Physics Faq

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-------------------------A theoretical physics FAQ

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The up-to-date version of the theoretical physics FAQ is athttp://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html.

in a reorganized and clickable version.

The present document is an old ASCII version,in which the state of the FAQ on January 9, 2010 is frozen.

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Consider everything, and keep the good.

(St. Paul, 1 Thess. 5:21)

This document (a simple ASCII file) contains answers to some more orless frequently asked questions from theoretical physics. Currently,the FAQ contains 148 topics, grouped into 20 chapters, and filling over11000 lines of text (about half a megabyte), corresponding to a bookof about 220 pages. Starting in 2004, the topics were edited from myanswers to postings to the moderated newsgroup sci.physics.research(or, for some, translated from postings to the unmoderated Germannewsgroup de.sci.physik).

If you like the FAQ and/or found it useful, please link to it fromyour home page to make it more widely known.

If you spot errors or have suggestions for improvements,please write me (at [email protected]).

If you have questions, please post them to the moderated newsgroupsci.physics.research (http://www.lns.cornell.edu/spr)!

If you found this FAQ useful you are likely to benefit also fromreading our book

Arnold Neumaier and Dennis Westra,

Classical and Quantum Mechanics via Lie algebras,http://www.mat.univie.ac.at/~neum/papers/physpapers.html#QMLhttp://de.arxiv.org/abs/0810.1019


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Of course, the FAQ refers only to a tiny part of theoretical physics,namely to what I happened to discuss on sci.physics.research.The answers are only as good as my understanding of the matter.This doesn't mean that they are poor but probably that they arenot perfect. Many topics are discussed quite in detail, but this is

not a book, so don't expect completeness or comprehensiveness in anysense.

On topics where the physics community has not yet reached a consensus,my point of view is of course only one of the possibilities, and notalways the mainstream view, although I tend to discuss that view, too.In any case, I try to be accurate, consistent, and intelligible.

Happy Reading!

Arnold Neumaier

University of Viennahttp://www.mat.univie.ac.at/~neum/I like to see people grow

-----------------Table of Contents-----------------

The 21 topics in the initial version, posted there on April 28, 2004,have grown to 88 by January 1, 2005, to 116 by January 4, 2006,to 128 by January 3, 2007, to 140 by January 3, 2008, to 147 byJanuary 30, 2009, and are likely to grow further.

(A * indicates addition of a new topic, or large modification ofan old one, since January 30, 2009. Minor changes or additions toold topics are not indicated.)

The various topics can usually be read independently of each other;they are arranged into groups of loosely related topics.To read a particular entry, grep for its label, e.g., S2e.The labels may change with time as answers to further questionswill be added and old answers regrouped. So, to quote part of the FAQ,refer to the title of a section and not only to its label.

Abbreviations:QM = quantum mechanics, QFT = quantum field theory,QED = quantum electrodynamics, CCR = canonical commutation relations,s.p.r. = sci.physics.research (newsgroup).Strings like quant-ph/0303047 or arXiv:0810.1019 refer to electronicdocuments in the e-Print archive at

http://xxx.lanl.gov and mirror sites.p_0 and \p are the time and space part of a 4-vector p;the Minkowski inner product is always taken to be p^2=p_0^2-\p^2.

Chapter 1 (20 sections)S1a. What are bras and kets?S1b. Projective geometry and quantum mechanicsS1c. What is the meaning of the entries of a density matrix?

S1d. Postulates for the formal core of quantum mechanicsS1e. Open quantum systemsS1f. Interaction with a heat bath


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S1g. Quantum-classical mechanicsS1h. Can all quantum states be realized in nature?S1i. Modes and wave functions of laser beamsS1j. Classical and quantum tunnelingS1k. Quantization in non-Cartesian coordinatesS1l. Second quantizationS1m. When is an object macroscopic?

S1n. The role of the ergodic hypothesisS1o. Does quantum mechanics apply to single systems?*S1p. Dissipative dynamics and Lagrangians*S1q. How can QM be stochastic while the Schroedinger equation is not?*S1r. Measurement theory for real numbers*S1s. The classical limit of quantum mechanics*S1t. The classical limit via coherent states

Chapter 2 (10 sections)S2a. Lie groups and Lie algebrasS2b. The Galilei group as contraction of the Poincare groupS2c. Representations of the Poincare group, spin and gauge invariance

S2d. Forms of relativistic dynamicsS2e. Is there a multiparticle relativistic quantum mechanics?S2f. What is a photon?S2g. Particle positions and the position operatorS2h. Localization and position operators*S2i. Position operators in relativistic quantum field theoryS2j. Coherent states of light as ensembles

Chapter 3 (6 sections)S3a. What are 'bare' and 'dressed' particles?S3b. How meaningful are single Feynman diagrams?S3c. How real are 'virtual particles'?

S3d. What is the meaning of 'on-shell' and 'off-shell'?S3e. Virtual particles and Coulomb interactionS3f. Are virtual particles and decaying particles the same?

Chapter 4 (10 sections)S4a. How do atoms and molecules look like?S4b. Why are observable densities state-dependent?S4c. Are electrons pointlike/structureless?S4d. How much information is in a particle?S4e. Entropy and missing informationS4f. How real is the wave function?S4g. How real are Feynman's paths?S4h. Can particles go backward in time?S4i. What about particles faster than light (tachyons)?S4j. Do free particles exist?Chapter 5 (9 sections)S5a. QM pictures and representationsS5b. Inequivalent representations of the CCR/CARS5c. Why does QFT look so different from QM?S5d. Why is QFT based on a classical action?S5e. Why does the action only contain first derivatives?S5f. Why normal ordering?S5g. Why locality and causal commutation relations?S5h. Creation operators and rigged Hilbert space

S5i. Why Feynman diagrams?

Chapter 6 (8 sections)


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S6a. Nonperturbative computations in quantum field theoryS6b. The formal functional integral approach to QFTS6c. Functional integrals, Wightman functions, and rigorous QFTS6d. Is there a rigorous interacting QFT in 4 dimensions?S6e. Constructive field theoryS6f. The classical limit of relativistic QFTS6g. What are interpolating fields?

S6h. Hilbert space and Hamiltonian in relativistic quantum field theory*S6i. 2-dimensional quantum field theory

Chapter 7 (3 sections)S7a. What is the mass gap?S7b. Why can a bound state of massless quarks be heavy?S7c. Bound states in relativistic quantum field theory

Chapter 8 (9 sections)S8a. Why renormalization?S8b. Renormalization without infinities IS8c. Renormalization without infinities II

S8d. Renormalization and coarse grainingS8e. Renormalization scale and experimental energy scaleS8f. Dimensional regularizationS8g. Nonrelativistic quantum field theoryS8h. Nonrenormalizable theories as effective theoriesS8i. What about infrared divergences?

Chapter 9 (6 sections)S9a. Summing divergent seriesS9b. Is QED consistent?S9c. What about relativistic QFT at finite times?S9d. Perturbation theory and instantaneous forcesS9e. QED and relativistic quantum chemistry

S9f. Are protons described by QED?

Chapter 10 (13 sections)S10a. How are matrices and tensors related?S10b. Is quantum mechanics compatible with general relativity?S10c. Difficulties in quantizing gravityS10d. Renormalization in quantum gravityS10e. Hadamard states and their Hilbert spacesS10f. Why do gravitons have spin 2?S10g. What is the tetrad formalism?S10h. Energy in general relativityS10i. What happened to the aether?S10j. What is time?S10k. Time in quantum mechanicsS10l. Diffeomorphism invariant classical mechanicsS10m. The concept of ''Now''

Chapter 11 (7 sections)S11a. A concise formulation of the measurement problem of QMS11b. The double slit experimentS11c. The Stern-Gerlach experimentS11d. The minimal interpretationS11e. The preferred basis problemS11f. Master equation and pointer variablesS11g. Does decoherence solve the measurement problem?

Chapter 12 (6 sections)S12a. Which interpretation of quantum mechanics is most consistent?


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S12b. Which textbook of quantum mechanics is best for foundations?S12c. What is the role of quantum logic?S12d. Stochastic quantum mechanicsS12e. Is there a relativistic measurement theory?S12f. Quantum mechanics and dice


S13a. Random numbers and other random objectsS13b. What is the meaning of probabilities?S13c. What about the subjective interpretation of probabilities?S13d. Are probabilities limits of relative frequencies?S13e. How meaningful are probabilities of single events?S13f. Objective probabilitiesS13g. How probable are realizations of stochastic processes?S13h. How do probabilities apply in practice?S13i. Incomplete knowledge and statisticsS13j. Priors and entropy in probability theory


S14a. Theoretical challenges close to experimental dataS14b. Does the standard model predict chemistry?S14c. Is the result of a measurement a real number?S14d. Why use complex numbers in physics?

Chapter 15 (5 sections)S15a. How precise can physical language be?S15b. Why bother about rigor in physics?S15c. Justifying the foundations of a theoryS15d. Foundations, theory and experimentS15e. Theoretical physics as a formal model of reality


S16a. On progress in scienceS16b. How different are physical sciences and social sciencesS16c. Can good theories be falsified?S16d. What, then, distinguishes a good theory?S16e. When is a theory preferred to another one?S16f. What is a fact?S16g. Physics and experienceS16h. Modeling realityS16i. What is a system (e.g., an ideal gas)?S16j. When is a theory confirmed?S16k. What is real?S16l. How many angels fit onto the tip of a needle?

