EJT P E THEORETICAL JOURNAL OF PHYSICS - … · A. Z ANZI 3. An Accumulative Model for Quantum...

EJTP ELECTRONIC JOURNAL OFTHEORETICAL PHYSICS

Volume 12 Number 33November, 2015

http://www.ejtp.com E-mail:[email protected]

editorsIgnazio LicataAmmar Sakaji

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Ist edition: November 2015

Table of Contents

1. Preface ............................................................................................ iL. LICATA

2. Chameleonic Equivalence Postulate and Wave Function Collapse ....... 1A. ZANZI

3. An Accumulative Model for Quantum Theories...................................... 29

C. THRON

4. Spin Angular Momentum and the Dirac Equation ................................ 43R.A.CLOSE

5. An Introduction to Strict Quantization ................................................... 61J.M. VELHINHO

6. What is the Wave Function and Why is it used in Quantum Mechanics? ... 91Y.A. RYLOV

7. Analytic Solution of the Algebraic Equation Associated to the Ricci Tensorin Extended Palatini Gravity ................................................................. 109G.R.P.TERUEL

8. Testing the Everett Interpretation of Quantum Mechanics with Cosmology ..... 127A. BARRAU

9. Elliott Formula for Particle-hole Pair of Dirac Cone ............................... 135L.E.LOKOT

10. Hamilton-Jacobi Formulation of Supermembrane ................................. 149M.K. SROUR, M. ALWER, N. I.FARAHAT

11. Fractional Spin Through Quantum Affine Algebra $\hat A(n)$ and Quan-tum Affine Super-algebra $\hat A(n,m) $}............................................. 155M. MANSOUR, M. DAOUD

12. Nobel Lecture: Spontaneous Symmetry Breaking In Particle Physics: ACase of Cross Fertilization .................................................................... 171Y. NAMBU

Editor in Chief

Ignazio Licata

Foundations of Quantum Mechanics, Complex System & Computation in Physics and Biology, IxtuCyber for Complex Systems , and ISEM, Institute for Scientific Methodology, Palermo, Sicily – Italy

editor[AT]ejtp.info Email: ignazio.licata[AT]ejtp.info

ignazio.licata[AT]ixtucyber.org

Co-Editor

Ammar Sakaji

Theoretical Condensed Matter, Mathematical Physics International Institute for Theoretical Physics and Mathematics (IITPM), Prato, Italy. Amman-Jordan Tel:+962778195003 :+971557967946 Email: info[AT]ejtp.com info[AT]ejtp.info

Editorial Board

Gerardo F. Torres del Castillo

Mathematical Physics, Classical Mechanics, General Relativity, Universidad Autónoma de Puebla, México, Email:gtorres[AT]fcfm.buap.mx Torresdelcastillo[AT]gmail.com

Leonardo Chiatti

Medical Physics Laboratory AUSL VT Via Enrico Fermi 15, 01100 Viterbo (Italy) Tel : (0039) 0761 1711055 Fax (0039) 0761 1711055 Email: fisica1.san[AT]asl.vt.it chiatti[AT]ejtp.info

Francisco Javier Chinea

Differential Geometry & General Relativity, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Spain, E-mail: chinea[AT]fis.ucm.es

Maurizio Consoli

Non Perturbative Description of Spontaneous Symmetry Breaking as a Condensation Phenomenon, Emerging Gravity and Higgs Mechanism, Dip. Phys., Univ. CT, INFN,Italy

Email: Maurizio.Consoli[AT]ct.infn.it

Avshalom Elitzur

Foundations of Quantum Physics ISEM, Institute for Scientific Methodology, Palermo, Italy Email: Avshalom.Elitzur[AT]ejtp.info

Elvira Fortunato

Quantum Devices and Nanotechnology:

Departamento de Ciência dos Materiais CENIMAT, Centro de Investigação de Materiais I3N, Instituto de Nanoestruturas, Nanomodelação e Nanofabricação FCT-UNL Campus de Caparica 2829-516 Caparica Portugal

Tel: +351 212948562; Directo:+351 212949630 Fax: +351 212948558 Email:emf[AT]fct.unl.pt elvira.fortunato[AT]fct.unl.pt

Tepper L. Gill

Mathematical Physics, Quantum Field Theory Department of Electrical and Computer Engineering Howard University, Washington, DC, USA

Email: tgill[AT]Howard.edu tgill[AT]ejtp.info

Alessandro Giuliani

Mathematical Models for Molecular Biology Senior Scientist at Istituto Superiore di Sanità Roma-Italy

Email: alessandro.giuliani[AT]iss.it

Vitiello Giuseppe

Quantum Field Theories, Neutrino Oscillations, Biological Systems Dipartimento di Fisica Università di Salerno Baronissi (SA) - 84081 Italy Phone: +39 (0)89 965311 Fax : +39 (0)89 953804 Email: [email protected]

Richard Hammond

General Relativity High energy laser interactions with charged particles Classical equation of motion with radiation reaction Electromagnetic radiation reaction forces Department of Physics University of North Carolina at Chapel Hill, USA Email: rhammond[AT]email.unc.edu

Arbab Ibrahim

Theoretical Astrophysics and Cosmology Department of Physics, Faculty of Science, University of Khartoum, P.O. Box 321, Khartoum 11115, Sudan

Email: aiarbab[AT]uofk.edu arbab_ibrahim[AT]ejtp.info

Kirsty Kitto

Quantum Theory and Complexity Information Systems | Faculty of Science and Technology Queensland University of Technology Brisbane 4001 Australia

Email: kirsty.kitto[AT]qut.edu.au

Hagen Kleinert

Quantum Field Theory Institut für Theoretische Physik, Freie Universit¨at Berlin, 14195 Berlin, Germany

Email: h.k[AT]fu-berlin.de

Wai-ning Mei

Condensed matter Theory Physics Department University of Nebraska at Omaha,

Omaha, Nebraska, USA Email: wmei[AT]mail.unomaha.edu physmei[AT]unomaha.edu

Beny Neta

Applied Mathematics Department of Mathematics Naval Postgraduate School 1141 Cunningham Road Monterey, CA 93943, USA Email: byneta[AT]gmail.com

Peter O'Donnell

General Relativity & Mathematical Physics, Homerton College, University of Cambridge, Hills Road, Cambridge CB2 8PH, UK E-mail: po242[AT]cam.ac.uk

Rajeev Kumar Puri

Theoretical Nuclear Physics, Physics Department, Panjab University Chandigarh -160014, India Email: drrkpuri[AT]gmail.com rkpuri[AT]pu.ac.in

Haret C. Rosu

Advanced Materials Division Institute for Scientific and Technological Research (IPICyT) Camino a la Presa San José 2055 Col. Lomas 4a. sección, C.P. 78216 San Luis Potosí, San Luis Potosí, México Email: hcr[AT]titan.ipicyt.edu.mx

Zdenek Stuchlik

Relativistic Astrophysics Department of Physics, Faculty of Philosophy and Science, Silesian University, Bezru covo n´am. 13, 746 01 Opava, Czech Republic Email: Zdenek.Stuchlik[AT]fpf.slu.cz

S.I. Themelis

Atomic, Molecular & Optical Physics Foundation for Research and Technology - Hellas P.O. Box 1527, GR-711 10 Heraklion, Greece Email: stheme[AT]iesl.forth.gr

Yurij Yaremko

Special and General Relativity, Electrodynamics of classical charged particles, Mathematical Physics, Institute for Condensed Matter Physics of Ukrainian National Academy of Sciences 79011 Lviv, Svientsytskii Str. 1 Ukraine Email: yu.yaremko[AT]gmail.com yar[AT]icmp.lviv.ua

yar[AT]ph.icmp.lviv.ua

Nicola Yordanov

Physical Chemistry Bulgarian Academy of Sciences, BG-1113 Sofia, Bulgaria Telephone: (+359 2) 724917 , (+359 2) 9792546

Email: ndyepr[AT]ic.bas.bg ndyepr[AT]bas.bg

Former Editors:

Ignazio Licata, Editor in Chief (August 2015-)

Ignazio Licata, Editor in Chief (October 2009- August 2012)

Losé Luis López-Bonilla, Co-Editor (2008-2012)

Ammar Sakaji, Founder and Editor in Chief (2003- October 2009) and (August 2012- August2015).

EJTP 33 Issue Foreword: “A Purely Quantum Issue”

When Ammar Sakaji sent me the drafts, he characterized this EJTP number as “a

purely quantum Issue”. In fact, the interpretative debate about the foundation of QM

and their consequences in the complex theoretical landscape of contemporary Physics is

central in these pages .I want to mention briefly here some works.

Andrea Zanzi proposes a microscopic counterpart of Einstein’s Equivalence Principle as

a guideline towards QG (quantum Gravity) and a dictionary connecting the classical and

the quantum regime; it allows to interpret the collapse of the wave function as a genuine

QG effect.

The approach by Jose Velhinho to quantization procedure is a very fine generalization of

Weyl-Moyal procedure. A. Barrau has worked on a testing of Everett Theory, one of the

interpretations preferred by cosmologists. Semantic object are central also in Y. Rylov

paper on hydrodynamic quantum fluid: overturning the traditional ”analog” approach,

he shows as the wave function and the QM formalism are determined by the structure of

the classical stochastic dynamics.

I allow myself a special flag about the work of Christopher Thron, “An Accumulative

Model for Quantum Theories”, indeed this model has an “air family” with my work with

L. Chiatti about the emergence of locality from informational vacua. But the Thron

theory has peculiarities all its own and very original: particle detection is represented

as the outcome of a threeshold signal accumulation process which occurs in spacetime

augmented by an extra, non-spacetime dimension

The Quantum deformed algebras are the topics of M. Mansour and M. Daoud, a powe-

ful tool in Quantum Field Theories that promises elegant and unsuspected connections

between particle behaviors at first sight very different.

The old good Dirac equation - the most beautiful equation in Physics!-, it is always a

source of new theoretical perspectives ( R. Close, L. E. Lokot). Rounding out the Issue

two interesting works of M. Kh. Srour, M. and N. Alwer I.Farahat (Hamilton-Jacobi

Formulation of Supermembrane) and G. R. Perez Teruel on Ricci Tensor in Extended

Palatini Gravity.

We feel only right to dedicate this issue to Yoichiro Nambu(Tokyo, January 18, 1921 -

Osaka, July 5, 2015), Nobel fof Physics 2008.

Nambu’s work has always been marked by a great math skills coupled with a strong in-

tuition and a pragmatic sense of theoretical game. It is hard here to list its valuable and

original contributions to particle physics. We limit ourselves to remember the work on

the confinement of quarks and its contribution to one of the key concepts for the under-

standing of the physical world, the Spontaneous Symmetry Breaking. And we are pleased

to propose to our readers his Nobel talk (with the permission of Nobel Foundation and

AIP), dedicated just to the richness of this concept, whose fertility that goes far beyond

particle physics to the new themes in complex systems Physics.

Ignazio Iicata, EJTP Editor in Chief.

EJTP 12, No. 33 (2015) 1–28 Electronic Journal of Theoretical Physics

Chameleonic Equivalence Postulate andWave Function Collapse

Andrea Zanzi∗

Dipartimento di Fisica e Scienze della Terra, Universita di Ferrara - Italy

Received 22 December 2014, Accepted 1 April 2015, Published 25 August 2015

Abstract: A chameleonic solution to the cosmological constant problem and the non-

equivalence of different conformal frames at the quantum level have been recently suggested

[Phys. Rev. D82 (2010) 044006]. In this article we further discuss the theoretical grounds of that

model and we are led to a chameleonic equivalence postulate (CEP). Whenever a theory satisfies

our CEP (and some other additional conditions), a density-dependence of the mass of matter

fields is naturally present. Let us summarize the main results of this paper. 1) The CEP can

be considered the microscopic counterpart of the Einstein’s Equivalence Principle and, hence, a

chameleonic description of quantum gravity is obtained: in our model, (quantum) gravitation is

equivalent to a conformal anomaly. 2) To illustrate one of the possible applications of the CEP,

we point out a connection between chameleon fields and quantum-mechanical wave function

collapse. The collapse is induced by the chameleonic nature of the theory. We discuss the

collapse for a Stern-Gerlach experiment and for a diffraction experiment with electrons. More

research efforts are necessary to verify whether these ideas are compatible with phenomenological

constraints.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Quantum Gravity; Cosmological Constant; Chameleonic Equivalence Postulate

(CEP); Chameleon Fields

PACS (2010): 04.60.-m; 98.80.Es; 98.80.-k; 04.20.-q

1. Introduction

One of the main problems in modern cosmology is the cosmological constant (CC) one [1]

(for a review see [2]). Current observational data are compatible with a meV CC, however,

theoretical predictions of the CC are, typically, very different from the meV scale. Let us

start considering the Standard Model (SM) of particle Physics. It is common knowledge

that the Standard Model (SM) provides an extremely successful description of electroweak

∗ Email: [email protected], [email protected].

2 Electronic Journal of Theoretical Physics 12, No. 33 (2015) 1–28

and strong interactions up to (roughly) the TeV scale. On the one hand, if we decide

to choose the TeV scale as UV cut-off, the theoretical prediction of the vacuum energy

(which contributes to the CC) is already much larger than the observed value of the CC

(even if the TeV scale is much smaller than the Planck one). Therefore, a naturalness

problem must be faced: the meV scale is much smaller than typical particle Physics mass

scales. On the other hand, one might think that this is not a big problem, because the idea

of neglecting the vacuum energy in the SM is not unusual. After all, the vacuum energy

is important in problems with boundaries (for example Casimir energy) or in models

including gravitation (because energy gravitates) and the SM does not include gravity.

How can we describe gravity? Many theories of gravitation are available today, however,

the reference theory is Einstein’s General Relativity (GR). GR does not make a prediction

of the CC and we are free to set its value to zero as Einstein did (see for example [3] for

a discussion of this point). For this reason, the CC problem is not so acute granted that

we remain in the framework of classical gravity (in our example GR) and this remains

true even if we include a quantum description of the remaining interactions exploiting

the SM. Obtaining a reasonable CC from SM+GR may require a fine-tuning, but the CC

problem becomes really acute only when we move to the quantum gravity (QG) regime.

In this case, indeed, the CC is not a free parameter anymore and, moreover, we have

to understand how the vacuum energy gravitates on cosmological distances. Once this

task is accomplished, we have to find a mechanism to keep under control the contribution

to the CC of an infinite number of quantum diagrams (including graviton loops). One

crucial aspect of this analysis is the predictive nature of our QG theory with respect to the

CC. In other words: do we manage to obtain a prediction of the CC from our QG theory?

It is common knowledge that our chances to make a prediction in the SM are related to

the renormalizability of the theory: QED or QCD are not finite theories, nevertheless

they are predictive once the renormalization process is properly taken into account. The

situation is similar in QG: the CC problem is closely related to the renormalizability of

our QG theory. Needless to say, the CC problem is much more acute than the unpalatable

fine-tuning of the SM+GR scenario mentioned above.

In a recent paper [4] we proposed a stringy solution to the CC problem and we pointed

out the non-equivalence of different conformal frames at the quantum level. Among the

crucial elements of our analysis [4], we can mention, on the one hand, a splitting of all the

fields into a local fluctuating component and a global background one, on the other hand,

the peculiar ”chameleonic” behaviour of the Einstein-frame (E-frame) dilaton σ. Let us

start considering the element we mentioned last. In our model the E-frame dilaton belongs

to that group of scalar fields (introduced in the literature in [5, 6, 7]) coupled to matter

(including the baryonic one) and with an increasing mass as a function of the matter

density of the environment. In other words, the physical properties of this field vary

with the matter density of the environment and, therefore, it has been called chameleon.

This is one of the possible ways in which local tests of gravity can be faced. Among

the other possibilities already discussed in the literature, we mention: 1) symmetron

theories [8, 9, 10] (where fifth-forces are screened through a restoration of symmetry at

Electronic Journal of Theoretical Physics 12, No. 33 (2015) 1–28 3

high densities); 2) Galileon theories [11], where non-canonical kinetic terms reduce the

effective coupling to matter. Moreover, the chameleon mechanism can be considered

as a stabilization mechanism. Many other stabilization mechanisms have been studied

for the string dilaton in the literature. In particular, as far as heterotic string theory

is concerned, we can mention: the racetrack mechanism [12, 13], the inclusion of non-

perturbative corrections to the Kaehler potential [14, 15, 16], the inclusion of a downlifting

sector [17]. As far as moduli stabilization in heterotic-M-theory is concerned the reader

is referred to [18, 19] and related articles.

Remarkably, in our model [4], the E-frame dilaton parametrizes the amount of scale

symmetry of the system. Therefore, the chameleonic behaviour of the field guarantees a

scenario where scale invariance is abundantly broken locally (on short distance scales),

while, on the contrary, scale invariance is almost restored on cosmological distances and,

in particular, the E-frame CC is under control. Interestingly, this result is valid including

all quantum contributions and without fine-tuning of the parameters. Happily, in the

E-frame, the CC is under control and the dilaton is a chameleon, even if we consider

in the string frame (S-frame) a very large vacuum energy with a stabilized dilaton: the

necessary hierarchy between the S-frame CC and the E-frame one is produced after the

conformal transformation (see also [20]). For this reason we pointed out a non-equivalence

of different conformal frames at the quantum level and we selected the E-frame as the

physical one: in this model, the CC has a small positive value only in the E-frame (for

a detailed discussion of the conformal transformations from the S-frame to the E-frame

the reader is referred to [20]). In the model of references [4, 20, 21], there are a number

of consequences of this peculiar E-frame chameleonic scale invariance:

1) the E-frame CC is under control.

2) The concept of particle is re-examined [4, 21]: the string length is chameleonic (the

string mass is an increasing function of the matter density). Local particles are the

relevant degrees of freedom on short distances (i.e. locally) and they are small interacting

strings. On the contrary, global (background) particles are the relevant degrees of freedom

on very large cosmological distances and they are (almost non-interacting) cosmic strings

[21].

3) E-frame matter fields are chameleonic [21].

One relevant question is related to the theoretical origin of the model of references

[4, 20, 21]. This issue has been discussed, at least partially, in [22]: the model can be em-

bedded, to some extent, in heterotic-M-theory and, under certain assumptions about the

full M-theory action, the Casimir origin of the stabilizing potential for the S-frame dila-

ton has been pointed out in [22]. Moreover, the similarity of the dark matter lagrangian

with a Ginzburg-Landau lagrangian for extradimensional neutrino condensation has been

discussed in [22].

It seems worthwhile summarizing some open problems of our scenario for the CC

problem.

A) A more detailed description of the QG aspects of the model would be welcome. After

all, in [4] gravity is described through a metric and, hence, the reader might think that


our description of gravity is semi-classical at best. Needless to say, it would be better

to identify the fundamental pillars of our QG theory and a top-down approach would be

rewarding.

B) It would be interesting to investigate the peculiar features of a theory respecting the

fundamental principles of point (A).

C) The careful reader may be worried by the presence of multiple quantizations in [4] and

this issue should be further discussed. Particular attention should be dedicated to poten-

tial connections with the non-equivalence of different conformal frames at the quantum

level.

D) The renormalizability of our QG theory should be carefully discussed. This issue is

particularly relevant in connection to the CC problem.

E) In [4] a crucial step of our procedure was a conformal transformation from the string

frame to the Einstein one. It would be rewarding to understand whether this transfor-

mation is linked to some particular event of the cosmological evolution. Indeed, in this

case, the conformal transformation would be easily included in the model.

F) It would be interesting to understand whether our chameleonic model predicts a pecu-

liar pattern of soft terms. The breakdown of supersymmetry (SUSY) should be analyzed

carefully.

G) The connection between our model and string theory should be further investigated.

Particular attention should be dedicated to the fact that in the S-frame of [4] we are

considering a strongly coupled theory and, therefore, the lagrangian must be exact (i.e.

it must include all quantum loops).

In this paper we are going to analyze points A and B. The remaining points C-G will

not be discussed in this paper and they will be simply assumed to be solved. Here is a

summary of our results:

A) We are going to further analyze the theoretical grounds of the model of reference

[4, 20, 21, 22] and we are going to focus our attention on the chameleonic behaviour of

matter fields. In particular, we will identify some requirements that produce as a final

outcome a density-dependent mass of matter fields in a top-down approach. During this

analysis, we will formulate a chameleonic equivalence postulate (CEP) as one of those

requirements. In this way, the model of reference [4] is just one particular example of a set

of models, induced by the CEP, where the mass of matter particles is density-dependent.

Remarkably, the CEP is a microscopic counterpart of Einstein’s Equivalence Principle

and the gravitational aspects of the CEP are a consequence of the chameleonic nature of

our model (in this way, the gravitational aspects of the CEP should not be considered as

a postulate but as a principle). A new ”chameleonic” description of gravity is obtained:

quantum gravitation is equivalent to a conformal anomaly in our model. We suggest to

exploit the CEP as a guideline towards QG and we establish a ”dictionary” connecting

the classical with the quantum regime.

B) We will point out that chameleon fields provide elements which are useful to under-

stand the wave function collapse (for a review paper on the wave-function collapse the

reader is referred to [23]). The collapse is induced by the chameleonic nature of the theory


and, as we will see, it is related to a shift of the ground state of the theory. Interestingly,

in our model, the collapse of the wave-function is a QG effect.

As far as the organization of this article is concerned, in section 2 we discuss the CEP;

section 3 discusses the connection between chameleon theories and quantum-mechanical

wave-function collapse: in particular we will consider the model of reference [4] as an

example of one model satisfying our CEP and in this framework we will analyze the

collapse of the wave-function. Section 4 describes some gravitational aspects of the CEP.

Some concluding remarks are discussed in section 5. The appendix summarizes a few

elements about Bogoliubov transformations: these comments are useful to understand

the projection operator relevant for the collapse of the wave-function.

2. Chameleonic Equivalence Postulate

In the standard chameleonic literature, the Planck mass is typically constant and the

chameleonic action is written in the E-frame. Matter fields are conformally coupled to

the chameleon and the conformal factor guarantees the required competition between the

scalar potential and the matter branch. For example, a typical choice that can be found

in the literature is the exponential function ρmeβφ/Mp , where the matter density includes

the mass m0 of the matter field and β parametrizes the coupling strength. Basically the

exponential coupling with the scalar field can be interpreted as a mass-varying term for

matter particles.

Now, let us suppose we are given a stringy model matching the following require-

ments:

R1) the gravity part of the action can be summarized through a single scalar-tensor the-

ory where a dilaton field is present (let us call it φ in the S-frame and σ in the E-frame).

R2) The E-frame dilaton σ is chameleonic and it controls the strength of the couplings2.

R3) Chameleonic Equivalence Postulate - CEP: for each pair of vacua V1 and V2 al-

lowed by the theory there is a conformal transformation that connects them and such

that the mass of matter fields m0,V 1 (i.e. m0 evaluated in V1) is mapped to m0,V 2 (i.e.

m0 evaluated in V2). When a conformal transformation connects two vacua with a dif-

ferent amount of conformal symmetry, an additional term (in the form of a conformal

anomaly) must be included in the field equations and this additional term is equivalent to

the gravitational field.

In order to render the physical content of the CEP easier to understand, we will

proceed stepwise touching upon the requirements mentioned above. Point (1) is very

natural in a stringy model. Basically it is useful because we are simply getting rid of

more complicated situations like multi-scalar-tensor theories. Point (2) is crucial and,

in particular, it links the energy scale in the E-frame to the expectation value of the

E-frame dilaton. From reference [20] we infer that point (3) guarantees that the physical

2 This means, in particular, that the saturation mechanism for the couplings discussed in reference [24]

is not part of our approach (at least at this stage).


renormalized Planck mass is σ-dependent. Remarkably, even if we choose V1 and V2

in the same frame we can make a triangle V1-V3-V2 (where V3 belongs to a different

frame with respect to V1 and V2) by composing two different conformal transformations.

The first one connects V1 to V3 while the second one connects V3 to V2. The total

transformation connects V1 to V2 in a single conformal frame. Since the mass of matter

fields should be properly renormalized, σ-dependent couplings and a σ-dependent Planck

mass imply, through renormalization, σ-dependent masses of matter fields. Therefore, a

σ-dependence of the E-frame masses is naturally expected in a theory satisfying our three

conditions. The chameleonic (the mass is an increasing function of the matter density)

or anti-chameleonic (the mass is a decreasing function of the matter density) behaviour

of matter particles is model-dependent and must be checked separately. In general, in a

model where φ is not stabilized, a φ-dependent renormalized Planck mass is expected:

the renormalized Planck mass becomes a function of two variables (φ and σ).

For a discussion of the connection between CEP and QG, the reader is referred to

section 4..

In this paper we will be particularly interested in the lagrangian of [4] satisfying our

requirements (R1-R3). In other words we write the string frame lagrangian as

L = LSI + LSB, (2.1)

where the scale-invariant Lagrangian is given by:

LSI =√−g

(1

2ξφ2R− 1

2εgμν∂μφ∂νφ− 1

2gμν∂μΦ∂νΦ − 1

4fφ2Φ2 − λΦ

4!Φ4

). (2.2)

Φ is a scalar field representative of matter fields, ε = −1, (6 + εξ−1) ≡ ζ−2 � 1, f < 0

and λΦ > 0. One may write also terms like φ3Φ, φΦ3 and φ4 which are multiplied by

dimensionless couplings. However we will not include these terms in the lagrangian.

Happily, a φ4 term does not clash with the solution to the CC problem, because the

renormalized Planck mass in the IR region is an exponentially decreasing function of σ

(see also [20]).

To proceed further, let us discuss the symmetry breaking Lagrangian LSB, which

is supposed to contain scale-non-invariant terms, in particular, a stabilizing (stringy)

potential for φ in the S-frame. For this reason we write:

LSB = −√−g(aφ2 + b+ c1

φ2). (2.3)

Happily, it is possible to satisfy the field equations with constant values of the fields

φ and Φ through a proper choice (but not fine-tuned) values of the parameters a, b, c,

maintaining f < 0 and λΦ > 0. This lagrangian will be further discussed in the remaining

part of this article.

Interestingly, our CEP resembles the Equivalence Principle of quantum mechanics

[25, 26, 27, 28]. Let us add some comments. Recently, the Quantum Stationary Hamilton-

Jacobi equation (QSHJE) has been obtained in a top-down approach from the Equivalence


Postulate/Principle (EP) of quantum mechanics. Remarkably, the QSHJE is a non-

linear equation (like the chameleonic equations) and, once it is linearized, it produces the

Schrodinger equation. The potential connections between our CEP and the analysis of

[28] should be investigated.

