely be written as Q = ⊕k∈KQk where Qk ∈ L (B(Hk)).In fact Qk = PkQ, where Pk is the orthogonal projector onto Hk.

Let us focus on the problem of characterising the quantum probability measures in terms oftrace class operators and unit vectors up to phases.

Remark 4.10. We avoid here the use of already introduced terms like mixed states andpure states corresponding, in the case of absence of superselection rules, to unit vectors up tophases and positive unit-trace trace-class operators. This is because as we shall se shortly, thesemathematical objects are not yet one-to-one with extremal quantum probability measures andgeneric quantum probability measures. The physically safest viewpoint is to assume that quan-tum states are nothing but quantum probability measures.

It is possible to re-adapt Gleason’s result simply observing that a measure ρ on LR(H) as abovedefines a measure ρk on LRk(Hk) = L (Hk) by

ρk(P ) :=1

ρ(Pk)ρ(P ) , P ∈ L (Hk) ,

provided ρ(Pk) 6= 0. If dim(Hk) 6= 2 we can exploit Gleason’s theorem.

Theorem 4.3. Let H be a complex separable Hilbert space and assume that the von Neumannalgebra R in H satisfies (S1) and (S2), so that the decomposition (4.15) in coherent sectors isvalid where we suppose dimHk 6= 2 for every k ∈ K. The following facts hold.

(a) If T ∈ B1(H) satisfies T ≥ 0 and tr(T ) = 1, then ρT ∈MR(H) if

ρT : LR(H) 3 P 7→ tr(TP ) .

(b) For ρ ∈MR(H) there is a T ∈ B1(H) satisfying T ≥ 0 and tr(T ) = 1 such that ρ = ρT .

(c) If T1, T2 ∈ B1(H) satisfy the same hypotheses as T in (a), then ρT1 = ρT2 is valid if andonly if PkT1Pk = PkT2Pk for all k ∈ K, Pk being the orthogonal projector onto Hk.

(d) A unit vector ψ ∈ H defines an extreme measure if and only if it belongs to a coherentsector.More precisely, a measure ρ ∈MR(H) is extremal, if and only if there is k0 ∈ K, ψ ∈ Hk0with ||ψ|| = 1 such that

ρ(P ) = 0 if P ∈ L (Hk), k 6= k0 and ρ(P ) = 〈ψ|Pψ〉 if P ∈ L (Hk0)

Proof. (a) is obvious from Proposition 3.10, as restricting a state ρ on L (H) to LR(H) we stillobtain a state as one can immediately verify. Let us prove (b). Evidently, every ρ|L (Hk) is a

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positive measure with 0 ≤ ρ(Pk) ≤ 1. We can apply Gleason’s theorem finding Tk ∈ B(Hk)with Tk ≥ 0 and tr(Tk) = ρ(Pk) such that ρ(Q) = tr(TkQ) if Q ∈ L (Hk). Notice also that||Tk|| ≤ ρ(Pk) because

||Tk|| = supλ∈σp(Tk)

|λ| = supλ∈σp(Tk)

λ ≤∑

λ∈σp(Tk)

dλλ = tr(Tk) = ρ(Pk) .

If Q ∈ LR(H), Q =∑kQk, where Qk := PkQ ∈ L (Hk), QkQh = 0 if k 6= h and thus, by

σ-additivity,ρ(Q) =

∑k

ρ(Qk) =∑k

tr(TkQk)

since Hk ⊥ Hh, this identity can be rewritten as ρ(Q) = tr(TQ) provided T := ⊕kTk ∈ B1(H).It is clear that T ∈ B(H) because, if x ∈ H and ||x|| = 1 then, as x =

∑k xk with xk ∈ Hk,

||Tx|| ≤ ∑k ||Tk|| ||xk|| ≤∑k ||Tk||1 ≤

∑k ρ(Pk) = 1. In particular ||T || ≤ 1. T ≥ 0 because

each Tk ≥ 0. Hence |T | =√T ∗T =

√TT = T via functional calculus, and also |Tk| = Tk.

Moreover, using the spectral decomposition of T , whose PVM commutes with each Pk, oneeasily has |T | = ⊕k|Tk| = ⊕kTk. The condition

1 = ρ(I) =∑k

ρ(Pk) =∑k

tr(TkPk) =∑k

tr(|Tk|Pk)

is equivalent to say that tr |T | = 1 using a Hilbert basis of H made of the union of bases in eachHk. We have obtained, as wanted, that T ∈ B1(H), T ≥ 0, tr(T ) = 1 and ρ(Q) = tr(TQ) for allQ ∈ LR(H).(c) The proof straightforwardly follows from LRk(Hk) = L (B(Hk)), because Rk = B(Hk) and,evidently, ρT1 = ρT2 if and only if ρT1L (Hk)= ρT2L (Hk) for all k ∈ K.(d) It is clear that if ρ encompasses more than one component, ρ|L (Hk) 6= 0 cannot be extremalbecause is, by construction, a convex combination of other states which vanishes in some of thegiven coherent subspace. Therefore only states such that only one restriction ρ L (Hk0 ) does

not vanish may be extremal. Now (a) of Proposition 3.11 implies that, among these states, theextremal ones are precisely those of the form said in (d) of the thesis.

Remark 4.11.(a) Take ψ =

∑k∈K ckψk where the ψk ∈ Hk are unit vectors and also suppose that ||ψ||2 =∑

k |ck|2 = 1. This vector induces a state ρψ on R by means of the standard procedure (whichis nothing but the trace procedure with respect to Tψ := 〈ψ| 〉ψ)

ρψ(P ) = 〈ψ|Pψ〉 P ∈ LR(H) .

In this case however, since PPk = PkP and ψk = Pkψk we have

ρψ(P ) = 〈ψ|Pψ〉 =∑k

∑h

ckch〈ψk|Pψk〉 =∑k

∑h

ckch〈Pkψk|PPhψk〉 =∑k

∑h

ckch〈ψk|PkPPhψh〉

=∑k

∑h

ckch〈ψk|PPkPhψk〉 =∑k

∑h

ckch〈ψ|PPkψ〉δkh =∑k

|ck|2〈ψk|Pψk〉 = tr(T ′ψP )

198

whereT ′ψ =

∑k∈K|ck|2〈ψk| 〉ψk

We conclude that the apparent pure state described by the vector ψ and the apparent mixedstate described by the operator T ′ψ cannot be distinguished, just because the algebra R is toosmall to make a difference. Actually they define the same probability measure at all, i.e. thesame quantum state, and this is an elementary case of (c) in the above theorem with T1 = 〈ψ| 〉ψand T2 = T ′ψ. This discussion, in the language of physicists is often stated as follows:No coherent superpositions ψ =

∑k∈K ckψk of pure states ψk ∈ Hk of different coherent sectors

are possible, only incoherent superpositions∑k∈K |ck|2〈ψk| 〉ψk are allowed.

(b) It should be clear that the one-to-one correspondence between extremal quantum mea-sures and atomic elementary observables (one-dimensional projectors) here does not work. Con-sequently, notions like probability amplitude must be handled with great care. In general, how-ever, everything goes right if staying in a fixed superselection sector Hk where the said corre-spondence exists.

(c) We leave to the reader the easy proof of the fact that Luders-von Neuman’s postulateon the post-measurement state (see Section 3.4.7) can be stated as it stands also in the pres-ence of superselection rules, independently from the particular trace-class operator T we use todescribe a quantum probability measure ρ: the post measurement probability measure ρ′ doesnot depend on the chosen operator representation of ρ. It is worth also stressing that, sincethe PVM of an observable in R (or affiliated to R) commutes with the central projectors Pkdefining the superselection sectors Hk, if an extreme quantum state is initially represented by avector belonging to a sector Hk, there is no chance to get out from that sector by means of asubsequent masurement process of any observable in R.

Example 4.3. Coming back to Example 4.2, states (probability measures over LR(H)) whereQ takes the value −e with probability 1 are properly said states of electrons. States wherethe observable Q takes the value +e with probability 1 are properly said states of positrons.However, as soon as we perform a measurement of Q, its value cannot subsequently changedue to further measurements of other observables, since all physically meaningful observablescommute with Q and the postulate of collapse leaves the state in the initial eigenspace of Q.This means that, once the charge has been observed, and the particle results to be an electronor a positron, from that moment on it is impossible to put the system in a state where the valueof Q is not defined and the particle is in a superposition electron-positron. It would be stillpossible in principle to put the system into a similar superposed state in view of time evolution.This is not the case however, since the electrical charge observable is a constant of motion as aconsequence of the law of conservation of electrical charge.

4.3.4 Quantum states in the general case of R ( B(H).

Let us finally focus on the notion of state in the completely general setup R ( B(H). We canadopt the definition of state in terms of quantum probability measures over LR(H), so that

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states form the convex body MR(H). In particular, positive unit-trace trace-class operatorsT ∈ B(H), still represent σ-additive probability measures of MR(H) due to Proposition 3.10.Obviously T and T ′ := V TV −1, where V ∈ GR, define the same measure because

tr(AV TV −1) = tr(V −1AV T ) = tr(AV −1V T ) = tr(AT ) , if A ∈ R.

So that there are many ways to describe the same state in terms of trace-class operators and ameaningful definition of pure quantum state is again provided by an extreme element of MR(H)if any, rather than unit vectors. Everything we said so far is valid also for Abelian superselec-tion rules. A difference with the case of a non-Abelian gauge group may arise considering theconverse question.Can all probability measures be written in terms of positive positive unit-trace trace-class opera-tors in the generic case R ( B(H)?The answer is partially positive: under some constraints on the structure of R in terms of its de-composition of definite-type von Neumann algebras, Gleason’s theorem still holds (see [Dvu92]).

To conclude, we stress that a more general definition of quantum state, the notion of algebraicstate, is at our disposal when we think of R as a unital C∗-algebra as we shall better see in the lastchapter. Algebraic states are linear functionals ω : R→ C such that ω(I) = 1 and ω(A∗A) ≥ 0if A ∈ R. Positive unit-trace trace-class operators T on a given von Neumann algebra R inducealgebraic states this way ωT (A) := tr(TA) if A ∈ R. These states are called the normal statesof R and they form the so called folium of R. From the standpoint of the measure theoryon LR(H), algebraic states are additive, but not necessarily σ-additive, probability measures onLR(H).

4.4 Appendix. On the von Neumann algebra of a PVM again

This section is devoted to prove Proposition 4.6, which we restate here for author’s convenience.

Proposition 4.6. Let P : Σ(X) → L (H) be a PVM over the measurable space (X,Σ(X))taking values on L (H). If H is separable,

PE | E ∈ Σ(X)′′ =ß∫

XfdP

∣∣∣∣ f ∈Mb(X)

™.

If H is not separable, the above statement is however valid if replacing = for ⊃.

Proof. First of all, observe that the von Neumann algebra generated by the initial ∗-closed setPE | E ∈ Σ(X) is the same as the von Neumann algebra generated by the unital ∗-algebraAP of the finite complex linear combinations of the elements of PE | E ∈ Σ(X). Accord-ing to (1) Exercise 4.1, PE | E ∈ Σ(X)′′ is therefore nothing but the strong closure of AP .Since

∫X fdP ∈ B(H) if f ∈ Mb(X), the integral can be always computed as strong limit

of elements of AP , according to (c) Proposition 2.25, by approximating f with a bounded

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sequence of simple functions point wise converging to f . Summing up, we necessarily have∫X fdP

∣∣ f ∈Mb(X)⊂ PE | E ∈ Σ(X)′′ = A′′P . So we have to establish the converse inclu-

sion. More precisely, we have to prove that, if∫X sndPψ → Aψ as n → +∞ for every ψ ∈ H,

some A ∈ H, and for a given sequence of simple functions sn ∈ Mb(X), then A =∫X fdP for

some f ∈Mb(X). A lemma is useful to this end.

Lemma 4.1. Let P : Σ(X)→ L (H) be a PVM over the measurable space (X,Σ(X)) takingvalues on the lattice of orthogonal projectors over the Hilbert space H. There exist

(i) a set of orthonormal vectors ψnn∈N with N of any cardinality and, in particular, finiteor countable when N is separable,

(ii) a corresponding set Hnn∈N of mutually orthogonal closed subspaces of H, such that theHilbert direct decomposition holds H =

⊕n∈N Hn, and that PE(Hn) ⊂ Hn for every n ∈ N

and every E ∈ Σ(X),

(iii) a corresponding set of isometric surjective operators Un : Hn → L2(X,µ(P )ψnψn

).

Proof. Take ψ1 ∈ H with ||ψ1|| = 1 and focus on the map V1 : L2(X,µ(P )ψ1ψ1

) → H defined as

V1f :=∫X fdPψ1 for f ∈ L2(X,µ

(P )ψ1ψ1

). According to (a) and (b) Proposition 2.27, this mapis linear and is also isometric (thus injective) in view of (d) Theorem 2.6. Therefore it alsopreserves the scalar product as a consequence of the polarization identity. The image of the

map is evidently the subspace H1 := ∫X fdPψ1 | f ∈ L2(X,µ

(P )ψ1ψ1

) ⊂ H. This subspace isclosed. Indeed, if H1 3 V (fn) → φ ∈ H as n → +∞, the sequence of the fns must be Cauchy

because V1(fn)n∈N converges and V1 is isometric. Therefore f ∈ L2(X,µ(P )ψ1ψ1

) exists such that

fn → f , because L2(X,µ(P )ψ1ψ1

) is complete. Since V1 is continuous it being isometric, V1(f) = φ

so that φ ∈ H1 which is consequently closed. The map U1 := V −11 (computing the inverse of V1

restricting its codomain to its image) is that argued to exist in (ii) for n = 1. Finally observethat PE(H1) ⊂ H1 as an immediate consequence of (b) Proposition 2.25 and (c) Proposition

2.27: PE∫X fdPψ1 =

∫X fχEdPψ1 ∈ H1 noticing that, obviously, fχE ∈ L2(X,µ

(P )ψ1ψ1

) if f ∈L2(X,µ

(P )ψ1ψ1

). If H1 ( H we can fix ψ2 ∈ H⊥1 with ||ψ2|| = 1 repeating the procedure, finding a

corresponding isometric surjectve map U2 : H2 → L2(X,µ(P )ψ2ψ2

), with H2 ⊂ H a closed subspace

satisfying H2 ⊥ H1 and PE(H2) ⊂ H2 for every E ∈ Σ(X). And so on with ψ3 ∈ (H1 ∪ H2)⊥ ifnecessary. A standard application of Zorn’s lemma proves the thesis. In case H is separable, Nmust be finite or countable, because the cardinality of the set of orthonormal vectors ψnn∈Ncannot exceed that of a Hilbert basis, since ψnn∈N is or can be completed to a Hilbert basis.

Let us come back to the main proof, assuming N = N since H is separable by hypothesis andthe case of N finite is a trivial subcase. So, suppose that

∫X skdPψ → Aψ as k → +∞ for every

ψ ∈ H, some A ∈ H, and for a given sequence of simple functions sk ∈ Mb(X). Consequently

∫X skdPψk∈N is Cauchy in H so that skk∈N is Cauchy in L2(X, dµ

(P )ψψ ) because of (d) Theorem

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2.6. In particular, what asserted must be true for ψ =∑n∈N

1√2nψn, which belongs to H as the

series converges because∑n∈N

12n = 2 and the orthonormal vectors ψn form or can be completed

to a Hilbert basis of H. From (ii) in the Lemma, in particular PE(Hn) ⊂ Hn, we have that

0 ≤ µ(P )ψψ (F ) =

∞∑n∈N

1√2nψn

∣∣∣∣∣∣PF ∑m∈N1√2m

ψn

∫=∑n∈N

1

2n〈ψn|PFψn〉 =

∑n∈N

1

2nµ

(P )ψnψn

(F ) ≤ 2 ,

where we have used µ(P )ψnψn

(X) = ||ψn||2 = 1. Since skk∈N is Cauchy in L2(X, dµ(P )ψψ ), there

is a function f ∈ L2(X, dµ(P )ψψ ) such that sk → f as k → +∞ with respect to L2(X, dµ

(P )ψψ ).

Furthermore [Rud86], there is a subsequence, we indicate with the same symbol skk∈N for the

sake of semplicity, that converges µ(P )ψψ -a.e. to f . In view of µ

(P )ψnψn

(F ) ≤ 2nµ(P )ψψ (F ), we also have

that sk → f simultaneously in L2 sense and a.e. for each of the measures µ(P )ψnψn

. In particular

f ∈ L2(X, dµ(P )ψnψn

). It is now natural to compare the operators A and∫X fdP , since both are

limit of the sequence of the∫X sndP . Let us focus on a space Hn as defined in the Lemma above.

Since Mb(X) is dense in L2(X, dµ(P )ψnψn

), we conclude that the subspace Mn := U−1n (Mb(X)) is

dense in Hn. However Mn ⊂ D(∫X fdP ) because D(

∫X fdP ) = φ ∈ H |

∫X |f |2dµ

(P )φφ < +∞.

Indeed, if φ =∫X gdPψn for g ∈ Mb(X), we have µ

(P )φφ (F ) = 〈

∫X gdPψn|PF

∫X gdPψn〉 =∫

F |g|2dµ(P )ψnψn

so that∫X |f |2dµ

(P )φφ =

∫X |f |2|g|2dµ

(P )ψnψn

≤ ||g||∞∫X |f |2dµ

(P )ψnψn

< +∞ and henceφ ∈ D(

∫X fdP ) as said. This is not the end of the story, since we also have, for φ ∈ Mn,∫

X fdPφ = Aφ. In fact we have∫X skdPφ→

∫X fdPφ because ((d) Theorem 2.6)∣∣∣∣∣∣∣∣∫

X(sk − f)dPφ

∣∣∣∣∣∣∣∣2 =

∫X|sk−f |2dµ

(P )φφ =

∫X|sk−f |2|g|2dµ

(P )ψnψn

≤ ||g||2∞∫X|sk−f |2dµ

(P )ψnψn

→ 0

for k → +∞, and also∫X skdPφ→ Aφ by hypothesis. Now consider the just established identity∫

XfdPφ = Aφ , ∀φ ∈Mn .