Chapter 17 (8 sections)S17a. How to get information from sci.physics.researchS17b. How to get your work publishedS17c. How to respond to critical referee's reportsS17d. How to sell your revolutionary ideaS17e. Useful background, online lecture notes, etc.S17f. Stories about physicistsS17g. Other physics FAQs*S17h. Naming in science

Chapter 18 (5 sections)S18a. What is the meaning of 'self-consistent'?S18b. What is a vector?


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S18c. Learning quantum mechanics at age 14S18d. Research at age 16S18e. Are there indefinite Hilbert spaces?Chapter 19 (1 section)S19a. God and physics

Chapter 20 (1 section)S20a. Acknowledgments

Since March 1, 2005, there is also a related FAQ in German language,Ein Theoretische Physik FAQhttp://www.mat.univie.ac.at/~neum/physik-faq.txt

where I describe some more topics which I have not translated.(Among other topics, it discusses a new interpretation of quantummechanics, which I call the 'consistent experiment interpretation'.It gives a new meaning to the foundations of physics, less paradoxthan the conventional interpretations. I expect to have soon an

English version of it.)

----------------------------S1a. What are bras and kets?----------------------------

In the language of linear algebra, kets psi> are just column vectorspsi (for systems with finitely many levels only; each component givesthe amplitude for the corresponding level), and the corresponding

bras


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This allows them to write not only psi(x) = , but alsopsi(x)^* = ^* = .

The price to be paid is that inner products are no longer well-definedin general; for example, is infinite. They say, x> is notnormalizable and mean that it is not in the Hilbert space ofwell-behaved pure states.

Caution: Physicists often use different bases which may cause confusingnotation. For example


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straight rails of a railway track...

Thic can be extended to higher dimensions. n-dimensional affine geometrycan be respresented by rays through 0 in n+1 dimensional space, and canbe completed there to a projective geometry, in which the vectorsubspaces are the geometrical objects. In Hilbert space one cannotcount anymore dimensions, but otherwise everything is similar.

Since, in quantum mechanics, state vectors are only defined up to aphase (even when normalized), they correspond uniquely to rays= 1-dimensional subspaces in Hilbert space. Hence quantum mechanics isintrinsically projective.

------------------------------------------------------------S1c. What is the meaning of the entries of a density matrix?------------------------------------------------------------

Density matrices are a convenient way of describing states of quantumsystems in contact with an environment. (State vectors = wave functionsare appropriate only for isolated systems at zero absolute temperature,though they can be used in an approximate way in thermally isolatedcontexts. But contact with an environment means positive temperature.)If the quantum system has only a finite number n of levels,the density matrix is an n x n matrix; otherwise it isa linear operator on Hilbert space (but nevertheless called a matrix).

The real use for density matrices is to compute expectations = trace (rho f)

for quantities f of interest. Indeed, rho is just a collection of

numbers enabling one to calculate these expectations.The fact that the constant 1 must have expectation 1 leads to therestriction that

sum_k rho_kk = trace rho = 1.Apart from that, rho must be a Hermitian, positive semidefinite matrix,to satisfy the requirements of statistics. (See quant-ph/0303047 fordetails.) For small systems, all such density matrices can indeed beapproximately realized in practice.

Since diagonal entries of a semidefiniteness are always nonnegative,the p_k:=rho_kk are nonnegative numbers summing to 1 and thus look likeprobabilities. What the components mean depends on the basis used.In particluar, if the basis consists of eigenstates of a Hamiltonian,and the eigenvalues E_k are all nondegenerate, a diagonal elementrho_kk can be interpreted as the probability that upon measuring theenergy of the system one will find the value E_k.

If f is a function of the Hamiltonian H, and the basis used consists ofeigenstates k> of H, with Hk>=E_kk> then the density matrix rhohas entries rho_jk = . If one now calculates the expectationof a function f(H), the equation f(H)k>=f(E_k)k> implies that

= trace (rho f(H)) = sum_k = sum_k = sum_k f(E_k)= sum_k rho_kk f(E_k).

If we average the results f(E) of a number of measurements of the

energy, where the energy E_k is measured with probability p_k,we get

= sum_k p_k f(E_k).


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Thus, to match the expectations no matter which function we areaveraging, we need to take p_k=rho_kk. This gives the claimedprobability interpretation of the diagonal entries.

Off-diagonal elements have no simple interpretation.Usually one does not look at off-diagonal elements at all, but theyare important in intermediate steps of calculations.

Close to absolute zero temperature, and assuming the absence ofdegeneracy, (but also in certain other, well prepared nearlyisolated systems), quantum state have the property that all columnsof the density matrix are nearly parallel to a wave function psithat is conventionally normalized to have norm 1,

psi^*psi=1.(In Dirac language, this says =1; see the FAQ entry for brasand kets.). This vector psi, which is clearly determined only up to acomplex number of absolute value 1, is called the wave vector(or, in infinite dimensions, the wave function) of the state.

Idealizing this situation, one describes such quantum systems by statesin which all columns of the density matrix are exactly parallel to somenonzero wave vector psi. (Such matrices are called rank 1 matrices;the wave vector, also referred to as a wave function, is definedonly up to a phase factor.)Then the k-th column is a multiple c_k psi of psi. The fact that rhois Hermitian forces each row to be a multiple of psi^*. But this impliesthat c_k is a multiple of phi^*_k, so that rho is a multiple ofpsi psi^*. Since psi is normalized, the multiplication factor is justthe trace, and since the trace is 1 we find

rho = psi psi^* for any rank 1 density matrix.If we now calculate the probability of measuring the energy E_k, we find

p_k = rho_kk = = = ,

and since is just the complex conjugate of ,we end up with

p_k = ^2.This is Born's squared amplitude formula for calculating probabilities.Thus one sees that the traditional wave vector calculus is just aspecial case of the density matrix calculus, appropriate (only) forthe study of tiny, well-prepared nearly isolated systems and forsystems close to zero absolute temperature. For the study of ordinarymatter under ordinary conditions, one needs to represent statesby density matrices.

Everything that is done with wave vectors can also be done withdensity matrices, or equivalently with the associated expectationmapping. Indeed, everything becomes simpler that way, much closerto classical mechanics, and much less weird-looking.See quant-ph/0303047 for an exposition of the foundations of quantummechanics (including the probability interpretation, uncertaintyrelations, nonlocality, and Bell's theorem) in terms of expectations.

--------------------------------------------------------S1d. Postulates for the formal core of quantum mechanics--------------------------------------------------------

Quantum mechanics consists of a formal core that isuniversally agreed upon (basically being a piece of mathematics


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with a few meager pointers on how to match it with experimentalreality) and an interpretational halo that remains highly disputedeven after 80 years of modern quantum mechanics. The latter is thesubject of the foundations of quantum mechanics; it is addressedelsewhere in this FAQ. Here I focus on the formal side.

As in any axiomatic setting (necessary for a formal discipline),

there are a number of different but equivalent sets of axiomsor postulates that can be used to define formal quantum mechanics.Since they are equivalent, their choice is a matter of convenience.

My choice presented here is the formulation which gives mostdirect access to statistical mechanics, which is the main tool forreal life applications of quantum mechanics. The relativistic caseis outside the scope of the present axioms. Thus the followingdescribes nonrelativistic quantum statistical mechanics in theSchroedinger picture. (The traditional starting point is insteadthe special case of this setting where all states are assumed to bepure.)

There are six basic axioms:

A1. A generic system (e.g., a 'hydrogen molecule')is defined by specifying a Hilbert space K whose elementsare called state vectors and a (densely defined, self-adjoint)Hermitian linear operator H called the _Hamiltonian_ or the _energy_.

A2. A particular system (e.g., 'the ion in the ion trap on thisparticular desk') is characterized by its _state_ rho(t)at every time t in R (the set of real numbers). Here rho(t) is aHermitian, positive semidefinite (trace class) linear operator on Ksatisfying at all times the conditions

trace rho(t) = 1. (normalization)A state is called _pure_ at time t if rho(t) maps K to a 1-dimensionalsubspace, and _mixed_ otherwise.

A3. A system is called _closed_ in a time interval [t1,t2]if it satisfies the evolution equation

d/dt rho(t) = i/hbar [rho(t),H] for t in [t1,t2],and _open_ otherwise. (hbar is Planck's constant, and is often setto 1.) If nothing else is apparent from the context,a system is assumed to be closed.

A4. Besides the energy H, certain other (densely defined, self-adjoint)Hermitian operators (or vectors of such operators) are distinguishedas _observables_.(E.g., the observables for an N-particle system conventionally includefor each particle a involved several 3-dimensional vectors:the _position_ xâ, _momentum_ pâ, _orbital_angular_momentum_ Lâand the _spin_vector_ (or Bloch vector) sigmaâ of the particle withlabel a. If u is a 3-vector of unit length then u dot pâ, u dot Lâand u dot sigmaâ define the momentum, orbital angular momentum,and spin of particle a in direction u.)