3. Chameleonic wave function collapse

Let us start from 2.1. For a discussion of the parameters of the model, the reader is

referred to [20]. It seems worthwhile pointing out that a better fit with the Casmir

energy of references [18, 22] can be obtained with a potential of the form (A,B and C

are constants):

VSB =A

φ2+B +

C

φ, (3.1)

which, after dilaton stabilization, corresponds to a constant once again.

When we perform a conformal transformation to the E-frame (for a detailed discussion

see [20]), we can rewrite the lagrangian (2.2) as

L∗ =√−g∗

(1

2R∗ − 1

2gμν∗ ∂μσ∂νσ + L∗matter

), (3.2)

where L∗matter turns out to be

L∗matter = −1

2gμν∗ DμΦ∗DνΦ∗ − e2

d−2d−1

ζσ(ξ−1f

4M2

pΦ2∗ +

λΦ4!

Φ4∗) (3.3)

and Dμ = ∂μ + ζ∂μσ.

The chameleonic competition can be obtained in various ways. Let us summarize

them explicitly.

• We can exploit a conformal-anomaly-induced interaction vertex between dilaton and

matter (see [4] and related references). The coupling is basically ρM × σ, where the

matter density in [4] satisfies ρM ∝ e−4ζσ. The careful reader may be worried

by this exponential damping of the matter density because, even if we manage to

construct one ground state, this is not enough to render operative the chameleon

mechanism. Indeed, whatever will be the matter density we choose, it must be

possible to construct a corresponding ground state (for the ”table of this room”, for

the ”air of this room”...). On the other hand, we cannot change the parameters of

the model, because this modifies the lagrangian and it is our intention to write the

lagrangian once and for all. Hence, an exponential damping of the matter density

is, at first sight, a phenomenological problem of [4]. Happily, however, as we will

show below, it is possible to construct a chameleonic minimum exploiting gauge

fields and in this paper we will parametrize the matter density following [4]. It

seems worthwhile pointing out that, once the parameters of the model are fixed, the

matter density is determined by σ. A shift in the matter density is related to a shift

of the dilatonic mass: if we increase ρM , the amount of scale symmetry is smaller,


therefore, the mass-protection mechanism induced by the symmetry is less effective

than before and σ becomes heavier. As already mentioned in [4], the dilaton is a

chameleon in this model and this remains true even in the absence of a competition

between the bare potential and the matter density: σ is coupled to the trace of

the energy-momentum tensor and its mass is an increasing function of the matter

density because of symmetry reasons. Now we discuss the chameleonic competition

obtained through the gauge fields.

• Let us start considering photons and let us consider the variation with respect to σ

of the relevant part of our E-frame lagrangian, namely, a dilatonic potential term

V (σ) and the F 2-term of photons. We can write:

δS =

∫d4x

√−g∗{−V,σ (σ)δσ − 1

16π

∂α−1g

∂σFμνF

μνδσ}

=

∫d4x

√−g∗{−V,σ (σ)δσ − 1

16π

∂α−1g

∂σ2(B2 − E2)δσ}

(3.4)

where we introduced explicitly the electric and magnetic fields through the formula

FμνFμν = 2(B2 − E2). (3.5)

We define3

ργ =B2 − E2

8π. (3.6)

If we require δS = 0 we thus have the following term in the equation of motion for

σ:

−V,σ (σ) + 1

α2(σ)

∂α

∂σργ. (3.7)

Therefore, as far as the contribution of radiation to the effective dilatonic potential

is concerned, the correct chameleonic competition between the two branches of the

curve is present, granted that the electromagnetic coupling αg(σ) is a decreasing

function of σ. This is a natural requirement in our model and it is part of the

requirement R2. A similar conclusion can be obtained for the other gauge fields

and, in particular, for gluons. Therefore, inside standard matter, the presence of the

chameleonic minimum is guaranteed by the competition with the F 2-term.

The chameleonic behaviour of matter fields [21] is useful to justify the ansatz about

the collapse of the wave-function in (non-relativistic) quantum mechanics. We will focus

our attention on electrons and we will discuss separately

1) the role of non-linearities in a standard diffraction experiment with electrons;

2) the role of non-linearities in a Stern-Gerlach experiment;

3) the wave-function collapse in both experiments just mentioned above.

3 Do not confuse ργ ∝ B2 − E2 with the electromagnetic energy density (proportional to the sum

B2 + E2).


3.A Breakdown of the superposition principle in a electronic diffraction

experiment

Let us consider a standard diffraction experiment with electrons where we produce a

diffraction pattern on a screen and let us describe the results of this experiment through

non-relativistic quantum mechanics. It is common knowledge that the position of one

electron is well-defined a posteriori, namely, after the interaction electron-screen has

taken place. Needless to say, this fact is compatible with the standard postulate of (non-

relativistic) quantum mechanics regarding the collapse of the wavefunction of the electron

(which is part of the so-called Copenhagen interpretation of the quantum formalism, see

for example [29]). To the best of our knowledge this postulate is still an open problem (at

least to a certain extent) for the scientific community. Remarkably, in the non-relativistic

quantum theory the collapse is supposed to take place instantaneously.

Now we come back to our model for a chameleonic dilaton.

Remarkably, in our proposal, the collapse of the wave-function is related to a vacuum

shift (i.e. to an energy density shift) in the chameleonic theory. This point we mentioned

last needs to be further elaborated.

It is well-known that in non-relativistic quantum mechanics the ket | α > which represents

a physical state can be written as a sum (i.e. a superposition) of position eigenvectors,

namely (for the 1-dimensional case):

| α >=∫dx | x >< x | α >, (3.8)

where < x | α > is the wave-function of the system and |< x | α >|2 dx is the proba-

bility that the particle is detected in a small interval dx around x. Needless to say, the

superposition principle plays a crucial role when we write | α > as a sum of kets and

this principle rests on the linearity of the wave equation. Let us now move to quan-

tum field theory (QFT). Interestingly, in QFT the expectation value of a scalar matter

field Φ∗ (squared) is related to the particle number density. Therefore, in our diffrac-

tion experiment with electrons, the number density of the electrons on the screen can

be parametrized, on the one hand, with the wave-function squared in the non-relativistic

formalism, on the other hand, with the (expectation value of the) matter field squared

in the relativistic formalism. Hence, we suggest in this way a connection between the

non-relativistic wave-function and the relativistic quantum matter field4 (this connection

will be further discussed in paragraph 3.C.3). The linearity of the theory can be checked,

on the one hand, in a non-relativistic formalism, by looking at the linear nature of the

wave-equation and, on the other hand, in a relativistic formalism, by looking at the linear

nature of the field equation. The relevant question is therefore:

what can we say about the linearity of our theory inside and outside the screen? Once

a certain matter density is fixed, the chameleonic vacuum is well-defined and we can

4 The reader may be interested in a historical introduction to QFT that can be found, for example, in

[30].


linearize the dilatonic theory around this vacuum: no matter whether we consider the

behaviour of the dilaton inside or outside the screen, we reasonably assume that the

theory can be well-approximated by linear equations obtained through harmonic approx-

imations around the correct chameleonic vacuum (the ”inside one” or the ”outside one”).

Hence, the next question is:

what can we say about the linear nature of our theory during a transition between two dif-

ferent chameleonic vacua (namely from the ”outside vacuum” to the ”inside vacuum”)?

In this last case, strong non-linearities are expected in the field equations (the harmonic

approximation is not valid anymore) and the superposition principle is broken for a short

(but non-vanishing) interval of time. We infer that the superposition 3.8 cannot be valid

anymore during the entrance of the electron in the screen (i.e. when the position mea-

surement takes place) and, therefore, one single position eigenvector might be selected

among the set {x} which we considered in the integration (i.e. the wave-function might

collapse). This chameleon-induced breakdown of the superposition principle corresponds

to a matter-density shift (i.e. to the entrance of the electron-string in the screen).

3.B Breakdown of the superposition principle in a Stern-Gerlach experi-

ment

Naturally the postulate regarding the collapse of the wave-function must be applied to all

measurement of (non-relativistic) quantum mechanics. In other words, whatever will be

the hermitian operator whose spectrum we are interested in, the effect of the measurement

must be described by this ansatz. Therefore, the diffraction example mentioned above is

not general enough: it is very easy to imagine different quantum measurement where the

collapse of the wave-function takes place in an environment with small matter density,

for example a Stern-Gerlach (SG) experiment.

Let us discuss, therefore, a SG experiment with, for example, electrons. When the

electrons are outside the magnet we have ργ = 0. When the electrons enter inside the

magnet, a non-vanishing ργ is present, hence, the chameleonic competition between the

E-frame dilatonic potential and ργαg

increases the mass of the dilaton and reduces the

amount of scale-symmetry. In other words, when we perform the (spin) measurement,

we electromagnetically interact with the electrons, therefore, a shift in ργ is obtained.

In our chameleonic model it is precisely this ργ-shift that induces (through the dilaton)

a breakdown of the superposition principle in a small transition region (whose size is

comparable with the string length) between the two chameleonic vacua 1) the B = 0

vacuum and 2) the B �= 0 vacuum. We now apply to the spinorial case the same discussion

developed for the position eigenvectors. In both chameleonic vacua (the B = 0 one and

the B �= 0 one) we reasonably assume that it is possible to linearize the theory, but

strong non-linearities are expected in the transition region. Before the entrance in the

magnet, we choose the ket which represents the electron as a superposition of spin-up


and spin-down kets, namely, with obvious notations:

| α >= | + > + | − >√2

, (3.9)

the theory is linear and the chameleonic vacuum is the B = 0 one. Inside the magnet, the

chameleonic vacuum is the B �= 0 one and the theory is linear once again. However, when

the electron-string enters the magnet (i.e. when the spin measurement takes place), we

have a transition between the two chameleonic vacua, hence, non-linearities are important

and, therefore, the superposition principle, which plays a crucial role in the sum 3.9

between up and down kets, is not valid anymore. The theory might choose for this reason

one single ket (i.e. the spinorial wave-function might collapse). In the next section we

will further discuss this ”selection mechanism”.

3.C Wave function collapse

Let us come back to the collapse. In our analysis, we managed to show that, given an

initial ket for the system (written as a superposition of various kets), during the transition

between two different chameleonic vacua, a breakdown of the superposition principle is

expected when non-linearities are properly taken into account. However, if our intention

is to justify the collapse through chameleons, this is not enough. For example, one crucial

question that is still waiting for an answer is: why the breakdown of the superposition

principle should imply the selection of one single ket?

Basically, in our approach we are discussing connections between different formalisms:

1) the classical formalism;

2) the quantum non-relativistic formalism;

3) the QFT formalism.

4) String theory.

We will proceed stepwise discussing (1) the origin of the projection operator respon-

sible for the collapse of the wave function and (2) our two previous experimental config-

urations, namely the SG and the diffraction experiment with electrons.

3.C.1 Projection operator

We point out that, during the jump between an initial and a final configuration σi ⇒σf (or Φ∗i ⇒ Φ∗f ), the vacuum energy is not conserved. The vacuum energy of the

system is chameleonic and, consequently, a non-unitary evolution might be present in the

chameleonic jump between two vacua5. To proceed further, we explore the possibility

that this potentially non-unitary evolution is connected with a non-unitary projection

operator. As already discussed in the literature [31], if we introduce in a model a set of

projection operators, we are splitting the Hilbert space in a set of subspaces, typically

5 As far as a connection between unitarity and energy conservation is concerned, the reader is referred

to the analysis of the scattering theory discussed in [30].


characterized by unitarily inequivalent representations of a certain algebra of observables.

Therefore, a connection between chameleon fields and superselection rules6 might take

shape.

The careful reader may be interested in a more detailed and formal discussion of the

projection operator in our model. To illustrate this point we exploit Bogoliubov transfor-

mations [32]. Let us start considering vanishing number density in the chameleonic model:

no particles. If our intention is to create particles, we must apply creation operators to

the vacuum. Needless to say, the creation operators are related to the quantum fields.

Once a non-negligible number of matter particles have been created7, the number density

(hence the local vacuum energy) is non-vanishing and the state of minimum energy is not

annihilated by all the annihilation operators. In other words, this chameleonic model is

telling us that ”in this room” we live in a state which is a minimum of the chameleonic

effective potential, but the matter density is non-vanishing. At this stage, our system

resembles the free electron gas model (FEGM) discussed in appendix, granted that we

compare (1) the electron of the FEGM with the matter particle of the chameleonic model

and (2) the Fermi energy of the FEGM with the non-vanishing local vacuum energy of

the chameleonic model. Now the matter density (hence the local vacuum energy) is non-

vanishing anymore and, therefore, if we want to create more particles we have to exploit

creation operators which are different with respect to the creation operators we started

with: the creation/annihilation operators we started with do not respect the Fock condi-

tion when the local vacuum energy is non vanishing. The two sets of operators are related

to each other by a Bogoliubov transformation (see the appendix). As already mentioned

above, creation/annihilation operators of matter particles are related to matter fields. We

infer that the two sets of quantum fields (pre and post measurement) are related to each

other by a Bogoliubov transformation. Happily, the general Bogoliubov transformation

is not unitary in the infinite volume limit (see [33]): this non-unitary transformation is,

in our proposal, the projection operator necessary for the collapse of the wave-function

in our model and, remarkably, it is connected to a variation of the vacuum energy.

To proceed further, we discuss an alternative derivation of the projection operator

exploiting projective representations of the conformal group. Indeed, the shift in the

vacuum energy is related to the conformal factor and therefore it must be possible to

summarize the presence of a superselection rule through the conformal factor. This idea

is fully compatible with the CEP because the jump between the two vacua (before/after

measurement) is performed through the composition of two different conformal transfor-

mations: (1) from the vacuum before the measurement to the string frame vacuum and

(2) from the string frame vacuum to the vacuum after the measurement.

6 A superselection rule guarantees, by definition, that the system occurs in states, for example, | ψ1 >

and | ψ2 > but never in a superposition a | ψ1 > +b | ψ2 >.7 This discussion can be repeated also for the magnetic field in the SG experiment, even if in that case

the chameleon mechanism is not due to the matter density. The presence of the magnetic field gives

a non-vanishing local vacuum energy and, once again, the state of minimum energy is not the state

annihilated by all the annihilation operators.


3.C.2 Projective representations of the conformal group

When a quantum measurement is performed, the dilaton is stabilized through the

chameleon mechanism. As already discussed in [4, 34], our dilaton σ is a pseudo-Nambu-

Goldstone boson of broken scale invariance. The mass of σ is related to an explicit

breakdown of scale invariance. The dilaton can get a mass in two ways in our model:

1) through the conformal anomaly induced σΦ2∗ term (as already mentioned above, even

if σΦ2∗ is exponentially damped, a shift in the matter density is related to a shift in the

amount of scale symmetry and, hence, of the dilatonic mass); 2) through the competition

with the F 2 term of gauge fields. Since both terms produce a chameleonic mass for the

dilaton (i.e. explicit breaking of scale invariance), we summarize both terms through

the presence of a central charge in the algebra of the conformal group and this is one

of the possible ways projective representations may present themselves (the other way is

topological). Indeed, it is common knowledge that the appearance of central charges is

the counterpart for the algebra of the presence of phases in a projective representation of

a group, namely, the elements T, T , etc. of a symmetry group are represented on physical

Hilbert space by unitary operators U(T ), U(T ), etc. which satisfy the composition rule

U(T )U(T ) = eiφ(T,T )U(T, T ), (3.10)

where φ is a real phase (for an introduction to projective representations the reader

is referred to [30]). The double conformal transformation mentioned in the previous

paragraph (from the vacuum before the measurement to the string frame and from the

string frame to the final vacuum after the measurement) corresponds to U(T )U(T ).

The relevant result, for our purposes, is the following. If it is possible to prepare the

system in a state represented by a linear combination, then the phases φ(T, T ) cannot

depend on which of these classes of states the operators act upon (for a proof of this

result the reader is referred again to [30], chapter 2). Equivalently, if the phases φ may

depend on the state, then there must be a superselection rule. In our model, the central

charge is a function of the fields and therefore it is related to the microscopic quantum

state of the local vacuum which, as already mentioned above, contains also the particles of

”this room” where the measurement is performed. Therefore, the phases of the projective

representation of the conformal group are functions of the various quantum observables

(mass, spin, angular momentum...). In other words, the various quantum numbers we

can measure are encoded into the corresponding fields (Φ∗, Aμ...) and the interaction

of those fields with σ will produce a dependence of the phase φ(σ) on the microscopic

quantum state of the theory (which includes the system we measure). In this way a

superselection rule is obtained when the conformal anomaly is non-negligible (i.e. when

the dilaton is stabilized) and, hence, the wave-function collapses through the chameleon

mechanism. In our proposal, when the measurement is performed, a certain amount of

energy is transferred to the system and, therefore, a field stabilization is expected through

the chameleon mechanism (after the measurement).

The careful reader may be worried by these ideas because it is well-known that it

may or it may not be possible to prepare a system in a linear combination of states, but


one cannot settle the question by reference to a symmetry principle, because whatever

will be the symmetry group we use, there will be always another group with identical

consequences except for the absence of superselection rules (see [30], page 90). However, to

the best of our knowledge, this remark we mentioned last does not apply with conformal

anomalies and, consequently, we still summarize the projection operator through the

presence of a non-trivial phase in a projective representation of the conformal group.

To proceed further, we will discuss the SG and the diffraction experiment.

3.C.3 Stern-Gerlach and diffraction experiment

• Non-degenerate case

As already mentioned above, we connect relativistic fields to non-relativistic wave-

functions. The purpose of this section is to render this connection more precise. The

connection is based on the number density of particles np. On the one hand, in non-

relativistic quantum mechanics np is parametrized by the absolute value squared of

the wave-function and, on the other hand, in a QFT language, it is parametrized

by the expectation value (squared) of the field (at least for a scalar field). Remark-

ably, expectation values of quantum operators (quantum fields) are spatial integrals

weighted by the integration volume (i.e. spatial averages). Let us now consider the

SG configuration and let us suppose we measure the z-component of the spin opera-

tor. One problem must be faced with our chameleon-induced collapse. If we justify

the collapse exploiting the chameleon mechanism, when the particles escape from

the magnetic field, the chameleon mechanism is not operative, the dilaton is not

stabilized and the solution to the chameleonic field equation is expected to be the

same we had before the measurement (namely, before the entrance into the magnet).

On the other hand, it is well known that, after a SG measurement is performed, the

particles are split into up and down particles (no linear combinations of up and down

are allowed) and this result is valid even if we start with a linear combination of up

and down kets before the measurement. Hence, if we prepare the system of electrons

in a linear combination of up and down kets and we let them enter into the magnetic

field, the question is: why the particles escaping from the magnet (in the absence of

magnetic field) are not described by the same ket they had before the entrance into

the magnetic field (namely by the linear combination of up and down kets)? This

question is relevant if our intention is to support the connection between quantum

fields and non relativistic wave function mentioned above. The answer is that the

number density of electrons outside the magnetic field doesn’t know anything about

the spin state. We can have up or down or a linear combination of up and down

kets, but the number density is the same. Hence, the expectation value of the field

is (roughly) the same we had before the measurement.

Remarkably, whenever we interpret an expectation value (squared) as a number

density, we are implicitly considering a macroscopic average of the field because, in

order to talk about number density, we must consider a large number of particles and,

hence, macroscopic length scales. The situation is reminiscent of thermodynamics


(to a certain extent), where a single value of a macroscopic averaged quantity (like

the temperature for example) is compatible with a large number of microscopic

quantum states. These microscopic quantum states are encoded in our model into

the quantum fields (it appears in a QFT language with the expansion of a field

in terms of creation and annihilation operators), while the averaged macroscopic

quantities are related to the number density (to the expectation values of the fields).

These comments are supporting the idea that the superselection rules, already ob-

tained through projective representations of the conformal group, should be related

to the number density. This comment must be further elaborated. Let us suppose

that the up and down states have different energy: this seems to be very reasonable

in the SG configuration. In our SG example, inside the magnet, formally we write

H | + >= E+ | + > (3.11)

H | − >= E− | − >, (3.12)

where E+ �= E−. When the field is sitting at the minimum (inside the magnet),

there is no dynamical evolution. Therefore, we suggest to connect this stabilized

field configuration where the expectation value is constant with a stationary state

of the non-relativistic formalism. In our SG example, the system is forced to choose

either the up or the down ket, because a linear combination of the two states would

not be stationary inside the magnet: it would correspond to a non-stationary wave-

function inside the magnet, it would give a non-trivial time evolution to the absolute

value squared of the wave function, this would force the macroscopic number density

to change in time and this would clash with the constant expectation value of the

stabilized field. In other words, a linear combination of kets is forbidden after the

measurement: it is not compatible with the environment which stabilizes the fields

and, hence, it is forbidden because it is not a stationary (stabilized) state.

• Degenerate case

Typically, when the quantum system is symmetric, degenerate states occur. Let us

now consider the diffraction experiment with electrons and let us suppose we are

equipped with an axially-symmetric experimental set-up. In other words, we con-

sider an initial electron beam (i.e. a plane wave) and we study the diffraction of the

electrons through a circular hole. The expected diffraction pattern will respect the

circular symmetry (i.e. it corresponds to a set of degenerate states), but the single

electron on the screen will break the symmetry. Where does this symmetry breaking

come from? First of all, let us come back to our chameleon fields. We point out that

the lagrangian is rotationally symmetric. What can we say about the vacuum? If

we manage to show that the vacuum is not invariant under rotations, we can claim

that the rotational symmetry is spontaneously broken. This is exactly the path that

we are going to follow here: spontaneous breakdown of rotational symmetry will be

useful to justify the diffraction pattern on the screen.

Naturally the matter density defines the vacuum through the chameleon mechanism.

Some comments are necessary to proceed further.


1) Even if we fix the value of the matter density, the length scale compatible with

that particular value of ρm is not unique. In particular, the matter density of the

screen is the correct matter density on macroscopic length scales (e.g. the length of

the screen), but also on microscopic length scales (e.g. the atomic radius).

2) As already mentioned above (see [21]), the string length of the electron is a de-

creasing function of the matter density. Before the measurement is performed, if the

matter density is small enough, the string length of the electron in the beam is larger

than the atomic radius and the relevant matter density during the measurement is

roughly 1 g/cm3. After the entrance in the screen, the string becomes shorter, but

its length must be always compatible with a matter density of (roughly) 1 g/cm3.

3) Whatever will be the Feynman diagram we consider, after the various quantiza-

tion steps in the E-frame, the UV cut-off of the theory is provided by the string mass

[21].

Now, to the point. What is the UV cut-off of the theory inside the screen (after the

measurement)? In other words, what is the shortest possible length scale compatible

with a matter density of, basically, 1 g/cm3? The answer is: the atomic radius. Re-

markably, the string length inside the screen is comparable with the atomic radius

which, needless to say, is comparable to the de Broglie wavelength of the atomic

electrons.8 Therefore, the UV cut-off inside the screen is basically the inverse of the

atomic radius.

On atomic length scales, namely on length scales similar to the UV cut-off of

the theory, rotational symmetry is broken: this is the origin of the spontaneous

breakdown of rotational symmetry inside the screen. To illustrate this point, let us

connect the E-frame lagrangian of [4] to the Olive-Pospelov lagrangian [9]:

Sφ =

∫d4x

√−g{

−M2Pl

2R +

M∗2

2∂μφ∂μφ− V (φ)

−BF (φ)

4FμνF

μν +∑

j=n,p,e

[ψjiD/ψj −Bj(φ)mjψjψj]

}. (3.13)

In other words, we replace the anomaly-induced coupling of [4] between σ and

the matter density (where the matter fields were represented by a scalar field Φ∗)with a coupling between σ and a matter density represented by a spinor field (like in

the Olive-Pospelov approach). To proceed further, we connect the spinorial matter

fields to the wave-function following chapter 14 of [30] and we write the renormalized

matter density as:

mjψj(< σ >)ψj(< σ >) � mj < ΦN | ψ†j | Φ0 >< Φ0 | ψj | ΦN > +BR + ...

= mju†NuN + BR + ... = mj | uN |2 +BR + ..., (3.14)

8 An interesting line of development will try to interpret the probability waves of quantum mechanics

as waves on quantum strings.


where | Φ0 > is the vacuum, | ΦN > is an atomic state vector, ”BR” represents the

contribution of the counterterm (the backreaction, see [4]) and the dots represent

the terms of the form mj < ΦN | ψ†j | Φi >< Φi | ψj | ΦN > (| Φi > is a set

of orthonormal states that is not complete because the vacuum is missing). We

infer that the matter density is related to the wave-function squared. Needless to

say, the contribution of atomic electrons is not rotationally invariant and, therefore,

the matter density shares the same property. Consequently, the amount of scale

invariance is slightly modified whenever a rotation is performed: the final result is

a non-symmetric renormalized local vacuum energy. We infer that the vacuum does

not respect rotational symmetry inside the screen and, hence, rotational symmetry

is spontaneously broken inside the screen.

There are a number of consequences of this fact. A single point on the screen

selects a direction in space. The connection between a non-relativistic formalism

and the relativistic one pointed out above, suggests that the selection of one par-

ticular direction in this case, which takes place after the collapse, is analogous to

the selection of one particular direction by a bended rod with circular cross section

placed vertically on a table and pushed with a (strong enough) force over it (the

classical example discussed in QFT textbooks when introducing spontaneous sym-

metry breaking, see for example [35]).

When a large number of electrons is considered, the symmetric pattern is restored

and the same behaviour is shown when a rod is bended a very large number of times

(the corresponding quantum vacuum is symmetric once again).

Interestingly, our approach is reminiscent of environment-induced decoherence [31,

36, 37, 38]. Another interesting line of development will try to clarify whether our loss of

unitarity in the theory is restored at a more fundamental level and will study potential

connections with the information problem in black-holes. As already mentioned above, a

detailed phenomenological analysis is definitely required.

4. CEP and quantum gravity

Let us discuss some gravitational aspects of the CEP.

The theory of Special Relativity (SR) is based on an invariance principle: the laws

of Nature are invariant under Lorentz transformations. In other words, whatever will

be the inertial frames we consider, we are free to perform a Lorentz transformation that

connects two inertial systems and this transformation cannot change the equations of the

theory. Needless to say, SR is not a theory of gravity. If our intention is to describe

gravity in a relativistic way, we can consider GR and the Equivalence Principle (EP)

plays a crucial role. The EP is telling us that inertia is equivalent to gravitation. It is

common knowledge that whenever we perform a transformation that brings us from an

inertial frame to a non-inertial one, additional terms will be present in the equations of


the theory. A non-relativistic example is given by the Newton equations: in non-inertial

frames the Newton equations acquire some additional terms (the inertial forces). This

idea is valid also in GR. When we perform a general coordinate transformation in GR,

some ”additional terms” (the metric and the connection) will be present in the equations

of the theory and these terms are exploited by Einstein to describe the gravitational field

in harmony with the EP. GR is a classical theory of gravity.