Since Mn is dense in Hn, the operator∫X fdP is closed ((b) in Theorem 2.6) and A is continuous,

it immediately arises that the identity is valid for every φ ∈ Hn. In particular, Hn ⊂ D(∫X fdP ).

By linearity, the said identity is true also when φ is a finite linear combinations of elements in⊕n∈NHn. Since these linear combinations are dense in H, the same argument exploited above

proves that ∫XfdPφ = Aφ , ∀φ ∈ H .

In particular∫X fdP = A ∈ B(H), so that f must be P -essentially bounded ((a) Proposition

2.25). By definition of || ||(P )∞ , we can modify f over a P -zero-measure set, obtaining a function

f1 ∈ Mb(X) giving rise to the same integral∫X f1dP =

∫X fdP = A. In summary, every

A ∈ PE | E ∈ Σ(X)′′ can be written as A =∫X f1dP for some f1 ∈ Mb(X), ending the

proof.

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Chapter 5

Quantum Symmetries

The notion of symmetry in Quantum Theory is quite abstract. There are actually at least threedistinct ideas, respectively due to Wigner, Kadison and Segal [Sim76]. Here, we focus on thefirst pair only and a fourth type which naturally arises form our formulation of the quantumtheory. An exhaustive discussion appears in [Lan17], where six different notions of quantumsymmetry are even introduced and their equivalence is discussed.

5.1 Quantum Symmetries according to Kadison and Wigner

Generally speaking, symmetries are supposed to mathematically describe some concrete trans-formation acting either on the physical system or on the instruments used to analyze the system.From a very general viewpoint a symmetry is an active transformation on either the quantumsystem or, by duality, on the observables representing physical instruments, such that

(1) the transformation is bijective in the sense that

(a) given a symmetry, every state of the system or observable representing instruments(according to the used notion of symmetry) can be reached varying the inital stateor observable, through the transformation;

(b) every symmetry transformation admits an inverse;

(2) the transformation is requested to preserve some mathematical structure of the space of thestates or of the space of observables, making the distinction between the various notionsof symmetry.

Very unfortunately, an intrinsically different notion of gauge symmetry exists in the literature.A gauge symmetry is not a symmetry in the sense presented above. A symmetry acts on thephysical system explicitly changing its state or the (observables representing the) instruments.Vice versa, a gauge symmetry is a mathematical transformation which does not change anythingwhich is directly related to measurements: states of the system or the instruments. An examplefor a system with algebra of observables R is the action of elements U of the whole commutant

203

group GR of R (the group of unitary operators in R′) over the quantum probability measures onLR(H) describing the states of a quantum system as discussed in Section 4.3.2. The quantumstates associated to the two measures ρ and ρ(U ·U−1) cannot be distinguished when acting onLR(H) simply because UPU−1 = P for every P ∈ LR(H) as already observed from a slightlydifferent perspective in Section 4.3.4.The idea of gauge symmetry is however technically very useful. In some fundamental theories theinitial relevant algebra of operators F is larger (in the von Neumann algebra framework is B(H)itself) than the (von Neumann) algebra of observables R. The latter is defined just requiringthat it is the von Neumann algebra made of the operators in F commuting with a suitable faithfulstrongly-continuous representation U of a certain compact group G named the global gauge groupof internal symmetries: R = U ′. (As a consequence U ⊂ GR and U ′ = G′R = R.) We havealready seen this procedure at work in the first part of Sect.4.3.2. For instance dealing with spinorfields one encounters operators, in particular spinor field operators, which cannot be interpretedas observables (or complex combinations of observables) because they violate some fundamentalphysical requisite (typically causality relations) ascribed to meaningful observables. However,other operators constructed out of spinor field operators (typically currents) are observables. Away to select observables among the larger algebra F is requiring that operators representing(linear combinations of) observables, thus defining the von Neumann algebra R, are fixed underthe action of a suitable compact group G – in this case Abelian and coinciding with U(1) –represented in terms of unitary operators belonging in the whole commutant group GR of Raccording to the discussion in Sect.4.3.2. R turns out to be a sum of irreducible von Neumannalgebras Rk = B(Hk) over an orthogonal sum of sectors Hk decomposing H. As seen in Section4.3.2, the procedure is however general and works also if the commutant group is non-Abelian asit is the case in chromodynamics where the used group is G = SU(3) (color). In this juncture, Ris a sum of factors Rk defined over a orthogonal sum of sectors Hk which are invariant under theaction of G. In this view, internal symmetries (i.e. different form those induced by geometry ofthe spacetime) are not symmetries at all, since they do not act on observables by definition (see[Haa96] for further discussion related to locality in particular and the so-called DHR analysis ofsuperselection rules in algebraic formulation).

5.1.1 Wigner, Kadison symmetries and ortho-automorphism symmetries

We henceforth consider a quantum system described in the Hilbert space H. We assume thateither H is the whole Hilbert space in the absence of superselection charges or it denotes a singlecoherent sector when Abelian superselection rules occur. Let S (H) and Sp(H) respectivelyindicate the convex body of the quantum states (positive, unit-trace, trace-class operators ) andthe subset of pure quantum states (orthogonal projectors over one-dimensional subspaces), ev-erything referred to the considered sector if it is the case.A first pair of notion of symmetry can be defined when looking at the space of the states. Sincestates are actually better pictured in terms of probability measures over L (H), the definitionsabove are physical meaningful when the afore-mentioned measures are faithfully described bytrace class operators according to Gleason theorem. This is the case when H is separable with

204

dimension 6= 2. However, since the notions we go to introduce are mathematically consistentalso relaxing these physically crucial hypotheses, we state the following definition in the generalcontext.

Definition 5.1. If H is a Hilbert space, we have the following types of symmetries.

(a) A Wigner symmetry is a bijective map

sW : Sp(H) 3 〈ψ| 〉ψ → 〈ψ′| 〉ψ′ ∈ Sp(H)

which preserves the probabilities of transition. In other words

|〈ψ1|ψ2〉|2 = |〈ψ′1|ψ′2〉|2 if ψ1 , ψ2 ∈ H with ||ψ1|| = ||ψ2|| = 1 .

(b) A Kadison symmetry is a bijective map

sK : S (H) 3 T → T ′ ∈ S (H)

which preserves the convex structure of the space of the states. In other words

(pT1 + qT2)′ = pT ′1 + qT ′2 if T1, T2 ∈ S (H) and p, q ≥ 0 with p+ q = 1.

Remark 5.1. We observe that the first definition is well-posed even if unit vectors define purestates just up to a phase, as the reader can immediately prove, because transition probabilitiesare not affected by that ambiguity.

An apparently different approach to define symmetries focuses on elementary observables ofL (H) instead of states of S (H). Here symmetries are viewed as active transformations pre-serving the lattice structure of the elementary observables. From a practical viewpoint, thesesymmetries are interpreted as some sort of reversible active actions on the instruments usedto perform measurement procedures. These transformations must preserve the logical relationsbetween elementary propositions on the system.

Definition 5.2. If H is a Hilbert space, a symmetry of elementary observables is amap h : L (H)→ L (H) satisfying

(i) h is bijective,

(ii) h(P ) ≥ h(Q) if P,Q ∈ L (H),

(iii) h(I − P ) = I − h(P ) if P ∈ L (H)

205

A similar map is named ortho-automorphism of L (H).

Remark 5.2.(a) It is easy to prove that an ortho-automorphism h : L (H)→ L (H) preserves the structure

of orthocomplemented complete lattice in view of the given definition. In particular

(i) h(0) = 0 and h(I) = I,

(ii) h(∨j∈JPj) = ∨j∈Jh(Pj), h(∧j∈JPj) = ∧j∈Jh(Pj) for every family Pjj∈J ⊂ L (H).

Also, h−1 : L (H)→ L (H) is evidently an ortho-automorphism.(b) As the reader can straightforwardly prove, a symmetry of elementary observables induces

a Kadison symmetry by duality if Gleason’s theorem 3.5 is valid. In fact, if T ∈ S (H) and h isan ortho-automorphism, then

ρT,h : L (H) 3 P 7→ tr(Th(P )) ∈ [0, 1]

is a probability measure over L (H), the proof being trivial and relies on the fact that h preservesthe lattice structures. Therefore there exists exactly one T ′h ∈ S (H) such that

ρT,h(P ) = tr(T ′hP ) for every P ∈ L (H).

By construction, s(h)K : T 7→ T ′h preserves the convex structure of S (H). Indeed,(

s(h)K (pT1 + qT2)

)(P ) = tr ((pT1 + qT2)h(P )) =

(ps

(h)K (pT1) + qs

(h)K (T2)

)(P ) .

Since P ∈ L (H) is arbitrary,

s(h)K (pT1 + qT2) = ps

(h)K (pT1) + qs

(h)K (T2) .

Finally,(s

(h)K

)−1= s

(h−1)K .

(c) Symmetries of the three types do exist. If U : H→ H is a unitary operator, the maps

s(U)W : Sp(H) 3 〈ψ| 〉ψ 7→ 〈Uψ| 〉Uψ ∈ Sp(H) ,

s(U)K : S (H) 3 T → UTU−1 ∈ S (H)

andh(U) : L (H) 3 P 7→ U−1PU ∈ L (H)

are respectively a Wigner symmetry, a Kadison symmetry and a ortho-automorphism of L (H).

Furthermore s(U)K is induced by h(U) according to remark (b) if Gleason’s theorem holds.

(d) When Abelian superselection rules occur, a more general notion of symmetry exist. Thatis defined between different superselection sectors. For instance a bijective map form L (Hk) andL (Hh) with k 6= h preserving the orthocomplemented lattice structure, or similar maps betweenthe spaces of the states S (Hk) and S (Hh) preserving the convex structure or even a bijectivemap between Sp(Hk) and Sp(Hh) preserving the probability transitions. A typical example ofthese symmetries swapping superselection sectors is the charge conjugation symmetry. We shallnot discuss this sort of symmetries (see however [Mor18]). The reader can easily extend to thesecases the theory developed below.

206

5.1.2 Theorems by Wigner, Kadison and Dye

Though the three presented definitions are evidently of different nature, they lead to the samemathematical object, as established by a triple of characterization theorems we quote into aunique statement. We need a preliminary definition.

Definition 5.3. Let H,H′ be Hilbert spaces. A map U : H → H′ is said to be an antiunitary operator if it is surjective, isometric and

U(ax+ by) = aUx+ bUy

when x, y ∈ H and a, b ∈ C.

Notice that if U : H → H′ is anti unitary, then 〈Ux|Uy〉 = 〈x|y〉′ for x, y ∈ H in view of thepolarization identity.We come to the afore-mentioned theorems.

Theorem 5.1.Let H 6= 0 be a Hilbert space. The following facts hold.

(a) [Wigner’s theorem] For every Wigner symmetry sW , there is an operator U : H → H,which can be either unitary or anti unitary – depending on sW if dim(H) 6= 1, otherwiseboth choices are permitted – such that

sW : 〈ψ| 〉ψ 7→ 〈Uψ| 〉Uψ , ∀〈ψ| 〉ψ ∈ Sp(H) . (5.1)

For dimH > 1, U and U ′ are associated to the same sW if and only if U ′ = eiaU for a ∈ R.

(b) [Kadison’s theorem] For every Kadison symmetry sK , there is an operator U : H→ H,which can be either unitary or anti unitary – depending on sK if dim(H) 6= 1, otherwiseboth choices are permitted – such that

sK : T 7→ UTU−1 , ∀T ∈ S (H) . (5.2)

For dimH > 1, U and U ′ are associated to the same sK if and only if U ′ = eiaU for a ∈ R.

(c) [Dye’s theorem (most elementary version)] If H is separable with dim(H) 6= 2, forevery ortho-automorphism h : L (H) → L (H) there is an operator U : H → H, which canbe either unitary or anti unitary – depending on h if dim(H) 6= 1, otherwise both choicesare permitted – such that

h : P 7→ U−1PU , ∀P ∈ L (H) . (5.3)

For dimH > 1, U and U ′ are associated to the same h if and only if U ′ = eiaU for a ∈ R.

207

(d) U : H→ H, either unitary or anti unitary, simultaneously defines a Wigner symmetry (thesame also replacing U for eiaU with a ∈ R), a Kadison symmetry or a ortho-automorphismby means of (5.1), (5.2), and (5.3) respectively.

Proof. (d) is trivial. The proof of existence of U in the case (a) is difficult and can be found,e.g., in [Sim76, Var07, Lan17, Mor18]. The proof of existence for (b) arises from that for (a)and can be found in [Sim76, Lan17, Mor18]. The proof of (c) is immediate consequence of theexistence for the case (b) and the discussion in (b) Remark 5.2.Let us come to the uniqueness issues. If dimH = 1, the corresponding U of a given symmetry canbe taken arbitrarily unitary or anti unitary. The proof is direct and can be obtained identifyingH with C. The fact that, for dimH > 1, U is fixed up to a phase, is proven as follows. Supposethat U and V are both unitary or both anti unitary and define the same symmetry. With thethree kinds of symmetry, we have UPU−1 = V PV −1 for P a orthogonal projector onto a one-dimensional subspace P = 〈ψ| 〉ψ that can be viewed simultaneously as an element of Sp(H),S (H), and L (H). So V −1UP = PV −1U . As a consequence V −1Uψ = aψψ for some complexaψ ∈ H. If dimH > 1, there are at least ψ,ψ′ ∈ H orthogonal and with unit norm. Hence

aψψ + aψ′ψ√2

= V −1Uψ + ψ′√

2= aψ+ψ′√

2

ψ + ψ′√2

.

Consequently

Åaψ+ψ′√

2

− aψ′ãψ′ = −

Åaψ+ψ′√

2

− aψãψ . Since the vectors are orthonormal, the

only possibility is that the numerical factors vanish. In particular aψ′ = aψ. If N ⊂ H is aHilbert basis, we therefore have that V −1Uψu = au for every u ∈ N and a unique constanta ∈ C. Therefore

V −1Uφ = V −1U∑u∈N〈u|φ〉u =

∑u∈N〈u|φ〉au = aφ ∀φ ∈ H .

Since V −1U is unitary, |a| = 1 and U = aV .An analogous argument proves that, for dimH > 1, U and V must be both unitary or both antiunitary. In fact, if it were not the case, the above reasoning would prove that the anti unitaryoperator V −1U , for every Hilbert basis N , acts as V −1Uu = aNu with u ∈ N and aN ∈ C.However, defining the new Hilbert basis N ′ whose elements are the same as those of N , exceptfor an element u′0 ∈ N ′ that is defined as u′0 := iu0, we would find a contradiction: if u 6= u0 wehave aN ′u = V −1Uu = aNu, but also iaN ′u0 = aN ′u

′0 = V −1Uu′0 = V −1Uiu0 = −iV −1Uu0 =

−iaNu0. So that we would have aN ′ = aN = −aN implying aN = 0. This is not possible becauseV −1U would be the zero operator whereas it is isometric by hypothesis and H 6= 0.

Remark 5.3. If Abelian superselection rules are present, quantum symmetries are describedsimilarly with unitary or anti unitary operators acting in a single coherent sector or also swappingdifferent sectors [Mor18].

208

5.1.3 Action of symmetries on observables and their physical interpretation

If a unitary or antiunitary operator V represents a (Kadison or Wigner) symmetry s, it has anaction on observables, too. If A is an observable (a selfadjoint operator on H), we define thetransformed observable along the action of s as

s∗(A) := V −1AV . (5.4)

Obviously D(s∗(A)) = V (D(A)). This action is the dual action on an observable of a Kadi-son/Wigner symmetry. There is another similar action, the inverse dual action

s∗−1(A) := V AV −1 . (5.5)

Again D(s∗−1(A)) = V (D(A)). It is evident that these definitions are not affected by theambiguity of the arbitrary phase in the choice of V when s is given. Moreover, according with(j) in Proposition 2.32, the spectral measure of s∗(A) is

P(s∗(A))E = V −1P

(A)E V = s∗(P

(A)E )

as expected and this is nothing but the ortho-automorphism induced by the same unitary opera-tor U . (Instead s∗−1 is the associated inverse ortho-automorphism.) So, acting on an observableA by means of a symmetry is completely equivalent to perform the same action on the elemen-tary observables of its PVM P (A). This fact is in perfect agreement with the physical idea,mathematically supported by the spectral theorem, that an observable (a selfadjoint operator)contains the same physical information as that of its PVM.

The meaning of the inverse dual action s∗−1 on observables should be evident. The proba-bility that the observable s∗−1(A) produces the outcome E when the state is s(T ) (namely

tr(P(s∗−1(A))E s(T ))) is the same as the probability that the observable A produces the outcome

E when the state is T (that is tr(P(A)E T )). In other words, changing simultaneously and coher-

ently observables and states nothing changes. Indeed

tr(P(s∗−1(A))E s(T )) = tr(V P

(A)E V −1V TV −1) = tr(V P

(A)E TV −1)

= tr(P(A)E TV −1V ) = tr(P

(A)E T ) .

So, the inverse dual action of a Kadison/Wigner symmetry on observables is a transformationof them such that it cancels the direct action of the symmetry on the states. As an examplethink of a translation along the z axis of an isolated quantum system. In an inertial referenceframes at least, it should be cancelled by a translation of the orgin of the axes along the z axis.

The meaning of the dual action s∗ on observables is similarly clear. This operation on theobservables (keeping fixed the states) produces the same result as the direct action of s on thestates (keeping fixed the observables).

tr(P(s∗(A))E T ) = tr(V −1P

(A)E V T ) = tr(P

(A)E V TV −1) = tr(P

(A)E s(T )) .

209

As an example, think of a translation along the z axis of an isolated quantum system in an in-ertial reference frame. As far as, for instance, measurement of position are concerned, it shouldbe equivalent to a translation of the orgin of the axes along the z axis along the opposite direction.

Example 5.1.(1) Fixing an inertial reference frame, the pure state of a quantum particle is defined, up tophases, as a unit-norm element ψ of L2(R3, d3x), where R3 stands for the rest three space ofthe reference frame. The group of isometries IO(3) of R3 equipped with the standard Euclideanstructure acts on states by means of symmetries the sense of Wigner and Kadison. If

(R, t) : R3 3 x 7→ Rx+ t ∈ R3

is the action of the generic element (R, t) of IO(3) on x ∈ R3, where R ∈ O(3) and t ∈ R3, theassociated quantum (Wigner) symmetry s(R,t)(〈ψ| 〉ψ) = 〈U(R,t)ψ| 〉U(R,t)ψ is completely fixedby the unitary operators U(R,t). They are defined as

(U(R,t)ψ)(x) := ψ((R, t)−1x) = ψ(R−1(x− t)) , x ∈ R3 , ψ ∈ L2(R3, d3x) , ||ψ|| = 1 .