A5. For any particular system, one associates to every vector Xof observables with commuting components a time-dependent monotonelinear functional _t defining the _expectation_

_t:=trace rho(t) f(X)of bounded continuous functions f(X) at time t.This is equivalent to a multivariate probability measure dmu_t(X)


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(on a suitable sigma algebra over the spectrum spec(X) of X)defined by

integral dmu_t(X) f(X) := trace rho(t) f(X) =_t.

A6. Quantum mechanical predictions amount to predicting properties(typically expectations or conditional probabilities)of the measures defined in axiom A5 given reasonable assumptions

about the states (e.g., ground state, equilibrium state, etc.)

Axiom A6 specifies that the formal content of the theory is coveredexactly by what can be deduced from axioms A1-A5 withoutanything else added (except for restrictions defining the specificnature of the state), and hence says that Axioms A1-A5 are complete.

The description of a particular closed system is therefore given bythe specification of a particular Hilbert space in A1, thespecification of the observable quantities in A4, and thespecification of conditions singling out a particular class ofstates (in A6). Everything else is determined by the theory and

hence is (in principle) predicted by the theory.

The description of an open system involves, in addition, thespecification of the details of the dynamical law. (For the basics,see the entry 'Open quantum systems' in this FAQ.)

In addition to these formal axioms one needs a rudimentaryinterpretation relating the formal part to experiments.The following _minimal_interpretation_ seems to be universallyaccepted.

MI. Upon measuring at times t_l (l=1,...,n) a vector X of observables

with commuting components, for a large collection of independentidentical(particular) systems closed for times t


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Interpretational axioms necessarily have this form, since they mustassume some unexplained common cultural background for perceivingreality. (This is even true in pure mathematics, since the languagestating the axioms must be assumed to be common cultural background.)

Everything beyond MI seems to be controversial. In particular,already what constitutes a measurement of X is controversial.(E.g., reading a pointer, different readers may get marginallydifferent results. What is the true pointer reading?)

On the other hand there is an informal consensus on how toperform measurements in practice. Good foundations including agood measurement theory should be able to properly justify thisinformal consensus by defining additional formal concepts thatbehave within the theory just as their informal relatives withthe same name behave in reality.

In complete foundations, there would be formal objects in themathematical theory corresponding to all informal objects discussedby physicists, such that talking about the formal objectsand talking about the real objects is essentially isomorphic.We are currently far from such complete foundations.

Although much of traditional quantum mechanics is phrased in terms ofpure states, this is a very special case; in most actual experimentsthe systems are open and the states are mixed states. Pure statesare relevant only if they come from the ground state of aHamiltonian in which the first excited state has a large energy gap.Indeed, assume for simplicity that H has discrete spectrum. In an

orthonormal basis of eigenstates psi_k,f(H) = sum_k f(E_k) psi_k psi_k^*

for every function f defined on the spectrum. Setting the Boltzmannconstant to 1 to simplify the formulas, the equilibrium density isthe canonical ensemble,

rho(T) = 1/Z(T) exp(-H/T) = sum_k exp(-E_k/T)/Z(T) psi_k psi_k^*.(Of course, equating this ensemble with equilibrium in a closed systemis an additional step beyond our axiom system, which would requirejustification.) Taking the trace (which must be 1) gives

Z(T) = sum_k exp(-E_k/T),and in the limit T -> 0, all terms exp(-E_k/T)/Z(T) become 0 or 1,with 1 only for the k corresponding to the states with least energyThus, if the ground state psi_1 is unique,

lim_{T->0} rho(T) = psi_1 psi_1^*.This implies that for low enough temperatures, the equilibrium stateis approximately pure. The larger the gap to the second smallestenergy level, the better is the approximation at a given nonzerotemperature. In particular (reinstalling the Boltzmann constant kbar),if the energy gap exceeds a small multiple of E^* := kbar T theapproximation is good.

States of simple enough systems with a few levels onlycan often be prepared in nearly pure states, by realizing a sourcegoverned by a Hamiltonian in which the first excited state has a muchlarger energy than the ground state. Dissipation then brings the

system into equilibrium, and as seen above, the resulting equilibriumstate is nearly pure.


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To see how the more traditional setting in terms of theSchroedinger equation arises, we consider the case of a closedsystem in a pure state rho(t) at some time t.

If psi(t) is a unit vector in the range of the pure state rho(t)then psi(t), called the _state_vector_ of the system is determined

up to a phase, and one easily verifies thatrho(t) = psi(t)psi(t)^*.

Remarkably, under the dynamics for a closed system specified in theabove axioms, this property persists with time (only) if the systemis closed, and the state vector satisfies the Schroedinger equationi hbar psi(t) = H psi(t)

Thus the state remains pure at all times.

Moreover, if X is a vector of observables with commuting componentsand the spectrum of X is discrete, then the measure from axiom A5is discrete,

integral dmu(X) f(X) = sum_k p_k f(X_k)

with nonnegative numbers p_k summing to 1, commonly called_probabilities_.Moreover, associated with the p_k are eigenspaces K_k such that

X psi = X_k psi for psi in K_k,and K is the direct sum of the K_k. Therefore, every state vector psican be uniquely decomposed into a sum

psi = sum_k psi_k with psi_k in K_k.psi_k is called the _projection_ of psi to the eigenspace K_k.

A short calculation using axiom A5 now reveals that for a pure staterho(t)=psi(t)psi(t)^*, the probabilities p_k are given by theso-called _Born_rule_

p_k = psi_k(t)^2, (*)

where psi_k(t) is the projection of psi(t) to the eigenspace K_k.

Deriving the Born rule (*) from axioms A1-A5 makes it completelynatural, while the traditional approach starting with (*)makes it an irreducible rule full of mystery and only justifiableby its agreement with experiment.

-------------------------S1e. Open quantum systems-------------------------

Open quantum systems are usually modelled in a stochastic wayto account for the unpredictability of the measurement process.(Note that a measurement is any non-negligible interaction with theenvironment, whether or not it is observed by something deservingthe name 'detector' or 'observer').

In the simplest setting in which states can be assumed tobe pure and measurements occur at definite, a priori known timesand have a negligible duration, an open quantum system is a discretestochastic process with values psi(t) in the Hilbert space of statevectors, normalized to norm 1. Between two consecutive measurements,

the system is assumed to be closed.

Thus between two consecutive measurements at times t' and t''>t',


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the normalized state psi(t) evolves according to the Schroedingerequation

i hbar psidot = H psi,so that

psi(t''-0)= P psi(t'+0), P = exp (i/hbar (t'-t'')H). (1)(In the interaction picture, H=0 and psi remains constant betweenmeasurements.)

A measurement at time t is assumed to happen in infinitesimal timeand replaces psi(t-0) independent of other measurements withprobability p_s by

psi(t+0)= P_s psi(t-0)/p_s if p_s>0, (2)where the P_s are linear operators determined by the experimentalarrangement, satisfying the relation

sum_s P_s^*P_s = 1, (3)and

p_s=P_s\psi(t-0)^2 (4)guarantees that psi(t+0) remains normalized. Clearly the p_s arenonnegative and by (3), they sum up to 1 (since psi(t-0) is normalized).

(For measurements with more than countably many possible outcomes,one must replace the probabilities by probability densities and thesums by integrals.)Thus this is a well-defined stochastic process.

A von-Neumann measurement of a self-adjoint linear operator Acorresponds to the special case where P_s is an orthogonal projectorto the eigenspace corresponding to the eigenvalue a_s of A(respective to the set of eigenvalues corresponding to the s-thinterval in a partition of the continuous spectrum of A.)

If the measurement at different times has the same (or different)

nature, the P_s at these times are the same (or different).It is possible to introduce 'empty measurements' at arbitraryintermediate times with a trivial sum over a singleton s, where P_s=1.

For continuous measurements (where the open system cannot be consideredclosed at all but a discrete number of times), one needs to takea continuum limit of the above description. Depending how one takesthe limit, one gets quantum diffusion processes or quantum jumpprocesses. In this case, the density matrix for the associateddeterministic expectation evolves according to a Lindblad dynamics.

Realistic measurements (i.e. those taking into account the unavoidableuncertainty) are not modelled by von-Neumann measurements, but ratherby positive operator valued measures, short POVMs. These are wellexplained in

http://en.wikipedia.org/wiki/POVM

For more on real measurement processes (as opposed to thevon-Neumann measurement caricature treated in typical textbooksof quantum mechanics), see, e.g.,

V.B. Braginsky and F.Ya. Khalili,Quantum measurement,Cambridge Univ. Press, Cambridge 1992

---------------------------------S1f. Interaction with a heat bath


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---------------------------------

Quantum mechanics in the presence of a heat bath requires the useof density matrices. Instead of the usual von-Neumann equation

rhodot = rho \lp H(for \lp see the section on 'Quantum-classical correspondence'),the dynamics of the density matrix is given by a dissipative version

of it,rhodot = rho \lp H + L(rho)

usually associated with the name of Lindblad. Here L(rho)is a linear operator responsible for dissipation of energy tothe heat bath; it is not a simple commutator but can havea rather complex form.

To get the Lindblad dynamics from a Hamiltonian description ofsystem plus bath, one uses the projection operator formalism.The clearest treatment I know of is in

H Grabert,Projection Operator Techniques in Nonequilibrium

Statistical Mechanics,Springer Tracts in Modern Physics, 1982.The final equations for the Lindblad dynamics are (5.4.48/49)in Grabert's book.