Let us now discuss the quantum regime and let us start with a question: how is it

possible to describe a complete absence of gravity? Our GR-based intuition tells us that

we must remove all the masses and all the energy sources (including vacuum energy!).

The 4D cosmological (IR) vacuum discussed in this paper provides one ground state where

gravity is basically absent. If our intention is to switch on a (small) gravitational field

we can add a source in the form, for example, of a massive (or even massless) particle

and the amount of conformal symmetry will be slightly reduced. For example, if we add

an electron, the related contribution to the matter density will reduce the amount of

conformal symmetry while, if we add a photon, the coupling to the F 2-term will reduce

the amount of conformal symmetry. In this way, whenever we modify the gravitational

field, we obtain a chameleonic shift of the ground state. In other words, in our model,

the chameleon mechanism is telling us that the gravitational field is described by the

conformal anomaly in harmony with the CEP. Our CEP connects different vacua where

a different degree of conformal symmetry is present. We suggest to exploit the CEP as a

guideline towards the QG regime. In particular, let us construct the following dictionary

for QG:

• let us replace the inertial frame of Einstein’s theories with a conformal ground state

in our chameleonic model. Let us write the connection between the two models in

this way: inertial frame → conformal ground state.

• Non-inertial frame → non-conformal ground state.

• General coordinate transformation → conformal transformation.

• Metric and connection → conformal anomaly.

In this way, the ”dictionary” mentioned above creates a connection between classical

and QG: A) conformal transformations in QG are analogous to general coordinate trans-

formations in classical gravity; B) the cocycle in QG is analogous to the metric and the

connection in classical gravity. The CEP is the microscopic counterpart of the EP.

A few comments are in order.

1) Remarkably, our description of gravity through a conformal anomaly is telling us how

to deal with vacuum energy in QG. A shift in the vacuum energy can be summarized by

a conformal anomaly through the chameleon mechanism.

2) Another question is related to the holographic description of the model. As already dis-

cussed by [39] in the framework of quantum N-portrait, whenever we consider AdS/CFT

correspondence, the central charge of the CFT is the occupation number of AdS gravi-

tons. Our model is developed from the standpoint of heterotic-M-theory and the 5D bulk

is (almost) AdS. In other words, our CEP is giving us an ”additional term” to the field

equations, namely the cocycle (the conformal anomaly), and this term is related to the


occupation number of gravitons in 5D. Once again, conformal anomaly is gravity.

3) As already discussed in [40], the question ”why gravitation should obey the EP?” does

not find an answer inside GR. On the contrary, in our chameleonic model, we can con-

sider the equivalence of gravitation and conformal anomaly as an outcome of our analysis.

Therefore, in our model, (the gravitational aspects of) the CEP should not be considered

as a postulate, but as a final result. A fast exponential damping of the matter density and

the related problems in obtaining a minimum of the effective potential do not clash with

our conclusions because we can exploit the shift in the amount of conformal symmetry:

as already mentioned above, if we add one electron, the amount of conformal symmetry is

slightly smaller and, therefore, the vacuum energy and the mass of σ are (slightly) larger

or, in other words, a chameleonic shift is obtained.

4) Let us come back for a moment to the wave function collapse. From the elements we

gathered, we infer that the collapse of the wave-function is a QG effect.

5) Needless to say, scale invariance plays a key role in our model. The σ-dependence of

the amount of symmetry is even more important: we have an abundantly broken sym-

metry in the UV region that becomes restored in the IR. Therefore, a natural question

is: do we have similar theories in the literature? Interestingly, see for example [41] for a

recent discussion, heterotic string compactifications can be conveniently described in the

language of gauged linear sigma models (GLSMs). GLSMs are not conformal, because

in two dimensions the gauge couplings and some kinetic terms have non-vanishing mass

dimension, but it is believed that in the IR limit the theory flows to a conformal model.

A promising line of development will investigate potential connections between our model

and the papers [41, 42, 43, 44, 45].

6) Our description of QG resembles, to a certain extent, QCD. For example, we know

that the mass of a baryon is related to ΛQCD (and hence to a conformal anomaly) and it is

due, to a large extent, to the presence of strong interaction (the quark masses give a very

small contribution). In other words, the conformal anomaly plays a crucial role also in the

low-energy description of strong interaction. Moreover, in our model, the Planck mass

is a condensation scale (see also [22]) and dilaton stabilization (fixing the value of the

dilaton) is a gauge-fixing for our QG theory analogously to gauge-fixing in non-abelian

gauge theories. Moreover, if we exploit S-duality in our scenario, our QG theory is ex-

pected to be free in the deep UV region. We will further analyze these issues in the future.

An interesting line of development will analyze the microscopic nature of backreaction

(the counterterm in our model), but now we already know that backreaction is equivalent

to a conformal anomaly and, in other words, to gravity, because backreaction changes

the amount of conformal symmetry of the ground state. A good starting point for a more

detailed and formal analysis is given by [46, 47, 45]. In particular, in [46], the cocycle

S(g1) − S(g2) is written as a Liouville action (g1 and g2 are two metrics related to each

other by a conformal transformation).

Remarkably, the vacuum energy in our model gives a repulsive contribution to the

particles, because the vacuum energy is minimized for small number densities. This


contribution might be relevant because we do not fine tune the scale of the potential

to small values. An interesting line of development will further analyze this aspect of

our chameleonic approach to QG. We will try to understand whether it is possible to

reinterpret our gravitational theory through entropic arguments (a good starting point

would be [48]). For example, we could try to study the free expansion of a gas of particles

(or a gravitational collapse problem) taking into account also the chameleonic behaviour

of the vacuum energy. A repulsive vacuum energy should be present also in the Sun and

in the Earth and might give us some interesting phenomenological signature of chameleon

fields.

5. Conclusions and possible lines of development

In this paper we discussed these points:

A) We further analyzed the theoretical grounds of the model of reference [4, 20, 21, 22]

and we focused our attention on the chameleonic behaviour of matter fields. In particu-

lar, we identified some requirements that produce as a final outcome a density-dependent

mass of matter fields in a top-down approach. During this analysis, we formulated a

chameleonic equivalence postulate (CEP) as one of those requirements. In this way, the

model of reference [4] is just one particular example of a set of models, induced by the

CEP, where the mass of matter particles is density-dependent. Remarkably, the CEP is

a microscopic counterpart of Einstein’s Equivalence Principle and the gravitational as-

pects of the CEP are a consequence of the chameleonic nature of our model (in this way,

the gravitational aspects of the CEP should not be considered as a postulate but as a

principle). A new ”chameleonic” description of gravity is obtained: quantum gravitation

is equivalent to a conformal anomaly. We suggested to exploit the CEP as a guideline

towards QG and we established a ”dictionary” connecting the classical with the quantum

regime.

B) We pointed out that chameleon fields provide elements which are useful to understand

the wave function collapse (for a review paper on the wave-function collapse the reader

is referred to [23]). The collapse is induced by the chameleonic nature of the theory and

it is related to a shift of the ground state of the theory. Interestingly, in our model, the

collapse of the wave-function is a QG effect.

A detailed phenomenological analysis of the entire model is required to test these

ideas.

Some comments are in order.

1) One problem might arise if we apply the Veneziano’s mechanism of reference [24, 21]

and we saturate the fundamental couplings. It remains to be seen whether the many

copies of the SM clash with the chameleon-induced collapse of the wave-function. Another

comment is in order. As already mentioned above, in this model the E-frame dilaton must

be large enough in order to generate reasonable values of matter density but, at the same

time, σ must be in the non-perturbative regime to make the Veneziano’s mechanism


operative. We should check that the chameleonic collapse of the wave function passes the

dangerous phenomenological tests related to the variation of fundamental couplings. A

detailed phenomenological analysis is necessary. At the moment, we simply point out that

the mass of matter fields is planckian and, hence, there is a nice cancellation between GN

and the Planck mass (squared) in the non-relativistic formula of the gravitational force

between two matter particles.

2) As far as the chameleonic competition between the potential and the radiation term

is concerned, we notice that E2 and B2 enter into the effective potential with opposite

signs and, in particular, only the magnetic field corresponds to a stabilizing chameleonic

contribution. This minus sign and its potential consequences should be further discussed.

Among the possible lines of development we can mention:

a) As far as the chameleon-induced collapse of the wave function is concerned, a detailed

phenomenological analysis is definitely required to test these ideas. It would be interest-

ing to understand whether the theory restores unitarity at a more fundamental level.

b) The ansatz about the collapse of the wave function enters in every single measurement

process of quantum mechanics, therefore, our chameleon-induced collapse together with

the CEP can be considered as a step towards the construction of a chameleonic quantum

mechanics. These issues should be investigated. For example, one possibility would be

to reanalyze quantum neutron interferometry experiments in the presence of a magnetic

field from the standpoint of this chameleonic model (for a non-chameleonic discussion of

these experiments the reader is referred to [49] and related references). We will also try to

establish deeper connections between our proposal and references [23, 28, 29]. Moreover,

we will try to find an answer to this question: is it possible to introduce in the theory a

chameleonic Planck constant h = h(σ)?

c) The chameleonic behaviour of matter particles should be further analyzed and partic-

ular attention should be dedicated to its potential phenomenological signatures.

d) Potential connections between chameleons and Black-Holes’ physics should be ana-

lyzed starting from [31, 23, 50, 51, 52, 36, 53, 54, 55, 56].

e) Potential connections between our model and references [36, 28, 33, 37, 38, 31, 57, 52,

58, 59, 60, 61, 62] should be investigated.

f) Another interesting line of development will analyze potential connections between

chameleon fields and fractals.

g) An interesting future research direction will try to clarify whether probability waves

of quantum mechanics can be interpreted as waves on quantum strings.

h) It would be interesting to construct a stabilizing potential for the S-frame dilaton

exploiting an extension of the Seiberg-Witten [63, 64] theory to D=5 N=2 Supergrav-

ity. Happily, for this theory, gravitational quantum corrections have been analyzed in

[65, 66, 67] where a D=5 N=2 Seiberg-Witten theory coupled to gravity has been dis-

cussed in the framework of heterotic theory and therefore the resulting potential is exact

(a condition which must be satisfied by our dilatonic potential in the S-frame, because we

are in the strong coupling regime of string theory and hence perturbation theory cannot

be exploited). However, particular attention is necessary with the contribution of mas-


sive string states. A promising line of development will further discuss the connection

between our model and references [65, 66, 67].

i) A future research direction will search for potential connections between the three

quantization steps of our proposal and references [68, 69, 70, 71].

l) It would be interesting to understand whether chameleonic QG is able to tell us why

Nature obeys the Principle of Inertia in classical physics. During this analysis, potential

connections between our model and Mach’s principle should be investigated.

m) It would be interesting to understand whether we can construct a chameleonic gravi-

ton. Perhaps reference [72] might be useful.

Acknowledgements

Special thanks are due to Dieter Luest, Pieralberto Marchetti, Antonio Masiero, Marco

Matone, Marios Petropoulos, Massimo Pietroni and Augusto Sagnotti. They were a great

source of inspiration during the development of this article.

I thank also Ignatios Antoniadis, Marco Bochicchio, Denis Comelli, Michele Redi,

Mario Tonin and Roberto Volpato for useful comments and discussions.

A Bogoliubov transformation and free electron gas

In this appendix we touch upon some useful formulas regarding Bogoliubov transforma-

tions and their application to the free electron gas. For a more detailed discussion the

reader is referred to [33].

Let us start considering a certain number N of electrons in a cubic box of side L.

Whatever will be the electron we choose, the wave vector k can take discrete values

ki = (2π/L)ni, ni = 0, 1, 2.... The ground state of the system is the lowest energy state

compatible with the Pauli principle. Remarkably, the Fermi energy EF is related to the

number density n = N/V and, in particular, we can write

EF =�2

2m(3π2n)2/3. (A.1)

In this system, the state of minimum energy is not the state annihilated by all the

annihilation operators. In other words, the ground state (i.e. the state of minimum

energy) does not satisfy the Fock condition with respect to the electron annihilation

operators and, indeed, for k < kF we have:

a(k, s)Ψ0 �= 0, (A.2)

where k is the momentum and s is a spin variable. This implies that a and a∗ do not

describe the elementary excitations of the free electron gas. Interestingly, it is possible to

perform a Bogoliubov transformation in order to introduce new creation and annihilation

operators such that the ground state Ψ0 satisfies the Fock condition for the new operators.


In general a Bogoliubov transformation can be written as

Ak = ukak − vka∗−k (A.3)

A∗k = u∗ka∗k − v∗ka−k. (A.4)

In this picture the excitation of an electron from the ground state (k < kF ) to a state

with k > kF is described by the creation of (1) an excited state above the Fermi surface

with energy E(k)−EF and (2) the creation of a hole (i.e. an unoccupied state inside the

Fermi sphere) with energy | E(k) − EF |. A general Bogoliubov transformation in the

infinite volume limit is non-unitary (see for example [33]).

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An Accumulative Model for Quantum Theories

Christopher Thron∗

Department of Mathematics, Texas A& M University - Central Texas, 1001 LeadershipPlace, Killeen, TX 76549, United States

Received 18 May 2015, Accepted 10 July 2015, Published 25 August 2015

Abstract: For a general quantum theory that is describable by a path integral formalism,

we construct a mathematical model of an accumulation-to-threshold process whose outcomes

give predictions that are nearly identical to the given quantum theory. The model is neither

local nor causal in spacetime, but is both local and causal is in a non-observable path space.

The probabilistic nature of the squared wavefunction is a natural consequence of the model.

We verify the model with simulations, and we discuss possible discrepancies from conventional

quantum theory that might be detectable via experiment. Finally, we discuss the physical

implications of the model.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Quantum Theory; Quantum Mechanics; Pre-Space; Born Rule; Signal Processing;

Threshold Process; Path Integral; Causality

PACS (2010): 03.65.-w; 03.70.+k; 11.10.-z; 04.60.-m; 31.15.xk

1 Introduction

The paradoxical, apparently indeterministic nature of quantum theory has prompted

numerous attempts to provide a deterministic, causal basis for the theory. One possible

approach is to admit the possibility of causes outside of space and time. Bohm and

Hiley take this point of view, and identifies spacetime events as “unfoldings” of a more

fundamental “implicate order” that is manifested within “pre-space” ([1],[2]). Frescura

and Hiley, building on this foundation, have developed an algebraic representation of

pre-space dynamics [3]. A somewhat different tack is taken in [4], which essentially

proposes a conceptual model of pre-space and a local, deterministic (but statistically

random) dynamics within that pre-space that produces quantum particle transmission

when “unfolded”. The model derives from an analogy with signal detection in wireless

∗ Email:[email protected].


communications: particle detection is represented as the outcome of a signal accumulation

process which occurs in spacetime augmented by an extra, non-spacetime dimension

(referred to as the a-dimension). The quantum wavefunction corresponds to in-phase

and quadrature-phase components of an amplitude and phase-modulated carrier signal

field that is present throughout spacetime augmented by the a-dimension. The location of

particle detection is determined when an accumulated signal reaches a threshold (so that

attaining the threshold effects the “unfolding”). The paper derives the Born probability

rule is a mathematical consequence: however, the paper gives no explanation of the origin

or formation of the carrier signal field required for the model.

The current paper provides a more comprehensive interpretation of quantum proba-

bilities than [4] by taking a related, but somewhat different approach. The approach is

based on the observation that both quantum mechanics and quantum field theory may be

derived from a path integral formalism. We conjecture that path integrals correspond to

a universal physical process which essentially performs a numerical integration. As in the

previous paper, this process unfolds in a non-spacetime dimension, and the observable

universe is the outcome of the process upon attaining a threshold.

The paper is organized as follows. Section 2 presents a simplified preliminary mathe-

matical model which illustrates the basic model structure. We demonstrates the model’s

ability to generate quantum probabilities both theoretically and with simulations. Sec-

tion 3 gives a more detailed model which is designed to conform more closely with the

hypothesized physical processes involved. Results of model simulations are also pre-

sented. Section 4 discusses the possibility of experimental verification of the model; and

Section 5 gives a summary discussion. For the sake of completeness, the Matlab/Octave

source code used in the simulations is given in Section 6.

2 Preliminary model

Let U represent the space of all possible configurations of the observable universe. We

emphasize that any u ∈ U expresses the entire configuration of the universe over all

times, not just its configuration at a single time. We do not need to specify whether we

are employing a quantum-mechanical or field-theoretic representation of the universe’s

configuration space – our argument does not depend on the specific nature of U .

In both quantum-mechanical or the field-theoretic representations of U , the wavefunc-

tion can be expressed in terms of a path integral Ψ : U → C of the form:

Ψ(u) ≡ 1

|Γu|∑γ∈Γu

eiS(γ), (1)

where Γu is a space of paths corresponding to the configuration u, and S(γ) is the action

associated with the path γ. Here we have used summation notation to facilitate the

connection with simulations that we will describe later. We shall suppose that |Γu| isindependent of u, so that |Γu| = |Γ|/|U| where Γ ≡ ∪uΓu. We also suppose that the

{Γu}u∈U are disjoint, which implies that for every γ ∈ Γ there exists a unique uγ ∈ Usuch that γ ∈ Γuγ .


The path integral is associated with a probability distribution:

PS(u) ≡ |Ψ(u)|2∑v∈U |Ψ(v)|2 . (2)

The fact that this probability is written in terms of a summation (or integral) suggests

that some sort of accumulation process could be involved. The main purpose of this paper

is to show that such an interpretation is indeed feasible, and provides a simple, plausible

explanation of the hidden dynamics that give rise to quantum theories. Preliminarily, we

note that our interpretation must address two issues:

• Why is probability obtained from a squared complex amplitude?

• What physically corresponds to the division in (2)?

In the following, we give what we believe to be satisfactory answers to these two questions.

We define an accumulation process as follows. Given the sequence of paths γ1, γ2, . . .

in Γ, we define an accumulated amplitude AK (K ∈ Z+) as:

AK ≡ ΣKk=1e

iS(γk). (3)

One possible interpretation of each factor eiS(γk) is as the phasor representation [5] of

an oscillation (of unknown frequency) which depends on γk. The summation then cor-

responds to the complex amplitude of a harmonic oscillator (with the same frequency)

that is successively perturbed by these oscillations.

Although we are using discrete notation, the sequence {γk} should be thought of as

a discrete approximation of a path-valued function of a continuous index, corresponding

to a continuously-varying path within the space Γ of all possible paths. The continuous

index corresponds the the a-dimension introduced in [4]: and the variation within Γ

corresponds to an evolutionary process within this dimension which uniformly samples

Γ over the long term. Note that as γk varies, the corresponding state of the universe

uk ≡ uγk also varies. In the process we will define, the accumulated amplitude grows to

reach a fixed threshold at a particular index K, at which point uK gives the configuration

of the observable universe.

In order to obtain the probabilities (2) via this process, we impose additional condi-

tions on the sequences {γk} and {uk} as follows.

(a) There exists N 1 andM 1 such that ukNM+1 = ukNM+2 = . . . = u(k+1)NM , ∀k ∈Z≥0;

(b) For each k ∈ Z≥0, the sequence {γkN+1, γkN+2, . . . , γ(k+1)N} uniformly samples ΓukN;

(c) The sequence {uNM , u2NM , . . .} is mixing [6] and uniformly samples U .

These conditions correspond to a situation where {γk} varies throughout Γ such that

the sequence {γk} uniformly samples each Γu that it visits before passing on to the next

Γu. In this simple model, the dwell time within each Γu visited is the constant N : in

our subsequent model, this assumption will be relaxed. The significance of M will be

explained later.

Let ηk (k = 1, 2, . . .) be a sequence of independent, identically distributed (i.i.d.)


complex-valued random variables with zero mean and finite variance, and define:

A′K =

K∑k=1

η�k/N�eiS(γk). (4)

Finally, given Θ > 0, we define the threshold index as the random variable:

KΘ ≡ min(k||A′k| < Θ and |A′

k| ≥ Θ). (5)

Given the above conditions and definitions, we have the following result:

Proposition : As N,M, θ → ∞, we have

P (uKθN√M= u) → PS(u). (6)

In other words, the probability distribution on U at the stopping time defined by attaining

the threshold θN√M agrees with the probability distribution (2) obtained from the path-

integral formalism.

The proof of this proposition is similar to that given in [4]. Notice that (4) can be

rewritten as

A′KN

θN√M

=1

θN√M

K∑k=1

ηk

(N∑

n=1

eiS(γ(k−1)N+n)

)(7)

−→N→∞

|U|θ|Γ|√M

K∑k=1

ηkΨ(u�k/M�) (8)

=|U|θ|Γ|

⎛⎝K/M�∑

k=1

Ψ(uk)( 1√

M

M∑m=1

η(k−1)M+m

)+Ψ(u�k/M�)

( 1√M

K∑m′=MK/M�+1

ηm′)⎞⎠ .

(9)

The proof is based on the fact thatA′KN

θN√M

can be approximated in distribution as a

Brownian motion B(a) in C with absorbing boundary at |z| = 1, where a ≡ Kθ2NM

. For

any fixed a, near the boundary the probability density of an absorbing Brownian motion is

proportional (to first order) to the distance from the boundary. This can be used to show

that for any K, the probability P (KθN√M = K|uK = u) is approximately proportional

to E[|ηKΨ(u)|2], which is proportional to |Ψ(u)|2. Since for P (uK = u) is independent of

u when 1 � K < KθN√M , it follows that P (KθN

√M = K & uKθN

√M= u) is proportional

to |Ψ(u)|2, and summing over K gives the desired result.

Figure 1 shows the results of simulations of the model specified by conditions (a)–

(c) and equations (4)–(5). The simulations were performed on a discrete system with 11

possible states. To shorten computational time, the simulation was based on equation (8)

rather than performing the full computation (7) on a path-by-path basis. The random

variables {uNM , u2NM , . . .} referred to in (c) were generated uniformly randomly. The

curves show the difference between the simulated probabilities and actual probabilities

for two different probability distributions |Ψ|2, for different values of the threshold θ. The


errors are shown on the y-axis, versus the actual probability values which are shown on

the x-axis. As θ increases, the errors decrease: for θ = 40, the maximum error is under 5

percent. The pattern of error apparently depends on the type of probability distribution

being modeled. However, in both cases the larger probabilities are underestimated, and

there is a range of intermediate probabilities that are overestimated. These phenomena

may possibly enable an experimental test of the model: this possibility is explored further

in Section 4.

Fig. 1 Deviations of computed probabilities from quantum values, for simulated preliminaryaccumulation model with θ = 10, 20, 30, 40 and M = 10000, where {ηk} are i.i.d. standardnormal random variables. Each simulation was run 100,000 times. All simulations used 11configurations u. For the figure at left, |ψ(uj)| ∝ j, (j = 0, . . . , 10), while for the figure at right,|ψ(uj)|2 ∝ j (j = 0, . . . , 10).

3 Refined model

The process we have presented above has some seemingly artificial features:

• Why should {uk} remain constant for intervals of size MN?

• What is the physical significance of the ηk’s?

As to the first point, instead of supposing that {uk} remains constant on intervals of size

N , we may suppose that {uk} varies slowly with k, so that

p(uk+1 �= uk) = O(

1

N

). (10)

Supposing that {γk}k=1.2.3.... is generated by a Markov process, it is reasonable to suppose

that residence times in each u state visited are (approximately) i.i.d. geometrical random

variables. This is because under reasonable conditions, hitting times in Markov chains

are asymptotically exponentially distributed [7]. (The geometrical distribution is the

discrete analog of the exponential distribution.) Accordingly, we may modify the model

by replacing the constant M with a geometrically-distributed random variable with the

same mean.


As to the second point, we must recognize that we have failed to account for the fact

that in practice we never measure the state of the entire universe, but only a subsystem.

So we must take into account the effect of variations in the external system during

the accumulation process. Accordingly we let Ω be the possible states of the measured

subsystem, while Ω′ denote the possible states of the universe external to the measured

subsystem. Thus we may represent any element u ∈ U uniquely as u = (w,w′), wherew ∈ Ω and w′ ∈ Ω′.

We suppose that any path in Γ can be factored into a part for Ω and a part for Ω′:more precisely, that there are path spaces C and C ′ respectively such that any γ ∈ Γ can

be decomposed as γ = (c, c′) where c ∈ C, c′ ∈ C ′, and such that uγ = (wc, w′c′). We define

Cw ≡ {c|wc = w}, and suppose (as in the simple model) that |Cw| is independent of w ∈ Ω,

so that |Cw| = |C|/|Ω| ∀w. We similarly define C ′w′ , and suppose |C ′w′ | = |C ′|/|Ω′| ∀w′.Finally, we suppose that the action S is additive: S(γk) = S(ck) + S(c′k). From this it

follows that we may write:

Ψ(u) = Ψ((w,w′)) = ψ(w)φ(w′), (11)

where

ψ(w) ≡ |Ω||C|

∑c∈Cw

eiS(c); φ(w′) ≡ |Ω′||C ′|

∑c′∈Cw′

eiS(c′). (12)

We may also rewrite (3) as

AK ≡ ΣKk=1e

iS(ck)eiS(c′k). (13)

We now postulate the existence of a Markov chain {(c1, c′1), (c2, c′2), . . .} that satisfies the

following properties. Define inductively a sequence of random times {Xk} such that

X0 ≡ 1; Xk+1 ≡ min(j|w′j �= w′Xk).

We suppose the Markov chain has transition probabilities such that wj �= wj+1 =⇒w′j �= w′j+1. This supposition reflects the assumption that the external state varies more

rapidly than the observed state, which is reasonable since the external state is much,

much larger and has many more possibilities for variation. In this case, it is possible

to define inductively a sequence of random times {Zk} such that Z0 = 1 and Zk+1 ≡min(j|wXj

�= wXZk). According to these definitions, the state w′ does not change on

each time interval [Xk,Xk+1 − 1], and the state w does not change on each time interval

[XZk,XZk+1−1]. We also suppose the paths vary much faster than the states, so that the

space CwXkis uniformly sampled on the time interval [Xk,Xk+1 − 1].