The fact that the Lebesgue measure is invariant under IO(3) immediately proves that U(R,t) isisometric and U(R,t) is also unitary because it is surjective, as it admits U(R,t)−1 as right inverse.It is furthermore easy to prove that, with the given definition,

U(I,0) = I , U(R,t)U(R′,t′) = U(R,t)(R′,t′) , ∀(R, t), (R′, t′) ∈ IO(3) . (5.6)

(2) The so-called time-reversal transformation classically corresponds to inverting the sign ofall the velocities of the physical system. It is possible to prove [Mor18] (see also (3) Exercise 5.5below) that, in QM and for systems whose energy is bounded below but not above, the time-reversal symmetry cannot be represented by unitary transformations, but only anti unitary. Inthe most elementary situation as in (1), the time reversal symmetry is defined (up to phases)by means of the anti unitary operator

(Tψ)(x) := ψ(x) , x ∈ R3 , ψ ∈ L2(R3, d3x) , ||ψ|| = 1 .

(3) According to the example in (1), let us focus on the subgroup of IO(3) of displacementsalong x1 parametrized by u ∈ R,

R3 3 x 7→ x+ ue1 ,

where e1 denotes the unit vector in R3 along x1. For every value of the parameter u, we indicateby su the corresponding (Wigner) quantum symmetry, su(〈ψ| 〉ψ) = 〈Uuψ| 〉Uuψ with

(Uuψ)(x) = ψ(x− ue1) , u ∈ R .

The the inverse dual action of this symmetry on the observable Xk turns out to be

s∗−1u (Xk) = UuXkU

−1u = Xk − uδk1I , u ∈ R .

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5.2 Groups of quantum symmetries

As in (1) in the example above, one deals very often in physics with groups of symmetries. Inother words, there is a certain group G, with unit element e and group product ·, and oneassociates each element g ∈ G to a symmetry sg (if Kadison or Wigner is immaterial here, inview of Theorem 5.1). In turn, sg is associated to an operator Ug, unitary or anti unitary. Thisassociation however is ambiguous because we are free to change these operators with arbitraryphases. This section is devoted to study this sort of representations.

5.2.1 Unitary(-projective) representations of groups of quantum symmetries

Consider a map G 3 g 7→ Ug, where G is a group which is supposed to represent a group ofsymmetries on quantum system described in the Hilbert space H, with dimH > 1, and theseactions are practically implemented by the unitary operators Ug ∈ B(H). We know that theoperators Ug can be changed by factors given by arbitrary phases keeping the symmetry associ-ated to each Ug. It would be nice to fix these operators Ug, by possibly changing their arbitraryphases, in order that the map G 3 g 7→ Ug be a unitary representation of G on H.

Definition 5.4. If G is a group, a group homomorphism G 3 g 7→ Ug from G to the group ofunitary operators in the Hilbert space H is called unitary representation of G on H.Equivalently, a unitary representation G 3 g 7→ Ug is such that the following identities

Ue = I , UgUg′ = Ug·g′ , U−1g = U∗g , ∀g, g′ ∈ G (5.7)

are valid.

Identities (5.6) found in (1) Example 5.1 shows that unitary representations of group of sym-metries exist. However, generally speaking, the requirement (5.7) does not hold. What is onlyguaranteed by physics, if G is a group of quantum symmetries, is that every Ug is unitary (oranti unitary but we stick here to the first case only) and that Ug·g′ equals UgUg′ just up tophases:

UgUg′U−1g·g′ = ω(g, g′)I with ω(g, g′) ∈ T for all g, g′ ∈ G. (5.8)

(As usual, T := z ∈ C | |z| = 1.) For g = g′ = e this identity gives in particular

Ue = ω(e, e)I . (5.9)

The numbers ω(g, g′) are called multipliers. They cannot be completely arbitrary, indeedassociativity of composition of operators (Ug1Ug2)Ug3 = Ug1(Ug2Ug3) yields the identity

ω(g1, g2)ω(g1 · g2, g3) = ω(g1, g2 · g3)ω(g2, g3) , ∀g1, g2, g3 ∈ G , (5.10)

which also implies, for suitable choices of g1, g2, g3 (the reader prove it),

ω(g, e) = ω(e, g) = ω(g′, e) , ω(g, g−1) = ω(g−1, g) , ∀g, g′ ∈ G . (5.11)

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All that leads us to the following important definition.

Definition 5.5. If G is a group, a map G 3 g 7→ Ug – where the Ug are unitary operators inthe Hilbert space H – is named a unitary-projective representation of G on H if (5.8) holdsfor a map ω : G×G→ T satisfying (5.9) and (5.10). Moreover,

(i) two unitary-projective representation G 3 g 7→ Ug ∈ B(H) and G 3 g 7→ U ′g ∈ B(H) aresaid to be equivalent if U ′g = χgUg, where χg ∈ U(1) for every g ∈ G. That is the sameas requiring that there are numbers χh ∈ U(1), if h ∈ G, such that

ω′(g, g′) =χg·g′

χgχg′ω(g, g′) ∀g, g′ ∈ G , (5.12)

where ω(g, g′)I = UgUg′U−1g·g′ and ω′(g, g′)I = U ′gU

′g′U′−1g·g′ ;

(ii) a unitary-projective representation with ω(e, e) = ω(g, e) = ω(e, g) = 1 for every g ∈ G issaid to be normalized.

A unitary-projective representation of groups G 3 g 7→ Ug ∈ B(H) acts both on states T ∈ S (H)and on elementary observables P ∈ L (H) (and also on observables as already discussed). Theaction on states is

S (H) 3 T 7→ UgTU−1g ∈ S (H) for every g ∈ G . (5.13)

We have two possible actions on elementary observables according to the dual action

S (H) 3 P 7→ h′g(P ) := U−1g PUg ∈ L (H) for every g ∈ G , (5.14)

or the inverse dual action

S (H) 3 P 7→ hg(P ) := UgPU−1g ∈ L (H) for every g ∈ G . (5.15)

Notice that changing the phase of Ug does not affect the action on states and observables. Sothese actions are invariant under equivalence transformations of unitary-projective representa-tions. Both actions on elementary observables have a physical meaning as discussed in Section5.1.3 and the choice between dual or inverse dual depends on physical convenience. However,from a pure mathematical viewpont, the maps G 3 g 7→ hg and G 3 g 7→ h′g have differentproperties. As the reader immediately proves, the following facts hold.

(1) The inverse dual action G 3 g 7→ hg is a representation of G in terms of ortho-automorphisms of L (H). In other words, every hg is an ortho-automorphisms of L (H)and also

he = id , hghg′ = hg·g′ .

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(2) The dual action G 3 g 7→ h′g is instead a left representation of G in terms of ortho-automorphisms of L (H). In other words, every hg is an ortho-automorphisms of L (H)and also

h′e = id , h′gh′g′ = h′g′·g .

(notice the reversed order of g and g′.)

Evidently, if G is Abelian also the dual action can be viewed as a standard representation.

Remark 5.4.(a) It is easily proved that every unitary projective representation is always equivalent to

a normalized representation. It is sufficient re-defining U ′g := χgUg with χe = ω(e, e)−1 andχg = 1 otherwise, and taking the general identity ω′(g, e) = ω′(e, g) = ω′(g′, e) into account.

(b) To be equivalent is evidently an equivalence relation between unitary-projective repre-sentations. It is clear that two projective unitary representations are equivalent if and only ifthey are made of the same Wigner (or Kadison) symmetries since these symmetries disregardthe arbitrary phases in front of the unitary operators describing them.

5.2.2 Representations including anti unitary operators

Up to now, we only considered the case where the operators Vg of a unitary-projective repre-sentation are properly unitary. We may however wonder if it is possible to construct a mapG 3 g 7→ Vg where the operators Vg, which we assumed to represent quantum symmetries overthe Hilbert space H with dimH > 1, are anti unitary, or also unitary and anti unitary, and thegroup operation are preserved up to phases as in (5.7). Notice that the nature unitary or antiunitary of Vg is therefore fixed by the corresponding g (since it defines the quantum symmetry)and Theorem 5.1 is valid. If every g ∈ G can be written as g = h · h for some h depending ong or, more generally, every g ∈ G can be written as a finite product of elements g1, . . . , gn suchthat each of them can be written as gk = hk · hk, then all the operators Ug must be unitary. Infact, Vg = ω(h, h)−1VhVh is necessarily linear no matter if Uh is linear or anti linear.The argument above is valid in particular if G is a connected Lie group1 [Mor18]. This is because(a) there is a sufficiently small neighbourhood O of the neutral element such that g ∈ O has theform g = exp(tgTg) for some Tg ∈ g (the Lie algebra of G) and tg ∈ R, so that h = exp((tg/2)Tg),furthermore (b) every g ∈ G can be written as a product of finite elements g1, . . . , gn ∈ O. Asa matter of fact, there are generalized unitary-projective representations where anti unitary op-erators show up. These representations can be treated as particular cases. For instance, whenrepresenting the complete (non-connected) Poincare Lie group P for quantum systems with non-negative squared mass and non-negative energy, the time-reversal symmetry is necessarily antiunitary. We stress that the time reversal operation does not belong to the connected componentof P including the identity element.

1A Lie group is a second-countable Hausdorff real-analytic manifold, locally homeomorphic to Rn, and equippedwith group operations which are smooth. Real analyticity can be equivalently replaced by smoothness [Mor18].

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Dealing with unitary-projective representations of groups of quantum symmetries, in this workwe stick to the case of unitary operators only.

5.2.3 Unitary-projective representations of Lie groups and Bargmann’s the-orem

As stressed above, given a unitary-projective representation, a technical problem is to check ifit is equivalent to a unitary representation, because unitary representations are much simplerto handle. This is a difficult problem [Var07, Mor18] which is tackled especially when G isa topological group or more strongly a Lie group (see [Mor18] for a quick summary on thesemathematical notions and [NaSt82] and [Var84] for classical treatises stressing on the analyticstructure of Lie groups, and [HiNe13] for a complete up-to-date modern report dealing with thesmooth structure). In these cases the representation satisfies the following physically naturalcontinuity property. It refers to the transition probability of two pure states, which is a physicallymeasurable quantity.

Definition 5.6. A unitary-projective representation G 3 g 7→ Ug of the topological group Gon the Hilbert space H is said to be continuous if the map

G 3 g 7→ |〈ψ|Ugφ〉|

is continuous for every ψ, φ ∈ H.

Remark 5.5. In case of superselection rules, continuous symmetries representing a connectedtopological group cannot swap different coherent sectors when acting on pure states [Mor18] fortopological reasons.

A well-known co-homological condition assuring that every unitary-projective representation ofLie groups is equivalent to a unitary one is due to Bargmann [BaRa84, Mor18].

Theorem 5.2. [Bargmann’s criterion] Let G be a connected and simply connected (realfinite dimensional) Lie group with Lie algebra g. Every continuous unitary-projective represen-tation of G in a Hilbert space H is equivalent to a strongly-continuous unitary representation ofG on H if, for every bilinear antisymmetric map Θ : g× g→ R such that

Θ([u, v], w) + Θ([v, w], u) + Θ([w, u], v) = 0 , ∀u, v, w ∈ g (5.16)

there is a linear map α : g→ R such that

Θ(u, v) = α([u, v]) for all u, v ∈ g. (5.17)

Remark 5.6. The condition is equivalent to require that the second cohomology groupH2(g,R) is trivial.

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Example 5.2. Let us prove that the connected simply-connected group SU(2) satisfies thehypotheses of Bargmann’s Theorem 5.2. As is well known (e.g., see [HiNe13]), SU(2) is connectedand simply connected. We must prove that the condition (5.17) is satisfied. The Lie algebrasu(2) of SU(2) is made by all of anti Hermitian 2×2 matrices. As a real vector space, it is threedimensional and, in particular, admits a basis T1, T2, T3 of anti Hermitian matrices given byTk := − i

2σk. Therefore [Ta, Tb] =∑3c=1 εabcTc, where εabc ∈ R is completely antisymmetric for

a, b, c ∈ 1, 2, 3 and ε123 = 1. Now consider Θ : su(2)× su(2)→ R bilinear and antisymmetric.It is completely determined by the numbers Θab := Θ(Ta, Tb) = −Θba. In fact, considering apair of generic vectors u =

∑3a=1 taTa and v =

∑3b=1 sbTb, we have

Θ(u, v) = Θ

(3∑

a=1

taTa,3∑b=1

sbTb

)=

3∑a=1

3∑b=1

tasbΘab .

By direct inspection one sees that, as Θab = −Θba, we also have Θab =∑3c=1 αcεcab, where

α1 = Θ23, α2 := Θ31, α3 := Θ12. Finally observe that, defining α : su(2)→ R as follows

α

(3∑

a=1

taTa

):=

3∑a=1

αata , with αa := α(Ta),

we have

α

([3∑

a=1

taTa,3∑b=1

sbTb

])=

3∑a=1

3∑b=1

tasbα ([Ta, Tb]) =3∑

a,b,c=1

tasbεabcα (Tc) =3∑

a,b,c=1

tasbεabcαc .

Now, notice that∑3c=1 εabcαc =

∑3c=1 εcabαc, so that

α([u, v]) = α

([3∑

a=1

taTa,3∑b=1

sbTb

])=

3∑a,b,c=1

tasbαcεcab

=3∑a,b

tasbΘab = Θ

(3∑

a=1

taTa,3∑b=1

sbTb

)= Θ(u, v) .

We have proved that (5.17) is satisfied Θ(u, v) = α([u, v]), for all u, v ∈ su(2). We stress thatwe have not even imposed the constraint (5.16)

Θ([u, v], w) + Θ([v, w], u) + Θ([w, u], v) = 0 , ∀u, v, w ∈ su(2) ,

since this identity is automatically true in our case as the reader can prove.

Remark 5.7. The hypothesis of simple connection in Bargmann’s theorem is not so fun-damental. If G is a connected Lie group which is not simply connected, every continuousunitary-projective representation G 3 g 7→ Vg can be viewed as a continuous unitary-projective

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representation of the universal covering ‹G of G (which has the same Lie algebra) taking ad-vantage of the canonical projection homomorphism π : ‹G → G (which is a surjective Lie-grouphomomorphism and a local Lie-group isomorphism) as follows‹G 3 h 7→ Uh := Vπ(h) .

Notice also that if V is irreducible, U is irreducible as well, since irreducibility depends on theimage of U and V which are identical. By definition, ‹G is connected and simply connectedand if remaining Bargmann’s hypotheses are true, U turns out to be unitarisable. In this case,knowing all (irreducible) strongly-continuous unitary representations of ‹G, we also know up toequivalence all (irreducible) continuous unitary-projective representations of G.

Example 5.3. The discussion above contains the reason why, though not all irreduciblecontinuous unitary-projective of the connected but non-simply connected group SO(3) are uni-tarisable and annoying phases show up, they however can be obtained as proper irreduciblestrongly-continuous unitary representation of SO(3)’s universal covering SU(2) (which satisfiesBargmann’s hypotheses as seen in Example 5.2).Let us briefly analyse the structure of the arising representations. Since (e.g., see [Mor18])ker(π) = ±I where π : SU(2) → SO(3) is the canonical projection homomorphism, twocases are possible for a given irreducible unitary representation SU(2) 3 g 7→ Ug. Observethat U−IUg = U−I·g = Ug·(−I) = UgU−I for every g ∈ SU(2). Since the representationis irreducible, Schur’s lemma (Theorem 4.2) implies U−I = χIB(H) for some χ ∈ T. AsIB(H) = UI = U−I·(−I) = χ2IB(H) we conclude that there are only two possibilities, eitherU−I = IB(H) or U−I = −IB(H). Now assume to consider all irreducible strongly-continuousunitary representations U : SU(2)→ B(H).

(1) If U−I = IB(H), then SU(2) 3 g 7→ Ug can be seen as an irreducible unitary representationSO(3) 3 R 7→ VR as well, where VR := Uπ−1(R). This is a well-posed definition sinceπ−1(R) = ±gR, but U−gR = U−IgR = U−IUgR = UgR . Notice that SO(3) 3 R 7→ Uπ−1(R)

is also strongly-continuous if U is, because SO(3) is homeomorphic to SU(2)/ker(π) withthe quotient topology2 and V π = U . These unitary representations of SU(2) are saidinteger-spin representations.

(2) If U−I = −IB(H) the picture is different. In this case, VR := Uπ−1(R) would be ill-definedbecause π−1(R) = ±gR, but UgR = −U−gR . However, choosing one element of SU(2)among ±gR for every given R, we obtain this way a unitary-projective representation ofSO(3) whose multiplicators take values in ±1. The so-defined map V : SO(3)→ B(H)satisfies |〈ψ|Vπ(g)φ〉| = |〈ψ|Ugφ〉| and, varying g ∈ SU(2), the latter function is continuous.By definition of quotient topology, as SO(3) is homeomorphic to SU(2)/ker(π) the mapSO(3) 3 R 7→ |〈ψ|VRφ〉| is continuous. Hence, V : SO(3) → B(H) is continuous as aunitary-projective representation. These irreducible representations of SU(2) are calledsemi-integer-spin representations.

2Hence A ⊂ SU(2)/ker(π) = SO(3) is open if and only if π−1(A) ⊂ SU(2) is open.

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Due to Remark 5.7, all possible irreducible continuous unitary-projective representation of SO(3)are constructed this way up to equivalence and necessarily stay in one of the two classes definedabove. The (semi-integer-spin) unitary-projective representations of SO(3) are often interpretedas many-valued unitary representations.As observed in Section 5.3.1, all strongly-continuous unitary representations of SU(2) are directsums of irreducible strongly-continuous finite-dimensional unitary representations of SU(2) inview of Peter-Weyl’s theorem. So, considering irreducible representations is not reductive.It is finally important to stress that the use of the proper unitary representations of SU(2)is only due to mathematical convenience, but there is no physical reason to prefer them tounitary-projective representations of SO(3) where multipliers show up. The physical group ofsymmetries is SO(3), not SU(2), and the action of SO(3) on states and observables is not af-fected by multipliers as is evident form (5.13)-(5.15).