--------------------------------S1g. Quantum-classical mechanics--------------------------------

Quantum mechanics and classical mechanics are very close relatives.

There are analogous objects for everything of relevance inclassical and quantum statistical mechanics.

Observable f:classical - real phase space function f(x,p)quantum - Hermitian linear operator or sesquilinear form f

Lie product f \lp g:read \lp as 'Lie', and visualize it as inverted, stylized L;Macro for LaTeX:\def\lp{\mbox{\Large$\,_\urcorner\,$}}

classical: f \lp g = {g,f} in terms of the Poisson bracketquantum: f \lp g = i/hbar [f,g] in terms of the commutator

The Lie product is bilinear in the arguments and satisfiesf \lp g = - g \lp ff \lp gh = (f \lp g)h + g(f \lp h) (Leibniz)f \lp (g \lp h) = (f \lp g) \lp h + g \lp (f \lp h) (Jacobi)

Invariant measure:classical - integral f := integral dxdp f(x,p)quantum - integral f := trace f

Integrability: integral f finitequantum integrable f trace class

Partial integration formula:integral f \lp g = 0.

Dynamics: df/dt = X_H f := H \lp f with Hermitian Hcanonical transformations = mappings exp(tX_H) with Hermitian H

Liouville's theorem says thatintegral f = integral exp(tX_H)f

The infinitesimal form of this is the partial integration formula.


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State rho:classical - real integrable phase space function rho(x,p)>=0quantum - Hermitian positive semidefinite trace class operator rhoboth normalized to integral rho = 1.

expectation of f in state rho: = integral rho f

--------------------------------------------------S1h. Can all quantum states be realized in Nature?--------------------------------------------------

No. Many mathematically conceivable states do not exist in Nature,for example, that of water at an absolute temperature of zero.

Quantum mechanics does not demand that all states are realizable.For a number of tiny systems with a few levels, all states arerealizable with reasonable precision. However, the larger the system

the fewer states are realized.

The number of states realized at a given time of very large systemssuch as human beings or galaxy clusters is even so small that itcan be approximately counted!

--------------------------------------------S1i. Modes and wave functions of laser beams--------------------------------------------

The physical state described by a typical laser beam is a state withan indeterminate number of photons, since it is usually not aneigenstate of the photon number operator. This essentially means thatin a beam, a certain number of photons cannot be meaningfully asserted;instead, one has a meaningful photon density, referred to as the beamintensity.

Thus the traditional N-particle picture does not apply.Instead one has to work in a suitable Fock space.

The Maxwell-Fock space is obtained by 'second quantization' of the modespace H_photon, consisting of all mode functions, i.e., solutions A(x)of the free Maxwell equations, describing a classical backgroundelectromagnetic field in vacuum. H_photon may be thought of as thesingle photon Hilbert space, in analogy to the single electron Hilbertspace of solutions of the Dirac equation. (However, following up onthis analogy and calling A(x) a wave function leads to confusion lateron, and is best avoided.)

Actually, because of gauge invariance, the situation is slightly morecomplicasted, and best described in momentum space. The Maxwellequations reduce in Lorentz gauge, partial dot A(x) = 0, topartial^2 A(x)=0, whence the Fourier transform of A(x) has the formdelta(p^2) Ahat(p), and Ahat(p) must satisfy the transversalitycondition

p dot Ahat(p) = 0.By gauge invariance, only the coset of Ahat(p) obtained by addingarbitrary multiples of p has a physical meaning, reflecting the


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transversal nature of the free electromagnetic field.This coset construction is needed to turn the space of modesinto a Hilbert space H_photon with invariant inner product

= integral Ahat(p) dot Bhat(p) Dp,where

Dp = d\p/p_0 = dp_1 dp_2 dp_3/p_0,is the Lorentz invariant measure on the photon mass shell,

0 < p_0 = \p = sqrt(p_1^2+p_2^2+p_3^2)(negative frequencies are discarded to get an irreduciblerepresentation of the Poincare group).Indeed, without the coset construction, the inner product is onlypositive semidefinite, hence gives only a pre-Hilbert space.

Each (sufficiently nice) mode function A(x) gives rise to a coherentstate A>> in the Maxwell-Fock space, to an associated annihilationoperator

a(A) = integral Ahat(p) a(p) Dp,where a(p) is the QED annihilation operator for a photon with

momentum p, and to the corresponding creation operator a^*(A) = a(A)^*.

The annihilation and creation operators a(A) and a^*(A) produce asingle-mode Fock subspace consisting of all A,psi>, where psi is theunnormalized wave function of a harmonic oscillator; psi^2 is theintensity of the beam.The coherent state itself corresponds to the normalized vacuum stateof the harmonic oscillator, A>> = A,vac>. If psi is a Hermitepolynomial H_k, A,psi> is an eigenstate of the photon number operatorwith eigenvalue k, and one has a k-photon state.

The Maxwell-Fock space is the closure of the space spanned by allthe A,psi> together (and indeed, already the closure of the space

spanned by all A>>). This space is the pure electromagnetic fieldsector of QED, describing a physical vacuum, i.e., a region of theuniverse where matter is absent though radiation may be present.

In optics experiments, laser beams are often idealized by ignoringtheir extension perpendicular to the transmission direction. Then eachbeam can be described by some A,psi>. In particular, for amonochromatic beam, A is a plane wave, A(x)=A_0 exp(-i p dot x).Of course, this matches the original approximation that we have abeam only with a grain of salt, since a plane wave is not normalized.

A coherent pair of laser beams obtained by splitting is described bya superposition A_1,psi_1> + A_2,psi_2> of the two beams.

Beams of thermal light (such as that from the sun) and pairs ofbeams created by independent sources, cannot be described by wavefunctions alone, but need a density formulation. A single light beamis then described (in the same idealization) by a mode A and a densitymatrix rho in a single-mode Fock space, while k light beams aredescribed by k modes A and a density matrix rho in a k-mode Fock space.

In many treatments, the modes are left implicit, so that one worksonly in the k-mode Fock space. This simplifies the presentation, buthides the connection to the more fundamental QED picture.For a thorough study of the latter, see the bible on quantum optics,

L. Mandel and E. Wolf,Optical Coherence and Quantum Optics,Cambridge University Press, 1995.


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------------------------------------S1j. Classical and quantum tunneling------------------------------------

Consider a particle in an external potential.Assume the potential is everywhere finite, locally constant and positivenear the origin, and decays to zero far away.

There is no force, when the motion is deterministic and classical.

In practice, however, the classical, deterministic setting is anapproximation only, and the particle makes random motions.Thus it moves away from the origin and will sooner or later reachthe nonconstant part of the potential. With low probability p,it will even escape over any barrier; roughly, log p is proportional

to the negative barrier height. For details, you mightwish to consult my paperA. Neumaier,Molecular modeling of proteins and mathematical prediction ofprotein structure,SIAM Rev. 39 (1997), 407-460.http://www.mat.univie.ac.at/~neum/papers/physpapers.html#protein

and the references there.

Quantum mechanically, there is always a probability of escaping toinfinity, without assuming any approximations. This is calledtunneling.

In both cases, once the particle is in the infinite region,the probability that it returns is zero.

Thus a positive potential drives a particle in the long run off toinfinity (though, in case of a high barrier, one has to wait a longtime). In particular, in the classical case one also has a form of(stochastic) tunneling.

Thus it is justified to refer to a potential such as the above asrepelling. However, no one would object if you call a potentialrepelling _only_ in the neighborhood of a strict local minimizer, i.e.,close to a metastable state.

Of course, a golf ball sitting on top of a flat hill will not movedown the hill; because of friction it remains in a metastable state.

Thus the above is an idealization. But most of physics is idealized,and the language is also somewhat idealized (and, as actually used bypeople, not even completely precise).

----------------------------------------------

S1k. Quantization in non-Cartesian coordinates----------------------------------------------


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Textbook quantization rules assume (often silently, without warning)Cartesian coordinates. The rules derived there are based oncanonical commutation rules and are invalid for systemsdescribed in other coordinate systems.

In particular, a Hamiltonian alone does not have a physical meaningsince it can be quite arbitrarily transformed by coordinate

transformations. The Hamiltonian needs to be combined with thecorrect Poisson bracket to yield the correct dynamical equations.Only if the classical Poisson bracket satisfies the canonicalcommutation rules, the quantum mechanics is obtained by imposingcanonical commutation rules on the commutators.

The standard quantization procedure assumes that the symplectic formunderlying the Hamiltonian description has the standard formp dq - q dp. Under a coordinate transformation, the symplectic formchanges into something nonstandard, and naive quantization giveswrong results.

To get correct results, one has to take account of the correctsymplectic structure, more precisely of the Poisson bracket definedby it. This is most naturally done in a differential geometricsetting, in terms of symplectic manifolds and Poisson manifolds.

To proceed, one must quantize a symplectic (or a Poisson) manifoldtogether with a Hamiltonian defined on it.This combination is invariant under coordinate transformationsand hence has a coordinate-independent geometric meaning.

How to quantize Hamiltonians on a symplectic (or a Poisson) manifoldis the subject of geometric quantization, about which there is asignificant literature.