Based on the Markov chain described in the previous paragraph, we may formulate

the following model assumptions:

(A) There exists a N 1 and a sequence {ξ1, ξ2, . . .} of i.i.d. geometrically-distributed

random variables with E[ξk] = N , such that w′XK+1 = w′XK+2 = . . . = w′XK+ξk∀K ∈

Z≥0, where X0 ≡ 0 and XK ≡ ∑Kk=1 ξk, K ≥ 1;


(B) There exists a M 1 an a sequence {ζ1, ζ2, . . .} of i.i.d. geometrically-distributed

random variables with E[ζk] =M , such that wXZK+1 = wXZK

+2 = . . . = wXZK+1∀K ∈

Z≥0, where Z0 ≡ 0 and ZK ≡ ∑Kk=1 ζk, K ≥ 1;

(C) For eachK ∈ Z≥0, the sequences {c′XK+1, c′XK+2, . . . , c

′XK+ξk

} and {cXK+1, cXK+2, . . . , cXK+ξk}uniformly sample C ′XK+1

and C, respectively;

(D) The sequences {c′XK+1, c′XK+2, . . . , c

′XK+ξk

} and {cXK+1, cXK+2, . . . , cXK+ξk} are statis-

tically independent;

(E) The sequences {w′X1, w′X2

, . . .} and {wXZ1, wXZ2

, . . .} are mixing, and uniformly sam-

ple Ω and Ω′ respectively.Following these assumptions, we may compute:

A′ZK

θN√M

=1

θN√M

K−1∑k=0

Zk+1−1∑m=Zk

Xm+1∑n=Xm+1

eiS(cn)+S(c′n)

≈ |Ω||Ω′|θ√M |C||C ′|

K−1∑k=0

Zk+1−1∑m=Zk

ξm+1

Nψ(wXm+1)φ(w

′Xm+1

) (14)

=|Ω||Ω′|θ|C||C ′|

K−1∑k=0

(ψ(wXZk+1

) · 1√M

Zk+1−1∑m=Zk

ξm+1

Nφ(w′Xm+1

)

)

=|Ω||Ω′|θ|C||C ′|

K∑k=1

(ψ(wXZk

) · 1√ζk

ζk∑m=1

ηm,k

), (15)

where the approximation holds for large N and

ηm,k ≡√ζkM

(ξZk−1+m

N

)φ(w′XZk−1+m

). (16)

Notice the similarity between (9) and (15). Instead of a summation over M , there is a

summation over ζk, which has expectation M . Within this summation, instead of the

mean-zero i.i.d. random variables {ηk}, we now have {ηk,m} given by the complicated

expression (16). By assumption, the variables ζk/M and ξZk−1+m/N are independent, and

have expectation 1; while the additional complex factor φ(w′XZk−1+m

)will vary randomly

with mean zero as the process evolves. If we assume that {ηk,m} are (approximately) i.i.d.

mean-zero random variables, then (15) and (9) are virtually identical, except that ζk in

(15) replaces M in (9). However, E[ζk] = M ; and conditioning on the different possible

values of ζk, we may obtain the same result that the probability density for wKΘis given

by |ψ(w)|2.Figure 2 shows results of simulations of the refined model specified in (A)-(E). A

system with 31 discrete states was simulated, and the states’ probabilities were chosen

according to the sinusoidal wavefunction shown in the picture. The transition between

states w was determined according to a Markov chain that produced a mean dwell time

of M , followed by a transition to one of the four nearest-neighbor states with equal

probability 1/4. Parameters used were M = 625 and θ = 10. The figure shows very close


agreement between quantum-theoretic probabilities and those obtained from simulation.

Deviations are shown in more detail in Figure 3 for different values of M and θ. Small

|ψ|2’s are consistently overestimated, and large |ψ|2’s are underestimated. Deviations

between simulation and quantum theory decrease with increasing M and θ, so that the

model probabilities apparently converges to quantum-theoretic values as M, θ → ∞.

Fig. 2 (Left) Sinusoidal “wavefunction” used in simulation. 31 configurations were used withprobabilities as shown. (Right) Simulation results compared to theory for θ = 10,M = 625.Computed probabilities are based on 10 million repetitions.

Fig. 3 Deviations of computed probabilities from quantum values for simulated adjusted accu-mulation model, for different values of the accumulation length M and threshold parameter θ(as specified in the figure titles). All computed probabilities are based on 10 million repetitions.

4 Proposed Experimental Test

In the above model, quantum probabilities are generated by an accumulative process

which essentially performs a stochastic approximation to the quantum path integrals. In

the previous section we showed that finite values of θ and M introduced deviations from


quantum-theoretical probabilities. In both cases, the deviations are positive for small

probabilities, but negative for large probabilities.

Another possible source of numerical error, which we did not model in the simulation,

results from the approximation

1

ξm

Xm+ξm∑n=Xm+1

eiS(cn) ≈ ψ(wXm+1), (17)

which was used in (14). If we suppose there is a random error of constant variance

ε2 in this approximation, then by carrying through the computations it can be shown

that probabilities turn out to be proportional to |ψ(w)|2 + ε2 rather than |ψ(w)|2. This

produces a deviation from theoretical probabilities that decreases linearly with increasing

probability density. So the deviations from quantum-theoretic probabilities due to this

effect reinforce the deviations already discussed.

We may conclude that numerical approximation effects should introduce a deviation

from quantum-theoretic probabilities that for larger probabilities decreases roughly lin-

early with increasing probability density. Unfortunately, since the parameters of the

process are not directly accessible, it is not possible to predict the size of the deviations.

5 Discussion

The above model provides a conceptually simple solution to many conundrums of quan-

tum theory. It accounts for all quantum paradoxes, since it yields the same probabilities

as quantum theory (to a close approximation). It requires no distinction between observer

and observed, because the probabilistic significance of the wavefunction is a consequence

of the model, rather than an extraneous assumption that is added to match theory with

experiment. In particular, the “measurement problem” is no longer a problem: what is

perceived as a “collapse of the wavefunction” corresponds to the fact that one particular

state of the universe is selected as a result of the thresholding process.

Our model gives a very different perspective on several seemingly “evident” aspects

of the universe. Physical causality is attributed to correlation: causes and effects are

correlated outcomes of an inaccessible process that occurs outside of spacetime. The

Big Bang is not accounted as the “origin” of the universe, because it also is part of

the outcome of an extra-dimensional process which produces past, present, and future

together as an entirety. (The model thus seems to imply that the universe will have finite

duration.) The vacuum is not a “boiling, bubbling brew of virtual particles and fields

wildly fluctuating in magnitude,”[8] as quantum field theory seems to imply, but only

appears so because of the accumulation process through which the observable universe is

actualized.

We mentioned in the introduction that the “pre-space” approach proposed and de-

veloped by Bohm, Hiley, and others bears some similarity to our approach. However,

our portrayal of pre-space is radically different from that envisioned by Bohm et. al.


This difference is clearly seen when we compare the analogies that we use to explain our

respective notions of pre-space.

The original inspiration for our model was the example of a cellular phone which

accumulates a pilot signal broadcast by a base station as the phone is carried about

by the user. When the signal accumulation reaches a certain threshold, a detection is

logged. The user’s location at the moment of detection is determined by the process of

signal accumulation–but no comprehensive record of his past motion may be seen in the

final outcome. Still, the outcome reflects the process in that the detection location is

more likely to be at a location where the signal is strong. In other words, the legacy of

the process of signal accumulation is seen in the probability distribution of the observed

outcome.

On the other hand, Bohm in [1] describes an experiment in which a droplet of dye

is introduced into a viscous fluid, the fluid is stirred, and the process is repeated several

times. When the fluid is stirred in the reverse direction, the droplets reappear one by one.

These droplets represent the unfolded order that is evidenced in spacetime events. Fresca

and Hiley take this illustration as a jumping-off point in their portrayal of quantum

processes in terms of successive enfolding and unfolding. Thus spacetime events are

conceived as manifestations of an ongoing process. Clearly this is very different from our

description of a process from which the entire history of the universe springs full-blown

into existence, like Athena emerging from the head of Zeus.

Finally, we consider our proposed model in the context of the overall development

of theoretical physics. Physics has historically progressed by means of analogies which

have been proposed, explored, and pushed to their limits. For example, Maxwell’s equa-

tions were originally motivated by an analogy between electromagnetic fields and local

displacements within an incompressible fluid medium due to stresses and strains[9]. But

as electromagnetic theory developed, the limitations of this analogy became increasingly

apparent–to the extent that it is scarcely mentioned in university courses on electromag-

netism, and only a few vestiges may be seen in some of the terminology (such as stress

tensor). Another important analogy (that has captured the popular imagination) is the

idea that gravity bends space. This foundational idea motivated Einstein to look to dif-

ferential geometry for mathematical formulations of the theory. The inability of general

relativity to deal with quantum mechanics shows that the analogy can only go so far. The

same could be said for Rutherford’s planetary model of the atom. More germane to the

subject of this paper, the analogy between the statistical-mechanical partition function

and the expression (1) was a key motivation for Euclidean quantum field theory[10].

Historically, the analogies used in theoretical physics have in general been taken from

nature, as seen in the above examples. In contrast, our analogy comes from wireless

communications technology. We suggest that in view of its explosive development, tech-

nology may become a rich new source of analogies for physicists. Conversely, fundamen-

tal physics may increasingly suggest technological innovations–not necessarily through

direct application of the physics, but rather through analogical similarities between the

two regimes.


It is our hope that further exploration of the analogy presented in this paper may lead

to additional physical insights. Along these lines, we mention briefly some possibilities

suggested by the model we have developed:

• Feynman path integrals are notorious for yielding important physical results de-

spite lacking a mathematical rigorous foundation. Our model suggests making a

correspondence between paths and possible states of the universe. This line of at-

tack could lead to a less problematic mathematical characterization of these path

integrals.

• Although formula (1) is based on an action, so far we have said nothing about the

action, nor the fields that determine its value. Our analogy with signal processing

suggests that there may be a relationship between the various types of quantum

fields and signal modulations.

• The model is designed to give an account of observed probability distributions for

quantum events. However, so far we have not really defined event. Certainly this

has something to do with the configuration of the fields involved: and perhaps this

also may be understandable in terms of a signal-based representation of the fields.

Although we hope that our model will be a source of insight, we recognize that even in the

best case, the analogy that we have suggested will have proscribed limits. Nonetheless,

if this conceptual model proves to be accurate, it has profound implications for how we

may regard the world around us, and how we regard ourselves as “free agents” within it.

6 Simulation Code

The following Octave/Matlab code was used for the simulation in Figures 2 and 3.

% Parameters

clear all;

nsim = 10000000; % # simulations

nconfig=1; % # configs simulated

Theta_fac = 10; % Theta increment

Theta_fac0=Theta_fac; % Orig. theta

Ncfg = 31; % Number of internal configs

n_acc_mean = 625; % M interval

max_jump = 2; % For Markov -- max jump

acc_mean0 = n_acc_mean; % Orig. M

p = 1/n_acc_mean;

Theta = Theta_fac*sqrt(n_acc_mean); % Rescaled threshold

% Arrays to store results

Counts = zeros(Ncfg,1);

Q = [];

% Create measurable configurations


Psi = cos((0:1:Ncfg-1)/(Ncfg)*2*pi)’;

Prob = abs(Psi).^2;

Prob = Prob / sum(Prob);

% Computations

for jj = 1:1:nconfig % Loop over configurations

for ii = 1:1:nsim % Perform simulations

A = 0;

this_cfg = randi([0,Ncfg-1]); % Choose current w

% Accumulate:

while abs(A) < Theta % Until threshold is attained

this_cfg = mod(this_cfg + sign(randn)*randi([1,max_jump]),Ncfg);

Ptmp = Psi(this_cfg+1);% Amplitude

while rand() > p

Rtmp = randn()+1i*randn();

A = A + Rtmp*Ptmp;

if abs(A)>Theta % If pass threshold, the break and record w

break

end

end

end

Counts(this_cfg+1) = Counts(this_cfg+1)+1; % record w

end

Q = [Q Counts/sum(Counts)] % Summary results for this config

Theta_fac = Theta_fac + Theta_fac0; %Increment theta

Theta = Theta_fac*sqrt(n_acc_mean);

end

Prob = abs(Psi).^2;

Prob = Prob / sum(Prob); %Normalized, sorted probabilities (for theory)

plot(Prob,Q - Prob*ones(1,nconfig),’*’);

Acknowledgments

Thanks to Johnny Watts for help in the preparation of this paper for publication. Thanks

also to Ignazio Licata for pointing out several useful references.

References

[1] Bohm D, Wholeness and the Implicate Order, Routledge & Kegan Paul, 1980.

[2] Bohm D and Hiley B. J., “Generalization of the Twistor to Clifford Algebras as a


basis for Geometry”, Rev. Braiasileira de Fisica, Volume Especial, Os. 70 anos deMario Schonberg (1984), 1-26.

[3] Frescura F. A. M and Hiley B. J., “Algebras, Quantum Theory and Pre-space”,Revista Brasileira de Fisica (v. especial) p. 49-86; ISSN 0374-4922 (Jul 1984).

[4] Thron C and Watts J, “A Signal Processing Interpretation of Quantum Mechanics”,The African Review of Physics 8, (2013), http://www.aphysrev.org/index.php/aphysrev/issue/view/29/showToc.

[5] Scharf L (n.d.)“Phasors: Phasor Representations of Signals”, openstax cnx, Retrieved February 15,2015 from http://cnx.org/contents/733a0289-e9b2-40b2-bd0b-77666c1ba5f6@7/Phasors:_Phasor_Representation.

[6] Mixing (mathematics) (n.d.). In Wikipedia. Retrieved January 28,2015, from http://en.wikipedia.org/wiki/Mixing_(mathematics)#Mixing_in_stochastic_processes.

[7] Aldous D, “Markov Chains with Almost Exponential Hitting Times,” StochasticProcesses and their Applications 13 (1982), 305–310.

[8] Krauss L, A Universe from Nothing, Simon & Schuster, January 2012.

[9] Hall G,“Maxwell’s electromagnetic theory and special relativity”, PhilosophicalTransactions of the Royal Society A, Volume: 366 Issue: 1871 (May 2008).

[10] Guerra F, “Euclidean Field Theory,” arXiv:math-ph/0510087v1 (February 7 2008).


Spin Angular Momentum and the Dirac Equation

Robert A. Close∗

Department of Physics, Clark College, 1933 Fort Vancouver Way, Vancouver, WA98663, USA

Received 26 March 2015, Accepted 10 June 2015, Published 25 August 2015

Abstract: Quantum mechanical spin angular momentum density, unlike its orbital counterpart,

is independent of the choice of origin. A similar classical local angular momentum density may

be defined as the field whose curl is equal to twice the momentum density. Integration by parts

shows that this spin density yields the same total angular momentum and kinetic energy as

obtained using classical orbital angular momentum. We apply the definition of spin density to

a description of elastic waves. Using a simple wave interpretation of Dirac bispinors, we show

that Dirac’s equation of evolution for spin density is a special case of our more general equation.

Operators for elastic wave energy, momentum, and angular momentum are equivalent to those

of relativistic quantum mechanics.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Angular Momentum; Dirac Equation; Elasticity; Quantum Mechanics; Solid; Spin

PACS (2010): 03.65.Pm; 03.65.Ta; 11.10.Ef; 11.15.Kc; 14.60.Cd; 46.05.+b

1 Introduction

The spin angular momentum of elementary particles is often puzzling to students because

it is not obviously related to an angular velocity. The particle velocity is taken to be the

de Broglie wave velocity, and since the waves propagate in vacuum there is presumably

nothing else to rotate. However, it is possible to interpret spin as a property of waves. [1]

And it is well known that elastic waves in solids have two types of momentum: that of

the medium and that of the wave (see e.g. Ref. [2]). Clearly there must also be two types

of angular momentum in an elastic solid: ”spin” associated with rotation of the medium,

and ”orbital” associated with rotation of the wave. However, spin angular momentum is

not normally considered to be a classical physics concept.

A standard part of undergraduate physics education is the definition of angular mo-

mentum density as r × p , where r is the radius vector and p = ρu is the momentum

∗ Email: [email protected].


density (ρ = mass density, u = velocity). An obvious shortcoming of this definition is

that it depends on the arbitrary choice of origin of the coordinates. Hence the computed

angular momentum is not a local property of the physical system. Another limitation of

this definition is that for a given field of angular momentum density L = r × p , there is

no simple way to reconstruct the associated rotational momentum density. For example,

∇ × (r × p) = ekεijkεjlm∂i(rlpm) = ek(δklδim − δilδkm)∂i(rlpm)

= ek∂i(rkpi − ripk) = ek(−2pk + rk∂ipi − ri∂ipk)

= −2p + r∇ · p − r · ∇p . (1)

Even for incompressible momentum fields, the last term containing derivatives of p is

problematic.

Coordinate-independent descriptions of rotational dynamics can be traced back to the

nineteenth century. [3] MacCullagh modeled light as rotationally elastic shear waves in an

isotropic medium with shear modulus μ and displacements a(r, t), taking μ (∇ × a)2 /2

as the energy density. [4] Requiring stationary variations of the associated Lagrangian

yields the equation for elastic shear waves with speed c =√μ/ρ :

ρ∂2t a = −μ∇ × (∇ × a) . (2)

In terms of rotations, the force density is proportional to the curl of a torque density,

which itself is proportional to the infinitesimal rotation angle (∇ × a)/2. Heaviside

similarly interpreted force density as minus the curl of torque density. [5] We wish to

clarify these definitions and extend them to arbitrarily large rotations.

It is not a simple matter to describe rotational motion in an elastic solid. The standard

treatment of elastic waves (e.g. Ref. [6]) assumes infinitesimal derivatives of displacement

∂iaj, and decomposes them into symmetric strain ((∂iaj + ∂jai)/2) and anti-symmetric

rotation ((∂iaj − ∂jai)/2) tensors. The strain tensor may be regarded as the deviation

(to first order) from a rigid rotation. Stress is assumed to be proportional to strain. For

density ρ and elastic constants λ and μ, the resultant equation of evolution of displacement

is

ρ∂2t a = (λ+ 2μ)∇(∇ · a) − μ∇ × (∇ × a) . (3)

The two terms on the right side of this equation describe compressional and shear

waves, respectively. The equation for shear waves is of course equivalent to MacCullagh’s

equation for light (Eq. 2).

Feynman analyzed the shear resulting from variations of rotation angle ϕz along an

axis (z), obtaining the formula: [7]

∂2t ϕz = c2∂2zϕz . (4)

Equation (3) is valid only for infinitesimal displacements. Generalization to finite

rotations (rotational shear waves) destroys the neat separation between irrotational and

incompressible waves. The basic difficulty is that finite rotations have zero divergence


of velocity, but non-zero divergence of displacement. For example, rotation in the x− y

plane by angle ϕ yields displacement (ax, ay) of⎡⎢⎣axay

⎤⎥⎦ =

⎛⎜⎝ cosϕ− 1 − sinϕ

sinϕ cosϕ− 1

⎞⎟⎠

⎡⎢⎣xy

⎤⎥⎦ =

⎡⎢⎣x(cosϕ− 1) − y sinϕ

x sinϕ+ y(cosϕ− 1)

⎤⎥⎦ . (5)

The divergence of displacement is ∂xax+∂yay = 2(cosϕ−1), which is not zero in general.

The theory of elastic waves could be improved by including higher-order deriatives,

[8] but this does not solve the fundamental limitation to small displacements. Instead

we use a different approach based on velocity rather than displacement. This approach

leads directly to the concept of spin density.

According to Helmholtz’s Theorem, any vector field may be decomposed into irro-

tational and incompressible components (see e.g. Ref. [9]). Since shear waves are in-

compressible, it is more natural to describe them in terms of rotational (incompressible)

velocity rather than displacement. Recent work by this author attempts to utilize an-

gular momentum and torque densities in place of momentum and force densities as the

fundamental variables for rotational shear waves. [10–12] This description of elastic waves

results in equations quite similar to those of relativistic quantum mechanics, thereby pro-

viding a tangible basis for understanding quantum mechanical spin angular momentum.

In this paper we derive the relationships between spin density and the usual classical

angular momentum in Section 2, apply the concept of spin density to a rigidly rotating

cylinder in Section 3, and analyze elastic waves in Section 4.

2 Angular Variables

The relationship between coordinate-dependent and independent descriptions of angular

variables may be seen as follows. Consider a locally rigid rotation with angular velocity

w around the z-axis. The velocity is given by u = −r × w , and the differential velocity

is du = −dr × w . Solving for wz yields wz = ∂xvy = −∂yvx = (∇ × uϕ)/2, which is the

usual definition of vorticity.

We desire a similar local spin density S whose curl is proportional to linear mo-

mentum. This definition would make the motion explicitly rotational (incompressible),

distinguishing it from compressible (irrotational) motion.

Simply comparing equations L = r × p and u = −r × w , we might expect the

relationship to be ρu = −∇ × S/2. However, the angular momentum density must be

the same sign as vorticity in order to have positive kinetic energy density in the form

of (1/2)w · S . It must also fall to zero at infinity in order to have finite total angular

momentum. These conditions require an angular momentum density maximal at the axis

of rotation and decreasing with increasing radius. This requires dS = −dr × ρu , or

ρu = +1

2∇ × S . (6)


Force and torque should be similarly related:

f = +1

2(∇ × τ) . (7)

Fig. 1 Proportionality Between Force and the Curl of Torque.

The reason that the sign differs from that predicted from the relationship between w

and u is that in the equation u = −r × w , the velocity u is the circulating vector field,

whereas in the equation τ = r × f , the force f is the circulating vector field. Fig. 1 shows

that at the center of the counter-clockwise torque loop the force points toward the reader,

consistent with the positive curl of torque. Hence the positive signs in Eqs. (6) and (7)

are correct.

Assuming differentiable functions and boundary terms of zero when integrating by

parts, the total angular momentum is given by:

J =

∫(r × p)d3r =

1

2

∫r × (∇ × S)d3r

=1

2ek

∫(εijkriεlmj∂lSm)d

3r =1

2ek

∫(δklδim − δkmδil)(ri∂lSm)d

3r

=1

2ek

∫(ri∂kSi − ri∂iSk)d

3r = −1

2ek

∫([∂kri]Si − [∂iri])Skd

3r

=

∫Sd3r . (8)


The total kinetic energy is similarly:

K =1

2

∫ρu2d3r =

1

8ρ

∫[∇ × S ] · [∇ × S ]d3r

=1

8ρ

∫∂iSj∂iSj − ∂iSj∂jSi]d

3r = − 1

8ρ

∫[Sj∂i∂iSj − Sj∂j∂iSi]d

3r

= − 1

8ρ

∫S · ∇2S − S · ∇[∇ · S ]d3r = 1

8ρ

∫S · [∇ × ∇ × S ]d3r

=1

2

∫w · Sd3r . (9)

Notice that the last step requires the positive sign in Eq. (6).

Hence the definition of spin density given in Eq. (6) yields the same total angular

momentum and kinetic energy as the conventional definition r ×p . The new definition is

independent of the choice of origin, is defined only by the motion in a local neighborhood,

and completely determines the rotational momentum density p(r).

Next we show that spin density may be used to describe ordinary rigid rotations.

3 Rigid Rotation

Fig. 2 A Rotating Cylinder.

We will use spin density to describe a cylinder aligned with the z-axis and rotating

rigidly with angular velocity w0 (Fig. 2). The non-zero variables are

Sz = ρw0[R2 − r2] for r ≤ R; zero for r > R; (10)

uφ =1

2ρ

∂

∂rSz = rw0 for r ≤ R; zero for r > R ; (11)

wz =1

2r

∂

∂rruφ = w0 [1 −Rδ(r −R)/2] for r ≤ R; zero for r > R . (12)

The reader may verify that the delta-function in the vorticity yields the correct velocity

jump at the boundary r = R.


The total angular momentum per unit height is

J = 2π

∫ R

0

Szrdr = 2π

∫ R

0

ρw0[R2 − r2]rdr

=1

2πρR4w0 =

1

2MR2w0

= Iw0 . (13)

where we have used the mass per unit height M = ρπR2 and moment of inertia per unit

height I =MR2/2.

The kinetic energy per unit height is

K =1

2

∫w · S rdrdφ = π

∫ R

0

w0 [1 −Rδ(r −R)/2] ρw0[R2 − r2]rdr

= πρw20

[R4

2− R4

4

]=MR2

4w2

0 =1

2Iw2

0 . (14)

Thus we see that spin density correctly describes rigid rotation of a cylinder about

its axis, yielding the usual expressions for total angular momentum and kinetic energy.

However, orbital angular momentum is likely simpler for describing arbitrary motion

of rigid bodies. The main application for spin density is in continuous media where

incompressible motion may be described as the curl of a vector potential.

4 Application to Elastic Waves

The velocity defined by Eq. (6) is explicitly divergence-free, making this a natural way to

describe shear waves. Previous attempts have demonstrated that rotational shear waves

share many properties with relativistic quantum mechanics. [10–12] Here we derive the

wave equation for rotational shear waves and clarify the relationship to the Dirac equation.

4.1 Wave Equation

The equation of evolution for rotational shear waves is derived by relating torque density

to the rate of change of spin density:

τ =dS

dt= ∂tS + u · ∇S − w × S . (15)

The final two terms subtract the contributions of convection and rotation to the partial

time derivative of S(r, t), since these result from motion of the medium rather than

torque. The right-hand side of the equation is called the ”total” time derivative of S

since it describes the change in angular momentum density of a moving piece of the solid.

We introduce an angular potential Q defined by

∂tQ = Q = S . (16)


The relationship between S and u implies that for infinitesimal motion

1

2ρ∇ × Q ≈ a ; (17)

1

4ρ∇ × ∇ × Q ≈ ϕ . (18)

The usual equation for infinitesimal shear waves is therefore equivalent to:

1

2ρ∇ × (∂2tQ + c2∇ × ∇ × Q) = 0 . (19)

and Feynman’s Eq. (4) for one-dimensional torsion waves is equivalent to:

1

4ρ∇ × ∇ × (∂2tQ − c2∇[∇ · Q ]) = 0 . (20)

Hence for infinitesimal motion, the torque density is:

τ = c2∇[∇ · Q ] − c2∇ × ∇ × Q = c2∇2Q . (21)

This expression for torque density is based on the usual assumption of a linear rela-

tionship between stress and infinitesimal strain. However, we will simply take it to be

the defining characteristic of the solid medium for arbitrary motion. Notice that we are

not limited to small displacements because Q(r, t) is simply a time integral of S(r, t) at

each fixed point r.

We now have a consistent description of rotational variables:

S = Q ; (22)

ρu = (∇ × S)/2 ; (23)

w = (∇ × u)/2 ; (24)

τ = c2∇2Q . (25)

Setting the total time derivative of angular momentum equal to torque and rearrang-

ing terms yields the equation of evolution: [11]

∂2tQ − c2∇2Q + u · ∇Q − w × Q = 0 . (26)

The first two terms of this equation describe a wave-like response to torques in the

medium. The last two terms in Eq. (26) represent nonlinear corrections due to finite

motion of the medium. If the nonlinear terms do not cancel, they must be in phase with

the other terms. Replacing the nonlinear terms by M2Q yields :

∂2tQ − c2∇2Q +M2Q = 0 . (27)

If the coefficientM is a constant, then this is a vector Klein-Gordon equation. In general,

however, M could be a function of position.