In the general case, Bargmann’s theorem is not valid even ifG is connected and simply connected,so that non-unitarisable unitary-projective representations do exist and one has to deal withthem. There is nevertheless an overall way to circumvent the technical problem viewing them asunitary representations of another group. Given a unitary-projective representation G 3 g 7→ Ugwith multiplier ω, let us put on U(1)×G the group structure arising by the product

(χ, g) (χ′, g′) = (χχ′ω(g, g′), g · g′)

and indicate by Gω the obtained group. The map

Gω 3 (χ, g) 7→ χUg =: V(χ,g)

is a unitary representation of Gω. If the initial representation is normalized, Gω is said to be acentral extension of G by means of U(1)(= T) [Var07, Mor18]. Indeed, the elements (χ, e),χ ∈ U(1), commute with all the elements of Gω and thus they belong to the centre of thegroup. It is possible to prove that, with a suitable topology (different fron the product one ingeneral), Gω acquires the structure of a topological group if G is topological, and Lie if G is Lie[Var07, Mor18].These types of unitary representations of central extensions play a remarkable role in physics.Sometimes Gω with a particular choice for ω is seen as the true group of symmetries at quantumlevel, when G is the classical group of symmetries.

5.2.4 Inequivalent unitary-projective representations and superselection rules

The notion of equivalence given in (5.12) can be extended to pairs of unitary-projective repre-sentations G 3 g 7→ Ug ∈ B(H) and G 3 g 7→ U ′g ∈ B(H′) defined in different Hilbert spaces Hand H′. Again, they are said to be equivalent if there is an assignment G 3 g 7→ χg ∈ T suchthat (5.12) is valid for the respective multipliers.A such pair of unitary-projective representations, once we have re-arranged the multipliers so

217

that they are identical, can be summed together giving rise to an overall unitary-projectiverepresentation on the Hilbert space K := H⊕ H′,

G 3 g 7→ Ug ⊕ U ′g ∈ B(H⊕ H′) .

The obtained map G 3 g 7→ Ug ⊕ U ′g is a well-behaved unitary-projective representation: if themultipliers ω and ω′ of U and U ′ respectively are equal, then for any g, h ∈ G,

(Ug ⊕ U ′g)(Uh ⊕ U ′h) = UgUh ⊕ U ′gU ′h = ω(g, h)Ug·h ⊕ ω′(g, h)U ′g·h = ω(g, h)ÄUg·h ⊕ U ′g·h

ä.

If, conversely, the representations are not equivalent, it is impossible re-arranging the phases todefine an overall unitary-projective representation over the entire space K and G cannot be inter-preted as a symmetry group for a quantum system described on K (through a unitary-projectiverepresentation which reduces to U and U ′ over the subspaces H and H′).There is however a way out when suitable Abelian superselection rules occur (Section 4.3.1).Sometimes, it happens that the Hilbert space of the system H is a direct orthogonal sumH =

⊕j∈J Hj of closed subspaces and these subspaces are invariant under respective unitary-

projective representations G 3 g 7→ U(j)g ∈ B(Hj) of a common group G of quantum symmetries.

If some pairs of these unitary-projective representations are not equivalent, there is no globalaction of the group over the entire Hilbert space defined as the sum of the representations,since as already observed this sum cannot define a unitary-projective representation. So, if His the Hilbert space of the system, that is every orthogonal projector P ∈ L (H) represents anelementary observable of the system, the group G cannot be directly interpreted as a group ofsymmetries. If however each Hj is a superselection sectors or, more weakly, is a Hilbert sum ofsuperselection sectors, then the orthogonal projectors representing observables belong to the lat-tice LR(H) of the von Neumann algebra R of the observables of the system (see Sect.4.3.1) andhave the form P = ⊕j∈JPj , where Pj ∈ L (Hj). In this case, a global action of G is permittedand is defined as

hg : ⊕j∈JPj 7→ ⊕j∈Jh(j)g (Pj) = ⊕j∈JU (j)

g PjU(j)−1g

This is action is not induced by a unitary-projective representation of G over H, but it workswell anyway as a representation of G made of automorphisms of LR(H). In fact, the differentphases arising when composing the representations of different elements g, g′ cancel each other

hg(hg′ (⊕j∈JPj)

)= ⊕j∈JU (j)

g U(j)g′ PjU

′(j)∗g U (j)∗

g = ⊕j∈Jω(j)(g, g′)ω(j)(g, g′)U(j)g·g′PjU

(j)∗g·g′

= ⊕j∈JU (j)g·g′PjU

(j)∗g·g′ = hg·g′ (⊕j∈JPj) .

There are two important examples of this situation dealing with continuous unitary-projectiverepresentations discussed below.

Example 5.4.(1) A superselection rule arises when representing the group of spatial rotations SO(3). Allthese representations can be seen as continuous unitary-projective representations of SU(2) ac-cording to Example 5.3. The continuous irreducible unitary-projective representations of this

218

group are divided into two equivalence classes in accordance with the value of an observable ofthe quantum system, the total squared angular momentum J2. The spectrum of this observableis a point spectrum and their eigenvalues are ~j(j + 1) where j = 0, 1/2, 1, 3/2, 2, . . . Everyeigenspace of J2 is invariant and irreducible (or is a direct sum of irreducible closed subspaceswhere J2 attains the same value) for the action of a suitable unitary-projective representationof SO(3). All irreducible representations associated with j = 0, 1, 2, . . . are pairwise equivalent(also with different values of j of the said type) and are also proper strongly-continuous unitaryrepresentations of SO(3) because they are integer-spin representations according to Example5.3. All irreducible representations associated with j = 1/2, 3/2, 5/2, . . . are similarly pairwiseequivalent, but the representations of the first class are not equivalent to those of the second classwhich is made of semi-integer-spin representations according to Example 5.3. A superselectionrule occurs splitting the Hilbert space into two sectors, one is the sum of irreducible closed sub-spaces associated to integer values of j and the other is the sum of irreducible closed subspacesassociated to semi-integer values of j. It is possible to associate a superselection charge to thisstructure according to the discussion in Sect.4.3.1. For instance, associating the eigenvalue 0to the space of semi-integer j and the eigenvalue 1 to the space of integer j. Obviously, thissuperselection rule may be accompanied by further compatible rules (e.g., electrical charge su-perselction rule) producing a finer structure of sectors.

(2) Another important case of superselection rule related to inequivalent unitary-projectiverepresentations of the (universal covering of the) Galileian group G – the group of coordinatetransformations between inertial reference frames in classical physics, viewed as active transfor-mations. As clarified by Bargmann [Mor18], the only physically relevant continuous unitary-projective representations of G in QM are just the ones which are not equivalent to unitaryrepresentations! Furthermore there is an infinite set of inequivalent classes of these representa-tions. The multipliers embody the information about the mass m of the system, as they takethe form ωm(g, g′) = eimf(g,g′) with f : G × G → R an universal smooth function. Differentvaluess m ∈ (0,+∞) produce inequivalent continuous unitary-projective representations. Thisphenomenon, according to the discussion above, gives rise to a famous superselection structurein the Hilbert space of quantum systems admitting the Galileian group as a symmetry group,known as Bargmann’s superselection rule (see [Mor18] for a summary). There, the superse-lection charge can be defined as the mass of the system provided their values are discrete. Inother words, different superselection sectors are labelled by different eigenvalues m of the mass,thinking of the mass as a proper quantum observable, a selfadjoint operator M . Differentlyfrom the electric charge, however, the eigenvalues of the mass are not proportional to a givenelementary mass m0. As a consequence, no compact global gauge group can be introduced todescribe this Abelian superselection rule (see Section 4.3.2) if we want to use the mass operatorM (divided for some unit of mass) as superselection charge Q appearing in the exponent of(4.10): A representation of the non-compact Abelian group R – R 3 r 7→ eirM – can be howeverexploited according to beginning of Section 4.3.2. As before, this mass superselection rule maybe accompanied by further compatible superselction rules producing a finer structure of sectors.

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5.2.5 Continuous unitary-projective and unitary representations of R

An important consequence of Bargmann’s theorem is the following crucial result which presentthe notion of strongly-continuous one-parameter unitary group as a central tool in QuantumTheory. This theorem can be proved independently from Bargmann’s theorem [Mor18], but theproof is quite technical.

Theorem 5.3. Let γ : R 3 r 7→ Ur be a continuous unitary-projective representation of theadditive topological group R on the Hilbert space H. The following facts hold.

(a) γ is equivalent to a strongly-continuous unitary representation R 3 r 7→ Vr of R on H.

(b) A strongly continuous unitary representation R 3 r 7→ V ′r is equivalent to γ if and only if

V ′r = eicrVr for some constant c ∈ R and all r ∈ R.

Proof. (a) Let us view the connected simply-connected topological additive group R as a Liegroup (with the same topology) embedded in GL(2,R) made of all the 2× 2 matrices

Ar :=

ñ1 r0 1

ôwith r ∈ R. To this end, observe thatñ

1 a0 1

ô ñ1 b0 1

ô=

ñ1 a+ b0 1

ôR 3 r 7→ Ar ∈ GL(2,R) is a continuous injective group homomorphism which is also a homeo-morphism on its image. The two groups are therefore isomorphic as topological groups. Further-more the set of matrices Ar is a closed subgroup of GL(2,R), so that (due to Cartan’s theorem)they form a Lie subgroup of GL(2,R). In this picture, the Lie algebra of R is R itself, moreprecisely it is the one-dimensional real subspace of the Lie algebra of GL(2,R)

Ta :=

ñ0 a0 0

ôfor a ∈ R. In fact this is the linear space obtained by taking the derivative at the origin of adifferentiable curve r 7→ Ar such that A0 = I. The commutator in the Lie algebra of R, is therestriction of the one of GL(2,R): by direct inspection, [Ta, Tb] = TaTb − TbTa = 0. Since theLie algebra of R is one-dimensional as it coincides with R itself as a vector space, the uniqueantisymmetric map Θ : R × R → R is the zero map. So, Bargmann’s condition is satisfiedtrivially for the Lie group R.(b) If R 3 t 7→ Vt is strongly-continuous and c ∈ R, evidently R 3 t 7→ V ′t := eictVt is stillstrongly-continuous and is equivalent to the same unitary-projective representation of V . Let usprove the converse result. Suppose that V ′ and V are strongly-continuous unitary representation

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obtained out of the continuous unitary-projective representation U of R. As a consequence,V ′t = χ(t)Vt for some map χ : R → T = z ∈ C | |z| = 1 in particular. If 0 6= x ∈ H,we also have χ(t)〈x|x〉 = 〈x|V−tV ′t x〉 = 〈Vtx|V ′t x〉. Therefore χ is continuous. The identityV ′t V

′t′ = χ(t)χ(t′)VtVt′ implies V ′t+t′ = χ(t)χ(t′)Vt+t′ , that is χ(t + t′)Vt+t′ = χ(t)χ(t′)Vt+t′ , so

that χ(t + t′) = χ(t)χ(t′) since Vt+t′ is invertible. There exist only one type of continuousfunctions χ = χ(t) satisfying χ(t + t′) = χ(t)χ(t′) for all t, t′ ∈ R, the exponential functions ofthe form χ(t) = ebt. Since |χ(t)| = 1, it must eventually be χ(t) = eict for some c ∈ R.

The above unitary representations of R include the strongly-continuous one-parameter unitarygroups already encountered in Proposition 2.33, where we treated an apparently particular case.

Definition 5.7. If H is a Hilbert space, V : R 3 r 7→ Vr ∈ B(H), such that

(i) Vr is unitary for every r ∈ R

(ii) V0 = I and VrVs = Vr+s for all r, s ∈ R,

is called one-parameter unitary group. It is called strongly-continuous one-parameterunitary group if in addition to (i) and (ii) we also have

(iii) V is continuous referring to the strong operator topology: Vrψ → Vr0ψ for r → r0 andevery r0 ∈ R and ψ ∈ H.

Remark 5.8.(a) It is evident that, in view of the group structure, a one-parameter unitary group R 3

r 7→ Ur ∈ B(H) is strongly continuous if and only if is strongly continuous for r = 0. In fact,

||Urψ − Usψ|| = ||U−s(Urψ − Usψ)|| = ||Ur−sψ − ψ||

and r → s implies r − s→ 0.(b) It is a bit less evident, but equally true, that a one-parameter unitary group R 3 r 7→

Ur ∈ B(H) is strongly continuous if and only if it is weakly continuous at r = 0. Indeed, if V isweakly continuous at r = 0, for every ψ ∈ H, we have

||Urψ − ψ||2 = ||Urψ||2 + ||ψ||2 − 〈ψ|Urψ〉 − 〈Urψ|ψ〉 = 2||ψ||2 − 〈ψ|Urψ〉 − 〈Urψ|ψ〉 → 0

for r → 0.It is possible to find other apparently even weaker conditions actually equivalent to strong con-tinuity as discussed in [Mor18].

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5.2.6 Stone’s theorem

Theorem 5.3 establishes that, dealing with continuous unitary projective representation of theadditive topological group R, one can always reduce to work with proper strongly-continuousone-parameter unitary groups. The one-parameter unitary group which are not strongly con-tinuous are not so many in separable Hilbert spaces in view of the following result due to vonNeumann (for a proof see, e.g., [Sim76, Mor18]).

Theorem 5.4. If H is a separable complex Hilbert space and V : R 3 r 7→ Vr ∈ B(H) is a one-parameter unitary group, then V is strongly continuous if and only if the maps R 3 r 7→ 〈ψ|Urφ〉are Borel measurable for all ψ, φ ∈ H.

Let us come to a celebrated result due to Stone which characterizes strongly-continuous one-parameter unitary group. We already knows that, if A is a selfadjoint operator in a Hilbertspace, Ut := eitA, for t ∈ R, defines a strongly-continuous one-parameter unitary group as es-tablished in Proposition 2.33 (and Proposition 2.34). The result is remarkably reversible.

Theorem 5.5. [Stone theorem] Let R 3 t 7→ Ut ∈ B(H) be a strongly-continuous one-parameter unitary group in the Hilbert space H. The following facts hold.

(a) There exists a selfadjoint operator A : D(A)→ H such that

Ut = eitA , ∀t ∈ R . (5.18)

(b) If (5.18) is valid for some selfadjoint operator A, then

D(A) =

ßψ ∈ H

∣∣∣∣ ∃ limt→0

1

t(Ut − I)ψ ∈ H

™, Aψ = −i lim

t→0

1

t(Ut − I)ψ . (5.19)

(c) The selfadjoint operator A satisfying (5.18) is unique and is called the (selfadjoint in-finitesimal) generator of U .

(d) Ut(D(A)) = D(A) for all t ∈ R and

AUtψ = UtAψ if ψ ∈ D(A) and t ∈ R.

Proof. We have to prove (a), (b) and (c), since (d) has been established in Propositions 2.33and 2.34. We need a technical lemma to go on.

Lemma 5.1. Let H be a Hilbert space, f ∈ Cc(R) and ψ ∈ H. If R 3 t 7→ Vt ∈ B(H) is aweakly-continuous map such that ||Vt|| < K for all t ∈ R and some K < +∞ then the followingfacts hold.

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(a) There is a unique vector, denoted by∫R f(t)Vtψdt, such that≠

φ

∣∣∣∣∫Rf(t)Vt ψdt

∑=

∫Rf(t)〈φ|Vtψ〉dt for all φ ∈ H.

(b) For every B ∈ B(H),

B

∫Rf(t)Vtψdt =

∫Rf(t)BVtψdt .

(c) The estimate is valid, ∣∣∣∣∣∣∣∣∫Rf(t)Vtψdt

∣∣∣∣∣∣∣∣ ≤ ∫R|f(t)|||Vtψ||dt .

(d) If g ∈ Cc(R) and a, b ∈ C, then∫R

(af(t) + bg(t))Vtψdt = a

∫Rf(t)Vtψdt+ b

∫Rg(t)Vtψdt .

Proof. (a) By hypothesis, H 3 φ 7→∫R f(t)〈φ|Vtψ〉dt is well defined as the integrand function is

continuous and compactly supported. This map is anti linear in φ and also continuous because,taking advantage of Cauchy-Schwartz inequality, |

∫R f(t)〈φ|Vtψ〉dt| ≤

∫R |f(t)||〈φ|Vtψ〉|dt ≤

||φ||||ψ||K∫R |f(t)|dt. Riesz’ lemma therefore implies that it can be written as H 3 φ 7→ 〈φ|ψV,f,t〉

for a unique ψV,f,t ∈ H. By definition,∫R f(t)Vtψdt := ψV,f,t.

(b) Observe that R 3 t 7→ BVt ∈ B(H) is weakly continuous and ||BVt|| ≤ ||B||K, so∫R f(t)BVtψdt is well defined. From (a)≠

φ

∣∣∣∣B ∫Rf(t)Vt ψdt

∑=

≠B∗φ

∣∣∣∣∫Rf(t)Vt ψdt

∑=

∫Rf(t)〈B∗φ|Vtψ〉dt =

∫Rf(t)〈φ|BVtψ〉dt .

Using (a) again, we conclude that B∫R f(t)Vtψdt =

∫R f(t)BVtψdt.

(c) From (a) used twice and exploiting again Cauchy-Schwartz inequality in the penultimatepassage,∣∣∣∣∣∣∣∣∫

Rf(t)Vtψdt

∣∣∣∣∣∣∣∣2 =

∣∣∣∣≠∫Rf(s)Vsψds

∣∣∣∣∫Rf(t)Vt ψdt

∑∣∣∣∣ =

∣∣∣∣∫Rf(t)

≠∫Rf(s)Vsψds

∣∣∣∣Vtψ∑ dt∣∣∣∣=

∣∣∣∣∫R

∫Rf(s)f(t)〈Vsψ|Vtψ〉dsdt

∣∣∣∣ ≤ ∫R

∫R|f(s)||f(t)|||Vsψ||||Vtψ||dsdt =

Å∫R|f(t)|||Vtψ||dt

ã2

.

The proof of (d) is evident from (a) and linearity of the scalar product in the right argument.