------------------------S1l. Second quantization------------------------

Second quantization is a way of writing the quantum mechanics ofindistinguishable particles in such a way that it makes statisticalmechanics calculations easy and makes everything look like field theory.

One starts with a distinguished vacuum state vac> and a family ofannihilation operators a(x) whith their adjoints, the creationoperators a^*(x), satisfying the canonical commutation relations (CCR)

[a(x),a(y)]=[a^*(x),a^*(y)]=0,[a^(x),a^*(y)]=delta(x-y).

(This is for Bosons; for Fermions one has instead canonicalanticommutation relations, CAR, and everything below gets additionalminus signs in certain places.)

A pure (permutation symmetric) N-particle state with wave functionpsi(x_1:N) is written in 2nd quantization as

psi = integral dx_1:N psi(x_1:N) a^*(x_1:N) vac>,

hence the corresponding density matrixrho = psi psi^*

takes the form


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rho = integral dx_1:N dy_1:N rho(x_1:N,y_1:N),where rho(x_1:N,y_1:N) is the rank one operator

psi(x_1:N)psi^*(y_1:N)a^*(x_1:N)vac>


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If one accepts the vague terminology to avoid talking always aboutlimits, one can give the following definition (which reflects thesubjectivity in the qualification about the modeling accuracy):

In statistical mechanics, all macroscopic observables are ensembleaverages. Thus, formally, a "macroscopic observable" is the expectationof a space-time dependent field operator which remains constant

within the modeling accuracy under changes in space and timesmaller than the modeling accuracy.

---------------------------------------S1n. The role of the ergodic hypothesis---------------------------------------

Statistical mechanics textbook often invoke the so-called ergodichypothesis (assuming that every phase space trajectory comes

arbitrarily close to every phase space point with the same values ofall conserved variables as the initioal point of the trajectory)to derive thermodynamics from the foundations. However, textbookstatistical mechanics gives only a gross simplification of thepower of thermodynamics. The ergodic hypothesis is not needed to makethermodynamics valid. Indeed, the ergodic hypothesis is invalid inmany cases - namely always when the system needs additional variablesto be thermodynamically described.

This is the case for fluids near the critical point, for finite objectsat their surfaces, for systems with interfaces, for metastable states,for molecular systems in the absence of chemical reactions (here thenumber of molecules of each species is conserved), etc.

But this does not invalidate thermodynamics - the latter only requiresthat a sufficiently large set of macroscopic variables (in the abovesense) is included in the list of thermodynamic variables.Indeed, traditional thermodynamics accounts for molecules, surfacetension, metastability, etc., without any change to the formalism.

Probably the ergodic hypothesis, restricted to a limited piece of asubmanifold of the phase space with fixed values of the macroscopicvariables (whether conserved or not) is ''roughly'' equivalent to thecompleteness of the set of distinguished macroscopic observables,in the sense that every other macroscopic observable can be definedin terms of the distinguished ones. But ...

1. It is the latter property (only) which can be checked experimentally:Completeness holds if and only if the properties of the system understudy are indeed predicable by the thermodynamics of the distinguishedobservables. Experiment (or experience), together with simplicity ofthe description, decides in _all_ practical situations what is the setof distinguished observables.Indeed, we refine a model whenever we discover significant deviationsfrom the thermodynamical behavior of a previous simpler model.Thus thermodynamics takes the form of a setting for describingmaterial properties to which any successful description has to conform

by axiomatic decree.

2. The ergodic hypothesis can be proved only for extremely simple


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systems. In particular, these systems must conform to classicalmechanics - there is no simple quantum version of ergodic dynamics.Moreover, there are many classical systems which are chaotic only inpart of their phase space - they are probably not ergodic, as thenumber of conserved quantities depends on where in the phase space oneis.

3. Thermodynamics applies also for nearly conserved quantities, wherethe ergodic argument becomes vague; conversely, near ergodicity (up tothe model accuracy) is enough to make a thermodynamic descriptionvalid. In particular, thermodynamics applies near a critical pointwhere there cannot be an ergodic argument since there is no extraconserved quantity but an order parameter is needed to give a correctdescription. (At which distance from the critical point should oneignore the order parameter? Ergodic arguments have nothing to say here.)

4. There are studies about the nonergodic behavior of supercooledliquids, e.g., Phys. Rev. A 43, 1103 - 1106 (1991).

Thus I think it is best to ignore the ergodic hypothesis as a means forexplaining statistical mechanics, except in some simple model cases.It should have no deeper relevance than the hard sphere model of amonatomic gas (which has been shown to be ergodic, I believe).

----------------------------------------------------S1o. Does quantum mechanics apply to single systems?----------------------------------------------------

It is clear phenomenologically that statistical mechanics (and hencequantum mechanics) applies to single systems like a particular cup oftea, irrespective of what the discussions about the foundations ofphysics say (see many other entries in this FAQ). Thus statisticalmechanics and quantum mechanics do not only apply - as is oftenclaimed - to large ensembles of independently and identically preparedsystems; when the system is large enough (i.e., macroscopic),a _single_ system is enough.(For smaller single systems, see the entry''How do atoms and molecules look like?'' in the present FAQ.)In classical statistical mechanics, the traditional bridge betweenthe ensemble view and thermodynamics (which clearly applies to singlesystems) is the ergodic hypothesis. But there is not enough timein the universe to explore more than an extremely tiny region of theabout 10^25-dimensional phase space of the cup of tea to explain thesuccess of the thermodynamical description by ergodicity.

In quantum mechanics, the situation is even worse - usually it is noteven attempted here to bridge the gap.

The best treatment I know of the foundational problemsinvolved in classical statistical mechanics is in the book

L. Sklar,Physics and Chance,

Cambridge Univ. Press, Cambridge 1993.but it does not present a solution. Other sources are not better inthis respect.


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My own solution is the ''thermal interpretation'' ofphysics, discussed to some extent in Chapter 7 of the book

Arnold Neumaier and Dennis Westra,Classical and Quantum Mechanics via Lie algebras,Cambridge University Press, to appear (2009?).http://www.mat.univie.ac.at/~neum/papers/physpapers.html#QML

arXiv:0810.1019and in my recent slides

A. Neumaier,Classical and quantum field aspects of light,http://www.mat.univie.ac.at/~neum/papers/physpapers.html#lightslides

andA. Neumaier,Optical models for quantum mechanics,http://www.mat.univie.ac.at/~neum/papers/physpapers.html#optslides

and explored in more detail in my GermanEin Theoretische Physik FAQhttp://www.mat.univie.ac.at/~neum/physik-faq.txt

under the name ''consistent experiment interpretation''

The key idea is that mathematical expectation has two differentinterpretations in physics, one as average over a large number ofcases, and the other as a means of defining observables. That thetwo interpretations have the same mathematical properties is thereason they have been confused in the past. The thermal interpretationseparates them neatly and thus gets rid of most of the confusingaspects of the foundations of physics.

-----------------------------------------

S1p. Dissipative dynamics and Lagrangians-----------------------------------------

Any system of ordinary differential equations can be broughtinto an artificial Lagrangian form, by first rewriting it in firstorder form

F(q,q')=0doubling the degrees of freedom by introducing conjugate variables p,and then considering the Lagrangian

L(p,q)= p^T F(q,q').In particular, this provides a Lagrangian formulation of dissipativesystems, such as the damped harmonic oscillator

m q'' + c q' + k q = 0 (m,c,k >0)Unfortunately, the Hamiltonian in such a formulation hasnothing to do with the physical energy

E = (m q'^2 + k q^2)/2

The same holds for various other representations for the dampedharmonic oscillator found in the literature.Lagrangians for the damped harmonic oscillator go back toH. Bateman, Phys. Rev. 38, 815-819 (1931); the treatise

P.M. Morse and H. Feshbach,Methods of Theoretical PhysicsMacGraw-Hill, Boston 1953

discusses the procedure in Chapter 3 in terms of 'mirror images'

= additional dynamical variables needed to absorb the missing energy,and remarks on p 313:

''The introduction of the mirror image ... is probably too artificial


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a prcedure to expect to obtain much of physical significance fromit.''

And indeed, the book doesn't make use of it anywhere.

Having a formal Lagrangian or Hamiltonian is no virtue in itself.In particular, for a _quantum_ system, the Hamiltonian _must_ be the

energy. Playing around with alternative Lagrangians and Hamiltoniansmay be amusing, but does not produce relevant physics.

Since dissipative equations (like the diffusion equation or the dampedharmonic oscillator) describe open systems (where energy is lost to anunspecified environment), they cannot be described by a Schroedingerequation.

Classically, dissipative systems are described by stochasticdifferential equations (and their equivalent deterministicFokker-Planck equations) or master equations;

the diffusion equation is the particular case of a Fokker-Planckequation for Brownian motion.

Quantum mechanically, dissipative systems are described by stochasticSchroedinger equations or, corresponding to the Fokker-Planck level,by quantum Liouville equations with Lindblad terms. This gives correctphysics in a dissipative environment. Many quantum optical systemsare directly modeled on the Lindblad level, where the terms have anunderstandable and experimentally verifiable meaning independent ofany underlying more microscopic model.

An important recent example is that of photons on demand,M. Keller, B Lange, K Hayasaka, W Lange and H Walther,

A calcium ion in a cavity as a controlled single-photon source,New Journal of Physics 6 (2004), 95.