4.2 Dirac Equation

Richard Feynman referred to the Dirac equation as a simple and beautiful one ”...which

no one has really been able to understand in any direct fashion.” [13] Dirac’s equation is

admittedly difficult to manipulate since the wave function has four complex components.

However, the following analysis based on Ref. [11] shows that the Dirac equation may be

interpreted simply as a factorization of an ordinary second-order vector wave equation.

Consider a single-polarization wave propagating in one-dimension with amplitude (not

displacement) of a(z, t). If the wave equation is

∂2t a = c2∂2za , (28)

then the general solution is

a = aB(ct+ z) + aF (ct− z) (29)

where aB(z, t) and aF (z, t) are arbitrary functions that propagate along the axis in the

backward and forward directions, respectively. The two directions of wave propagation

are clearly independent states, and they are separated in space by a 180◦ rotation. Thisproperty is the fundamental characteristic of spin one-half functions. Generalization to

three dimensional space should therefore yield a Dirac wave function.

To demonstrate this, we write the wave equation as a matrix equation. The two wave

solutions form an array: ⎡⎢⎣aBaF

⎤⎥⎦ . (30)

Noting that temporal and spatial derivatives vary only by a factor of ±c, the wave

equation becomes

∂t

⎡⎢⎣aBaF

⎤⎥⎦+

⎛⎜⎝−1 0

0 1

⎞⎟⎠ c∂z

⎡⎢⎣aBaF

⎤⎥⎦ = 0 . (31)

We have now reduced the second-order scalar wave equation to a first-order matrix

equation. The next step is a bit unusual. We further divide each component of the wave

into positive and negative regions (aB = aB+ − aB− and aF = aF+ − aF−). Now each of

the four wave components (aB+, aB−, aF+, aF−) is positive-definite, and only one of the

components may be non-zero for each propagation direction. Some caution is warranted

because these components may have discontinuous derivatives at sign transitions, but we

will ignore that issue here. In higher dimensionality positive and negative components

can coexist, indicating a different polarization direction.

We arrange the components in the following order, corresponding to the chiral repre-


sentation of the Dirac wave function: ⎡⎢⎢⎢⎢⎢⎢⎢⎣

aB+

aF−

aF+

aB−

⎤⎥⎥⎥⎥⎥⎥⎥⎦. (32)

We may now write the time derivative of a as a matrix product:

a =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

a1/2B+

a1/2F−

a1/2F+

a1/2B−

⎤⎥⎥⎥⎥⎥⎥⎥⎦

T ⎛⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 −1

⎞⎟⎟⎟⎟⎟⎟⎟⎠

⎡⎢⎢⎢⎢⎢⎢⎢⎣

a1/2B+

a1/2F−

a1/2F+

a1/2B−

⎤⎥⎥⎥⎥⎥⎥⎥⎦= ψTσzψ (33)

where σz is the Dirac matrix for the z-component of spin. The temporal and spatial

derivatives have the same sign for backward-propagating waves and opposite signs for

forward-propagating waves. The spatial derivative is therefore given by:

c∂za = −

⎡⎢⎢⎢⎢⎢⎢⎢⎣

a1/2B+

a1/2F−

a1/2F+

a1/2B−

⎤⎥⎥⎥⎥⎥⎥⎥⎦

T ⎛⎜⎜⎜⎜⎜⎜⎜⎝

−1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 1

⎞⎟⎟⎟⎟⎟⎟⎟⎠

⎡⎢⎢⎢⎢⎢⎢⎢⎣

a1/2B+

a1/2F−

a1/2F+

a1/2B−

⎤⎥⎥⎥⎥⎥⎥⎥⎦= −ψTγ5ψ (34)

where γ5 is the Dirac matrix for chirality in the chiral representation. If the amplitude

(a) represents rotation angle, then positive and negative chirality (∂za) are analogous to

left- and right-handed threads on a screw. Wave velocity (v) is obtained by combining

the two matrices used above:

vψ = c

⎛⎜⎜⎜⎜⎜⎜⎜⎝

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 −1

⎞⎟⎟⎟⎟⎟⎟⎟⎠

⎡⎢⎢⎢⎢⎢⎢⎢⎣

a1/2B+

a1/2F−

a1/2F+

a1/2B−

⎤⎥⎥⎥⎥⎥⎥⎥⎦= cγ5σzψ . (35)

The one-dimensional scalar wave equation may be written in the form:

∂t[ψTσzψ] + c∂z[ψ

Tγ5ψ] = 0 . (36)


Other matrices may be inserted between the wave functions, resulting in the following

corresponding expressions, each of which equals zero for the wave solutions:

∂t[ψTσzψ] + c∂z[ψ

Tγ5ψ] = ∂2t a− c2∂2za; (37)

∂t[ψTψ] + c∂z[ψ

Tγ5σzψ] = ∂t|∂taF | + ∂t|∂taB| + c2[∂z|∂zaF | − ∂z|∂zaB|]; (38)

∂t[ψTγ5σzψ] + c∂z[ψ

Tψ] = c[∂t|∂zaF | − ∂t|∂zaB| + ∂t|∂zaF | + ∂t|∂zaB|]; (39)

∂t[ψTγ5ψ] + c∂z[ψ

Tσzψ] = ∂t[−c∂za] + c∂z[∂ta] . (40)

Generalization to three dimensions is straightforward. The 3-D generalization of

∂z∂zaz utilizes geometric algebra:

∇(∇a) = ∇(∇ · a + i∇ × a)

= ∇(∇ · a) − ∇ × (∇ × a) . (41)

The one-dimensional Dirac wave functions are real with positive-definite components.

Generalization to three dimensions requires complex components and additional matrices.

The one-dimensional wave equation has the bispinor form:

ψT{σz∂tψ + cγ5∂zψ

}+ Transpose = 0 . (42)

We can separate a common factor of ψ†σz:

ψ†σz{∂tψ + cγ5σz∂zψ

}+ Transpose = 0 . (43)

For arbitrary components and derivatives this becomes:

ψ†σj{∂tψ + cγ5σi∂iψ

}+ h.c. = 0 (44)

where ”h.c.” stands for ”hermitian conjugate”.

The matrices σj are the Dirac spin matrices:

σx =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 0

1 0 0 0

0 0 0 1

0 0 1 0

⎞⎟⎟⎟⎟⎟⎟⎟⎠

; (45)

σy =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

0 −i 0 0

i 0 0 0

0 0 0 −i

0 0 i 0

⎞⎟⎟⎟⎟⎟⎟⎟⎠

; (46)


σz =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 −1

⎞⎟⎟⎟⎟⎟⎟⎟⎠. (47)

The matrices cγ5σj are the Dirac velocity matrices. The matrix γ5 was defined in Eq.

(34).

Expanding the spatial derivative term in Eq. (44) yields the 3-D generalization of the

wave equation (37):

0 = ∂t[ψ†σψ

]+ c∇ [

ψ†γ5ψ] − ic

{[∇ψ†] × γ5σψ + ψ†γ5σ × ∇ψ}= ∂2t a − c2∇(∇ · a) + c2∇ × (∇ × a) . (48)

Eq. (40) similarly generalizes to:

0 = ∂t[ψ†γ5ψ] + c∇ · [ψ†σψ] = ∂t[−c∇ · a] + c∇ · [∂ta] ; (49)

Eqs. (38) and (39) are not easily generalized to vector equations, but in terms of

bispinors they become:

0 = ∂t[ψ†ψ] + c∇ · [ψ†γ5σψ] (50)

0 = ∂t[ψ†γ5σψ] + c∇[ψ†ψ] . (51)

The foregoing analysis results in the following identifications between vectors and

bispinors:

∂ta ≡ [ψ†σψ

]; (52)

[∇ · a] ≡ − [ψ†γ5ψ

]; (53)

c2 {∇ × ∇ × a} ≡ −ic{[∇ψ†] × γ5σψ + ψ†γ5σ × ∇ψ} ; (54)

0 = ic∇ · {[∇ψ†] × γ5σψ + ψ†γ5σ× ∇ψ} . (55)

These identifications provide seven constraints on the eight free parameters of the

complex Dirac bispinor: three for the first, one for the second, two for the third (since

a curl has only two independent components), and one for the fourth. There is also an

arbitrary overall phase factor.

The last identification simply states that the divergence of a curl is zero. This con-

dition is necessary for consistency. Note that if we attempt to define the curl as a single

term (i.e. c∇×a = ψ†γ5σψ, resulting in a ”-” sign in Eqs. [54] and [55]), then it becomes

impossible to write a Dirac equation for ∂tψ because there is no common factor of ψ†σjas in Eq. (44).

We now apply a similar interpretation of the Dirac wave function in terms of spin

density (correcting a sign error in Ref. [11]):


S = ∂tQ ≡ 1

2

[ψ†σψ

]; (56)

c∇ · Q ≡ −1

2

[ψ†γ5ψ

]; (57)

c2 {∇ × ∇ × Q} ≡ − ic

2

{[∇ψ†] × γ5σψ + ψ†γ5σ × ∇ψ} ; (58)

0 =ic

2∇ · {[∇ψ†] × γ5σψ + ψ†γ5σ× ∇ψ} . (59)

In terms of bispinors, the rotational wave equation (26) is

0 = ∂t[ψ†σψ

]+ c∇ [

ψ†γ5ψ] − ic

{[∇ψ†] × γ5σψ + ψ†γ5σ × ∇ψ}+u · ∇ [

ψ†σψ] − w × [

ψ†σψ]. (60)

For comparison, the Dirac equation for a free electron (with M = mec2/h) is

∂tψ + cγ5σ · ∇ψ + iMγ0ψ = 0 . (61)

Multiplying this equation by ψ†σj on the left and adding the hermitian conjugate

yields:

∂t[ψ†σjψ

]+c∂j

[ψ†γ5ψ

]+ icεijk

{[∂iψ

†] γ5σkψ − ψ†γ5σk∂iψ}= 0 . (62)

This is equivalent to:

∂t[ψ†σψ

]+ c∇ [

ψ†γ5ψ] − ic

{[∇ψ†] × γ5σψ + ψ†γ5σ × ∇ψ} = 0 . (63)

Using our definitions, this is just the wave equation. It differs from our equation for the

evolution of spin angular momentum density in an elastic solid only by the two nonlinear

terms. This is interesting because many researchers have attempted to obtain particle-like

solutions from the Dirac equation by adding nonlinear terms to Dirac’s original equation.

[14–21]

It is also instructive to write the equation for elastic waves in Dirac form. Expanding

the derivatives yields

ψ†σj

[∂tψ + cγ5σ · ∇ψ + u · ∇ψ + w · iσ

2ψ

]+ h.c. = 0 . (64)

The Hermitian conjugate wave function ψ† may be regarded as an independent variable

(it may be combined with the original wave function to separate the real and imaginary

parts). Validity for arbitrary ψ† requires the terms in brackets to sum to zero. This yields

the equation

∂tψ + cγ5σ · ∇ψ + u · ∇ψ + iw · σ2ψ + iχψ = 0 (65)

where χ may be any operator with the property

Re{ψ†σjiχψ

}= 0 . (66)

Dirac’s mass term is an example. However, we will set χ to zero, assuming that mass

(along with any missing nonlinear terms in the Dirac equation) is derived from the con-

vection and rotation terms.


4.3 Lagrangian and Hamiltonian

Now we construct a Lagrange density L . Lagrange’s equation of motion for a field

variable ψ is

∂t∂L

∂ [∂tψ]+

∑j

∂j∂L

∂ [∂jψ]− ∂L

∂ψ= 0 . (67)

A similar equation holds with ψ† replacing ψ. It is possible to construct a Lagrangian

with no derivatives of ψ†, in which case the equation of motion is simply ∂L /∂ψ† = 0.

The nonlinear terms contain two factors of ψ†. In the rotation term, these may be

interchanged using integration by parts, so a factor of 1/2 is required in the Lagrangian.

Integration by parts of the convection term yields a term containing ∇ · u , which is zero.

Therefore the factor of ψ† in u does not contribute to the equation of evolution. The

Lagrangian is therefore

L = iψ†∂tψ + ψ†cγ5σ · i∇ψ + u · ψ†i∇ψ − 1

2w · ψ†σ

2ψ . (68)

This Lagrangian is not real, but real-valued quantities may be regarded as the real part

of complex expressions.

The conjugate momentum to the field ψ is pψ:

pψ =∂L

∂ [∂tψ]= iψ† . (69)

The Hamiltonian is

H = pψ∂tψ − L = −ψ†cγ5σ · i∇ψ − u · ψ†i∇ψ +1

2w · ψ†σ

2ψ . (70)

We recognize the last term in the Hamiltonian as the kinetic energy density K =

w · S/2. The first term involves only spatial derivatives, so we propose that it represents

elastic potential energy. The second term represents convection of gradients by the motion

of the medium. Since shear waves are transverse, this motion is perpendicular to the wave

velocity (determined by the matrix γ5σ). Therefore this term can be non-zero only if

the wave velocity is not parallel to −ψ†i∇ψ (which we shall see is the wave momentum).

We hypothesize that this term integrates to zero, not contributing to the total energy. A

prior attempt by this author to incorporate this term into the kinetic energy blurred the

distinction between wave propagation and motion of the solid medium. [12]

The Hamiltonian operator is defined by i∂tψ = Hψ, with

H = −cγ5σ · i∇ψ − u · i∇ψ +1

2w · σ

2ψ . (71)

4.4 Dynamical Variables

The Hamiltonian is a special case (T 00 ) of the energy-momentum tensor:

T μν =

∂L

∂ [∂μψ]∂νψ − L δμν . (72)


Notice that in the Lagrangian, the kinetic energy term is negative. Therefore the

conjugate momenta computed from the Lagrangian will also have the opposite sign of

physical quantities. The dynamical (or wave) momentum density pi is

pi = −T 0i = − ∂L

∂ [∂tψ]∂iψ = −ψ†i∂iψ . (73)

The wave angular momentum density is likewise

L = − ∂L

∂ [∂tψ]∂ϕψ = −iψ†∂ϕψ = −iψ†

∂ri∂ϕ

∂iψ = −r × ψ†i∇ψ . (74)

This expression assumes a particular origin for the axis of rotation of the angle ϕ, in

contrast to the coordinate-independent spin angular momentum. One could attempt

to express orbital angular momentum density as the field whose curl is twice the wave

momentum density, but we will not pursue that here.

For total momentum and angular momentum, we must combine the wave and medium

contributions (p and q, respectively):

P = p + q = −ψ†i∇ψ +1

2∇ × ψ†

σ

2ψ ; (75)

J = L + S = −r × ψ†i∇ψ + ψ†σ

2ψ . (76)

The expression for total momentum density was previously obtained by Ohanian using a

symmetrized energy-momentum tensor. [1]

Interestingly, we could have obtained the results of Eqs. (75) and (76) by treating

either velocity u or vorticity w as an independent variable in the Lagrangian above.

Rewriting the kinetic energy density as ρu2/2, the negative of the conjugate momentum

would be

P = −∂L∂u

= −ψ†i∇ψ + ρu = −ψ†i∇ψ +1

2∇ × ψ†

σ

2ψ . (77)

And if u includes a rotational component r × w , then the negative of the conjugate

angular momentum would be (treating S as a function of w )

J = −∂L∂w

= −r × ψ†i∇ψ + ψ†σ

2ψ . (78)

These results are identical to Eqs. (75) and (76).

When interpreting angular derivatives, the reader should be cautious to distinguish

between active and passive rotations. The differential ∂ϕψ = r × ∇ψ refers to passive

rotation, or rotation of the point of evaluation of the function. Active rotations rotate

the function along with the evaluation point, as described by the operator Uϕ satisfying

the equations: [22]

∂ϕUϕψ = −i(L + S)ψ = −r × ∇ψ − iσ

2ψ ; (79)

Uϕψ = exp {−i(L + S) · ϕ}ψ . (80)

Thus we see that classical spin density applied to elastic waves yields equations and

operators very similar to relativistic quantum mechanics.


5 Discussion

We have shown that classical spin density, which was originally derived as an interpreta-

tion of quantum mechanical spin density, is consistent with the usual classical description

of arbitrary rotations. Spin density is therefore an important concept for a unified under-

standing of both classical and quantum physics. In particular, rotational elastic waves

share properties of both classical and quantum systems.

Elastic waves have played an important role in the study not only of solids, but also

of light and matter. Physicists have attempted to describe the universe as a solid since

the 18th century, when Thomas Young explained polarization of light as analogous to

shear waves. Young’s idea was further developed by the likes of Fresnel, Navier, Cauchy,

Rayleigh, Heaviside, Green, Thomson (Lord Kelvin), Riemann, Boussinesq, and many

others. [3] An elastic solid model was the basis for MacCullagh’s original derivation of an

equation for light. [4] Maxwell developed the equations for electromagnetism by modeling

a lattice of elastic cells, and questioned, ”... what if these molecules, indestructible as

they are, turn out to be not substances themselves, but mere affections of some other

substance?” [23]

Unfortunately, introductory physics textbooks typically dismiss the idea of a universal

wave medium, saying it was disproven by Michelson and Morley. That is of course

nonsense, as Lorentz-invariant equations such as MacCullagh’s and Maxwell’s are quite

commonly derived for a medium carrying a wave. What aether-drift experiments proved

is that Earth does not move through space like a rock through water (or through oobleck,

the corn starch solution that behaves like a solid for rapid vibrations but like a liquid for

slower processes).

We now know that matter propagates through the vacuum in a wave-like manner.

Equations describing these waves, such as the Dirac equation, may be interpreted as de-

scribing dynamics as well as probabilities. [24,25] Although the properties of matter can

be described without reference to an ”aether”, such models can still be useful for illu-

minating relationships between physical quantities. Dirac himself held this view, writing

”It is necessary to set up an action principle and to get a Hamiltonian formulation of the

equations suitable for quantization purposes, and for this the aether velocity is required.”

[26] Recently, there have been several investigations of solid crystalline models of the

vacuum. [27–30]

Given the similarity between classical and quantum equations, it is interesting to

ponder what the universe would be like if the vacuum were an elastic solid. Elementary

particles would have to be standing or particle-like waves, subject to the wave uncertainty

principle. Special relativity would be a consequence of the Lorentz invariance of the wave

solutions, and not a property of the space-time in which the waves propagate. [31] The

spatial reflection of any solution would also be a solution, so every particle would have an

anti-particle that behaves like its mirror image. [32] Measurements would have to change

the standing wave configuration from one stable state to another, implying quantization of

measurement. Tension induced by twisting of the elastic medium would increase density


and decrease wave speed, similar to the way the presence of matter decreases wave speed

in general relativity. Hence gravity could be described by an index of refraction. [33–35]

In short, an elastic solid universe would be similar in many ways to the one we live

in. And although some properties of matter may be impossible to explain using such a

classical model, spin angular momentum is not one of them.

6 Conclusions

Classical spin angular momentum density is the field whose curl is equal to twice the

momentum density for incompressible (rotational) motion. Compared with the usual

classical definition of angular momentum density as r × p , spin density is a local and

complete description of rotational motion that yields the same total angular momentum

and kinetic energy. A rotating cylinder constitutes a simple example for the application of

spin density. Using spin density to describe elastic waves yields equations similar to those

of fermions with identical operators for energy, momentum, and angular momentum.

References

[1] Hans C. Ohanian, ”What is spin?” Am. J. Phys. 54 (6), 500-505 (1986).

[2] Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1958), p. 306.

[3] Edmund Whittaker, A History of the Theories of Aether and Electricity, Vol. I(Philosophical Library, New York,1951).

[4] James MacCullagh, ”An essay towards a dynamical theory of crystalline reflexionand refraction”, Trans. Roy. Irish Acad. xxi , 17-50 (1846).

[5] Oliver Heaviside, ”The Rotational Ether in its Application to Electromagnetism,”Electrician, xxvi (1891).


[7] Richard P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures onPhysics, Vol. II (Addison-Wesley, Reading,1963), p 38-8.

[8] Hagen Kleinert, Gauge Fields in Condensed Matter, Vol. II Stresses and Defects(World Scientific, New Jersey, 1989) pp. 1240-1264.


[10] Robert A. Close, ”Torsion waves in three dimensions: Quantum Mechanics with aTwist,” Found. Phys. Lett. 15 , 71-83 (2002).

[11] Robert A. Close, ”A classical Dirac bispinor equation,” in Ether Space-time &Cosmology, vol. 3, edited by M. C. Duffy and J. Levy ( Apeiron, Montreal, 2009),pp. 49-73.

[12] Robert A. Close, ”Exact description of rotational waves in an elastic solid,” Adv.Appl. Clifford Algebras 21 , 273-281 (2011).


[13] Richard P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures onPhysics, Vol. I (Addison-Wesley, Reading,1963), p. 20-6.

[14] Werner Heisenberg, Introduction to the Unified Field Theory of Elementary Particles(Interscience Publishers, 1966).

[15] Hiroshi Yamamoto, ”Spinor soliton as an elementary particle,” Progress of TheoreticalPhysics, 58 (3), 1014-1023 (1977).

[16] A. F. Ranada, ”Classical nonlinear Dirac field models of extended particles,” inQuantum Theory, Groups, Fields, and Particles, edited by A. O. Barut (Reidel,Amsterdam, 1983) pp. 271-288.

[17] Wilhelm Fushchych and Renat Zhdanov, Symmetries and Exact Solutions ofNonlinear Dirac Equations, (Mathematical Ukraina, Kyiv, 1997).

[18] Ying-Qiu Gu, ”Some properties of the spinor soliton,” Adv. Applied Clifford Algebras8(1),17-29 (1998).

[19] C. Sean Bohun and Fred I. Cooperstock, ”Dirac-Maxwell solitons,” Phys. Rev. A60 (6), 4291-4300 (1999).

[20] Attilio Maccari, ”Nonlinear field equations and solitons as particles,” EJTP 3(10),39-88 (2006).

[21] Jian Xu, Sihong Shao, and Huazhong Tang, ”Numerical methods for nonlinear Diracequation,” J. Comp. Phys. 245 ,131-149 (2013).

[22] Leonard I. Schiff, Quantum Mechanics, Third Edition (McGraw-Hill, New York,1968) p. 198.

[23] James C. Maxwell, ”Introductory lecture on experimental physics”, in The ScientificPapers of James Clerk Maxwell, vol. II, edited by W.D. Niven (Dover, New York,1965), pp. 241-255.

[24] Takehiko Takabayasi, ”Relativistic Hydrodynamics of the Dirac Matter,” Suppl. Prog.Theor. Phys. 4(1):1-80 (1957).

[25] David Hestenes, ”Local observables in the Dirac theory,” J. Math. Phys. 14 (7):893-905 (1973).

[26] Paul A. M. Dirac, (1952). ”Is There an Aether?” Nature 169 , p. 702.

[27] Hagen Kleinert, ”Gravity as theory of defects in a crystal with only second-gradienteasticity,” Ann. Phys. 44 , 117 (1987).

[28] Hagen Kleinert and Jan Zaanen, ”World nematic crystal model of gravity explainingthe absence of torsion,” Phys. Lett. A 324 , 361 (2004).

[29] Marek Danielewsky, ”The Planck - Kleinert crystal,” Z. Naturforsch. 62a , 564 568(2007).

[30] Ilja Schmelzer, ”A condensed matter Interpretation of SM fermions and gauge fields,”Found. Phys. 39 (1), 73-107 (2009).

[31] Robert A. Close, The Wave Basis of Special Relativity (Verum Versa, Portland, OR,2014).

[32] Robert A. Close, ”The mirror symmetry of matter and antimatter,” Adv. Appl.Clifford Algebras 21 , 283-295 (2011).


[33] Fernando de Felice, ”On the gravitational field acting as an optical medium,” Gen.Relat. Gravit. 2, 347-357 (1971).

[34] Philip C. Peters, ”Index of refraction for scalar, electromagnetic, and gravitationalwaves in weak gravitational fields,” Phys. Rev. D 9, 2207-2218 (1974).

[35] James C. Evans et al., ”Matter waves in a gravitational field: An index of refractionfor massive particles in general relativity,” Am. J. Phys. 69 , 1103-1110 (2001).


An Introduction to Strict Quantization

J. M. Velhinho∗

Faculdade de Ciencias, Universidade da Beira InteriorR. Marques D’Avila e Bolama, 6201-001 Covilha, Portugal

Received 10 April 2015, Accepted 10 June 2015, Published 25 August 2015

Abstract: We present a short review of the approach to quantization known as strict

(deformation) quantization, which can be seen as a generalization of the Weyl-Moyal

quantization. We include examples and comments on the process of quantization.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Quantization; Strict Quantization; Deformation; generalized Weyl-Moyal

quantization

PACS (2010): 03.65.-w; 03.70.+k; 11.10.-z; 45.50.Dd; 03.50.-z;02.40.Gh; 02.10.-v; 02.40.Ky

1 Introduction

This brief review is my modest contribution to a field and a set of ideias that strongly

influenced my view on the process of quantization. I hope that this introduction to the

subject of strict quantization and the examples in it can be of help, and motivate both

theoretical and mathematical physicists to this beautiful and highly developed field. My

personal impression is that there is still a whole lot to explore in this area, in particular

concerning extensions of the formalism to the realm of infinite dimensional degrees of

freedom. In fact, it seems clear that quantum field theory could benefit greatly from

a strict quantization approach. Concrete steps have been taken in this direction (see

e.g. [1–3] and also [4] for a recent application in quantum gravity), in what is for sure a

rather promising area in mathematical physics.

This brief survey focus on finite dimensional examples exclusively, and is mostly in-

spired by (parts of) two beautiful books, namely Folland’s Harmonic Analysis on Phase

Space [5] and Landsman’s synthesisMathematical Topics Between Classical and Quantum

Mechanics [6]. Another major source of inspiration was Varilly’s book [7]. Concerning

broader fields, such as Poisson geometry, groupoids or deformation of algebra structures,



we follow also [8, 9], as well as [10]. An effort is made, hopefully sucessful, to keep the

discussion short and as pedagogical as possible, just enough to give the reader a grasp of

what strict quantization is all about and to motivate him/her to this field.

This work is organized as follows. In the remaining of the present section we give a very

brief introduction to the problem of quantization. In section 2 we review the formalism

of classical mechanics and prepare for the strict quantization approach. In section 3

we review the Weyl-Moyal quantization, which is the prototype of a strict deformation

quantization. In section 4 we present what is essentially Landsman’s definition of strict

quantization. In the following sections we present methods for the construction of strict

quantizations which, in one way or another, generalize the Weyl-Moyal quantization

process. First, in section 5 we review the notions of smooth groupoids and associated

convolution algebras. Then Connes’ tangent groupoid and Landsman’s strict quantization

of cotangent bundles of Riemannian manifolds are discussed, in section 6. Finally, we

discuss the interesting case of the 2-torus, treated both by geometric quantization related

and by strict quantization methods.