Let us come back to the main proof.(a) We first construct a candidate generator for U over a special dense subspace D. Taking

advantage of Lemma 5.1, define the subspace D made of all finite linear combinations of functionsψf :=

∫R f(t)Utψdt for every f ∈ C∞0 (R) and ψ ∈ H, that, in view of (d) of the proved lemma

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coincides with the set of ψf itself. This subspace is dense in H. To prove it, observe that,specializing Vt := Ut − I in Lemma 5.1,

||ψf − ψ|| =∣∣∣∣∣∣∣∣∫

Rf(t)(Ut − I)ψdt

∣∣∣∣∣∣∣∣ ≤ ∫R|f(t)|||(Ut − I)ψ||dt ≤

∫R|f(t)|dt sup

t∈supp(f)||(Ut − I)ψ|| .

For every ε > 0, we can now define fε(x) := 1ε g(x/ε) where g ∈ C∞0 (R) satisfies supp(g) ⊂ [−1, 1]

and∫R gdt = 1, so that

∫R fεdt = 1 and supp(fε) ⊂ [−ε, ε]. Inserting this choice in the found

inequality,0 ≤ ||ψfε − ψ|| ≤ sup

t∈[−ε,ε]||(Ut − I)ψ|| .

As R 3 t 7→ Ut is strongly continuous and U0 = I, we obtain that ψfε → ψ as ε → 0 for everyψ ∈ H. Hence D is dense in H.Next we prove that the strong derivative of U at t = 0 can be computed on D. Let us assumes ∈ [−ε, ε] for some ε > 0. With ψf as above, K = [−a, a] such that supp(f) ⊂ [−a, a] for asufficiently large a > 0, and taking advantage of Lemma 5.1,

1

s(Us − I)ψf =

1

s(Us − I)

∫Kf(t)Utψdt =

1

s

∫Kf(t)Ut+sψdt−

1

s

∫Kf(t)Utψdt

=1

s

∫Kε

f(t− s)Utψdt−1

s

∫Kf(t)Utψdt =

1

s

∫Kε

f(t− s)Utψdt−1

s

∫Kε

f(t)Utψdt

=

∫Kε

f(t− s)− f(t)

sUtψdt . (5.20)

where Kε := [−a− ε, a+ ε] ⊃ K. Now, assuming that f is real, Larange’s theorem implies that∣∣∣f(t−s)−f(t)s

∣∣∣ = |f ′(ξt,s)| < C < +∞ where ξt,s ∈ Kε, and C does not depend on t and s as the

continuous function f ′ is bounded over the compact Kε. The result trivailly extends to the caseof f complex, decomposing it into real and imaginary part. Dominated convergence theoremproves that, for s→ 0,∣∣∣∣∣∣∣∣1s (Us − I)ψf − ψ−f ′

∣∣∣∣∣∣∣∣ =

∣∣∣∣∣∣∣∣∣∣∫Kε

Çf(t− s)− f(t)

s+ f ′(t)

åUtψdt

∣∣∣∣∣∣∣∣∣∣

≤∫Kε

∣∣∣∣∣f(t− s)− f(t)

s+ f ′(t)

∣∣∣∣∣ ||Utψ||dt = ||ψ||∫Kε

∣∣∣∣∣f(t− s)− f(t)

s+ f ′(t)

∣∣∣∣∣ dt→ 0 .

We can therehence define the operator ‹A : D → D ⊂ H by means of‹Aψf := −i lims→0

1

s(Us − I)ψf = −iψ−f ′ , (5.21)

linearly extended to finite linear combinations of ψf . Observe that

Uu(D) = D and Uu‹A = ‹AUu ∀u ∈ R . (5.22)

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The first identity arises form the definition of D and (b) in Lemma 5.1 and finally observingthat U−1

u = U−u. The second identity is an immediate consequence of the first identity, thedefinition of ‹A in (5.21), continuity of Uu and, again, (b) in Lemma 5.1.Let us now prove that ‹A is essentially selfadjoint. First observe that it is symmetric because itis densely defined and Hermitian:

〈ψg|‹Aψf 〉 =

≠ψg

∣∣∣∣−i lims→0

1

s(Us − I)ψf

∑= lim

s→0

≠i1

s(U∗s − I)ψg

∣∣∣∣ψf .∑= lim

s→0

≠i1

s(U−s − I)ψg

∣∣∣∣ψf .∑ = lims→0

≠−i 1

−s(U−s − I)ψg

∣∣∣∣ψf .∑ =

≠−i lim

s→0

1

s(Us − I)ψg

∣∣∣∣ψf .∑= 〈‹Aψg|ψf 〉 .

Concerning essentially selfadjointness of ‹A, we employ (b) Proposition 2.11 directly. Supposethat there exist φ± ∈ D(‹A∗) such that ‹A∗φ± = ±iφ±. As a consequence, using (5.22) and(5.21), if ψ ∈ D = D(‹A)

d

dt〈Utψ|φ±〉 = lim

s→0

≠1

s(Us − I)Utψ

∣∣∣∣φ±∑ = 〈i‹AUtψ|φ±〉 = 〈iUtψ|‹A∗φ±〉 = ±〈Utψ|φ±〉 .

We have obtained that R 3 t 7→ 〈Utψ|φ±〉 is continuously differentiable and satisfies the writtendifferential equation so that,

〈Utψ|φ±〉 = 〈U0ψ|φ±〉e±t = 〈ψ|φ±〉e±t ∀t ∈ R .

The left-most side is bounded as |〈Utψ|φ±〉| ≤ ||ψ||||φ±||||Ut|| = ||ψ||||φ±||, whereas the right-most side is unbounded unless 〈ψ|φ±〉 = 0. This identity must be true for every ψ ∈ D. Since Dis dense, we conclude that φ± = 0 and thus ‹A is essentially selfadjoint on D for (b) Proposition2.11 and we denote by A its unique selfadjoint extension.To conclude, we can define the strongly continuous one-parameter group of unitary operators R 3t 7→ eitA according to Proposition 2.33. We want to prove that, if ψ, φ ∈ D, then 〈φ|U−teitAψ〉 =〈φ|ψ〉. To this end it is sufficient to establish that

d

dt〈φ|U−teitAψ〉 =

d

dt〈Utφ|eitAψ〉 = 0 .

Indeed, the second derivative is, defining Vt := eitA and using the fact that D is invariant underUt, as demonstrated above, and under and Vt for Proposition 2.33 (since D ⊂ D(A)),

limh→0

1

h(〈Ut+hφ|Vt+hψ〉 − 〈Utφ|Vtψ〉) = lim

h→0

1

h(〈UhUtφ|VhVtψ〉 − 〈Utφ|Vtψ〉)

= limh→0

≠UhUtφ

∣∣∣∣1h(Vh − I)Vt ψ

∑+ limh→0

≠1

h(Uh − I)Utφ

∣∣∣∣Vtψ∑ = 〈Utφ|iAVtψ〉+ 〈iAUtφ|Vtψ〉

i〈AUtφ|Vtψ〉 − i〈AUtφ|Vtψ〉 = 0 ,

225

where we exploited the fact that A is selfadjoint and Proposition 2.34. We have obtained that〈φ|(U−teitA − I)ψ〉 = 0 for all t ∈ R, so that U−te

itA = I because φ, ψ ∈ D which is dense. Insummary, we have proved that Ut = eitA for every t ∈ R and a selfadjoint operator A, concludingthe proof of existence.

(b) Consider a strongly continuous one-parameter group of unitary operators Ut = eitA,where A is some selfadjoint operator. We already known that if ψ ∈ D(A), then it holds that−i limt→0

1t (Ut−I)ψ = Aψ from Proposition 2.34. We intend to prove that, if limt→0

1t (Ut−I)ψ

exists, then ψ ∈ D(A) and the limit coincides with iAψ. Let us define Bψ := limt→01t (Ut− I)ψ

for all ψ ∈ H such that the right-hand side exists. It is easy to se that B is linear and D(B) isa linear subspace which is dense since it includes D(A). Furthermore, exactly as we did for ‹A,we immediately obtain that B is Hermitian. So B is a symmetric extension of the selfadjointoperator A. Hence it must hold that B = A from (a) Proposition 2.8, concluding the proof.

(c) Suppose that Ut = eitB = eitA for all t ∈ R and a pair of selfadjoint operators A and B.Applying (5.19) we have D(A) = D(B) and Aψ = Bψ for every ψ ∈ D(A) = D(B). The proofis over.

5.2.7 Time evolution, Heisenberg picture and quantum Noether theorem

We come back to some issues already introduced in Section 2.6.3. However the viewpoint adoptedhere is the higest one, based on the general notion of quantum symmetry, and it will permit usto better justify several introduced notions.Consider a quantum system described in the Hilbert space H referred to an inertial referenceframe. Suppose that, physically speaking, the system is either isolated or interacts with someexternal stationary environment. With these hypotheses, temporal homogeneity occurs and thetime evolution of states is axiomatically described by a continuous symmetry, more precisely, bya continuous unitary-projective representation R 3 t 7→ Vt.In view of Theorems 5.3 and 5.5, this group is equivalent to a strongly-continuous one-parametergroup of unitary operators R 3 t 7→ Ut and there is a selfadjoint operator H, called the Hamil-tonian operator, such that (notice the sign in front of the exponent)

Ut = e−i~ tH , t ∈ R , (5.23)

where we have occasionally explicitly written the constant ~. In view of Theorems 5.3 and 5.5,V determines H up to additive real constants: the selfadjoint operator H + cI defines the samecontinuous symmetry V . H is usually identified with the energy of the system in the consideredreference frame and the constant c ∈ R can be fixed using some physical argument and theproblem must be studied case by case.Within this picture, if T ∈ S (H) is the state of the system at t = 0, the state at time t is

Tt = UtTU−1t .

If the initial state is pure and represented by the unit vector ψ ∈ H, the state at time t isψt := Utψ. In this case, as already mentioned in Section 2.6.3, if ψ ∈ D(H) we have that

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ψt ∈ D(H) for every t ∈ R in view of (b) and (d) in Theorem 5.5,

i~dψtdt

= Hψt . (5.24)

where the derivative is computed with respect to the topology of H. One recognises in Eq. (5.24)the general form of Schodinger equation. From now on we pass to assume ~ = 1.

Remark 5.9. It is possible to study quantum systems interacting with some external systemwhich is not stationary. In this case the Hamiltonian observable depends parametrically on timeas already introduced in Section 1.2.1. In these cases a Schrodinger equation is supposed todescribe the time evolution of the system giving rise to a groupoid of unitary operators [Mor18].We shall not enter into the details of this technical issue here.

Adopting the said physical point of view, observables do not evolve and states do. This frame-work is called Schrodinger picture. There is however another approach to describe timeevolution called Heisenberg picture. In that representation, states do not evolve in time, butobservables do according to the dual action (5.4) of the symmetries induced by Ut. In this sense,if A is an observable at t = 0, its evolution at time t is the observable

At := U−1t AUt .

Obviously D(At) = U−1t (D(A)) = U−t(D(A)) = U∗t (D(A)). As already observed in the case of

a generic dual action of a symmetry, according with (j) in Proposition 2.32 the spectral measureof At is

P(At)E = U−1

t P(A)E Ut

as expected. The probability that, at time t, the observable A produces the outcome E whenthe state is ρ at t = 0, can equivalently be computed both using the standard (Schrodinger)

picture, where states evolve as tr(P(A)E ρt), or Heisenberg picture where observables do, obtaining

tr(P(At)E ρ). Indeed

tr(P(A)E ρt) = tr(P

(A)E U−1

t ρUt) = tr(UtP(A)E U−1

t ρ) = tr(P(At)E ρ) .

The two pictures are completely equivalent to describe physics in non-relativistic quantumphysics. In relativistic quantum physics, QFT in particular, Heisenberg picture (covariantlyextended to include spatial translations) is preferable, in view of the existence of a plethora ofdifferent notions of time evolution. Heisenberg picture permits to give the following importantdefinition already introduced in Section 2.6.3.

Definition 5.8. In the Hilbert space H equipped with a strongly-continuous unitary one-parameter group representing the time evolution R 3 t 7→ Ut, an observable A is said to be aconstant of motion with respect to U , if At := U−1

t AUt does not depend on t, i.e. At = A0

227

for every t ∈ R.

The given definition can be improved considering a possible further temporal dependence alreadyin Schroedinger picture.

Definition 5.9. In the Hilbert space H equipped with a strongly-continuous unitary one-parameter group representing the time evolution R 3 t 7→ Ut, a family of parametrically time-depending observables A(t)t∈R is said to be a parametrically-dependent constant of mo-tion with respect to U , if At := U−1

t A(t)Ut does not depend on t, i.e. At = A0 for every t ∈ R.

The meaning of the definition (both cases) should be clear: even if the state evolves, the prob-ability to obtain an outcome E, measuring a constant of motion, remains stationary. Alsoexpectation values and standard deviations do not change in time.We are now in a position to state the corresponding of Noether theorem in QM.

Theorem 5.6. [Quantum Noether theorem] Consider a quantum system described in theHilbert space H equipped with a strongly continuous unitary one-parameter group representingthe time evolution R 3 t 7→ Ut. If A is an observable represented by a (generally unbounded)selfadjoint operator A in H, the following facts are equivalent.

(a) A is a constant of motion: At = A0 for all t ∈ R.

(b) The one-parameter group of symmetries generated by A, R 3 s 7→ e−isA is a group ofdynamical symmetries: It commutes with the time evolution

e−isAUt = Ute−isA for all s, t ∈ R . (5.25)

In particular it transforms the time evolution of a pure state into the evolution of (another)pure state, i.e. e−isA Utψ = Ut e

−isAψ.

(c) The dual action on observables (5.4) (or equivalently the inverse dual action (5.5)) of theone-parameter group of symmetries generated by A, R 3 s 7→ e−isA leaves H invariant.That is

e−isAHeisA = H , for all s ∈ R .

Proof. Suppose that (a) holds. By definition U−1t AUt = A. From Proposition 2.36, we have that

U−1t e−isAUt = e−isA which is equivalent to (b). If (b) is true, we have that e−isAe−itHeisA =

e−itH . Here, Proposition 2.36 yields e−isAHeisA = H. Finally, suppose that (c) is valid. AgainProposition 2.36 produces e−isAUte

isA = Ut, which can be rearranged into U−1t e−isAUt = e−isA.

Eventually, Proposition 2.36 leads to U−1t AUt = A which is (a), concluding the proof.

The theorem can be extended to parametrically depending on time observables A(t)tR.

Theorem 5.7. [Noether quantum theorem II] Consider a quantum system described in theHilbert space H equipped with a strongly continuous unitary one-parameter group representing

228

the time evolution R 3 t 7→ Ut. If A(t)t∈R is a class of observables represented by a (generallyunbounded) selfadjoint operator depending on t, the following facts are equivalent.

(a) A(t)t∈R is a parametrically depending on time constant of motion: At = A0 for all t ∈ R.

(b) The one-parameter groups of symmetries generated by every A(t), R 3 s 7→ e−isA(t) definea group of dynamical symmetries parametrically depending on time:

e−isA(t)Ut = Ute−isA(0) for all s, t ∈ R . (5.26)

In particular it transforms the evolution of a pure state into the evolution of (another)pure state, i.e. e−isA(t) Utψ = Ut e

−isA(0)ψ.

Proof. The proof is trivial exploiting Proposition 2.36. At = A0 means U−1t A(t)Ut = A(0)

which, in turn, implies U−1t e−isA(t)Ut = e−isA(0), namely e−isA(t)Ut = Ute

−isA(0). So (a) entails(b). The reasoning is reversible and from the last identity we obtain U−1

t A(t)Ut = A(0). So (b)imples (a).

At this juncture, a suitable version of (c) Theorem 5.6 could be also stated for the case of ob-servables parametrically depending on time, but, exactly as it happens in Hamiltonian classicalmechanics, it has a more complicated interpretation [Mor18].In physics textbooks the above statements are almost always stated using time derivatives andcommutators. This approach is cumbersome, useless and involves many subtle troubles withdomains of the relevant operators.

Example 5.5.(1) As we seen in Example 2.9, for the free particle in the rest space R3 of an inertial referenceframe, the momentum along x1 is a constant of motion as a consequence of translational invari-ance along that axis. It is assumed that the unitary group representing translations along x1 isUu with (Uuψ)(x) = ψ(x − ue1) if ψ ∈ L2(R3, d3x). The Hamiltonian is H = 1

2m

∑3j=1 P

2j . It

commutes with the one-parameter unitary group describing displacements along x1, because asone can prove the said groups is generated by P1 itself: Uu := e−iuP1 . Theorem 5.6 yields thethesis.

(2) An example of a parametrically time-depending constant of motion is the generator ofthe boost one-parameter subgroup along the axis n of transformations of the Galileian groupR3 3 x 7→ x+ tvn ∈ R3, where the speed v ∈ R is the parameter of the group. The generator is[Mor18] the unique selfadjoint extension of

Kn(t) =3∑j=1

nj(mXj |D − tPj |D) , (5.27)

the constant m > 0 denoting the mass of the system and D being the Garding or the Nelsondomain of the representation of (central extension of the) Galileian group as we will discuss later.

229

(3) In QM there are symmetries described by operators which are simultaneously selfadjoint andunitary, so they are also observables and can be measured. The parity or spatial reflection isone of them: (Pψ)(x) := ηψ(−x) for a particle described in L2(R3, d3x), where η = ±1 does notdepend on ψ. These are constants of motion (U−1

t PUt = P) if and only if they are dynamicalsymmetries (PUt = UtP). This phenomenon has no classical corresponding.

(4) The time-reversal symmetry, when described by an anti unitary operator T is supposedto satisfy: THT−1 = H. However, since it is antilinear, it gives rise to the identity (exercise)Te−itHT−1 = e+itTHT−1

, so that TUt = U−tT as physically expected. We stress that T is asymmetry, but it is not a dynamical symmetry. There is no conserved quantity associated withthis operator (it is not selfadjoint nor linear!).

Exercise 5.1.(1) Prove that if the Hamiltonian observable does not depend on time is a constant of motion.

Solution. In this case the time translation is described by Ut = eitH and, trivially, it com-mutes with Us. Noether theorem implies the thesis 2

(2) Prove that if σ(H) is bounded below but not above, the time reversal symmetry (if any)cannot be unitary.