There is no trace of a Lagrangian in the modeling, and indeed, auseful Lagrangian formulation does not exist - unless one extends thedynamics and explicitly includes the environment.

Of course, in theory, a dissipative system is thought to be acontracted version of a bigger conservative system which includesthe envoironment, and in simple situations, this theoretical view canindeed be substantiated.

If one models the dissipative environment explicitly, on gets abigger conservative system, not a dissipative system. Of course,this conservative system has a Hamiltonian or Lagrangian description,but it does not describe the dissipative system alone. When onecontracts it to the degrees of freedoms of the original system,one gets an integro-differential equation with memory, which is nolonger described by a physically meaningful Hamiltonian or Lagrangianframework.

The reduced dynamics takes the exact formm x''(t) + k x(t) = int_0^t G(s) x(t-s) ds + F(t).

with functions F(t) (the noise caused by the environment) and G(s)(the memory kernel) that depend on the state of the environment.

If the interaction is of the usual, dissipative nature then both F(t)and G(s) are extremely oscillating, even for intervals short comparedto the inverse frequency T of the oscillator. But the short time


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averages of the memory Kernel have an exponentially decaying bound ontheir size and become negligible after some relaxation time tau 0, recovering thetraditional equation for the damped harmonic oscillator, including astochastic force term. (Its size can be related to the dampingcoefficient and the temperature of the environment, a relation knownas the fluctuation-dissipation theorem.)

A thorough discussion of the reduction of microscopic conservativelarge systems to dissipative subsystems of interest is given in

H Grabert,Projection Operator Techniques in NonequilibriumStatistical Mechanics,Springer Tracts in Modern Physics, 1982

at a much more general level that also applies formany other dissipative systems.

There are cases where one needs to model the memory to capture theessence of the reduced dynamics. But in many cases, a simpler,

memory-free description is possible and adequate. One can remove thememory by employing a Markov approximation, and gets again adifferential equation, which defines the Lindblad (or, classicallally,the Focker-Planck) dynamics. Again, this is no longer described by aHamiltonian or Lagrangian framework.

In the extended formulation with explicit environment or with memory,already a simple damped harmonic oscillator becomes a huge andunwieldy dynamical system which is no longer equivalent to the dampedharmonic oscillator, but includes unwanted environment terms or memoryterms. In cases where one really needs to model the memory, the systemtherefore is no longer a damped harmonic oscillator. The latter isdescribed by a simple linear constant coefficient second orderdifferential equation for a single function, and has no memory.Its analysis is very simple, and compared to that any more detaileddescription is unwieldy.

In practice, the dissipative formulation therefore stands by itself(apart from lip service paid to a hypothesized more fundamentalconservative description).

The situation is similar to that in fluid dynamics. In theory, theNavier-Stokes equations (which are dissipative) should be derivable froma Lagrangian. Indeed, such derivations have been given, but only for

very simple model problems such as an ideal gas. However, there is nomicroscopic derivation of the Navier-Stokes equations in the practicallyinteresting case of water at room temperature...


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---------------------------------------------------------------------S1q. How can QM be stochastic while the Schroedinger equation is not?---------------------------------------------------------------------

The Schroedinger equation is a deterministic wave equation.But when we set up an experiment to measure either position ormomentum, we get uncertain, stochastic outcomes.

So - is quantum mechanics deterministic or stochastic?

One has to be careful in the interpretation of the foundations...

Fortunately, the same apparent paradox already occurs in classicalphysics; hence the paradox cannot have anything to do with the

peculiarities of quantum mechanics.

Indeed, a Focker-Planck equation is a deterministic partialdifferential equation. But when measuring a process modelled by it- such as the position of a grain of pollen in Brownian motion -,we get only probabilistic results. Now Focker-Planck equations areessentially equivalent to classical stochstic differential equations.

So - do they describe a deterministic or a stochastic process?

The point resolving the issue is that, both in stochstic differentialequations and in quantum mechanics, probabilities satisfy deterministic

equations, while the quantities observed to deduce the probabilitiesdo not.

Thus, in both cases, probabilities are deterministic ''observables''while the position of a grain of pollen in classical mechanics, orposition and momentum in quantum mechanics, ar not.

----------------------------------------S1r. Measurement theory for real numbers----------------------------------------

The standard textbook measurement theory says that the possiblemeasurement results in measuring an observable given by a Hermitianoperator A are its possible eigenvalues, with a probability densitydepending on the state of the system. This is part of the content ofBorn's rule, and counts as one of the cornerstones of theinterpretation of quantum mechanics.

But Born's rule gives only a very idealized account of measurementtheory, and gives no sufficient explanation for what is going on inmany nontrivial measurements.

The spectrum of the Hamiltonian of the electron of a hydrogen atom

has a discrete part, catering for its bound states. According to theidealized textbook measurement theory, a measurement of the energyof a bound state should produce an infinitely accurate value agreeing


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with one of the values in the (QED-corrected) Balmer (etc.) series.

But this is ridiculous. Repeated preparation and measurement of theposition of the ``same'' spectral lines (which provide these energymeasurements, relative to an appropriate zero of the energy) yieldsdifferent results, from which the energies themselves can be obtainedonly to a certain accuracy.

Thus Born's rule does not account for the interpretation of ameasurement of the energy of an electron. For similar reasons,measurements of particle masses or resonance energies do not revealthe exact values (which they should according to Born's rule) but onlyapproximations whose quality depends a lot on the way the measurementis done (an aspect that does not figure at all in Born's rule).

Measurements such as that of a particle lifetime or the integral crosssection of a particular reaction do not even have a natural associatedoperator of which the measurement result would be an eigenvalue.

The idealized textbook measurement theory based on Born's rule isappropriate only for the measurement of spin and related variablesthat result in recording decisions of finite information content.

Thus the measurement process as described by von Neumann (and copiedfrom there to numerous textbooks) is an unrealistic idealizationcompared with many (and probably most) real measurements.The latter are usually much better described by suitable POVMs(positive operator valued measures) rather than by Born's rule,which corresponds to PVMs (projection-valued measures), a special caseof POVMs in which the positive operators are in fact projections.

See Sections 7.3-7.5 of the book

A. Neumaier and D. Westra,Classical and Quantum Mechanics via Lie algebras,arXiv:0810.1019

for a realistic account of measurement theory not dependent onBorn's rule. The latter is derived there as a special case, togetherwith giving the condition in which it is applicable.

---------------------------------------------S1s. The classical limit of quantum mechanics---------------------------------------------

Classical mechanics is often seen as the formal limit hbar-->0 ofquantum mechanics. Strictly speaking, this cannot be true since hbaris a constant of nature, which is often even set to one to haveconvenient units. The classical limit really is the limit of largequantum numbers M (typically of mass, number of particles, or size ofangular momentum), when attention is limited to quantities whoseuncertainties are small compared to their expectations.

In these situations, the effect is similar to taking the limithbar --> 0. In these cases the relative uncertainties scale withsqrt(hbar/M), which becomes small if either hbar is made formallytiny or if M is large.

Indeed, a quantum system is essentially classical if its relevantquantities have uncertainties that are small compared to their


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expectations.

The relation between classical mechanics is most easily seen if --as in statistical mechanics -- quantum mechnaics is presented in termsof mixed states, which correspond to density matrices.(Almost all quantum mechanics applied to real systems not inthe ground state needs density matrices, since pure states are very

difficult to create and propagate unless a system is in the groundstate. Pure states describe only an idealized version of quantumreality, which in statistical mechanics appears as the approximationin the cold limit T-->0.)

Density matrices are intrinsically quantum mechanical.Nevertheless they exhibit very close analogies to classical densities.

Therefore everyone interested in the relations between classical andquantum mechanics is well-advised to look at both theories in thestatistical mechanics version, where the analogies are obvious, andthe transition from quantum to classical takes the form of a simple

approximation.

QM in the statistical mechanics version is almost as intuitive asclassical statistical mechanics. The only somewhat nonintuitive partis in both cases how to interpret probability. (This is already asevere problem in classical statistical mechanics, as the book byLaurence Sklar, Physics and Chance, explains in detail.)

A density matrix describes the stochastic behavior of a quantum systemin the same way as a density function describes the stochastic behaviorof a classical system. In both cases, if the system is nice enough thatthe stochastic uncertainties (square roots of variances) in the

quantities of interest are much smaller than the quantities themselves,one can form a deterministic approximation.

This deterministic approximation is given by a classical dynamicalsystem for the (expectations of the) quantities of interest.

Thus, in a sense, classical variables are simply expectations ofrelevant quantum variables with small uncertainty. Then (and only then)is a deterministic approximation adequate. The small uncertaintymakes these variables approximately predictable in each individualevent, and hence classical.

Classicality therefore develops whenever the uncertainties of thequantities of interest become small compared to their expectations.Of course, there is significant interest in quantum systems where thisdoes not happen, since these are decidedly non-classical, but quantumtheory gets its strange, counterintuitive feature only when oneconcentrates on these systems only.