The formalism of classical mechanics is based on a phase space (the states) and on

functions on that space (the observables), on the set of which two operations are defined:

the product of functions and the Poisson bracket. Quantum mechanics presents a similar

structure, keeping the duality states-observables. In fact, in quantum mechanics one deals

with operators (observables) in a Hilbert space (the states). The operation that combines

observables is now the operator product. One can extend further the analogy with the

classical structure, by means of two new operations, obtained by symmetrization and

antisymetrization of the operator product. The anticommutator ◦, or Jordan product, is

given by

A ◦B :=1

2(AB +BA) . (1)

Following Dirac [11], one defines also the quantum Lie bracket, or quantum Poisson

bracket [ , ]h by

[A,B]h := (AB −BA)/ih , (2)

where h = h/2π, with h being Planck’s constant. Of course, we have

AB = A ◦B +ih

2[A,B]h . (3)

It is tempting to interpret the Jordan product as the quantum equivalent of the product

of functions, together with Dirac’s quantum condition stating that the quantum bracket

is to be seen as the quantum correspondent of the Poisson bracket. In broad terms,

the transition from a classical description to a quantum description of a given system

- quantization - is obtained by means of a map Q from functions on phase space to

operators on a Hilbert space. In this process, one aims at preserving the relations between

observables as much as possible. It would therefore be natural to search for maps Q such

that

Q(fg) = Q(f) ◦ Q(g) , (4)


Q({f, g}) = [Q(f),Q(g)]h . (5)

However, it is well known that such maps cannot be found. Even Dirac’s condition (5)

(supplemented with natural requirements, see below) is impossible to fullfil exactly, except

for a restricted set of observables (and typically on cotangent spaces only). Conditions (4)

and (5) should therefore be replaced by weaker ones, still keeping the purpose of rigorously

implementing the physical requirement embodied by those relations. The straightforward

canonical quantization approach, launched by Dirac, consists in totally relaxing condition

(4) and implementing condition (5) exactly on a small, but sufficiently large, subalgebra

of obervables. Further crucial requirements must be added in this approach, such as

irreducibility (see e.g. [12] for a general discussion of canonical quantization).

The strict quantization approach [6, 13] proposes an asymptotic implementation of

conditions (4) and (5). It involves not a single map Q, but a family of maps, labeled by

a parameter which we will call h, although this is now a free parameter and no longer

the value of the physical constant. One then considers a family of maps Qh, satisfying

relations of the type

limh→0

(Qh(fg) − Qh(f) ◦ Qh(g)

)= 0 , (6)

limh→0

(Qh({f, g}) − [Qh(f),Qh(g)]h

)= 0 . (7)

Let us admit further that each of the Qh is injective and that, as a function of h, Qh and

Q−1h are continuous. In this case, the operator product induces a family of associative

operations on the functions in phase space:

f ∗h g := Q−1h

(Qh(f)Qh(g)

), (8)

which is a deformation of the standard product, in the sense that

limh→0

f ∗h g = fg , limh→0

(f ∗h g − g ∗h f)/ih = {f, g} . (9)

2 Classical Systems

Given a C∞ manifold P , let C∞(P) denote the associative algebra of C∞ complex func-

tions on P , equipped with the usual product and involution given by complex conjugation,

f �→ f .

Definition 1. A Poisson structure on a manifold P is a bilinear operation { , } on C∞(P)

such that:

(i) (C∞(P), { , }) is a Lie algebra.

(ii) {f, ·} is a derivation on C∞(P) for every f ∈ C∞(P).

(iii) {f , g} = {f, g}.A manifold with a Poisson structure is said to be a Poisson manifold.

Given local coordinates (x1, . . . , xn) in P , the Poisson bracket is given by

{f, g} = Πij(x)∂if∂jg , (10)


where ∂if := ∂f∂xi

and the real quantities Πij(x) constitute the components of a contravari-

ant antisymmetric (real) 2-tensor

Π = Πij(x)∂i ⊗ ∂j . (11)

The latter is said to be a Poisson tensor and satisfies the condition

Πli∂lΠjk +Πlj∂lΠ

ki +Πlk∂lΠij = 0 . (12)

Conversely, an antisymmetric (real) tensor Π that fulfils (12) defines a Poisson structure

{ , } by

{f, g} = Π(df, dg) . (13)

Straightforward examples of Poisson manifolds are provided by symplectic manifolds,

e.g. pairs (P , ω) where ω is a closed nondegenerate 2-form. The Poisson tensor is in this

case also nondegenerate. It is given by the inverse of the symplectic form, or explicitly

by

Π(iX(ω), iY (ω)) = ω(X, Y ) , ∀X, Y ∈ X (P) , (14)

where X (P) denotes the space of vector fields on P . In fact, the property dω = 0

guarantees that Π (14) satisfies the condition (12).

Poisson manifolds (P , { , }) are precisely the mathematical structures taken as mod-

els for finite dimensional classical systems, both in classical mechanics and in classical

statistical mechanics. The physical interpretation of the formalism stands on the notions

of physical state and physical observable and on the relations between them. In the defi-

nition of states and observables below we follow from the start an approach adapted to

the C∗-algebras formalism.

Definition 2. A physical state on a Poisson manifold (P , { , }) is a regular Borel prob-

ability measure on P . The atomic measures, identified with points in P , are said to be

pure states, whereas the remaining ones are called mixed states.

Concerning observables, one typically considers the set of all real C∞ functions on P .

In this respect let us note the following.

There is no inconvenient in considering complex functions, given that real functions

are recovered as the invariant subset under involution. Likewise, when considering

the problem of quantization we will work with complex algebras, taking into account

the so-called reality conditions. So, we will require that real functions belonging to

the classical algebra are mapped under quantization to self-adjoint elements of the

quantum algebra, which is the same as requiring that the process of quantization

maps the involution of functions to the natural involution of operators.

Aiming at the introduction of the C∗-algebra formalism, it is convenient to work

with a subspace of C∞(P) which is also contained on the C∗-algebra C0(P) of con-


tinuous functions vanishing at infinity2. If P is compact the behaviour at infinity is

not a question and one can take the full space C∞(P). In case P is only locally com-

pact, several choices appear possible, for instance C∞0 (P), the set of C∞ functions

vanishing at infinity, or C∞c (P), the set of C∞ functions with compact support.

Without claiming to establish what should be meant by phyical observable, let us adopt

the following definition.

Definition 3. A complete algebra of regular observables on a Poisson manifold (P , { , })is a subspace A(P) ⊂ C∞0 (P) such that:

(i) (A(P), { , }) is a Lie subalgebra of (C∞(P), { , }).(ii) A(P) is a dense ∗-subalgebra (with respect to the supremum norm) of C0(P).

As examples of such complete algebras one can mention C∞0 (P) and C∞c (P), which

are well defined in any circunstance. Nonetheless, other choices may be more convenient,

in a given particular situation. In any case, we will consider chosen a complete algebra

of regular observables A(P) such that C∞c (P) ⊆ A(P) ⊆ C∞0 (P).

Such algebras are complete in the following sense: condition (ii) in definition 3 guar-

antees that A(P) separates points in P , i.e. given x �= y em P , there exists f ∈ A(P)

such that f(x) �= f(y).

Let us clarify immediately the folowing. Although there are enough functions in

a complete algebra of regular observables to define local coordinates, by no means

such an algebra contains all observables of physical interest, if P is noncompact.

The obvious example is provided by the standard global coordinate functions in

P = T∗R. Turning to quantization, the strategy will be to start with a convenient

algebra of regular observables, seeking afterwards to extend the quantization map

to other observables of interest, if necessary.

The quantities with direct physical correspondence are the (real) values of the pairing

(μ, f) �→∫dμ(x)f(x) , (15)

for real functions f and states μ. Therefore, it is usually said that the description of

the system in terms of states is “dual” to the description in terms of observables. In the

context of definitions 2 and 3, this duality has a precise sense: the physical states show

up as a subset of the unit ball in the dual of A(P) (let us remind that A(P) is dense in

C0(P), and therefore the dual of A(P) coincides with the dual of C0(P)). To be precise,

let us introduce the following:

Definition 4. A linear functional ϕ on a ∗-algebra A is said to be positive if ϕ(a∗a) ≥0 ∀a ∈ A (we write ϕ ≥ 0 to denote that ϕ is positive). A positive linear functional on a

C∗-algebra is called a state (of the algebra) if ‖ϕ‖ = 1.

2 For locally compact X, C0(X) is the subset of those f ∈ C(X) with the property that for any ε > 0,

there is a compact set Kε ⊂ X such that |f(x)| < ε if x �∈ Kε. The C∗-norm of C0(X) is the supremum

norm.


Note that a positive linear functional on a C∗-algebra is necessarily continuous (see

e.g. [14]).

Proposition 1. Let X be a locally compact Hausdorff space. Then the set of states of

the algebra C0(X) can be identified with the set of regular Borel probability measures on

X.

The proof of this result follows from the Riez-Markov theorem for locally compact

spaces (see [15]), which identifies the dual of C0(X) with the set of finite complex regular

Borel measures in X, by means of the bijective correspondence

ϕ �→ μϕ : ϕ(f) =

∫fdμϕ , ∀f ∈ C0(X). (16)

The remaining nontrivial part of the proof consists in a typical functional analysis argu-

ment, showing that ‖ϕ‖ = μϕ(X), which we will not present here.

The physical states of the system (P , { , }) can therefore be seen as the states of

the algebra C0(P). The pure physical states (atomic measures) admit also important

characterizations in terms of the algebra C0(P).

Proposition 2. Given a locally compact Hausdorff space X, there is bijective correspon-

dence between the following sets:

(i) The set of atomic measures on X.

(ii) The set of nonnull linear functionals ϕ on the algebra C0(X) such that ϕ(ab) =

ϕ(a)ϕ(b) ∀a, b ∈ C0(X).

(iii) The set of irreducible representations of the algebra C0(X).

The correspondence between (i) and (ii) is well known and constitutes part of Gelfand’s

theorem. The correspondence with (iii) is easy to check: given that the algebra C0(X)

is commutative, its irreducible representations have dimension 1, and are therefore C∗-algebras morphisms, ϕ : C0(X) → C, i.e., belong to the set defined by (ii). On the other

hand, it is clear that each atomic measure gives rise to such a dimension 1 representation.

One can further show that the above sets defined by (i), (ii) and (iii) are equivalent

to the set of pure states of the algebra C0(X), with the following definition.

Definition 5. A state ϕ on a C∗-algebra is said to be pure if the conditions ϕ ≥ χ ≥ 0

can only be fulfiled with χ = tϕ, t ∈ [0, 1].


3 Weyl-Moyal Quantization

3.1 Quantization Map

Let us consider the phase space P = T ∗R, with local coordinates (q, p), and its canonical

symplectic structure, defined by the form

ω = dq ∧ dp . (17)

The associated Poisson tensor is:

Π = ∂q ⊗ ∂p − ∂p ⊗ ∂q , (18)

and therefore

{f, g} = ∂qf∂pg − ∂pf∂qg . (19)

We choose as algebra of observables A(P) the space S(T ∗R) ∼= S(R2) of Schwartz func-

tions on T ∗R ∼= R2. In this context, the Weyl-Moyal (W-M) quantization consists of

a family of linear maps Qh, h ∈ R+, from the Schwartz space to operators in L2(R).

Explicitly:

S(T ∗R) � f(q, p) �→ Qh(f) : (20)(Qh(f)ψ

)(q) =

∫dp

2πhe

iph(q−q′)f

(q + q′2

, p)ψ(q′)dq′, ψ ∈ L2(R). (21)

We start by showing that the maps Qh have image on the subset of Hilbert-Schmidt

operators. The operators Qh(f), f ∈ S(T ∗R), are in fact integral operators, of kernel

Kfh (q, q′) =

∫dp

2πhe

iph(q−q′)f

(q + q′2

, p), (22)

which is clearly well defined and belongs to S(R2). We get∫dqdq′|Kf

h (q, q′)|2 =

∫dqdq′

∫dp

2πhe

iph(q′−q)f

(q + q′2

, p)

·

·∫

dp′2πh

eip′h(q−q′)f

(q + q′2

, p′). (23)

With the new variables

v :=q + q′2

, w :=q − q′h

(24)

it follows that ∫dqdq′|Kf

h (q, q′)|2 =

∫dp

2πhdp′dvdw

2πeiw(p′−p)f(v, p)f(v, p′)

=1

2πh

∫dvdp|f(v, p)|2 < ∞ , (25)

for every f ∈ S(T ∗R), which shows that Qh(f) is an Hilbert-Schmidt operator.

The smallest C∗-subalgebra of B(L2(R)) that contains the imagem of Qh coincides

with the closure in the uniform topology of the set of Hilbert-Schmidt operators. This is


the space K(L2(R)) of compact operators. Let us adopt K(L2(R)) as common range of

the maps Qh.

It is straightforward to check that the quantization satisfies the reality conditions

Qh(f) = Q+h (f) , ∀h ∈ R

+, ∀f ∈ S(T ∗R) . (26)

In fact, it is obvious that the kernel Kfh (22) satisfies

K fh (q, q′) = Kf

h (q′, q) , (27)

which is equivalent to (26).

One can also show (see [6]) that the W-M quantization (21) satisfies the following

conditions:

(i) limh→0

‖Qh({f, g}) − [Qh(f),Qh(g)]h‖ = 0 . (28)

(ii) limh→0

‖Qh(fg) − Qh(f) ◦ Qh(g)‖ = 0 . (29)

(iii) The maps h �→ ‖Qh(f)‖ are continuous in R+, ∀f . (30)

(iv) limh→0

‖Qh(f)‖ = ‖f‖ (= sup|f |) . (31)

Condition (i) is the form in which Dirac’s quantization condition is implemented in this

formalism: the classical Lie structure is not exactly preserved at the quantum level, but it

is violated only by operators that tend to zero with h. Condition (ii) plays the same role

with respect to the multiplicative structure, ensuring that the algebraic relations between

observables are recovered in the limit h → 0. Conditions (iii) and (iv) establish the

continuity of the process and provide, together with (ii), some control over the spectrum

of the quantum operators. In particular, conditions (iii) and (iv) establish precisely the

continuity (near h = 0) of the spectral radius of the quantum operators.

Let us now clarify the relation between the W-M quantization and the usual canonical

quantization of the so-called Heisenberg algebra, i.e. the Lie algebra generated by the

coordinate functions q and p. In the Dirac quantization, the observables q and p are

mapped to operators q and p in L2(R), such that(qψ

)(q) = qψ(q) (32)(

pψ)(q) = −ihdψ

dq(q). (33)

These operators are unbounded, and therefore cannot be defined for all ψ ∈ L2(R). It is

standard procedure to restrict attention to the Schwartz subspace S(R) ⊂ L2(R), which

is dense, belongs to the domain of both q and p and furthermore remains invariant under

the action of both operators. If the action of Qh(f) (21) is restricted to vectors ψ ∈ S(R),we see immediately that Qh(f) remains well defined for a much larger class of functions

in T ∗R (see [5] for a detailed discussion). In particular, Qh(q) and Qh(p) are well defined

in S(R) and coincide with the operators q and p above. Thus, the W-M quantization

is an extension of the canonical quantization q and p of coordinate functions, to a large

class of observables f(q, p).


3.2 Positivity in the Weyl-Moyal Quantization

In this section we address the question of positivity in the Weyl-Moyal quantization.

Let us recall that a self-adjoint operator A is said to be positive if its expectation

values are nonnegative, i.e. if 〈ψ,Aψ〉 ≥ 0, ∀ψ. An equivalent condition is that the

spectrum of A is a subset of R+0 .

Given the physical interpretation of observables, it is clearly desirable that, under a

given quantization, positive classical observables (i.e. those that take only nonnegative

values) are mapped to operators which are themselves positive. We show next that such

condition is not fully satisfied in the W-M quantization.3 However, Heisenberg’s uncer-

tainty relation helps in clarifying the situation, showing why a weaker form of positivity

is physically acceptable.

For definitiness, let us consider the positive observables given by gaussian functions in

phase space, whose quantization is particularly simple. Let then fx0α,β denote the gaussian

function:

fx0α,β(q, p) = 2 e−

12

(q−q0)2

α e−12

(p−p0)2

β , (34)

with arbitrary x0 = (q0, p0) and α > 0, β > 0. The kernel of the associated operador

Qh

(fx0α,β

)(21) is easily found to be:

Kfx0α,β

h (q, q′) = χ(q)χ(q′)e− 18α

( 4αβ

h2−1)(q−q′)2 , (35)

where χ is an element of S(R) given by

χ(q) =

(2β

πh2

)1/4

e−14α

(q−q0)2e−ip0h

(q−q0). (36)

To address the question of positivity of Qh

(fx0α,β

)let us then consider the expectation

values 〈ψ,Qh

(fx0α,β

)ψ〉, ψ ∈ L2(R). We get

〈ψ,Qh

(fx0α,β

)ψ〉 = (37)

=

∫dqdq′ χ(q)ψ(q)e− 1

8α( 4αβ

h2−1)(q−q′)2χ(q′)ψ(q′) .

Let us prove that 〈ψ,Qh

(fx0α,β

)ψ〉 ≥ 0 ∀ψ if and only if αβ ≥ (

h/2)2. The conclusion is

obvious for αβ =(h/2

)2. For αβ >

(h/2

)2the conclusion follows from the fact that, in

this case, the gaussian function in the integrand can be written as the Fourier transform

of a gaussian measure. It remains to show that positivity fails for αβ <(h/2

)2, i.e. that

one can in this case find ψ ∈ L2(R) such that 〈ψ,Qh

(fx0α,β

)ψ〉 < 0. To prove it, let us

consider the family of vectors

ψσ(q) = (q − q0)e− 1

2σ(q−q0)2e

ip0h

q, (38)

3 See [5] and [6] for a general discussion.


with σ > 0. From (36) and (37) we obtain

〈ψσ,Qh

(fx0α,β

)ψσ〉 = (39)

=(

2βπh2

)1/2 ∫dq1dq2 q1q2 e

− 12( 1σ+ 1

2α)(q21+q22) · e− 1

2Θ(q21+q22−2q1q2),

where

q1 := q − q0, q2 := q′ − q0, Θ :=1

4α

(4αβ

h2− 1

). (40)

Let us choose σ such that1

σ+

1

2α+ 2Θ �= 0 . (41)

One can then write (39) as a gaussian integral in R2:

〈ψσ,Qh

(fx0α,β

)ψσ〉 =

(2β

πh2

)1/42π

D

∫dq1dq2(2π/D)

q1q2 e− 1

2(q1 q2)C−1

(q1q2

), (42)

where C is the 2 × 2 matrix such that

C−1 :=

⎛⎜⎝ 1

σ+ 1

2α+Θ −Θ

−Θ 1σ+ 1

2α+Θ

⎞⎟⎠ (43)

and

D := detC−1 =( 1σ+

1

2α

)( 1σ+

1

2α+ 2Θ

). (44)

The gaussian integral (42) is now trivial:∫dq1dq2(2π/D)

q1q2 e− 1

2(q1 q2)C−1

(q1q2

)= C12 =

Θ

D. (45)

Putting it all together we finally get

〈ψσ,Qh

(fx0α,β

)ψσ〉 =

(2β

πh2

)1/42π

D2Θ . (46)

Since one can obviously choose σ > 0 compatible with (41) and Θ < 0, it is clear that

the operator Qh

(fx0α,β

), associated with the positive observable fx0

α,β, is not positive for4αβh2 < 1.

It is interesting to analyse this lack of positivity in the W-M quantization in light of

Heisenberg’s uncertainty relations. Note first that the observable fx0α,β (34) with αβ =

(h/2)2 is mapped precisely to the projector (35) onto the quantum state χ (36). This

alows the semiclassical interpretation of fx0α,β with αβ = (h/2)2 as the “characteristic

function of the quantum state centered at x0”. Less peaked gaussian functions, i.e. with

αβ > (h/2)2 and therefore with a slow variation with respect to the quantum scale h/2

are “well quantized”, i.e. they are mapped to positive operators. When it comes down

to gaussian functions that probe regions of the phase space of area less then h/2 (which

is the lower limit of the uncertainty relations), positivity is lost. Note however that, for


αβ < (h/2)2, there is no physical reason to require correspondence between (in particular

the spectrum of) the operator Qh

(fx0α,β

)and the observable fx0

α,β. In fact, precisely because

those functions probe deep inside intrinsically quantum domains in phase space, they are

inaccessible to the classical observer. The operators Qh

(fx0α,β

)in question, if they are true

physical observables, which is questionable, certainly have no classical limit.

4 Strict Quantization

Given a physical system admiting a classical mechanics description, by quantization one

means finding a “corresponding” quantum description. By hypothesis, the system in ques-

tion exhibits, under certain physical conditions determined by the values of the physical

observables involved, a classical mechanical behaviour. The correspondence between the

classical model and the quantum model is established at this limit: the predictions of the

classical model should be a good approximation to the predictions of the “true quantum

theory” at the classical regime, i.e. when the system evolves subjected to classical physical

conditions.

Although reasonably clear from the conceptual point of view, the establishment of

the classical limit of a quantum theory is also a complex problem, given the substantial

differences between the formalisms of the two models, classical and quantum.

In this sense, the emphasis on algebraic aspects constitutes a step towards the formal

approximation of the two models, useful both in the question of the classical limit and in

the inverse problem, that of quantization.

Let us then consider a physical system, with which we associate a Poisson manifold

P and an Hilbert space H. As discussed in section 2, we assume as chosen a complete

algebra of regular classical observables A(P). Given that the quantum and the classical

model describe the same system, there should be a correspondence between functions

f ∈ A(P) and quantum operators, let’s say Q(f), having f a classical limit. One expects

the operators Q(f), f ∈ A(P), to be bounded, and therefore Q should be a map between

A(P) and B(H), required to be linear and real, i.e., Q(f) = Q(f)+. The algebra of

quantum observables is thus assumed to be B(H), which is obviously complete, in the

sense that it acts irreducibly on H.

In this context, the observer deals with two algebras of observables: the classical

algebra, fitting phenomena at the classical scale; and the quantum algebra, decribing, in

principle, phenomena at any scale. A viable and useful perspective consists in admitting

the existence of a continuous family of algebras, interpolating between the classical and

the fully quantum domains. This is in broad terms the quantization programme put

forward by Rieffel [13] and Landsman [6], leading to the following definition.

Definition 6. Let P be a Poisson manifold and A(P) a complete algebra of regular ob-

servables on P . Let I ⊂ R be a set containing zero as a limit point. A strict quantization

of A(P), labeled by I, is a family of pairs {(Ah,Qh)}h∈I , where each Ah is a C∗-algebraand each Qh is a linear map Qh : A(P) → Ah, with A0 = C0(P), Q0(f) = f, ∀f ∈ A(P),


such that:

(i) Qh(f) = Q+h (f) , ∀h ∈ I, ∀f ∈ A(P) .

(ii) limh→0 ‖Qh({f, g}) − [Qh(f),Qh(g)]h‖ = 0 .

(iii) limh→0 ‖Qh(fg) − Qh(f) ◦ Qh(g)‖ = 0 .

(iv) limh→0 ‖Qh(f)‖ = ‖f‖ .

Conditions (ii) to (iv) establish the sense in which the classical limit is understood

or, from the point of view of quantization, the conditions that the maps Qh should fulfil

in order to ensure correspondence with the classical theory. Condition (iii) replaces

the so-called von Neumann condition on the preservation of the multiplicative structure.

Condition (ii) is the implementation, in this formalism, of Dirac’s original ideia that the

quantum correspondent of the Poisson bracket is the quantum Lie bracket [ , ]h. Condition

(iv) gives some control over the spectral radius of the operators, ensuring in particular

that the quantum spectrum is not radically different from the classical spectrum.

The perfect example of a strict quantization is the Weyl-Moyal quantization. In this

particular case, the maps Qh : S(R) → K(L2(R)) are bijective and it is therefore possible,

using the inverse maps Q−1h , to transpose the multiplicative structures over to S(R), i.e.,

to define a family of C∗-products, say �h, on S(R), thus obtaining a deformation of the

commutative algebra A(P).

Definition 7. A strict quantization {(Ah,Qh)}h∈I is said to be a strict deformation

quantization if Qh(A0) is a subalgebra and the maps Qh are injective.

5 Smooth Qroupoids

Some interesting deformations of classical algebras, and in particular Landsman’s quan-

tization of the cotangent bundle of a Riemannian manifold, are naturally associated with

groupoid convolution algebras. We review here very briefly the necessary notions, follow-

ing [10] and [8].

A groupoid G can be seen as a generalization of the notion of group. In a group every

element can be combined with each other, i.e. there is a map G×G → G. In a groupoid

ones drops the hypothesis that the map is defined for every pair of elements; it is only

assumed the existence of a binary operation on a subset, say G(2), of G×G.

Definition 8. A groupoid is a (concrete) category G such that all the arrows in the

category have an inverse. The elements of the groupoid are the arrows of the category,

the composition of which defines the binary operation on the groupoid.

We present next some examples of groupoids. In what follows, we identify the set

ObjG of objects of the category with set of identity arrows and denote by G both the

category and the set MorfG of its morphisms, or arrows. We use still the following

notation: Hom [x, y] denotes the set of morphisms from x to y; s (resp. r) denotes the


map that applies g ∈ Hom [x, y] into x (resp. y).

Example 1 . A group is a category with the identity as the only object. The elements of

the group are the arrows of the category, composition being the group operation. Since

all arrows are invertible, a group is a groupoid.

Example 2 . A vector bundle (E, V,M,Π) with fiber V over a manifold M is a category

whose objects are the points of M . The arrows of the category are the elements of V at

each point of M , i.e., Hom [x, y] = ∅ if x �= y and Hom [x, x] = ΠxE ∼= V . Composition

of arrows is defined by vector sum in the fiber, i.e., (x,X)(x, Y ) = (x,X + Y ), x ∈ M ,

X, Y ∈ V .

Example 3 . Given a set X, the product X × X is a groupoid with the following

category structure: the objects of the category are the points of X, the arrows are the

elements of X × X, i.e. Hom [y, x] = {(x, y)}. The composition of arrows is given by

(x, y)(y, z) = (x, z).

Example 4 . Given a set X, a group Γ and a right action α : X×Γ → X, one can obtain

the semidirect product groupoid G = X >�αΓ as follows. The set of objects coincides

with X. The arrows constitute the set X×Γ, with (x, γ) ∈ Hom [αγ(x), x]. Combination

of arrow is given by: (x, γ1)(y, γ2) = (x, γ1γ2), if αγ1(x) = y. The inverse of the arrow

(x, γ) is (αγ(x), γ−1).