Solution. We look for an operator, unitary or anti unitary such that TUt = U−tT forall t ∈ R. If the operator is unitary, the said identity easily implies THT−1 = −H andtherefore, with obvious notation, σ(THT−1) = −σ(H). Proposition 2.14 immediately yieldsσ(H) = −σ(H) which is false if σ(H) is bounded below but not above. 2

(3) Referring to the spinless particle, prove that if V : L2(R3, d3x)→ L2(R3, d3x) is unitary andselfadjoint and satisfies

V XkV−1 = −Xk , V PkV

−1 = −Pk for k = 1, 2, 3,

then V = P with P defined in (3) Example 5.5.

Solution. If V and V ′ satisfies the written identities, then V −1V ′ commutes with Xk andPk for k = 1, 2, 3. According to (3) Example 4.1, V −1V ′ = cI for some c ∈ C. Since both Vand V ′ are selfadjoint, c ∈ R. Since both V and V ′ are unitary, then c ∈ T. Hence c = ±1. Toconclude, observe that P defined in (3) Example 5.5 satisfies the hypothesis. 2

(4) Referring to the spinless particle, prove that if T : L2(R3, d3x)→ L2(R3, d3x) is anti unitaryand satisfies

TXkT−1 = Xk , TPkT

−1 = −Pk for k = 1, 2, 3,

then (Tψ)(x) := ηψ(x) for every ψ ∈ L2(R3, d3x) and where η is a phase independent from ψ.

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Solution. First observe that (V ψ)(x) := ψ(x) satisfies the hypotheses as the reader im-mediately proves and ηV also satisfies the hypotheses for every fixed η ∈ T. If T is anotheranti unitary operator satisfying the hypotheses then TV −1 is unitary and commutes with Xk

and Pk for k = 1, 2, 3. Exactly as in the solution of the previous exercise, it must therefore beTV −1 = ηI for some η ∈ T, proving the assert. 2

5.3 More on strongly-continuous unitary representations of Liegroups

Lie groups of symmetry naturally arise in physics when considering the whole group of sym-metries for a given quantum system [BaRa84]. For instance, in classical physics the Lie (or-thochronous proper) Galileian group (where SO(3) is replaced by SU(2)) is assumed to be thegroup of continuous symmetry of every isolated quantum system studied in an inertial referenceframe. Actually no non-trivial, strongly-continuous unitary representations of the Galileiangroup exists and those used in quantum physics are strongly-continuous unitary representationsof a Lie group consisting of a central extension the Galileian group. This happens for rea-sons of physical and mathematical nature: the mass of the system is necessary to describe theaction of the boost in quantum physics and this piece of information is not contained in theGalileian group (but it can be included in the multipliers when constructing central extensions)and, mathematically speaking, Galileian group does not satisfy the co-homological condition inBargmann’s theorem. The Lie (orthochronous proper) Poincare group (where again SU(2) isused rather than SO(3)) replaces the Galileian group in the relativistic realm and their con-tinuous unitary-projective representations are instead always unitarisable because Bargmann’scondition is satisfied [BaRa84].From an abstract viewpoint, the general groups of symmetries of a quantum system – omittingdiscrete symmetries if any – have by definition the structure of a topological group. We can alwayssuppose that this group is connected restricting ourselves to the connected component includingthe identity. Assuming some further natural mathematical-physics hypotheses in addition tocontinuity of group operations: (1) Hausdorff topology3, (2) second countability, and (3) theexistence of local coordinate patches continuously pairwise compatible around every element ofthe group creating a local identification with Rn, then celebrated Gleason-Montgomery-Zippin’stheorem (see [Mor18] for a concise discussion) implies that the topological group is also a Liegroup [HiNe13] with respect to a unique differentiable (analytic) structure homeomorphic to theinitial topological manfold structure.It is worth stressing that the general group of continuous symmetry of a quantum system includesin particular time evolution as a subgroup (also different notions of time evolutions correspond-ing to different choices of the reference frame in the relativistic context).Sometimes these Lie groups can be represented in terms of proper unitary representations, inparticular when Bargmann’s theorem hypotheses are satisfied. If not, central extensions of them

3From the experimental viewpoint, Hausdorff topology means that we can distinguish two different elementsof the group even if our knowledge of them is affected by experimental errors.

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which have the structure of Lie groups can however be unitarily and strongly-continuously rep-resented [Var07, Mor18]. It is therefore not too restrictive limiting ourselves to consider onlystrongly-continuous unitary representations of Lie groups.As general reference texts on unitary and projective-unitary representations of topological andLie groups with relevance in physics, we mention [BaRa84] (though it is not always rigorouslywritten), [Var07] and, for a concise summary on some topics, [Mor18]. A quite complete math-ematical treatise on continuous representations (also of algebras) is [Schm90].

5.3.1 Strongly continuous unitary representations

Before we focus attention on strongly-continuous representations of Lie groups, we start fromstrongly-continuous representations of more general topological groups. Sometimes strongly-continuous representations are simply called continuous representations. This is due to thefollowing elementary result.

Proposition 5.1. If G is a topological group and G 3 g 7→ Ug ∈ B(H) is a unitaryrepresentation over the Hilbert space H, the following facts are equivalent.

(a) U is strongly continuous.

(b) U is strongly continuous at the neutral element.

(c) U is weakly continuous.

(d) U is weakly continuous at the neutral element.

Proof. (a) implies (b) trivially and (b) implies (a) because ||Ugx− Ufx|| = ||Uf−1·gx− I|| fromunitarity. Evidently (a) implies (d) which, in turn, entails (d). Finally (d) implies (a) because||Ugx−x||2 = 〈Ugx−x|Ugx−x〉 = ||Ugx||2−2Re〈x|Ugx〉+||x||2 = ||x||2−2Re〈x|Ugx〉+||x||2.

The theory of strongly-continuous unitary representations of topological groups is an impor-tant part of the theory of representations (see in particular [NaSt82] for a classical treatise onthe subject and [BaRa84] for physical applications). An important general result due to Peterand Weyl concerns Hausdorff compact groups which establishes in particular what follows (see[Mor18] for the complete detailed statement and the proof).

Theorem 5.8. [Peter-Weyl’s basic statement] Let G be a compact Hausdorff topologicalgroup – a compact Lie group in particular – and G 3 g 7→ Ug ∈ B(H) a strongly-continuousunitary representations over the Hilbert space H 6= 0.

(a) If U is irreducible, then H is finite dimensional.

(b) If U is not irreducible, then the orthogonal Hilbert decomposition H =⊕

k∈K Hk holds,where Hk are pairwise-orthogonal non-trivial closed subspaces of finite dimension, sep-arately invariant under U . Furthermore every map U Hk : Hk → Hk is an irreduciblerepresentation of G.

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This result applies in particular to compact Lie groups like SU(n) and SO(n), whose irreduciblestrongly-continuous unitary representations are therefore always finite dimensional. The theoryof the spin deals with strongly-continuous unitary irreducible representations of SU(2) which,as physicists know very well, are finite dimensional just due to Peter-Weyl’s theorem.

Another technical general result on strongly-continuous unitary representations of topologicalgroups is the following one, connecting irreducibility and separability of the Hilbert space. Asbefore, we state the proposition into a more general fashion which includes the case of Lie groups.

Proposition 5.2. Let G 3 g 7→ Ug ∈ B(H) be a strongly-continuous unitary representationof a separable topological group G – a Lie group in particular – in the Hilbert space H. If therepresentation is irreducible, then H is separable.

Proof. Let V ⊂ G be a dense countable set which exists because G is seprable. Next, pickout ψ ∈ H \ 0. Since every Ug : H → H is continuous, the closure H0 of the set of finitecomplex linear combinations of elements Ugψ for g ∈ G is invariant under the action of U .The representation is irreducible and H0 6= 0, so that H0 = H. To conclude, observe that, inview of the strong continuity of G 3 g 7→ Ug, every element in H0 is the limit of finite linearcombinations with rational (complex) coefficients of elements Uhψ where h ∈ V .If G is a Lie group, then it is in particular a second-countable topological space by definitionand therefore it is separable. In fact, if Bnn∈N is a topological basis of G – where we assumethat every Bn 6= ∅ – choose a bn ∈ Bn for every n ∈ N. C := bn | n ∈ N is countable and alsodense because every open neighborhood Og of every g ∈ G necessarily includes some Bp, so thatOg 3 bp ∈ C.

5.3.2 From Garding’s space to Nelson’s theorem

We henceforth restrict our study to Lie groups.

Remark 5.10. In the rest of the chapter we consider only the case of a finite-dimensional realLie group, G, whose Lie algebra is indicated by g endowed with the Lie bracket or commutator , .

A fundamental technical fact is that strongly-continuous unitary representations of (connected)Lie groups are associated with representations of the Lie algebra of the group in terms of(anti)selfadjoint operators. These operators are often physically interpreted as constants ofmotion (generally parametrically depending on time) when the Hamiltonian of the system be-longs to the representation of the Lie algebra. We want to study this relation between therepresentation of the group on the one hand and the representation of the Lie algebra on theother hand. First of all, we define the operators representing the Lie algebra.

Definition 5.10. Let G be a Lie group and consider a strongly continuous unitary represen-

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tation U of G over the Hilbert space H.If A ∈ g let R 3 s 7→ exp(sA) ∈ G be the generated one-parameter Lie subgroup. The self-adjoint generator associated with A

A : D(A)→ H

is the generator of the strongly continuous one-parameter unitary group

R 3 s 7→ UexpsA = e−isA

in the sense of Theorem 5.5.

The expected result is that these generators (with a factor −i) define a representation of the Liealgebra of the group. The utmost reason is that they are associated to the unitary one-parametersubgroups exactly as the elements of the Lie algebra are associated to the Lie one-parametersubgroups. In particular, we expect that the Lie parenthesis correspond to the commutator ofoperators. The technical problem is that the generators A may have different domains. Wetherefore look for a common invariant (because the commutator must be defined thereon) do-main, where all them can be defined simultaneously. This domain should embody all the amountof information about the operators A themselves, disregarding the fact that they are defined inlarger domains. In other words, we would like that the domain is a core ((3) Definition 2.6) foreach generator. There are several candidates for this space, one of the most appealing is the socalled Garding space.

Definition 5.11. Let G be a Lie group and consider a strongly continuous unitary represen-tation U of G over the Hilbert space H. If f ∈ C∞0 (G) and x ∈ H, define

x[f ] :=

∫Gf(g)Ugx dg (5.28)

where dg denotes the left-invariant Haar measure over G and the integration is defined in aweak sense exploiting Riesz’ lemma: since the anti linear map H 3 y 7→

∫G f(g)〈y|Ugx〉dg is

continuous, x[f ] is the unique vector in H such that

〈y|x[f ]〉 =

∫Gf(g)〈y|Ugx〉dg , ∀y ∈ H .

The subspace finitely spanned by all vectors x[f ] ∈ H with f ∈ C∞0 (G) and x ∈ H is called

Garding space of the representation and is denoted by D(U)G .

The subspace D(U)G enjoys very remarkable properties we state in the next theorem. In the

following Lg : C∞0 (G) → C∞0 (G) denotes the standard left-action of g ∈ G on complex-valuedsmooth compactly-supported functions defined on G:

(Lgf)(h) := f(g−1h) ∀h ∈ G , (5.29)

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and, if A ∈ g, XA : C∞0 (G) → C∞0 (G) is the smooth vector field over G (a smooth differentialoperator) defined as:

(XA(f)) (g) := limt→0

f (exp−tAg)− f(g)

t∀g ∈ G . (5.30)

so that that map

g 3 A 7→ XA (5.31)

defines a representation of g in terms of vector fields (differential operators) on C∞0 (G). Weconclude with the following theorem [Schm90, Mor18], establishing that the Garding space hasall the expected properties.

Theorem 5.9. Referring to Definitions 5.10 and 5.11, D(U)G satisfies the following properties.

(a) D(U)G is dense in H.

(b) If g ∈ G, then Ug(D(U)G ) ⊂ D

(U)G . More precisely, if f ∈ C∞0 (G), x ∈ H, g ∈ G, it holds

that

Ugx[f ] = x[Lgf ] . (5.32)

(c) If A ∈ g, then D(U)G ⊂ D(A) and furthermore A(D

(U)G ) ⊂ D(U)

G . More precisely

− iAx[f ] = x[XA(f)] (5.33)

(d) The map

g 3 A 7→ −iA|D

(U)G

=: u(A) (5.34)

is a Lie algebra representation in terms of anti symmetric operators defined on the common

dense invariant domain D(U)G . In other words, the said map is R-linear and, if , is the

Lie commutator of g, we have

[u(A), u(A′)] = u(A,A′) if A,A′ ∈ g.

(e) D(U)G is a core for every selfadjoint generator A with A ∈ g, that is

A = A|D

(U)G

, ∀A ∈ g . (5.35)

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We pass to tackle the inverse problem. So, suppose we are given a representation of a Lie algebrag in terms of anti symmetric operators defined in common invariant subspace of a Hilbert spaceH. We wonder whether or not it is possible to lift the representation to a unitary strongly-continuous representation of the unique simply connected Lie group G admitting g as its Liealgebra. This is a much more difficult problem solved by Nelson [Nel69] introducing anotherrelevant domain in the Hilbert space of the representation.Given a strongly continuous representation U of a Lie group G, there is in fact, another space

D(U)N with similar features to D

(U)G , introduced by Nelson (see, e.g., [Mor18]). This space results

to be more useful than the Garding space to build up the representation U by exponentiating

the Lie algebra representation. D(U)N consists by definition of vectors ψ ∈ H such that the map

G 3 g 7→ Ugψ is analytic in g, i.e. expansible in power series in (real) analytic coordinates around

any point of G. The elements of D(U)N are called analytic vectors of the representation U

and D(U)N is the space of analytic vectors of the representation U . It turns out that D

(U)N

is invariant for every Ug, g ∈ G and that D(U)N ⊂ D

(U)G (this result is by no means trivial and

arises as a consequence of a deep theorem known as Dixmier-Mallievin’s theorem [Mor18] which

implies that ψ ∈ D(U)G if and only if the map G 3 g 7→ Ugψ is smooth).

A remarkable relationship exists between analytic vectors in D(U)N and analytic vectors according

to Definition 2.8. Nelson proved the following important result [Schm90, Mor18], which implies

that D(U)N is dense in H, as we said, because analytic vectors for a selfadjoint operator are dense

(Exercise 2.13). An operator is introduced we call Nelson’s operator.

Proposition 5.3. Let G be a Lie group and G 3 g 7→ Ug a strongly-continuous unitary rep-resentation on the Hilbert space H. Take A1, . . . ,An ∈ g a basis and define Nelson’s operator

on D(U)G by

∆ := −n∑k=1

u(Ak)2 ,

where the iu(Ak) are, as before, the selfadjoint generators Ak restricted to the Garding domain

D(U)G . Then

(a) ∆ is essentially selfadjoint on D(U)G .

(b) Every analytic vector of the selfadjoint operator ∆ is analytic and is an element of D(U)N ,

in particular D(U)N is dense.

(c) Every vector in D(U)N is analytic for every self-adjoint operator iu(Ak), which is therefore

essentially selfadjoint in D(U)N by Nelson’s criterion (Theorem 2.5)

Equipped with the introduced notions, we can eventually state the well-known theorem of Nel-son which associates representations of the only simply connected Lie group with a given Lie

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algebra to representations of that Lie algebra.

Theorem 5.10. [Nelson’s theorem] Consider a real n-dimensional Lie algebra V of operators−iS – with each S symmetric on the Hilbert space H, defined on a common invariant subspaceD dense in H and V -invariant – with the usual commutator of operators as Lie bracket.Let −iS1, · · · ,−iSn ∈ V be a basis of V and define Nelson’s operator with domain D:

∆ :=n∑k=1

S2k .

If ∆ is essentially self-adjoint, there exists a strongly-continuous unitary representation

GV 3 g 7→ Ug

on H, of the unique connected simply-connected Lie group GV with Lie algebra V .U is uniquely determined by the fact that the closures S, for every −iS ∈ V , are the selfadjointgenerators of the representation of the one-parameter subgroups of GV in the sense of Definition5.10. In particular, the symmetric operators S are essentially selfadjoint on D.

The hypotheses of the stated theorem can be relaxed preserving the validity thesis (see [Mor18]also for some further results on the subject).

Exercise 5.2. Let A,B be selfadjoint operators in the Hilbert space H with a commoninvariant dense domain D where they are symmetric and commute. Prove that if A2 + B2 isessentially selfadjoint on D, then the spectral measures of A and B commute.

Solution. Exploit Nelson’s theorem noticing that A,B define the Lie algebra of the additiveAbelian Lie group R2 (which is connected and simply-connected) and that D is a core for A andB, since they are essentially selfadjoint therein by Nelson’s theorem as well.

Example 5.6.(1) Using spherical polar coordinates, the Hilbert space L2(R3, d3x) can be factorised as

L2([0,+∞), r2dr)⊗ L2(S2, dΩ) ,

where dΩ is the natural rotationally invariant Borel measure on the sphere S2 with unit ra-dius in R3, with

∫S2 1dΩ = 4π. In particular a Hilbert basis of L2(R3, d3x) is therefore made

of the products ψn(r)Y lm(θ, φ) where ψnn∈N is any Hilbert basis in L2([0,+∞), r2dr) and

Y lm | l = 0, 1, 2, . . . ,m = 0,±1,±2, . . .± l is the standard Hilbert basis of spherical harmonics

of L2(S2, dΩ) [BaRa84]. Since the function Y lm are smooth on S2, it is possible to arrange the

basis of ψn made of compactly supported smooth functions whose derivatives in 0 vanish atevery order, in order that R3 3 x 7→ (ψn · Y l

m)(x) are elements of C∞(Rn) (and therefore also of

237

S (R3)). Now consider the three symmetric operators defined on the common dense invariantdomain S (R3)

Lk =3∑

i,j=1

εkijXiPj |S (R3)

where εijk is completely antisymmetric in ijk and ε123 = 1. By direct inspection one sees that

[−iLk,−iLh] =3∑r=1

εkhr(−iLr)

so that the finite real span of the operators −iLk is a representation of the Lie algebra of thesimply connected real Lie group SU(2) (the universal covering of SO(3)). Define the Nelson’soperator L 2 :=

∑3k=1 L 2

k on S (R3). Obviously this is a symmetric operator. A well knowncomputation proves that

L 2 ψn(r)Y lm = l(l + 1) ψn(r)Y l

m .