For more details, see, e.g., Sections 7.3-7.5 ofA. Neumaier and D. Westra,Classical and Quantum Mechanics via Lie algebrashttp://de.arxiv.org/abs/0810.1019

--------------------------------------------S1t. The classical limit via coherent states


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--------------------------------------------

One method for producing classical mechanics from a quantum theory isby looking at coherent states of the quantum theory. The standard(Glauber) coherent states have a localized probability distribution inclassical phase space? whose center follows the classical equationsof motion when the Hamiltonian is quadratic in positions and momenta.

(For nonquadratic Hamiltonians, this only holds approximately overshort times. For example, for the 2-body problem with a 1/r^2interaction, Glauber coherent states are not preserved by the dynamics.In this particular case, there are, however, alternative SO(2,4)-basedcoherent states that are preserved by the dynamics, smeared overKepler-like orbits. The reason is that the Kepler 2-body problem --and its quantum version, the hydrogen atom -- are superintegrablesystems with the large dynamical symmetry group SO(2,4).)

In general, roughly, coherent states form a nice orbit of unit vectorsof a Hilbert space H under a dynamical symmetry group G with a

triangular decomposition, such that the linear combinations ofcoherent states are dense in H, and the inner product phi^*psi ofcoherent states phi and psi can be calculated explicitly in terms ofthe highest weight representation theory of G. The diagonal of theN-th tensor power of H (coding systems with N-fold quantum numbers)has coherent states phi_N (labelled by the same classical phase spaceas the original coherent states, and orresponding to the N-fold highestweight) with inner product

phi_N^*psi_N=(phi^*psi) Nand for N --> inf, one gets a good classical limit. For the Heisenberggroup, phi^*psi is a 1/hbar-th power, and the N-th power correspondsto replacing hbar by hbar/N. Thus one gets the standard classical limit.

Basic literature on relations between coherent states and the classicallimit, based on irreducible unitary representations of Lie groupsincludes the book

A. M. Perelomov,Generalized Coherent States and Their Applications,Springer-Verlag, Berlin, 1986.

and the paperL. Yaffe,Large N limits as classical mechanics,Rev. Mod. Phys. 54, 407--435 (1982)

Both references assume that the Lie group is finite-dimensional andsemisimple. This excludes the Heisenberg group, in terms of which thestandard (Glauber) coherent states are usually defined. However, theHeisenberg group has a triangular decomposition, and this suffices toapply Perelomov's theory in spirit. The online book

Arnold Neumaier, Dennis Westra,Classical and Quantum Mechanics via Lie algebras,http://lanl.arxiv.org/abs/0810.1019

contains a general discussion of the relations between classicalmechanics and quantum mechanics, and discusses in Chapter 16 theconcept of a triangular decomposition of Lie algebras and a summary ofthe associated representation theory (though in its present versionnot the general relation to coherent states).

For other relevant approaches to a rigorous classical limit, see the

online sourceshttp://www.projecteuclid.org/Dienst/Repository/1.0/Disseminate/euclid.cmp/110

3859040/body/pdf


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http://www.univie.ac.at/nuhag-php/bibtex/open_files/si80_SIMON!!!.pdfhttp://arxiv.org/abs/quant-ph/9504016http://arxiv.org/pdf/math-ph/9807027

--------------------------------

S2a. Lie groups and Lie algebras--------------------------------

Lie groups can be illustrated by continuous rigid motion of a ballwith painted patterns on it in 3-dimensional space. The Lie group ISO(3)consists of all rigid transformations.

A rigid transformation is essentially the act of picking the ball andplacing it somewhere else, ignoring the detailed motion in between andthe location one started.Special transformations are for example a translation in northerndirection by 1 meter, or a rotation by one quarter around the vertical

axis at some particular point (think of a ball with a string attached).'Rigid' means that the distances between marked points on the ballremains the same; the mathematician talks about 'preserving distances',and the distances are therefore labeled 'invariants'.

One can repeat the same transformation several times, or two differenttransformations and get another one - This is called the product ofthese transformations. For example, the product of a translationsby 1 meter and another one by 2 meters in the same direction gives oneof 1+2=3 meters in the same direction. In this case, the distances add,but if one combines rotations about different axes the result is nolonger intuitive. To make this more tractable for calculations,one needs to take some kind of logarithms of transformations - these

behave again additively and make up the corresponding Lie algebraiso(3) [same letters but in lower case]. The elements of the Lie algebracan be visualized as very small, or 'infinitesimal', motions.

General Lie groups and Lie algebras extend these notions to to moregeneral manifolds. A manifold is just a higher-dimensional versionof space, and transformations are generalized motions preservinginvariants that are important in the manifold. The transformationspreserving these invariants are also called 'symmetries', and theLie group consisting of all symmetries is called a 'symmetry group'.The elements of the corresponding Lie algebra are 'infinitesimalsymmetries'.

For example, physical laws are invariant under rotations andtranslations, and hence unter all rigid motions. But not only these:If one includes time explicitly, the resulting 4-dimensional spacehas more invariant motions or ''symmetries''.The Lie group of all these symmetry transformations is called thePoincar'e group, and plays a basic role in the theory of relativity.The transformations are now about space-time frames in uniform motion.Apart from translations and rotations there are symmetries called'boosts' that accelerate a frame in a certain direction, andcombinations obtained by taking products. All infinitesimal symmetriestogether make up a Lie algebra, called the Poincar'e algebra.

Much more on Lie groups and Lie algebras from the perspective ofclassical and quantum physics can be found in:


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Arnold Neumaier and Dennis Westra,Classical and Quantum Mechanics via Lie algebras,Cambridge University Press, to appear (2009?).http://www.mat.univie.ac.at/~neum/papers/physpapers.html#QMLarXiv:0810.1019

-----------------------------------------------------------S2b. The Galilei group as contraction of the Poincare group-----------------------------------------------------------

The group of symmetries of special relativity is the Poincare group.

However, before Einstein invented the theory of relativity,physics was believed to follow Newton's laws, and these have adifferent group of symmetries - the Galilei group, and itsinfinitesimal symmetries form the Galilei algebra.

Now Newton's physics is just a special case of the theory of relativityin which all motions are very slow compared to the speed of light.Physicists speak of the 'nonrelativisitic limit'.Thus one would expect that the Galilei group is a kind ofnonrelativistic limit of the Poincar'e group.

This notion has been made precise by Inonu. He looked at thePoincar'e algebra and 'contracted' it in an ingenious wayto the Galilei algebra. The construction could then be lifted tothe corresponding groups. Not only that, it turned out to be ageneral machinery applicable to all Lie algebras and Lie groups,and therefore has found many applications far beyond that for which

it was originally developed.

---------------------------------------------------------------------S2c. Representations of the Poincare group, spin and gauge invariance---------------------------------------------------------------------

Whatever deserves the name ''particle'' must move like a single,indivisible object. The Poincare group must act on the description ofthis single object; so the state space of the object carries aunitary representation of the Poincare group. This splits into a directsum or direct integral of irreducible reps. But splitting meansdivisibility; so in the indivisible case, we have an irreduciblerepresentation. Thus particles are described by irreducible unitaryreps of the Poincare group. Additional parameters characterizing theirreducible representation of an internal symmetry group = gauge

On the other hand, not all irreducible unitary reps of the Poincaregroup qualify. Associated with the rep must be a consistent and causalfree field theory. As explained in Volume 1 of Weinberg's book onquantum field theory, this restricts the rep further to those withpositive mass, or massless reps with quantized helicity.

Weinberg's book on QFT argues for gauge invariance fromcausality + masslessness. He discusses massless fields inChapter 5, and observes (probably there, or in the beginning


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of Chapter 8 on quantum electrodynamics) roughly the following:

Since massless spin 1 fields have only two degrees of freedom,the 4-vector one can make from them does not transform correctlybut only up to a gauge transformation making up for the missinglongitudinal degree of freedom. Since sufficiently long rangeelementary fields (less than exponential decay) are necessarily

massless, they must either have spin 2. See the tables of the particle data group, e.g., Delta(2950)(randomly chosen from http://pdg.lbl.gov/2003/bxxxpdf.html ).

R.L. Ingraham,Prog. Theor. Phys. 51 91974), 249-261,http://ptp.ipap.jp/link?PTP/51/249/

constructs covariant propagators and complete vertices for spin Jbosons with conserved currents for all J. See also

H Shi-Zhong et al.,Eur. Phys. J. C 42 (2005), 375-389http://www.springerlink.com/content/ww61351722118853/

-----------------------------------S2d. Forms of relativistic dynamics-----------------------------------

Relativistic multiparticle mechanics is an intricate subject,and there are no-go theorems that imply that the most plausible


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possibilities cannot be realized. However, these no-go theoremsdepend on assumptions that, when questioned, allow meaningfulsolutions. The no-go theorems thus show that one needs to be carefulnot to introduce plausible but inappropriate intuition into theformal framework.

To pose the problem, one needs to distinguish between kinematicaland dynamical quantities in the theory. Kinematics answers thequestion "What are the general form and properties of objects thatare subject to the dynamics?" Thus it tells one about conceivablesolutions, mapping out the properties of the considered representationof the phase space (or what remains of it in the quantum case).Thus kinematics is geometric in nature. But kinematics does not knowof equations of motions, and hence can only tell general (kinematical)features of solutions.