Let us now consider the introduction of a compatible differential structure on a

groupoid [10].

Definition 9. A smooth groupoid is a groupoid G such that:

(i) G, ObjG and the set G(2) ⊂ G × G of pairs of combinable arrows are smooth

manifolds.

(ii) The inclusion ObjG → G, the composition of arrows G(2) → G and the inversion of

arrows are smooth maps.

(iii) The maps r, s : G →ObjG are submersions.

One can now construct convolution algebras and finally a C∗-algebra associated with

a smooth groupoid and a family of measures, as follows.

Let then G be a smooth groupoid. For each x ∈ ObjG, consider the sets Gx :=

∪y∈ObjGHom [y, x] and Gx := ∪y∈ObjGHom [x, y], called r -fibre and s-fibre, respectively.

These fibres inherited a locally compact topology, induced from the manifold structure

of G [10].

For each g ∈ G there is a map Θg : Gs(g) → Gr(g), defined by Θg(g′) = gg′, which

establishes a bijection between Gs(g) and Gr(g) (since every arrow g is invertible).

A family of measures μx on the r -fibres Gx is said to be a Haar system if it satisfies the

compatibility conditions μr(g) =(Θg

)∗μ

s(g), ∀g ∈ G, where(Θg

)∗μ

s(g) is the push-forward

of the measure μs(g), with respect to the map Θg. Finally, note that the map g �→ g−1

establishes also a bijection between Gx and Gx, for every x ∈ ObjG. By means of these

bijections, a Haar system defines also a family of measures on the s-fibres Gx.


Definition 10. Let G be a smooth groupoid equipped with a Haar system of measures.

The following defines a convolution on the space of C∞ functions on G with compact

support:

(F1 � F2)(g) =

∫Gr(g)

F1(h)F2(h−1g)dμr(g)(h), F1, F2 ∈ C∞c (G). (47)

On the same space, an involution is defined by

F (g) �→ F (g−1) , F ∈ C∞c (G). (48)

The convolution algebra C∞c (G) admits natural ∗-representations, one per each x ∈ObjG. In fact, let us consider the Hilbert spaces L2(Gx, μx), x ∈ ObjG. The (involutive)

representations(L2(Gx, μx), πx

)are defined by

(πx(F )ψ

)(g) =

∫Gr(g)

F (h)ψ(h−1g)dμr(g)(h) , (49)

where F ∈ C∞c (G), ψ ∈ L2(Gx, μx) and g ∈ Gx.

Definition 11. Let G be a smooth groupoid with Haar system {μx}x∈ObjG. The asso-

ciated (reduced) C∗-algebra C∗r (G) is the completion of the convolution algebra C∞c (G)

with respect to the norm ‖F‖ := supx∈ObjG‖πx(F )‖.

Let us analyse again the previous examples, each of which groupoid is now equipped

with a natural differential structure.

Example 1a . In a group there is only one object, and therefore we have only one r -fibre

and one s-fibre, both coincident with the group itself. In this case a locally compact

topology is sufficient to construct the C∗-algebra, which coincides with the convolution

algebra on the group for a given Haar measure. Consider in particular the additive group

R with the Lebesgue measure: the obtained C∗-algebra is the Fourier transform of the

multiplicative algebra C0(R).

Example 2a . We consider the particular case of a tangent bundle TM of a n-dimensional

Riemannian manifold (M, g). Let us fix a local coordinate system (q1, . . . , qn) in M ,

colectively denoted by q. At each point q ∈ M the r -fibre Gq coincides with TqM , which

in turn can be identified with Rn by X = (X1, . . . , Xn) �→ ∑

X i∂qi . On the r -fibres we

consider the measure dμq(X) =√

detg(q)dnX, where dnX is the Lebesgue measure and

g is the metric. Convolution is given by integration on the fibre at each point of M :

(F1 � F2)(q,X) =

∫TqM

F1(q, Y )F2(q,X − Y )dμq(Y ) . (50)

The obtained C∗-algebra is isomorphic to the multiplicative algebra C0(T∗M), by Fourier

transform F on the fibre:

(FF )(q, ξ) :=∫TqM

e−iξXF (q,X)dμq(X), (51)


where (q, ξ) ∈ T ∗M , FF ∈ C0(T∗M).

Example 3a . Let (M, g) be a n-dimensional Riemannian manifold and consider the

groupoid M × M . Every r -fibre and s-fibre is isomorphic to M . On each fibre, we

consider the measure dμ(q) =√

detg(q)dnq, for some local coordinate system onM . The

convolution is

(F1 � F2)(q, q′) =

∫Gq

F1(q, q′′)F2

((q, q′′)−1(q, q′)

)dμ(q′′) (52)

=

∫M

F1(q, q′′)F2(q

′′, q′)dμ(q′′), (53)

where one can recognize immediately the convolution of kernels of integral operators in

L2(M,μ). In fact, the algebra C∗r (M ×M) is isomorphic to the algebra K(L2(M,μ)

)of

compact operators in L2(M,μ) [7].

Example 4a . In this case we analyse an example directly related to the Weyl-Moyal

quantization. Consider a family of actions αε of R on R, labeled by a real number ε. For

each ε the actions are αεy(x) = x + εy. Independently of ε, the r -fibres and s-fibres of

the semidirect product R > �αεR can be identified with R. Let us then introduce the

Lebesgue measure on each fiber. The convolution is then given by

(f � g)(x, y) =

∫Gx

f(x, z)g((x, z)−1(x, y)

)dz (54)

=

∫R

f(x, z)g((x+ εz,−z)(x, y))dz (55)

=

∫R

f(x, z)g(x+ εz, y − z)dz. (56)

Concerning the action of the elements of the algebra on the Hilbert spaces L2(Gu), let

us distinguish the cases ε = 0 and ε �= 0. For ε > 0 the action does not depend on the

s-fibre. It is defined on L2(R) by

(π(f)ψ

)(x) =

∫R

f(x, y)ψ(x+ εy)dy, ψ ∈ L2(R). (57)

For ε = 0 we get the representations{(πu, L

2(R))}

u∈R:((πuf)ψ

)(x) =

∫R

f(u, y)ψ(x− y)dy. (58)

We recognize for ε = 0 the Fourier transform (in the second variable) of the multiplicative

algebra C0(R×R). In the case ε > 0 the nontrivial action of R on R deforms the convolution

algebra in a way that corresponds precisely to the Weyl-Moyal deformation, see section

7.2 below.

6 Tangent Groupoid

This section is dedicated to Landsman’s quantization of the cotangent bundle of a Rie-

mannian manifold Q [16]. Although this construction can be described without reference


to Connes’ tangent groupoid [10], we adhere to the tangent groupoid perspective right

from the start, since it is quite natural, both from the geometric and the algebraic view-

points. We start by showing how the simplest case, that of Q = R, fits in this framework.

6.1 Weyl-Moyal Quantization Revisited

The Weyl-Moyal quantization admits a reformulation in terms of the so-called tangent

groupoid. As we will see shortly, the classical algebra and the quantum algebras appear

unified, as elements of the same algebra of functions on the tangent groupoid.

First note that the W-M quantization maps Qh (21) can be naturally split into three

distinct maps. Consider the Fourier transform

F : S(T ∗R) → S(TR) (59)

f(q, p) �→ f(q, v) =

∫dp

2πeipvf

(q, p

)(60)

and the representation π : S(R × R) → K(L2(R)) of S(R × R) functions as kernels of

integral operators. It is then clear that the maps Qh : S(T ∗R) → K(L2(R)) (21) are

obtained as the composition

Qh = π ◦ ϕh ◦ F , (61)

where the map ϕh : S(TR) → S(R × R) is defined by(ϕhf

)(x, y) =

1

hf(x+ y

2,x− y

h

). (62)

The importance of this decomposition is that it isolates the nontrivial map ϕh, in which is

effectively present the deformation of the algebraic structure. In fact, the Fourier trans-

form is a morphism from the multiplicative algebra S(T ∗R) to the convolution algebra

(with respect to the second variable) in S(TR). This is, in turn, the algebra naturally

associated with the groupoid structure of TR. As we have seen above, R × R is also a

groupoid, and the map π is precisely a morphism from the associated groupoid algebra

to the algebra K(L2(R)) of compact operators.

The W-M maps are therefore naturally decomposed into a couple of morphisms and

a map between groupoid algebras, the deformation ϕh:

S(T ∗R) Qh→ K(L2(R))

F ↓ ↑ πC∗r (TR)

ϕh→ C∗r ((R × R) × {h}) ,where C∗r (TR) denotes the C∗-algebra of the groupoid TR and C∗r ((R×R)×{h}) denotesthe C∗-algebra of the groupoid (R × R) × {h} ∼= R × R.

The crucial point is that the transformations ϕh are induced by an identification of

TR with R × R. Let us consider the map φ : TR → R × R defined by

TR � (q, v)φ�→ (q +

1

2v, q − 1

2v) . (63)


Consider still the family of maps φh : TR → (R × R) × {h} obtained from the previous

one:

φh(q, v) = φ(q, hv) = (q +h

2v, q − h

2v) . (64)

It is then clear that (ϕhf

)(x, y) =

1

hf(φ−1h (x, y)

). (65)

The maps φh (64) allow the construction of a manifold with boundary, the so-called

tangent groupoid [10], as follows. Consider first the product manifold G1R

:= (R ×R)×]0, 1]. This is also a groupoid: two elements (x, y, h) and (x′, y′, h′) can be combined

if and only if they belong to the same leaf (R × R) × {h}, i.e. if h = h′. The associated

C∗-algebra turns out to be C∗r (G1R) ∼= C0(]0, 1])⊗C∗r (R×R) ∼= C0(]0, 1])⊗K(L2(R)). It is

interesting to note that G1Rcan be seen as the space of secant lines to R, or more precisely

of finite difference operators. In fact, the elements (x, y, h) ∈ G1Rdefine elements of the

dual of the space C1(R) of differentiable functions in R, by

(x, y, h) �→ f(x) − f(y)

h, f ∈ C1(R) .

The closure of this open set in C1(R)∗is the tangent groupoid [10,17].

An explicit construction of the tangent groupoid, both as a manifold and as a groupoid,

is the folowing [10] (see also [7, 17]). Let us consider the union GR := G1R

∪ G2R, where

G2R:= TR. GR is a groupoid with the obvious structure of union of groupoids. The struc-

ture of manifold with boundary is defined by the maps φh (64), which give coordinates

in GR. In fact, as a manifold, the tangent groupoid GR is diffeomorphic to TR × [0, 1].

The announced coordinate system in GR is given by the transformation

Φ : TR × [0, 1] → GR (66)

such that

Φ(q, v, h) =

⎧⎪⎨⎪⎩

(q + h2v, q − h

2v, h) if h > 0

(q, v) if h = 0.

The transformation Φ is a diffeomorphism when restricted to TR×]0, 1] and maps the

boundary of TR × [0, 1] to the boundary of GR.

The C∗-algebra C∗r (GR) associated with the tangent groupoid GR is formed by pairs({kh}h∈]0,1], f), where kh ∈ C0(R × R), f ∈ C0(TR), subject to the continuity condition

at the boundary, i.e.

limh→0

kh(q +

h

2v, q − h

2v)= f(q, v) . (67)

This continuity condition is in fact a quantization condition, imposing that any element of

C∗r (GR) is a family of quantum operators having f(q, v) (or its inverse Fourier transform

f(q, p)) as a limit.

The above continuity condition at the boundary still leaves a great deal of freedom as

to the choice of quantum operators to be associated with a given classical observable, and


therefore the quantization maps are not fixed. However, the construction of the groupoid

itself suggests the construction of well determined linear maps S(TR) → C0(R × R).

Let then f(q, v) be an element of S(TR). Consider the element F := {fh}h∈[0,1] of

C∗r (TR × [0, 1]) given by fh = f , ∀h. The element 1h(Φ−1)∗F of C∗r (GR) provides then

the required quantization. The composition of this map with the Fourier transform and

the representation π finally gives the Weyl-Moyal quantization.

6.2 Cotangent Bundle of a Riemannian Manifold

Following Landsman [6,16], Connes [10] and also reference [17], we present in this section

the strict quantization of the most common type of phase space in physical applications,

which is the cotangent bundle T ∗Q of some Riemannian manifold Q. We start with

the construction of the tangent groupoid GQ, which generalizes the construction of the

previous section. We introduce first the algebraic structure of the tangent groupoid GQ,

followed by its manifold structure.

Let then (Q, g) be a n-dimensional Riemannian manifold, with metric g. The asso-

ciated tangent groupoid is a disjoint union of groupoids, GQ = ((Q × Q)×]0, 1]) ∪TQ,formed by the groupoid TQ and by a family of copies of the groupoid Q×Q. The tangentbundle TQ is a smooth groupoid, equipped with a Haar system of measures dμq on the

fibres TqQ:

dμq(v) = dnv√det g(q) , (68)

where (q, v) denotes local coordinates on TQ. The convolution algebra of TQ is deter-

mined by the expression

(f � g)(q, v) =

∫dμq(v′)f(q, v′)g(q, v − v′) . (69)

The product Q×Q is also a smooth groupoid, with measure

dμ(q) = dnq√

det g(q) (70)

on the fiber Q. The product (Q×Q)×]0, 1] is again a smooth groupoid, with the prod-

uct manifold structure and the following groupoid structure. The elements (x, y, h)

and (x′, y′, h′) can be combined if and only if h = h′ and y = x′, and in that case

(x, y, h)(y, y′, h) = (x, y′, h). The r -fibres and s-fibres of (Q × Q)×]0, 1] both coincide

with Q× {h} ∼= Q, and the convolution algebra is given by

(f � g)(x, y, h) =

∫dμ(z)f(x, z, h)g(x, y, h) . (71)

Finally, the disjoint union GQ = ((Q × Q)×]0, 1]) ∪TQ acquires a natural groupoid

structure, in the sense that elements of (Q×Q)×]0, 1] (resp. TQ) combine only amongst

themselves.


As in the previous section, which corresponds to Q = R, GQ becomes a smooth

groupoid [7,10,17] when equipped with the topology of a manifold with boundary. Gen-

eralizing the previous procedure, we present next a map from an open set U ⊂ TQ×]0, 1]

to GQ, which defines the boundary of GQ.

Let us fix a local coordinate system q on Q. This gives us also coordinates (q, q′) onQ×Q and a basis (∂q, ∂q′) on each space T(q,q′)(Q×Q). Consider the diagonal embedding

Δ : Q → Q × Q given by Δ(q) = (q, q). At each point Δ(q) the metric g ⊕ g on Q × Q

allows a decomposition of T(q,q)(Q×Q) vectors in tangent and normal parts, i.e.

T(q,q)(Q×Q) = Δ∗TqQ⊕ (Δ∗TqQ)⊥ ,

where Δ∗TqQ is the push-forward of the tangent space and (Δ∗TqQ)⊥ is its orthogonal

complement. The union ∪q∈Q(Δ∗TqQ)⊥ is a subbundle of the restriction of T (Q×Q) to

Δ(q), whose fibres are normal to Δ(q) at each point. This is the normal bundle associated

with Δ, hereafter denoted by NΔQ.

Clearly, the elements of Δ∗TqQ are of the form (Xq, Xq), with Xq ∈ TqQ and in

the same way the elements of (Δ∗TqQ)⊥ can be written in the form (Xq,−Xq), with

Xq ∈ TqQ. One can therefore build a map η : TQ → NΔQ, given by

η(q,Xq) =(Δ(q),

1

2Xq,−1

2Xq

). (72)

The transformation η can now be combined with the exponential map defined by normal

geodesics at Δ(q). Let then W1 be an open set in NΔQ where the exponencial map

is defined and let U1 ⊂ Q × Q denote the image of W1. Consider the transformations

φ : V1 → U1, φ = exp ◦η, where V1 := η−1(W1) ∈ TQ. Explicitly

TQ � (q,Xq)η�→

(Δ(q),

1

2Xq,−1

2Xq

)exp�→

(expq(

1

2Xq), expq(−

1

2Xq)

). (73)

Let us define still the maps

φh : Vh :=1

hV1 → V1

φ→ Uh∼= U1 × {h} (74)

TQ � (q,Xq) �→ (q, hXq) �→(expq(

1

2hXq), expq(−

1

2hXq), h

). (75)

Finally, consider the manifold TQ × [0, 1] with its product structure and its boundary

TQ × {0} ∼= TQ. Let U be the open set in TQ × [0, 1] defined by U :=(Xh∈]0,1]Vh

) ×(TQ× {0}), which contains the boundary. The transformation Φ : U → GQ defined by

Φ(q,Xq, h) =

⎧⎪⎨⎪⎩φh(q,Xq) if h > 0

(q,Xq) if h = 0

is a diffeomorphism when restricted to TQ×]0, 1] and maps TQ to TQ, thus defining a

coordinate system on an open set in GQ containing the boundary. This concludes the

description of the tangent groupoid GQ.


Let us then describe the quantization of the symplectic manifold P := T ∗Q. Considerfirst the measure

dμq(p) =dnp

(2π)n√det g(q)

(76)

on the fibres T ∗qQ of T ∗Q, where (q, p) is a local coordinate system. The Fourier transform

on the fibre maps functions f(q, p) on T ∗Q to functions f(q, v) on TQ:

f(q, v) :=

∫dμq(p) e

ipvf(q, p) . (77)

Let us adopt as a complete algebra of regular observables A(P) the subalgebra of the

functions f ∈ C∞0 (P) such that f ∈ C∞c (TQ).

The quantization maps are defined as follows. For a given observable f ∈ A(P), let

h(f) be a real number such that suppf ⊂ Vh(f). Then, for every h ≤ h(f),

Kfh := h−n

(φ−1h

)∗f (78)

is well defined and belongs to C∗r ((Q×Q)×{h}). The observable f is therefore quantized,

∀h ≤ h(f), by the operator Qh(f) ∈ K(L2(Q, μ)

):

(Qh(f)ψ

)(x) =

∫dμ(x′)Kf

h (x, x′)ψ(x′), ψ ∈ L2(Q, μ) . (79)

7 The Torus: a Case Study

The 2-torus T 2 provides a good test for any quantization scheme. Although rather inno-

cent looking, the torus possesses a set of characteristics that make it somewhat special.

To begin with, it is not a cotangent bundle, and it is compact, and therefore the physi-

cal expectation is that at the quantum level one will find only bounded observables and

moreover finite dimensional Hilbert spaces. There is, however, another characteristic

that distinguishes the torus from e.g. the two-sphere, with which it shares the above two

properties. In fact, the Poisson algebra C∞(T 2) of the torus does not seem to admit

any subalgebra that separates points (and contain the constant function 1), besides the

algebra C∞(T 2) itself (and dense subalgebras thereof). In particular, it is known that no

such finite dimensional subalgebra exists [18]. Thus, from the point of view of canonical

quantization, it seems that for T 2 one is forced to impose the Dirac condition (5) on the

whole Poisson algebra. But it is also known that no nontrivial finite dimensional Lie

representation of C∞(P) can be found, for any connected compact symplectic manifold

P [19]. On the other hand, any infinite dimensional representation of such a Poisson alge-

bra will include unbounded operators [20]. So, it seems that every conceivable Dirac-like

quantization of the torus will produce infinite dimensional Hilbert spaces and unbounded

observables, conflicting with natural physical expectations.

We discuss next a quantization of C∞(T 2) proposed by Gotay, which has the great

interest of proving that (irreducible) quantizations of full Poisson algebras can indeed

be found. It does not, however, avoids the above mentioned drawbacks. A modified


quantization, departing from the exact implementation of Dirac’s condition, is already

presented in the next section. This presentation follows geometric quantization methods,

although at some point a deformation is introduced. The same quantization is discussed

in section 7.2, this time showing that it is directly obtained from a group action such as

those discussed in example 4 of section 5.

7.1 Geometric Quantization of the Torus

Let (P , ω) be a symplectic manifold and { , } the corresponding Poisson bracket. Let S be

a Lie-subalgebra of(C∞(P), { , }), containing the constant function 1. By prequantization

of S it is meant a linear map Q from S to self-adoint operators on a Hilbert space, such

that Dirac’s condition

Q({f, g}) = [Q(f),Q(g)]h ∀f, g ∈ S (80)

is satisfied and

Q(1) = 1 , (81)

where 1 is the identity operator.

In general, Dirac’s condition can be achieved in the full algebra C∞(P): given that

Hamiltonian vector fields provide an (anti-)representation of the Poisson algebra, one

could just adopt the map Q(f) = −ihξf , where ξf is the Hamiltonian vector field defined

by f , acting on (an appropriate dense domain of) L2(P , ωn/n!). However, this type

of representation always leads to Q(1) = 0, which is not acceptable. The formalism

of geometric quantization (see e.g. [21–23]) corrects this aspect. In this formalism, the

quantization map is of the form

Q(f) = f − ihξf + θ(ξf ) ,

where dθ = ω. In general, the 1-form θ is defined only locally, with ξf + ihθ(ξf ) being

properly interpreted as a covariant derivative on a certain line bundle, which requires

h−1ω to be of integral cohomology class. (In this subsection h is Planck’s constant.)

Besides (80) and (81), a true canonical quantization is required to satisfy a further set

of mathematical physics conditions, one of the most proeminent being irreducibility (see

[12] for a thorough discussion). It is this irreducibility condition that typically calls for

the necessity of a polarization in geometric quantization, leading to a drastic reduction

of the algebra over which the quantization map is defined.

Nevertheless, in [24] (see also [12]) Gotay shows that a given prequantization of the

full algebra of smooth functions on the torus T 2 satisfies even the condition of being

irreducible. Thus, it seems that in this case polarization would not be necessary and

that a canonical quantization of all smooth observables has been achieved. Despite the

obvious interest of this result, there is a high price to pay for this full quantization, in

the sense that von Neumann’s condition (4) is badly broken (see [25]). We discuss next

Gotay’s proposal.


Let N ∈ N and consider the torus T 2 of area Nh, which we identify with R2/Z2

equipped with the symplectic form

ω = Nhdx ∧ dy , (82)

where we have introduced local coordinates (x, y). The Poisson algebra C∞(T 2) is there-

fore the algebra of C∞ periodic functions on R2. Gotay’s quantization, which is obtained

through geometric quantization methods, can be described as follows. Choosing the con-

nection θ = −Nhydx, the prequantum Hilbert space HN can be seen as the completion

of the space DN of complex C∞ functions in R2 such that,

φ(x+m, y + n) = e2πiNnxφ(x, y), ∀m,n ∈ Z , (83)

with respect to the inner product

〈φ, φ′〉 =∫[0,1]×[0,1]

dxdy φ φ′ . (84)

The prequantization map is given by

QN(f)φ = fφ− i

2πN

(∂f

∂y

(∂φ

∂x− 2πNiyφ

)− ∂f

∂x

∂φ

∂y

), ∀f ∈ C∞(T 2) . (85)

Being a prequantization, it is true ∀N that Dirac’s condition (5) is fulfilled for all observ-

ables in the algebra C∞(T 2). The case N = 1 is special in that irreducibility conditions

are satisfied [24]. Thus, it appears that Q1 above gives a bona fide canonical quantization

of all observables in a symplectic manifold. However, there is no control over the multi-

plicative structure of the algebra C∞(T 2), and therefore there is also no control over the

spectrum of the quantum operators Q1(f) (85). For instance, considering the classical

observables sin(2πx) and cos(2πx), one can show [25] that

Q21

(sin(2πx)

)+ Q2

1

(cos(2πx)

)= 1 + R , (86)

where R is an unbounded operator with no correspondence with any observable. (The

same happens with the functions sin(2πy) and cos(2πy).) In particular, the spectrum of

the quantum operators corresponding to the sinus and cosinus functions is the full line

R, and the correlation between the two functions is lost.

Let us now discuss a different quantization of the torus T 2 which we believe satisfies

appropriate physical requirements. This quantization can be introduced in a number of

ways (see [7,26–28]). Our treatment in this section is inspired in [26]. As we will see in the

next section, the same quantization appears naturally in the context of noncommutative

geometry.

Let us consider then the prequantizations QN (85). In the geometric quantization

formalism, the way to achieve a quantization starting from a given prequantization is to

restrict the action of observables to (covariantly) constant sections over a given polar-

ization [21, 22]. Let us then focus on the space of sections φ ∈ HN such that ∂∂yφ = 0


(note that with the connection θ = −Nhydx, the covariant derivative ∇y :=∂∂y

+ ihθ(

∂∂y

)coincides with ∂

∂x). Taking (83) into account, those sections satisfy(

1 − e2πiNx)ψ(x, 0) = 0 , (87)

and it is therefore clear that there are no nontrivial solutions in HN . There are, however

N independent generalized solutions, of the form

ψk = δ(x− k/N), k = 0, 1, . . . , N − 1 . (88)

In fact, the distributions ψk defined by:

ψk(φ) =

∫dx dy δ(x− k/N)φ(x, y) =

∫ 1

0

dy φ(k/N, y) (89)

are well defined on the dense space DN ⊂ HN and satisfy

ψk

(∂φ∂y

)= 0, ∀φ ∈ DN . (90)

(The appearance of distributional solutions is common in geometric quantization; the

fibres x = k/N are an example of so-called Bohr-Sommerfeld submanifolds [21, 22].) Let

us then choose the finite dimensional Hilbert space (isomorphic to CN) generated by the

N distributions ψk, with inner product 〈ψk, ψk′〉 = δkk′ , to be the quantum Hilbert space

associated with the 2-torus of area Nh. In general, the choice of a polarization selects a

restricted subalgebra of observables with a well defined action on the quantum Hilbert

space. In the present case, one can easily check that a quantum observable QN(f) is well

defined if and only if

ψk

(QN(f)

∂φ

∂y

)= ψk

(∇x

(∂2f∂y2

φ))

= 0, ∀k, ∀φ . (91)

It follows that the only functions f ∈ C∞(T 2) that are quantized by this process are the

ones which depend exclusively on x. These act simply by multiplication, i.e.