We conclude that L 2 admits a Hilbert basis of eigenvectors. Corollary 2.3 entails that L 2 isessentially selfadjoint. Therefore we can apply Theorem 5.10, infering that there is a stronglycontinuous unitary representation SU(2) 3M 7→ UM of SU(2) (actually it can be proved to bealso of SO(3) since U−I = I). The three selfadjoint operators Lk := Lk are the generators ofthe one-parameter of rotations around the corresponding three orthogonal Cartesian axes xk,k = 1, 2, 3. The one-parameter subgroup of rotations around the generic unit vector n, withcomponents nk, admits the selfadjoint generator Ln =

∑3k=1 nkLk. The observable Ln has the

physical meaning of the n-component of the angular momentum of the particle described inL2(R3, d3x). It turns out that, for ψ ∈ L2(R3, d3x),

(UMψ)(x) = ψ(π(M)−1x) , M ∈ SU(2) , x ∈ R3 (5.36)

where π : SU(2)→ SO(3) is the standard covering map. Equation (5.36) describes the action ofthe 3D rotation group on pure states in terms of quantum symmetries. This representation is,in fact, a subrepresentation of the unitary representation of IO(3) already seen in (1) Example5.1.

(2) Given a quantum system, a quite general situation is that where the quantum symmetriesof the systems are described by a strongly continuous representation V : G 3 g 7→ Vg on theHilbert space H of the system, and the time evolution is the representation of a one-parameterLie subgroup with generator H ∈ g. So that

Vexp(tH) = e−itH =: Ut .

This is the case, for instance, of relativistic quantum particles, where G is the special or-thochronous Poincare group, the semi-direct product SO(1, 3)+ n R4, (or its universal coveringwhere SO(1, 3)+ is replaced by SL(2,C)). Describing non-relativistic quantum particles, the

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relevant group G is an U(1) central extension of the universal covering of the (connected or-thochronous) Galileian group.In this situation, every element of g determines a constant of motion. There are actually twocases.

(i) If A ∈ g and H,A = 0, then the Lie subgroups exp(tH) and exp(sA) commute as, forexample, follows from Baker-Campbell-Hausdorff formula (see [NaSt82, Mor18], for instance).Consequently A is a constant of motion because Vexp(tH) = e−itH and Vexp(sA) = e−isA commute

as well and Theorem 5.6 is valid. In this case e−isA defines a dynamical symmetry in accordancewith the afore-mentioned theorem. This picture applies in particular, referring to a free particle,to A = Jn, the observable describing total angular momentum along the unit vector n computedin an inertial reference frame.

(ii) A bit more complicated is when, for A ∈ g, we find H,A 6= 0. Here we exploit Theorem5.7. In this situation, A defines a constant of motion in terms of selfadjont operators (observables)belonging to the representation of the Lie algebra of G. The difference with respect to theprevious case is that, now, the constant of motion parametrically depend on time. We thereforehave a class of observables A(t)t∈R in the Schrodinger picture, such that At := U−1

t A(t)Ut arethe corresponding observables in the Heisenberg picture. The equation stating that we have aconstant of motion is therefore At = A0.Exploiting the natural action of the Lie one-parameters subgroups on g, let us define the timeparametrised class of elements of the Lie algebra

A(t) := exp(tH)A exp(−tH) ∈ g , t ∈ R .

If Akk=1,...,n is a basis of g, it must consequently hold that

A(t) =n∑k=1

ak(t)Ak (5.37)

for some real-valued smooth functions ak = ak(t). By construction, the corresponding class ofselfadjoint generators A(t), t ∈ R, define a parametrically time dependent constant of motion.Indeed, since (exercise)

exp(s exp(tH)A exp(−tH)) = exp(tH) exp(sA) exp(−tH) ,

we have

−iA(t) =d

ds|s=0Vexp(s exp(tH)A exp(−tH)) =

d

ds|s=0Vexp(tH) exp(sA) exp(−tH)

=d

ds|s=0Vexp(tH)Vexp(sA)Vexp(−tH) = −iUtAU−1

t

Therefore, as pre-announced, we end up with a constant of motion parametrically dependent ontime,

At = U−1t A(t)Ut = U−1

t UtAU−1t Ut = A = A0 .

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In view of Theorem 5.9, as the map g 3 A 7→ A|D

(V )G

is a Lie algebra isomorphism, we can recast

(5.37) for selfadjoint generators

A(t)|D

(V )G

=n∑k=1

ak(t)Ak|D(V )G

(5.38)

(where D(V )G could be replaced by D

(V )N as the reader can easily establish, using Proposition 5.3

and Theorem 5.10). Since D(V )G (resp. D

(V )N ) is a core for A(t), it also hold that

A(t) =n∑k=1

ak(t)Ak|D(V )G

, (5.39)

the bar denoting the closure of an operator as usual. (The same is valid replacing D(V )G for

D(V )N .)

A relevant case, both for the non-relativistic and the relativistic framework is the selfadjointgenerator Kn(t) associated with the boost transformation along the unit vector n ∈ R3, therest space of the inertial reference frame where the boost transformation is viewed as an activetransformation. Indeed, referring to the Lie generators of (a U(1) central extension of theuniversal covering of the connected orthochronous) Galileian group, we find

h, kn = −pn 6= 0 ,

where pn is the generator of spatial translations along n, corresponding to the observable momen-tum along the said axis when passing to selfadjoint generators. The non-relativistic expression ofKn(t), for a single particle, appears in (5.27). For an extended discussion on the non-relativisticcase consult [Mor18]. A pretty physically complete discussion encompassing the relativistic caseappears in [BaRa84].

5.3.3 Selfadjoint version of Stone - von Neumann - Mackey Theorem

A remarkable consequence of Nelson’s theorem is a selfadjoint operator version of Stone-vonNeumann theorem usually presented in terms of unitary operators [Mor18], proving that an irre-ducible representation of CCRs always gives rise to the standard representation in L2(Rn, dnx).We state and prove below this version of the theorem, assuming the validity of the unitaryversion of the theorem (e.g., see [Mor18]).

Theorem 5.11. [Stone - von Neumann - Mackey Theorem] Let H be a Hilbert space andsuppose that there are 2n selfadjoint operators in H we indicate with Q1, . . . , Qn and M1, . . . ,Mn

such the following requirements are valid.

(1) There is a common dense invariant subspace D ⊂ H where the CCRs hold

[Qh,Mk]ψ = i~δhkψ , [Qh, Qk]ψ = 0 , [Mh,Mk]ψ = 0 ψ ∈ D , h, k = 1, . . . , n .(5.40)

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(2) The representation is irreducible, in the sense that there is no closed subspace K ⊂ H suchthat Qk(K ∩D(Qk)) ⊂ K and Mk(K ∩D(Mk)) ⊂ K for k = 1, . . . , n.

(3) The operator∑nk=1Q

2k|D +M2

k |D is essentially selfadjoint.

Under these conditions, there is a Hilbert-space isomorphism (a surjective isometric linear map)U : H→ L2(Rn, dnx) such that

UQkU−1 = Xk and UMkU

−1 = Pk k = 1, . . . , n (5.41)

where Xk and Pk respectively are the standard position (2.21) and momentum (2.22) selfadjointoperators in L2(Rn, dnx). In particular H results to be separable.If only (1) and (3) are valid, then H decomposes into an orthogonal Hilbert sum H = ⊕r∈RHrwhere R is finite or countable if H is separable, the Hr ⊂ H are closed subspaces with

Qk(Hr ∩D(Qk)) ⊂ Hr and Mk(Hr ∩D(Mk)) ⊂ Hr

for all r ∈ R, k = 1, . . . , n and the restrictions of all the Qk and Mk to each Hr satisfy (5.41)for suitable surjective linear isometric maps Ur : Hr → L2(Rn, dnx).

Proof. If (1) holds, the restrictions to D of the selfadjoint operators Qk, Mk define symmetricoperators (since they are selfadjoint and D is dense and included in their domains), also theirpowers are symmetric since D is invariant. If also (2) is valid, in view of Nelson’s theorem(since evidently the symmetric operator I|2D +

∑nk=1Q

2k|D + M2

k |D is essentially selfadjoint if∑nk=1Q

2k|D+M2

k |D is), there is a strongly continuous unitary representationW 3 g 7→ Vg ∈ B(H)of the simply connected 2n+ 1-dimensional Lie group W whose Lie algebra is defined by (5.40)(correspondingly re-stated for the operators −iI,−iQk,−iMk) together with [−iQh,−iI] =[−iMk,−iI] = 0, where the operator −iI restricted to D is the remaining Lie generator. Wis the Weyl-Heisenberg group [Mor18]. The selfadjoint generators of this representation arejust the operators Qk and Pk (and I), since they coincide with the closure of their restrictionsto D, because they are selfadjoint (so they admit unique selfadjoint extensions) and D is acore. If furthermore the Lie algebra representation is irreducible, the unitary representationis irreducible, too: If K were an invariant subspace for the unitary operators, Stone theoremwould imply that K be also invariant (in the sense of (2) in the hypotheses of the theorem)under the selfadjoint generators of the one parameter Lie subgroups associated to each Qkand Pk. This is impossible if the Lie algebra representation is irreducible as we are assuming.The standard version of Stone-von Neumann theorem [Mor18] implies that there is isometricsurjective operator U : H → L2(Rn, dnx) such that W 3 g 7→ UVgU

−1 ∈ B(L2(Rn, dnx)) isthe standard unitary representation of the group W in L2(Rn, dnx) generated by Xk and Pk(and I) [Mor18]. Again, Stone theorem immediately yields (5.41). The last statement easilyfollows from the standard form of Mackey’s theorem which completes Stone-von Neumann result[Mor18].

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The result a posteriori gives, in particular, a strong justification of the requirement that theHilbert space of an elementary quantum system, like a particle in non-relativistic quantummechanics, must be separable. Separability also arises from Proposition 5.2 in the relativisticcase when, following Wigner’s ideas, we think of elementary particles as described by strongly-continuous unitary representations of the (universal covering of the orthochronous special)Poincare Lie group.

5.3.4 Pauli’s Theorem

Physically meaningful Hamiltonian operators have spectrum bounded from below to avoid ther-modynamical instability. This fact prevents the definition of a “time operator” canonicallyconjugated with H following the standard way. This result is sometime quoted as Pauli’s theo-rem. As a consequence, the meaning of Heisenberg relations ∆E∆T ≥ ~/2 is different from themeaning of the analogous relations for position and momentum. It is however possible to definea sort of time osservable just extending the notion of PVM to the notion of POVM (positive val-ued operator measure) [Mor18]. POVMs are exploited to describe concrete physical phenomenarelated to measurement procedures, especially in quantum information theory [Bus03, BGL95].

Theorem 5.12. [Pauli’s Theorem] If the Hamiltonian operator σ(H) of a quantum systemis bounded below, there is no selfadjoint operator (time operator) T satisfying the standard CCRwith H and the hypotheses (1), (2), (3) of Theorem 5.11.

Proof. The couple H,T should be mapped to a corresponding couple X,P in L2(R, dx), or adirect sum of such spaces, by means of a Hilbert space isomorphism. In both cases the spectrumof H should consequently be identical to X’s one, namely is R. This fact is forbidden fromscratch.

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Chapter 6

A very quick Introduction to theAlgebraic Formulation

The fundamental theorem 5.3.3 of Stone-von Neumann and Mackey is stated in the jargon oftheoretical physics as follows:“all irreducible representations of the CCRs with a finite, and fixed, number of degrees of freedomare unitarily equivalent,”.The expression unitarily equivalent refers to the existence of the Hilbert-space isomorphism U ,and the finite number of degrees of freedom is the dimension of the Lie algebra spanned by thegenerators I,Xk, Pk.What happens then in infinite dimensions?Jumping from the finite-dimensional case to the infinite-dimensional one corresponds to pass-ing from Quantum Mechanics to Quantum Field Theory (possibly relativistic, and on curvedspacetime [KhMo15]). This is the case when dealing with quantum fields, where the 2n + 1generators

I,Xk, Pk , k = 1, 2, . . . , n

are replaced by a the identity operator and continuum of generators, the so-called quantum fieldoperators at fixed time and the conjugated momentum at fixed time:

I, φ(f), π(g)

which are smeared by arbitrary real-valued functions f, g ∈ C∞0 (R3;R) and, in this sense, onesays that there are infinite dimensions (a different operator for every different function) withrespect to the 2n+ 1 operators of the CCRs above. More precisely

C∞0 (R3;R) 3 f 7→ φ(f) , C∞0 (R3;R) 3 g 7→ π(g)

are linear maps associating test functions f, g to respective selfadjoint operators φ(f), π(g) de-fined on a dense invariant domainD of a Hilbert space H. These observables admit time evolutionand all the picture can be recast into a completely covariant way, but here we consider them

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at fixed time in a fixed Minkowskian reference frame. R3 is, in fact, the rest space of a givenreference frame in Minkowski spacetime. Those field operators satisfy commutation relations,called bosonic commutation relations (bCCR) similar to the ones of Xk and Pk (e.g., see[Haa96, Ara09, KhMo15]).

[φ(f), φ(f ′)] = [π(f), π(f ′)] = 0 , [φ(f), π(f ′)] = i

∫R3f(x)f ′(x)d3x ID .

Then the Stone–von Neumann theorem no longer holds. In this case, theoretical physicists saythat

“there exist irreducible non-equivalent bCCR representations”.In practice, there exist pairs of isomorphic ∗-algebras of field operators, the one generatedby I, φ(f), π(g) (i.e., made of finite complex linear combinations of finite products of theseoperators) in the Hilbert space H and the other generated by I ′, φ′(f), π′(g) in the Hilbert spaceH′ that admit no Hilbert space isomorphism U : H′ → H satisfying:

Uφ′(f) U−1 = φ(f) , Uπ′(g) U−1 = π(g) for any pair f, g ∈ C∞0 (R3).

Pairs of this kind are called unitarily inequivalent. The presence of non-equivalent representa-tions of one single physical system shows that a formulation in a fixed Hilbert space is fullyinadequate, at least because it insists on one fixed Hilbert space, whereas the physical system ischaracterized by a more abstract object: An algebra of observables which may be represented indifferent Hilbert spaces in terms of operators. These representations are not unitarily equivalentand none can be considered more fundamental than the remaining ones. We should abandon thestructure of Hilbert space in order to lay the foundations of quantum theories in broader gen-erality. This programme has been widely developed (see e.g., [BrRo02, Stro05, Haa96, Ara09]),starting from the pioneering work of von Neumann himself, and is nowadays called algebraicformulation of quantum (field) theories. Within this framework it was possible to formalise, forinstance, field theories in curved spacetime in relationship to the quantum phenomenology ofblack-hole thermodynamics.

6.1 Algebraic formulation

The algebraic formulation prescinds, anyway, from the nature of the quantum system and maybe stated also for systems with finitely many degrees of freedom as well [Stro05]. The newviewpoint, in the general case, relies upon two assumptions [Haa96, Ara09, Stro05, Mor18].

A1. A physical system S is described by its observables, viewed now as selfadjoint elementsin a certain C∗-algebra A with unit 11 associated to S.

A2. An algebraic state on AS is a linear functional ω : AS → C such that:

ω(a∗a) ≥ 0 ∀ a ∈ AS , ω(11) = 1 ,

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that is, positive and normalised to 1.

A is not seen here as a concrete C∗-algebra of operators (a von Neumann algebra for instance)on a given Hilbert space, but remains an abstract C∗-algebra. Physically, ω(a) is intepreted asthe expectation value of the observable a ∈ A in state ω.

Remark 6.1.(a) The introductory case of quantum fields discussed above gives rise to a ∗-algebra instead

of a C∗-algebra. There are two possibilities here, either going down to an algebraic formulationfor ∗-algebras (e.g., see [KhMo15]) or formally lifting the ∗-algebra of quantum fields to a properC∗-algebra. This second approach is based on the notion of Weyl C∗-algebra associated to thefield operators (see, e.g., [BrRo02]).

(b) A is usually called the algebra of observables of S though, properly speaking, the observ-ables are the selfadjoint elements of A only.

(c) Differently form the Hilbert space formulation, the algebraic approach can be adoptedto describe both classical and quantum systems. The two cases are distinguished on the base ofcommutativity of the algebra of observables AS : A commutative algebra is assumed to describea classical system whereas a non-commutative one is supposed to be associated with a quantumsystems.

(d) As already stated in Definition 2.10, the notion of resolvent set and spectrum of anelement a of a C∗-algebra A, with unit element 11, are defined analogously to the operator case[BrRo02, Mor18].

ρ(a) := λ ∈ C | ∃(a− λ11)−1 ∈ A , σ(a) := C \ ρ(a) . (6.1)

We know from Proposition 2.40 that

||a|| = supλ∈σ(a)

|λ| if a ∈ A is normal: a∗a = aa∗ . (6.2)

If a is not normal, a∗a is however selfadjoint and hence normal and the C∗-property ||a||2 =||a∗a|| permits us to compute ||a|| in terms of the spectrum of a∗a. As the spectrum is a com-pletely algebraic notion, we conclude that it is impossible to change the norm of a C∗-algebrapreserving its C∗ nature: a unital ∗-algebra admits at most one C∗-norm.Further important consequences straightforwardly arising from (6.1), (6.2), the C∗-property||a∗a|| = ||a||2, and Theorem 2.10 are listed below [Mor18].

Proposition 6.1. Let A and B be unital C∗-algebras and π : A → B a ∗-homomorphism(Definition 2.3). The following facts are valid.

(a) π(A) is a unital sub C∗-algebra of B.

(b) π is norm decreasing (||π(a)||B ≤ ||a||A), hence continuous.

(c) σB(π(a)) ⊂ σA(a) if a ∈ A, and = replaces ⊂ if π is injective.

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(d) π is isometric if and only if it is injective. In particular, ∗-isomorphisms of unital C∗-algebras are isometric.