In contrast, dynamics is based on an equation of motion (or anassociated variational principle) and answers the question 'What

characterizes the actual solution?', given appropriate initial orboundary conditions. Although the actual solution may not be availablein closed form, one can discuss their detailed properties and devisenumerical approximation schemes.

The difference between kinematical and dynamical is one of convention,and has nothing to do with the physics. By choosing the representation,i.e., the geometric setting, one chooses what is kinematical;everything else is dynamical.

Since something which is up to the choice of the person describingan experiment can never be distinguished experimentally, the physicsis unaffected. However, the formulas look very different in different

descriptions, and - just as in choosing coordinate systems - choosinga form adapted to a problem may make a huge difference for actualcomputations.

Dirac distinguishes in his seminal paperRev. Mod. Phys. 21 (1949), 392-399

three natural forms of relativistic dynamics, the instant form,the point form, and the fromt form. They are distinguished bywhat they consider to be kinematical quantities and what are thedynamical quantities.

The familiar form of dynamics is the instant form,which treats space (hence spatial translations and rotations)as kinematical and time (and hence time translation and Lorentz boosts)as dynamical. This is the dynamics from the point of view of ahypothetical observer (let us call it an 'instant observer')who has knowledge about all information at some time t (the present),and asks how this information changes as time proceeds.

Because of causality (the finite bound of c on the speed of materialmotion and communication), the resulting differential equationsshould be symmetric hyperbolic differential equations for which theinitial-value problem is well-posed.

Because of Lorentz invariance, the time axis can beany axis along a timelike 4-vector, and (in special relativity)space is the 3-space orthogonal to it. For a real observer,


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the natural timelike vector is the momentum 4-vector of the materialsystem defining its reference frame (e.g., the solar system).

While very close to the Newtonian view of reality, it involvesan element of fiction in that no real observer can get all theinformation needed as intial data. Indeed, causality implies thatit is impossible for a physical observer to know the present anywhere

except at its own position.

A second, natural form of relativistic dynamics is, according to Dirac,the point form. This is the form of dynamics in which a particularspace-time point x=0 (the here and now) in Minkowski space isdistinguished, and the kinematical object replacing space is,for fixed L, a hyperboloid x^2=L^2 (and x_0


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transformations preserving algebraic operations and the Poisson bracket,and quantum mechanically by unitary transformations preservingalgebraic operations and hence the commutator. This means that anystatement about a system in one of the forms can be translated intoan equivalent statement of an equivalent system in any of the otherforms.

Preferences are therefore given to one form over the other dependingsolely on the relative simplicity of the computations one wants to do.This is completely analogous to the choice of coordinate systems(cartesian, polar, cylindric, etc.) in classical mechanics.

For a multiparticle theory, however, the different forms and theneed to pick a particular one seem to give different pictures ofreality. This invites paradoxes if one is not careful.

This can be seen by considering trajectories of classical relativisticmany-particle systems. There is a famous theorem by

Currie, Jordan and SudarshanRev. Mod. Phys. 35 (1963), 350-375which asserts that interacting two-particle systems cannot haveLorentz invariant trajectories in Minkowski space. Traditionally,this was taken by mainstream physics as an indication that themultiparticle view of relativistic mechanics is inadequate,and a field theoretical formulation is essential.However, as time proceeded, several approaches to valid relativisticmulti-particle (quantum) dynamics were found (see the FAQ entry on'Is there a multiparticle relativistic quantum mechanics?'),and the theorem had the same fate as von Neumann's proof thathidden-variable theories are impossible. Both results are now simplytaken as an indication that the assumptions under which they were

made are too strong.

In particular, once the assumption by Currie, Jordan and Sudarshanthat all observers see the same trajectories of a system of interactingparticles is rejected, their no-go theorem no longer applies.The question then is how to find a consistent and covariant descriptionwithout this at first sight very intuitive property. But once it isadmitted that different observers see the same world but representedin different personal spaces, the formerly intuitive property becomesmeaningless. For objectivity, it is enough that one can consistentlytranslate the views of any observer into that of any other observer.Precisely this is the role of the dynamical Poincare transformations.

Thus nothing forbids an instant observer to observeparticle trajectories in its present space, or apoint observer to observe particle trajectories in its past hyperboloid.However, the present space (or the past hyperboloid) of two differentobservers is related not by kinematical transforms but dynamically,with the result that trajectories seen by different observers ontheir different kinematical 3-surface look different.Classically, this looks strange on first sight, althoughthe Poincare group provides well-defined recipes for translatingthe trajectories seen by one observer into those seen by anotherobserver.

Quantum mechanically, trajectories are fuzzy anyway, due to theuncertainty principle, and as various successful multiparticletheories show, there is no mathematical obstacle for such a description.


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The mathematical reason of this superficially paradoxical situationlies in the fact that there is no observer-independent definitionof the center of mass of relativistic particles, and the related factthat there is no observer-independent definition of space-timecoordinates for a multiparticle system.The best one can do is to define either a covariant position operator

whose components do not commute (thus definig a noncommutativespace-time), or a spatial position operator, the so-calledNewton-Wigner position operator, which has three commuting coordinatesbut is observer-dependent.(See the FAQ entry on 'Localization and position operators'.)

-------------------------------------------------------------S2e. Is there a multiparticle relativistic quantum mechanics?-------------------------------------------------------------

In his QFT book, Weinberg says no, arguing that there is no way toimplement the cluster separation property. But in fact there is:

There is a big survey by Keister and Polyzou on the subjectB.D. Keister and W.N. Polyzou,Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics,in: Advances in Nuclear Physics, Volume 20,(J. W. Negele and E.W. Vogt, eds.)Plenum Press 1991.www.physics.uiowa.edu/~wpolyzou/papers/rev.pdf

that covered everything known at that time. This survey was quotedat least 116 times, see

http://www.slac.stanford.edu/spires/find/hep?c=ANUPB,20,225

looking these up will bring you close to the state of the arton this.

They survey the construction of effective few-particle models.There are no singular interactions, hence there is no need forrenormalization.

The models are _not_ field theories, only Poincare-invariant few-bodydynamics with cluster decomposition and phenomenological termswhich can be matched to approximate form factors from experiment orsome field theory. (Actually many-body dynamics also works, but themany particle case is extremely messy.)They are useful phenomenological models, but somewhat limited;for example, it is not clear how to incorporate external fields.

The papers by Klink athttp://www.physics.uiowa.edu/~wklink/

and work by Polyzou athttp://www.physics.uiowa.edu/~wpolyzou/

contain lots of multiparticle relativistic quantum mechanics,applied to real particles. See also the Ph.D. thesis by Krassnigg at

http://physik.uni-graz.at/~ank/dissertation-f.html

(Other work in this direction includes Dirac's many-time quantumtheory, with a separate time coordinate for each particle; see, e.g.,

Marian Guenther, Phys Rev 94, 1347-1357 (1954)and references there. Related multi-time work was done under thename of 'proper time quantum mechanics' or 'manifestly covariant


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quantum mechanics', see, e.g.,L.P. Horwitz and C. Piron, Helv. Phys. Acta 48 (1973) 316,

but it does not reproduce standard physics, and apparently neverreached a stage useful to phenomenology.)

Note that in the working single-time approaches, covariance is alwaysachieved through a representation of the Poincare group on a

Hilbert space corresponding to a fixed time (or another 3D manifold inspace-time), rather than through multiple times.Thus the whole theory has a single time only, whose dynamics isgenerated by the Hamiltonian, the generator H=P_0 of the Poincare group.(This is completely analogous to the nonrelativistic case,where multiparticle systems also have a single time only.)

The natural manifestly covariant picture is that of a vector bundleon Minkowski space-time, with a standard Fock space attached to eachpoint. An observer (i.e., formally, an orthonormal frame attached atsome space-time point) moves in space-time via the Poincare group,and this action extends to the bundle by means of the representation

defining the Fock space.

----------------------S2f. What is a photon?----------------------

According to quantum electrodynamics, the most accurately verifiedtheory in physics, a photon is a single-particle excitation of thefree quantum electromagnetic field. More formally, it is a state ofthe free electromagnetic field which is an eigenstate of the photon

number operator with eigenvalue 1.

The pure states of the free quantum electromagnetic fieldare elements of a Fock space constructed from 1-photon states.A general n-photon state vector is an arbitrary linear combinationsof tensor products of n 1-photon state vectors; and a general purestate of the free quantum electromagnetic field is a sum of n-photonstate vectors, one for each n. If only the 0-photon term contributes,we have the dark state, usually called the vacuum; if only the1-photon term contributes, we have a single photon.

A single photon has the same degrees of freedom as a classical vacuumradiation field. Its shape is characterized by an arbitrary nonzeroreal 4-potential A(x) satisfying the free Maxwell equations, which inthe Lorentz gauge take the form

nabla dot nabla A(x) = 0,nabla dot A(x) = 0,

expressing the zero mass and the transversality of photons. Thus forevery such A there is a corresponding pure photon state A>.Here A(x) is _not_ a field operator but a photon amplitude;photons whose amplitude differ by an x-independent phase factor arethe same. For a photon in the normalized state A>, the observableelectromagnetic field expectations are given by the usual formulasrelating the 4-potential and the fields,

=

= - partial \A(x)/partial x_0 - c nabla_\x A_0(x),and

= = nabla_\x x \A(x)


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