(e) If A ⊂ B is a unital sub C∗-algebra of B, then σA(a) = σB(a).

Properties (a),(b),(c),(d) apply in particular to ∗-representations (Definition 2.5) of unital C∗-algebras over Hilbert spaces: π : A→ B(H).

(e) A special case of unital C∗-algebra is a von Neumann algebra R in a Hilbert spaceH. It is worth stressing that the algebraic notion of state weaker than the Hilbert space onegiven in terms of positive unit-trace trace-class operators called normal states in the contextof C∗-algebras. A normal state T induces an associated algebraic state ωT : R → C definedas ωT (A) := tr(AT ) for A ∈ R. These algebraic states becomes σ-additive measures when re-stricted to LR(H). Conversely, generic algebraic states restricted to LR(H) givesrise to additivemeasures which are not necessarily σ-additive. (Notice that every algebraic state ω is necessarilycontinuous with respect to the uniform topology of B(H) as it immediately arises form the state-ment of the GNS theorem Theorem 6.1 below, however uniform continuity is not enough. Theappropriate continuity to assure σ-additivity would be that referred to strong operator topologyand this is not guaranteed.) This is reason why algebraic states over a von Neumann algebraare many more than normal states and the notion of algebraic state is less restrictive than thatof normal state.

The most evident a posteriori justification of the algebraic approach lies in its powerfulness[Haa96]. However there have been a host of attempts to account for assumptions A1 and A2and their physical meaning in full generality (see the study of [Emc72], [Ara09] and [Stro05] andespecially the work of I.E. Segal [Seg47] based on so-called Jordan algebras). Yet none seems tobe definitive [Stre07]. An evident difference with respect to the standard QM, where states aremeasures on the lattice of elementary propositions, is that we have now a complete identificationof the notion of state with that of expectation value. This identification would be naturalwithin the Hilbert space formulation, where the class of observables includes the elementaryones, represented by orthogonal projectors, and corresponding to “Yes-No” statements. Theexpectation value of such an observable coincides with the probability that the outcome of themeasurement is “Yes”. The set of all those probabilities defines, in fact, a quantum state of thesystem as we know. However, the analogues of these elementary propositions generally do notbelong to the C∗-algebra of observables in the algebraic formulation. Nevertheless, this is notan insurmountable obstruction. Referring to a completely general physical system and following

[Ara09], the most general notion of state, ω, is the assignment of all probabilities, w(A)ω (a), that

the outcome of the measurement of the observable A is a, for all observables A and all of valuesa. On the other hand, it is known [Stro05] that all experimental information on the measurement

of an observable A in the state ω – the probabilities w(A)ω (a) in particular – is recorded in the

expectation values of the polynomials of A. Here, we should think of p(A) as the observablewhose values are the values p(a) for all values a of A. This characterization of an observable istheoretically supported by the various solutions to the moment problem in probability measure

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theory. To adopt this paradigm we have thus to assume that the set of observables must includeat least all real polynomials p(A) whenever it contains the observable A. This is in agreementwith the much stronger requirement A1.

6.1.1 The GNS reconstruction theorem

It is possible to prove (e.g., see [Mor18]) that positivity of an algebraic state ω : AS 7→ C impliesits continuity and this result also immediately follows form Theorem 6.1 below. So, the set ofalgebraic states on AS is a convex subset in the dual A∗S of AS : if ω1 and ω2 are positive andnormalised linear functionals, ω = λω1 + (1− λ)ω2 is clearly still the same for any λ ∈ [0, 1].Hence, just as we saw for the standard formulation, we can define pure algebraic states as ex-treme elements of the convex body.

Definition 6.1. An algebraic state ω : A → C on the C∗-algebra with unit A is called apure algebraic state if it is extreme in the set of algebraic states. An algebraic state that isnot pure is called mixed.

Surprisingly, most of the entire abstract apparatus introduced, given by a C∗-algebra and a setof states, admits elementary Hilbert space representations when a reference algebraic state isgiven. This is by virtue of a celebrated procedure that Gelfand, Najmark and Segal invented[Haa96, Ara09, Stro05, Mor18]. Referring in particular to the notion of ∗-algebra representation(Definition 2.5), we have the following paramount result.

Theorem 6.1. [GNS reconstruction theorem] Let A be a C∗-algebra with unit 11 andω : A→ C a positive linear functional with ω(11) = 1. Then the following holds.

(a) There exist a triple (Hω, πω,Ψω), where

(1) Hω is a Hilbert space,

(2) πω : A→ B(Hω) a continuous (norm-decreasing) unital ∗-algebra representation,

(3) Ψω ∈ Hω has unit norm

such that:

(i) Ψω is cyclic for πω, namely πω(A)Ψω is dense in Hω,

(ii) 〈Ψω|πω(a)Ψω〉 = ω(a) for every a ∈ A.

(b) If (H, π,Ψ) satisfies (1),(2),(3),(i),(ii) for the said ω, then there exists a unitary operatorU : Hω → H such that

Ψ = UΨω and π(a) = Uπω(a)U−1 for any a ∈ A.

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Remark 6.2. The GNS representation πω : A → B(Hω) is a ∗-homomorphism and thusProposition 6.1 applies. Physically meaningful representations of the algebra of observables aregenerally expected to be faithful, i.e. injective (though non-injective cases exist and may havephysical interpretation). In this case, according to Proposition 6.1, πω is isometric and preservesthe spectra of the elements. A sufficient condition for having πω faithful is that ω(a∗a) = 0implies a = 0 (see, e.g. [Mor18]). If a ∈ A is selfadjoint πω(a) is a selfadjoint operator andits spectrum has the well-known quantum meaning. This meaning, in view of the property ofpermanence of the spectrum for πω faithful, can be directly attributed to the spectrum of a ∈ A:If a ∈ A represents an abstract observable, σ(a) is the set of the possible values attained by a.

As we initially said, it turns out that different algebraic states ω, ω′ give generally rise to uni-tarily inequivalent GNS representations (Hω, πω,Ψω) and (Hω′ , πω′ ,Ψω′): There is no isometricsurjective operator U : Hω′ → Hω such that

Uπω′(a)U−1 = πω(a) ∀a ∈ A .

The fact that one may simultaneously deal with all these inequivalent representations is a rep-resentation of the power of the algebraic approach with respect to the Hilbert space framework.However one may also focus on states referred to a fixed GNS representation. If ω is an alge-braic state on A, every statistical operator over the von Neumann algebra πω(A)′′ in the GNSrepresentation of ω – i.e. every positive, trace-class operator with unit trace T ∈ B1(Hω) –determines an algebraic state A 3 a 7→ tr (Tπω(a)) evidently. This is true, in particular, forΦ ∈ Hω with ||Φ|| = 1, in which case the above definition reduces to A 3 a 7→ 〈Φ|πω(a)Φ〉.

Definition 6.2. If ω is an algebraic state on the C∗-algebra with unit A, every algebraicstate on A obtained either from a every positive, trace-class operator with unit trace in a GNSrepresentation of ω, is called normal state of ω. Their set Fol(ω) is the folium of the algebraicstate ω.

To determine Fol(ω) one can use a fixed GNS representation of ω. In fact, as the GNS repre-sentation of ω varies, normal states do not change, as implied by part (b) of the GNS theorem.

6.1.2 Pure states and irreducible representations

We explain now how pure states are characterised in the algebraic framework. To this end wehave the following simple result (e.g., see [Haa96, Ara09, Stro05, Mor18].

Theorem 6.2. [Characterisation of pure algebraic states] Let ω be an algebraic state onthe C∗-algebra with unit A and (Hω, πω,Ψω) a corresponding GNS triple. Then ω is pure if andonly if πω is irreducible (πω(A)′ = CI that is equivalent to πω(A)′′ = B(Hω)).

The algebraic notion of pure state is in nice agreement with the Hilbert space formulation resultwhere pure states are represented by unit vectors (in the absence of superselection rules). Indeed

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we have the following proposition which make a comparison between the two notions.

Proposition 6.2. Let ω be a pure state on the C∗-algebra with unit A and Φ ∈ Hω a unitvector. Then

(a) the functionalA 3 a 7→ 〈Φ|πω(a)Φ〉 ,

defines a pure algebraic state and (Hω, πω,Φ) is a GNS triple for it. In that case, GNSrepresentations of algebraic states given by non-zero vectors in Hω are all unitarily equiv-alent.

(b) Unit vectors Φ,Φ′ ∈ Hω give the same (pure) algebraic state if and only if Φ = cΦ′ forsome c ∈ C, |c| = 1, i.e. if and only if Φ and Φ′ belong to the same ray.

The correspondence pure (algebraic) states vs. state vectors, automatic in the standard formu-lation, holds in Hilbert spaces of GNS representations of pure algebraic states, but in generalnot for mixed algebraic states. The following exercise focusses on this apparent problem.

Exercise 6.1.(1) Consider, in the standard (not algebraic) formulation, a physical system described on

the Hilbert space H and a statistical operator T ∈ B1(H) (not of the form 〈ψ| 〉ψ!). The mapωT : B(H) 3 A 7→ tr(TA) defines an algebraic state on the C∗-algebra B(H). By the GNStheorem, there exist another Hilbert space HωT , a representation πωT : B(H)→ B(HωT ) an unitvector ΨωT ∈ HωT such that

tr(TA) = 〈ΨωT |πωT (A)ΨωT 〉

for A ∈ B(H). Thus it seems that the initial mixed state has been transformed into a pure state!How is this fact explained?

Solution. There is no transformtion from mixed to pure state because the mixed state isrepresented by a vector, ΨωT , in a different Hilbert space, HωT . Moreover, there is no Hilbertspace isomorphism U : H → HωT with UAU−1 = πωT (A), so that U−1ΨωT ∈ H. In fact, therepresentation B(H) 3 A 7→ A ∈ B(H) is irreducible, whereas πωT cannot be irreducible (as itwould be if U existed), because the state T is not an extreme point in the space of non-algebraicstates, and so it cannot be extreme in the larger space of algebraic states.

(2) Consider T ∈ B1(H) with T ≥ 0 and Tr(T ) = 1 defining the algebraic state ωT : B(H)→C by means of ωT (A) := tr(TA). Construct explicitly a GNS triple of ωT , proving that πωT isreducible.

Solution. Spectrally decompose T =∑+∞j=1 pj〈ψj | 〉ψj , where pj > 0 and

∑+∞j=1 pj = 1.

Consider the Hilbert direct sum HωT :=⊕+∞

j=1 H, define ΨωT := ⊕+∞j=1√pjψj , and eventually

πωT (A) :=⊕+∞j=1 A. It is easy to prove that all requirements for a GNS triple of ω are satisfied.

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In particular, use the fact that APkψh = δkhAψh is dense in B(H), for every fixed k = h, whenA ∈ B(H) and where Pk is the orthogonal projector onto spanψk. Finally, every copy of Hin the space HωT :=

⊕+∞j=1 H is separately invariant under πωT (A) for all A ∈ B(H) so that the

representation is not irreducible.

6.2 Symmetries and Algebraic Formulation

The notion of symmetry, in the algebraic approach, is the direct generalization of the corre-sponding notion in the Hilbert space formulation. However here observables are more importantthan states and this fact shows up in the relevant definitions.

6.2.1 Symmetries and symmetry spontaneous breakdown

Definition 6.3. If a quantum physical system S is represented by a unital C∗-algebra ofobservables AS , an (algebraic) symmetry α : AS → AS is, by definition, either an automor-phism or an anti automorphism of AS . In other words α is respectively linear or anti linear andα(11) = α(11), α(ab) = α(a)α(b) and α(a∗) = α(a)∗ for a, b ∈ AS .

The definition reflects on states into a straightforward way.

Definition 6.4. An algebraic state ω : AS → C is said to be invariant under a symmetryα : AS → AS if

ω(α(a)) = ω(a) , for every a∗ = a ∈ AS .

(The requirement extends, by linearity or anti linearity according to the nature of α, to everyelement of AS .)

When an α-symmetric state exists, the action of α is unitarily implementable (by means of theinverse dual action of a Wigner-Kadison symmetry) in the GNS Hilbert space (see, e.g., [Mor18]).

Theorem 6.3. Suppose that α : AS → AS is a symmetry and the state ω : AS → C isinvariant under α. Then, referring to the GNS triple (Hω, πω,Ψω) of ω the following holds.

(1) α is (anti)unitarily implementable: there exists an isometric surjective map U : Hω → Hωwhich is linear or anti linear according to α, such that

Uπω(a)U−1 = πω(α(a)) ∀a ∈ A .

(2) UΨω = Ψω.

The (anti)unitary operator U is uniquely defined by (1) and (2).

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Proof. The requested operator is the unique map satisfying

Uπ(a)Ψω := πω(α(a))Ψω ∀a ∈ AS (6.3)

exploiting the fact that πω(AS)Ψω is dense in HS .

The algebraic approach permits us to deal with a physically important situation where a sym-metry exists only at the level of algebra of observables, but the symmetry is broken at thelevel of states. This situation is usually said spontaneously breakdown of symmetry. There areseveral interpretations of this idea (see [Lan17] for a wide up-to-date review on the subject and[Haa96, Stro08] for more specific results in relativistic local QFT). Generally speaking, spon-taneous breakdown of symmetry occurs when the C∗-algebra of observables AS admits asymmetry α described by an (anti)automorphism, but there is no state invariant under α in acertain class of states of physical relevance. In particular,

Definition 6.5. The symmetry α : AS → AS is said to be spontaneously broken by agiven algebraic state ω : A→ C, if ω is not invariant under α.

Notice that in this case α could be however implemented in the GNS representation of ω: (1) inTheorem 6.3 could be valid for some (anti)unitary operator U in Hω, though U does not satisfy(2). This possibility is considered within a stronger interpretation of the notion of spontaneouslybroken symmetry.

Definition 6.6. The symmetry α : AS → AS is said to be spontaneously broken by analgebraic state ω : A→ C in strong sense, if α cannot be implemented in the GNS representa-tion of ω: (1) in Theorem 6.3 is not valid for every (anti)unitary operator U in Hω (in particularω cannot be invariant under α due to Theorem 6.3).

Exercise 6.2. Consider an algebraic state ω : AS → C and an algebraic (linear) symmetryα : AS → AS. Prove that ω spontaneously breaks α in strong sense if and only if πω and πωαare not unitarily equivalent.

Solution. The thesis is equivalent to saying that α can be unitarily implemented in Hω if andonly if πω and πωα are unitarily equivalent. Let us prove this statement. First observe that therepresentation πωα α−1 : AS → B(Hωα) satisfies 〈Ψωα|πωα α−1(a)Ψωα〉 = ω(α(α−1(a)) =ω(a) so that (the remaining conditions being trivialy satisfied) (Hωα, πωαα−1,Ψωα) is a GNStriple for ω α. The final part of the GNS theorem proves that there is a surjective isometricoperator V : Hωα → Hω such that, in particular, V πωα(α−1(a))V −1 = πω(a) for all a ∈ AS .Namely, πωα(a) = V −1πω(α(a))V for all a ∈ AS . To conclude, observe that πω and πωα areunitarily equivalent if and only if there exist a surjective isometric operator U : Hωα → Hω withUπωα(a)U−1 = πω(a) for all a ∈ AS , that is equivalent to UV −1πω(α(a))V U−1 = πω(a) for alla ∈ AS . This is equivalent to saying that V −1U implements α in Hω.

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6.2.2 Groups of symmetries in algebraic approach

To conclude, we consider the case of a topological (Lie) group of symmetries. We only considerthe case of symmetries represented by linear C∗-algebra automorphisms for reasons analogous tothose of the Hilbert space theory discussed in Section 5.2.2. The notion of continuity is referredto the folium of an invariant state. Suppose that α : G 3 g 7→ αg associates every element g ofgroup with a unital C∗-algebra automorphism αg : AS → AS and that this map represents thegroup, i.e., αe = id and αg αg′ = αg·g′ . If the state ω : AS → C is α-invariant, in view of (6.3)in Theorem 6.3, there exists a unitary representation G 3 g 7→ Ug ∈ B(Hω) satisfying

(1) Ugπω(a)U−1g = πω(αg(a)) for every a ∈ A and every g ∈ G.

(2) UgΨω = Ψω for every g ∈ G.

If G is topological, the natural continuity requirement on α with respect to ω (which can beimmediately stated in terms of a topology induced by seminorms) is that

G 3 g 7→ ω (b∗αg(a)b) is continuos for every a∗ = a and b elements of AS . (6.4)

This is because the right-hand side satisfies

ω (b∗αg(a)b) = 〈πω(b)Ψω|πω(a)πω(b)Ψω〉

and this is, up to normalization (if ω(b∗b) 6= 0), the expectation value of the selfadjoint operatorπω(a) in a state represented by a unit vector in Hω. It is not difficult to prove [Mor18] that ifω αg = ω for every g ∈ G, the notion of continuity referred to ω written above is equivalent tothe apparently simpler one (but not directly related to physics)1

G 3 g 7→ ω (a∗αg(a)) is continuous for all a ∈ AS . (6.5)

The following result holds (e.g., see [Mor18]) whose proof easily arises from (6.3).

Theorem 6.4. Let ω : AS → C be an invariant state with respect to the representationG 3 g 7→ αg : AS → AS in terms of (linear) automorphisms. The following facts are valid.

(a) There is a unitary representation G 3 g 7→ Ug ∈ B(Hω) such that UgΨω = Ψω andUgπω(a)U−1

g = πω(αg(a)) for every g ∈ G and every a ∈ AS.

(b) Continuity of α with respect to ω according to the two equivalent notions (6.4) and (6.5)is equivalent to the strong continuity of G 3 g 7→ Ug ∈ B(Hω).

Remark 6.3. Suppose that G = R in the theorem above, ω is R-invariant, and αtt∈Ris continuous with respect to ω, then ω is said ground state if the selfadjoint generator Hof the associated unitary R-representation Ut = e−itH in the GNS representation of ω satisfiesσ(H) ⊂ [0,+∞). Observe that, from (a) above, HΨω = 0 so that 0 ∈ σp(H).

1As the reader easily proves using the GNS theorem, ω (a∗αg(a)) appearing in (6.5) can be in turn replacedby the apparently more general term ω (bαg(a)) for all a, b ∈ AS preserving the introduced notion of continuity.

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