QN

(f(x)

)ψk = f(k/N)ψk . (92)

The extension of the quantization to further observables requires a new look at the

quantization of functions g(y) ∈ C∞(T ).Following [26], let us consider the unitary operators given by VN(b), b ∈ R:(VN(b)φ

)(x, y) := φ(x+ b, y), φ ∈ HN , (93)

which are associated with translations x �→ x + b. Clearly, the operators VN(b) have a

well defined action in HN only for values of b of the form b = n/N , n ∈ Z, as follows from

(83). These finite translations correspond to maps between the Bohr-Sommerfeld leaves,

and we therefore get well defined unitary translation operators VN(n) on the quantum

Hilbert space:

VN(n)ψk := ψk−n (modN) . (94)


Taking into account the Weyl-Moyal quantization4 in T ∗R, one can look at the operators

VN(n) as the quantization of the functions e2πiny. This interpretation is further supported

by the commutation relations satisfied by these operators and the quantization of the

functions e2πimx, namely

VN(n)QN

(e2πimx

)= e2πimn/NQN

(e2πimx

)VN(n) . (95)

Let us then define a quantization map by

QN

(e2πi(mx+ny)

):= eπinm/NQN

(e2πimx

)VN(n). (96)

Concerning the Dirac rule we obtain in particular[QN

(e2πimx

), QN

(e2πiny

)] − ihQN

({e2πimx, e2πiny

})=

= 2i(mnπ/N − sin(mnπ/N)

)QN

(e2πi(mx+ny)

), (97)

which shows that, for large N , the Dirac condition is well approximated by slow varying

functions, i.e., such that the Fourier decomposition contains only components e2πi(mx+ny)

of frequencies m and n which are small compared to N .

7.2 C∗-Algebraic Quantization of the Torus

The Weyl-Moyal quantization map (21), section 3, can be immediately rewritten in the

form (Qh(f)ψ)(x) =

∫ (∫dp

2πe−ipyf

(x+

h

2y, p

))ψ(x+ hy)dy . (98)

When comparing (98) with expression (57), section 5, we see that Qh(f) coincides with

π(fh), where

fh(x, y) =

∫dp

2πe−ipyf

(x+

h

2y, p

). (99)

Here, fh should be considered as an element of R > �αhR. So, the quantum algebra

coincides precisely with C∗r (R >�αhR), and the quantization map is given by Qh = π◦∧h,

where ∧h(f) = fh.

In this perspective, the crucial step in the quantization process consists in the intro-

duction of the algebra C∗r (R >�αhR), associated with the nontrivial action of the tangent

vectors in configuration space.

Let us consider again the two dimensional torus T 2. The identification T 2 ∼= R2/Z2

gives also a bijection between the set of continuous functions on the torus and the set of

periodic continuous functions on R2:

f(x+m, y + n) = f(x, y), ∀m,n ∈ Z .

4 In T ∗R, the W-M quantization (21) of the function g(p) = eibp/h is in fact the translation operator in

L2(R): ψ(q) �→ ψ(q + b).


The partial Fourier transform:

F : f(x, y) �→ f(x, n) =

∫ 1

0

e−2πinyf(x, y) dy , (100)

further allows us to pass from functions on the torus to functions on S1 × Z, or periodic

functions on R×Z. For definiteness, let us consider as complete algebra of regular observ-

ables the subspace A(T 2) of those functions whose (total) Fourier transform possesses

only a finite number of nonzero coefficients. It follows that the partial Fourier trans-

form F (100) is an isomorphism between A(T 2) and the space A(S1 × Z) of finite linear

combinations of the functions Fmk in S1 × Z defined by:

Fmk(x, n) := e2πimxδnk . (101)

Let us consider the family of actions αh of Z on S1, parametrized by real numbers h ∈ [0, 1[

and defined by

αhn(x) = x+ nh mod 1 , n ∈ Z . (102)

With each of these actions, let us associate the semidirect product groupoid S1 >�αhZ,

as in examples 4 and 4a in section 5. Between two elements g1 = (x1, n1) and g2 = (x2, n2)

of S1 >�αhZ such that x2 = αhn1(x1) = x1 + n1h mod 1, the composition rule is given

by g1g2 = (x1, n1 + n2). The inverse of the element g = (x, n) is g−1 = (αhn(x),−n) =

(x + nh mod 1,−n). Finally, let us note that both r-fibres and s-fibres can be identified

with Z. In particular, given an object x0 ∈ Obj(S1 > �αhZ) ∼= S1, the corresponding

r-fibre is the set

Gx0 = {(x0, n), n ∈ Z} , (103)

and the corresponding s-fibre is

Gx0 = {(αhn(x0),−n), n ∈ Z} . (104)

The discrete structure of the fibres allows us to define the convolution algebra, which

we now describe. Consider then the space A(S1 × Z), whose elements we identify with

functions F : R × Z → C such that F (x +m,n) = F (x, n), m ∈ Z. The involution ∗ inA(S1 × Z) is defined by

F ∗(g) = F (g−1), or (105)

F ∗(x, n) = F (x+ nh,−n), F ∈ A(S1 × Z) . (106)

In particular for the basis elements Fmk (101) we get

F ∗mk = e2πimkhF−m−k . (107)

The convolution � is defined by

(F � G)(x, n) =∑m∈Z

F (x,m)G(x+mh, n−m) . (108)


A(S1×Z) is therefore an involutive algebra with identity, namely the function F00(x, n) =

δn0. In particular, the convolution (108) of basis elements leads to

Fmk � Fm′k′ = e2πim′khFm+m′ k+k′ . (109)

One can easily show that this algebra is generated by the two elements F10 e F01. Fur-

thermore, since F10 are F01 are unitary and satisfy the commutation relations

F01 � F10 = e2πihF10 � F01 , (110)

we conclude that the algebra in question is none other than the well know universal

rotation algebra Aroth , parametrized by h, which is precisely defined as the ∗-algebra

generated by two elements u and v subject to the relations u∗u = uu∗ = v∗v = vv∗ = 1

and vu = e2πihuv [29]. The rotation C∗-algebra Aroth is by definition the completion of

Aroth with respect to the norm

‖a‖ := sup{‖πa‖ : π is a representation of Aroth }, a ∈ Arot

h , (111)

and satisfies the folowing universality property [29]

Theorem 1. Let A be a C∗-algebra with two elements u′, v′ satisfying the same relations

as the generators u, v of Aroth . Then there exists a morphism ϕ : Arot

h → A such that v �→ v′

and u �→ u′. If h is irrational then ϕ is an isomorphism between Aroth and the smallest

closed subalgebra of A that contains u and v.

This result shows immediately that C∗r (S1 >�αhZ) is isomorphic to Arot

h when h is

irrational. In the h rational case, and for an arbitrary algebra A, the map ϕ of the above

theorem is not necessarily injective. In the present case, however, injectivity is clearly

ensured, and therefore C∗r (S1 >�αhZ) is isomorphic to Arot

h , ∀h.The structure of the rotation algebras Arot

h depends heavily on the value of h: for

h = 0 we recover, as expected, the algebra C(T 2) of continuous functions on the torus;

the irrational h case is extremely interesting from the point of view of noncommutative

geometry and has been extensively studied [7, 10, 27, 30]. Let us focus on the rational

(nonzero) h case, following [7,30]. As we will see shortly, the quantization (96) described

in the previous section will emerge here quite naturally.

Let us first note that two distinct values h, h′ such that h + h′ = 1 lead to the same

algebra, i.e. Arot1−h is isomorphic to Arot

h . Let then h = KN, with K,N ∈ N and K ≤ N/2.

Taking into account the convolution (109), one can easily check that the elements of

the form FmN kN , m, k ∈ Z, commute with the generators, and therefore belong to the

centre of the algebra. So, given any irreducible representation π0, the image π0(FmN kN)

of those elements must be proportional to the identity. It follows that the irreducible

representations of the algebra Aroth , with h = K/N as above, are finite dimensional, of

dimension N . Note also that, being finite dimensional, the irreducible representation is

unique (modulo unitary equivalence).


A convenient irreducible representation can be easily found, as follows. Let HN∼= C

N

be the Hilbert space generated by an orthonormal set of N vectors, say {v0, v1, . . . , vN−1}.In B(HN) consider unitary operators UN(K), VN(K) such that

UN(K)vk = e2πik/Nvk , (112)

VN(K)vk = vk−KmodN . (113)

We obtain immediately the commutation relations:

VN(K)UN(K) = e2πiK/NUN(K)VN(K) . (114)

The pair UN(K), VN(K) therefore satisfies the relations (110) corresponding to h = K/N ,

which shows that the ∗-morphism πK,N : Aroth → B(HN) given by

πK,N(F10) = UN(K) (115)

πK,N(F01) = VN(K) (116)

is well defined and is a representation, obviously irreducible, of ArotK/N .

Let us finally construct a family QK/N of quantizations of the 2-torus. As already sug-

gested, let us adopt as complete algebra of regular observables the subalgebra A(T 2) ⊂C(T 2) of those functions whose Fourier transform possesses only a finite number of

nonzero coefficients. Let then ∧K/N : A(T 2) → C∗r (S1 >�αhZ) denote the maps given by

f(x, y) �→ fK/N(x, n) =

∫ 1

0

e−2πinyf(x+ nK/2N, y) dy , (117)

which correspond to (99). For the elements e2πi(mx+ky) of the base we get simply

e2πi(mx+ky)∧K/N�−→ eπimkK/NFmk . (118)

The quantizations maps are then QK/N = πK,N ◦ ∧K/N , leading to

QK/N

(e2πi(mx+ky)

)= eπimkK/NUN(K)mVN(K)k ∈ B(HN). (119)

In particular for K = 1, this coincides with the quantization put forward in the previous

section, expressed namely in quantization rules (92), (94) and (96).

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[1] R. Honegger, A. Rieckers, L. Schlafer, SIGMA 4 (2008) 047.

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[5] G. B. Folland, Harmonic Analysis in Phase Space (Princeton University Press, 1989).

[6] N. P. Landsman, Mathematical Topics between Classical and Quantum Mechanics(Springer-Verlag, New York, 1998).

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[9] R. Loja Fernandes, Deformation Quantization and Poisson Geometry, ResenhasIME-USP 4 (2000) 327.

[10] A. Connes, Noncommutative Geometry (Academic Press, London, 1994).

[11] P. Dirac, The Principles of Quantum Mechanics (Oxford University Press, 4th ed.1967).

[12] M. J. Gotay, Obstructions to Quantization, in Mechanics: From Theory toComputation (Essays in Honour of Juan-Carlos Simo), J. Nonlinear Sci. Eds.(Springer, New York, 2000) 171.

[13] M. A. Rieffel, Deformation Quantization for Actions of Rd, Memoirs Am. Math. Soc.106 , Nr. 506 (1993).

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[15] M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. I (AcademicPress, 1980).

[16] N. P. Landsman, J. Geom. Phys. 12 (1993) 93.

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[18] M. J. Gotay, J. Grabowski, H. B. Grundling, Proc. Amer. Math. Soc. 128 (2000) 237.

[19] V. L. Ginzburg, R. Montgomery, Geometric Quantization and No Go Theorems,Banach Center Publications 51 (2000) 69.

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[25] J. M. Velhinho, Int. J. Mod. Phys. A 22 (1998) 3905.

[26] V. Aldaya, M. Calixto, J. Guerrero, Comm. Math. Phys. 178 (1996) 399.

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[28] M. Bordemann, E. Meinrenken, M. Schlichenmaier, Comm. Math. Phys. 165 (1994)281.

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[29] N.E. Wegge-Olsen, K-Theory and C*-algebras - a friendly approach (OxfordUniversity Press, 1993).

[30] N. F. Antonio, Algebras-C∗ de Rotacao : Propriedades Elementares e de Estrutura,Dissertacao de Mestrado (IST-UTL, Lisboa, 1998).


What is the Wave Function and Why is it used inQuantum Mechanics?

Yuri A. Rylov∗†

Institute for Problems in Mechanics, Russian Academy of Sciences,101-1, Vernadskii Ave., Moscow, 119526, Russia.

Received 15 March 2015, Accepted 10 May 2015, Published 25 August 2015

Abstract: It is shown that quantum mechanics can be presented as a hydrodynamic of some

quantum fluid. In this case the quantum mechanics ceases to be an axiomatic conception,

because the axiomatic object of QM (wave function) ceases to be an axiomatic object. In

the hydrodynamics the wave function is a method of an ideal fluid description. The quantum

mechanics becomes to be a classical dynamics of the stochastic particles. Problems of the

quantum mechanics interpretation disappear, because the interpretation is determined by the

mathematical formalism of classical dynamics.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Classical Dynamics of Stochastic Particles; κ-field; Classical Gas Dynamics; Wave

Function; Quantum Mechanics QM; QM interpretation

PACS (2010): 03.65.-w; 03.70.+k; 11.10.-z;67.10.Hk;45.50.Dd

1. Introduction

The problem of the quantum mechanics interpretation exists almost a century. There

are numerous versions of interpretations. It is connected with the fact, that quantum

mechanics is an axiomatic conception. It means that the quantum mechanics conception

contains an axiomatic object, defined only by its properties. This axiomatic object is

the wave function. Nobody knows, what is the wave function. One cannot interpret

an axiomatic conception exactly, because of indefinite axiomatic object. In the classical

mechanics there are no problems of interpretation, because the mathematical formalism

of classical mechanics admits one to interpret any physical phenomenon, described in

terms of classical mechanics.

∗ Email:[email protected].† Web site: http://gasdyn-ipm.ipmnet.ru/˜rylov/yrylov.htm.


It was shown that the Schrodinger equation describes a nonrotational flow of some

”quantum” fluid [1]. D.Bohm used this circumstance in his hydrodynamic interpretation

of quantum mechanics [2]. Unfortunately, the connection between the quantum mechanics

and hydrodynamics has been one-sided all the twentieth century. One could obtain

hydrodynamic description from the quantum mechanics, but one was not enable to obtain

the Schrodinger equation from hydrodynamical equations. The reason of such a situation

was the axiomatic object - wave function. The wave function is considered as a natural

attribute of quantum mechanics, whereas connection between the wave function and

hydrodynamics was not clear. Situation changed, when it has been shown, that the

wave function appears to be a method of description of any ideal (nondissipative) fluid

[3]. Description in terms of wave function is connected with conventional hydrodynamic

description in terms of hydrodynamic variables: density and velocity.

Due to this connection the wave function ceased to be an axiomatic object, Wave

function and spin turned to natural attributes of a classical dynamics of continuous

medium. The quantum mechanics, considered as a hydrodynamics, became to be a

classical dynamics, where there is no problem of interpretation. Besides, some force field

appears. It is responsible for quantum effects. In the quantum mechanics there is the

problem of uniting of nonrelativistic quantum principles with the principles of relativity

theory. In the quantum mechanics, considered as a hydrodynamics, such a problem is

absent, because the principles of quantum mechanics are absent. Instead there is a force

field κl, l = 0, 1, 2, 3, which is responsible for quantum effects.

A use of a fluid is a natural method of a stochastical particle description. There are

no dynamic equations for a single stochastic particle description. One can describe only

a mean motion of a sochastic particle. A use of the distribution function for description

of a stochastic particle motion is possible only for a nonrelativistic stochastic particle,

because the distribution function in the phase space of coordinates and momenta is a

nonrelativistic construction. However, the nonrelativistic quantum mechanics should be

considered as a relativistic conception, because random component of the particle velocity

may be relativistic, even if the regular component of the particle velocity is nonrelativistic.

For description of relativistic particles one needs to use a statistical ensemble. Sta-

tistical ensemble E [Sd] of deterministic particles Sd is a set of N (N → ∞) independent

particles Sd. E [Sd] is a fluidlike dynamical system. One can obtain dynamic equations

for Sd from dynamic equations for E [Sd]. On the contrary, one can obtain dynamic equa-

tions for E [Sd] from dynamic equations for Sd. Thus, descriptions in terms of E [Sd] and

in terms of Sd are equivalent, if Sd is a deterministic particle and there exist dynamic

equations for Sd.

Let now particles Sd interact between themselves via some force field κ. Then the

statistical ensemble E [Sd] ceases to be a statistical ensemble, because its elements S

are not independent particles. The particles Sd turn to interacting particles Sst. The set

E [Sst] ceases to be a statistical ensemble, but it remains to be a fluidlike dynamic system.

We shall use the term int-ensemble for the dynamic system E [Sst]. One cannot obtain

dynamic equations for Sst from dynamic equations for E [Sst]. It means that elements


Sst of int-ensemble E [Sst] are not deterministic particles. They are stochastic particles.

Dynamic equations for E [Sst] describe a mean motion of stochastic particles Sst.

A simple example of such a situation is an ideal gas. Molecules of ideal gas are

stochastic particles, and there are no dynamic equations for a single molecule inside a

gas. Motion of the ”gas particles” is described by equations of classical gas dynamics.

Any gas particle contains many molecules, and motion of a gas particle describes a mean

motion of the gas molecule. Interaction between the gas molecules is realized by the

molecular collisions. A single molecule (outside the gas) is a deterministic particle. There

are dynamic equations for a single molecule outside the gas. As to stochastic particles,

they are not deterministic particles, and there are no dynamic equations for a single

particle Sst. A use of int-ensemble E [Sst], where the stochastic particles Sst interact via

some force field κ is only a mathematical method, which enables to describe the mean

motion of Sst by means a int-ensemble E [Sst], which is a fluidlike dynamic system. Of

course, the kind of the particle stochastics depends on the force field κ. This circumstance

admits one to classify stochastic particles by the force field κ, which appears in dynamic

equations for int-ensemble E [Sst]. It admits one also to consider the force field κ as a

source of the particle stochasticity, although the κ-field appears as a mathematical means

of the stochastical particle description.

2. Clebsch Potentials

The complete system of hydrodynamic equations contains seven equations

∂ρ

∂t+∇ (ρv) = 0,

∂v

∂t+ (v∇) v = −∇p (ρ)

ρ(1)

∂ξ

∂t+ (v∇) ξ = 0 (2)

Three equations (2) describe the motion of the fluid particle in the given velocity field

v = v (t, x). They can be presented in the form

dx

dt= v (t, x) (3)

Solutions ξ = {ξ1 (t, x) , ξ2 (t, x) , ξ3 (t, x)} =const of (2) are three independent integrals

of (3). Indeed, due to (3) and (2) the expression

d

dtξ (t, x (t)) =

∂ξ (t, x (t))

∂t+

(dx

dt∇

)ξ (t, x (t)) = 0 (4)

vanishes. It means that ξ(t, x(t)) =const are integrals of (3). If ξ1 (t, x) , ξ2 (t, x) , ξ3 (t, x)

are three independent solutions of (2), equations (2) and (3) are equivalent. Variables ξ

are known as Lagrangian coordinates, which are constant along the fluid particle trajec-

tories according to their definition.

Four equations (1) form a closed subsystem of the system of dynamic equations (1),

(2). It is a reason, why usually one considers only four equations (1) as a system of


hydrodynamic equations, ignoring (2). One keeps in mind, that the ordinary equations

(3) can be solved relatively easy, if the velocity field v(t, x) is determined from (1).

Besides, in many hydrodynamic problems the trajectories of the fluid particles are not

interesting.

The complete hydrodynamic system (1), (2) may be integrated partly in the form

∂ρ

∂t+∇ (ρv) = 0, v =b0 (∇ϕ+ gα (ξ)∇ξα) (5)

∂ξ

∂t+ (v∇) ξ = 0 (6)

where gα (ξ), α = 1, 2, 3 are arbitrary functions, which are determined from the initial

conditions for v . The variable ϕ is a new variable, introduced instead of v . The quantity

b0 is an arbitrary constant. The second relation (5) is known as Clebsch potentials.

Clebsch obtained them for incompressible fluid [4, 5].

The wave function ψ = {ψα}, α = 1, 2, . . . , n is a n-component complex function. It

is constructed from Clebsch potentials by means of relations,

ψα =√ρeiϕwα(ξ), ψ∗α =

√ρe−iϕw∗α(ξ), α = 1, 2, . . . , n, (7)

ψ∗ψ ≡n∑

α=1

ψ∗αψα, (8)

where (*) means the complex conjugate, wα(ξ), α = 1, 2, . . . , n are functions of only

variables ξ. They satisfy the relations

− i

2

n∑α=1

(w∗α∂wα

∂ξβ− ∂w∗α∂ξβ

wα) = gβ(ξ), β = 1, 2, 3,n∑

α=1

w∗αwα = 1. (9)

The number n is such a natural number, that equations (9) admit a solution. In general

n may depend on the form of the arbitrary integration functions g = {gβ(ξ)}, β = 1, 2, 3.

Hydrodynamic equations, described in terms of Clebsch potentials or in terms of

the wave function, are rather bulky. Hydrodynamists do not use them, and such a

presentation of hydrodynamic equations is known slightly. The only known exclusion

takes place, when the internal energy E of a fluid has the form

E =1

2ρv2

dif , vdif = − �

2m∇ log ρ, E =

�2

8m2

(∇ρ)2

ρ(10)

If besides the fluid flow is nonrotational, one can set in (5) gα(ξ) = 0. In this case the

wave function is one-component, and choosing b0 = � in (5), hydrodynamic equations

in terms of the wave function appear to be linear. They coincide with the Schrodinger

equation.

Thus, we see, that the wave function is a method of description of any nondissipa-

tive fluid, but this fact was discovered only in 1999 [3], and this fact is not interesting

for hydrodynamists, because the internal energy depending on (∇ρ)2 is not used at a


description of usual fluids. As concerns physicists, dealing with quantum mechanics, a

linear dynamic equation in terms of the wave function is very attractive, even if it is not

known, what is the wave function.

But why is the quantum particle described as a continuous medium, which has infinite

number of the freedom degrees? The quantum particle is a nondeterministic (stochastic)

particle. Dynamic equations do not exist for a single stochastic particle. One can describe

only a mean motion of a stochastic particle. To describe the mean motion of a stochastic

particle, one considers an int-ensemble (gas) of stochastic particles, i.e. a set E [Sst] of

N (N → ∞) identical stochastic particles Sst. The int-ensemble E [Sst] is a continuous

medium at N → ∞, and there are dynamic equations for E [Sst].

For instance, a gas as a continuous medium is a dynamic system. Molecules of the

gas move stochastically, because of collisions. Equations of the gas dynamics describe

only a mean motion of molecules. Exact motion of molecules remains unknown at such

a description. To obtain a description of the mean motion of molecules, it is sufficient to

know only mean energy of molecules. In the case of ideal gas the mean energy of the gas

is E = 32kTρ, where T is the gas temperature, k is the Boltzmann constant, and ρ is the

gas density. In the case of quantum particle the mean energy is defined by the relation

(10), where vdif is the mean diffusion velocity. It means that the motion of a quantum

particle deflects from the rectilinear motion uniformly in all directions. In the case of a

gas a more detailed information on the velocity distribution (Maxwell distribution) is not

necessary for description of the mean motion of the gas molecules. The same is valid for

description of the mean motion of stochastic particles.

Note, that solving equations of the gas dynamics, one cannot determine the mean value

〈F (x, v)〉 of arbitrary function F (x, v) of coordinates and velocity of gas molecules. One

can determine only the mean value 〈f (x)〉 of the arbitrary function f (x) of coordinates,

and mean values of additive variables such as momentum p = mv , energy E = mv2/2,

and angular momentum L = p × x. For determination of other mean values, one needs

to know the distribution function f (x, v), which cannot be determined from the classical

equations of the gas dynamics. The same is valid in the case of quantum mechanics, when

one has the Schrodinger equation instead of the gas dynamics equations. The formula

for calculation of mean values in quantum mechanics

〈F (x, p)〉 =∫ψ∗ (x)F (x,− i�∇)ψ (x) dx (11)

is valid only for f (x), p , E = p2/2m, and L = p × x. However, according to principles

of quantum mechanics the formula (11) is considered to be valid for arbitrary function

F (x, p). The von Neumann’s theorem [6] on incompatibility of hidden variables with

principles of quantum mechanics is founded on the formula (11), which is considered to

be valid for arbitrary functions F .


3. Statistical Ensemble and Int-ensemble of Charged Particles

Let us consider statistical ensemble E [Sd] of deterministic charged particles Sd. The

action for E [Sd] has the form

E [Sd] : A [x] =

∫ξ0

∫Vξ

(−mc

√glkxlxk − e

cAlx

l)d4ξ, xi =

∂xi

∂ξ0(1)

where ξ = {ξ0, ξ1, ξ2, ξ3} are independent variables, and x = {x0 (ξ) , x1 (ξ) , x2 (ξ) , x3 (ξ)},x = x (ξ) are dependent variables. The quantity ξ0 is an evolutional parameter along the

world line of a particle. The quantity Al is the 4-potential of the electromagnetic field.

The electromagnetic field Al is an external field, and via electromagnetic field there are

no interaction between the particles.

As far as the particles are deterministic, there are dynamic equations for each single

particle. They are obtained as a result of variation with respect to xl

mcd

dξ0

glkxk (ξ)√

xs (ξ) xs (ξ)+e

c(∂lAk (x) − ∂kAl (x)) x

k (ξ) = 0, ξ =const (2)

Let us imagine that, particles interact via some force field κl, l = 0, 1, 2, 3, which

changes the particle mass m

m2 → M2 (x) = m2 +�2

c2(gklκ

kκl + ∂lκl), ∂l ≡ ∂

∂xl(3)

The force field κl, l = 0, 1, 2, 3 acts on the particle mass m, transforming it into effective

mass M . Here the κ-field κl = κl (x) = {κ0 (x) , κ1 (x) , κ2 (x) , κ3 (x)}.Introduction of the κ-field in the action (1) turns the deterministic particle Sd into a

stochastic particle Sst. The action (1) takes the form

E [Sst] : A [x , κ] =

∫ξ0

∫Vξ

(−mcK

√glkxlxk − e

cAlx

l)d4ξ, xi =

∂xi

∂ξ0(4)

K =M

m=

√1 + λ2 (κlκl + ∂lκl), λ =

�

mc, ∂l ≡ ∂

∂xl(5)

Here λ = �

mcis the Compton wave length. After introduction of interaction between par-

ticles the statistical ensemble E [Sd] ceases to be a statistical ensemble, because elements

(particles) of the statistical ensemble are to be independent by definition. Interaction of

particles violates their independence. The statistical ensemble turns to int-ensemble, i.e.

a set of identical interacting particles.

If one tries to obtain the action for a single particle, removing integration over d3ξ

one obtains

Sd : A [x , κ] =

∫ (−mcK

√glkxlxk − e

cAlx

l)dξ0, xi =

∂xi

∂ξ0(6)

The action (6) appears to be not well defined, because the K-factor (5) contains derivatives

of κ-field κl (x) in all directions of the space-time, whereas the action (6) admits only


derivatives along the world line. It means that the action (4) cannot describe a motion

of a single stochastic particle (if κl (x) �= 0). It can describe only an int-ensemble of

stochastic particles. However, if κl (x) ≡ 0, K-factor K ≡ 1 in (6), and the action (6)

becomes to be well defined. It can describe a single deterministic particle. In fact the

particles, described by the action (4), interact between themselves via the κ-field.

After introduction of interaction between the particles of E [Sd] the statistical ensemble

E [Sd] turns to int-ensemble. As we have mentioned, E [Sst] is not a statistical ensemble,

because particles Sst of the int-ensemble E [Sst] interact between themselves. Introduc-

tion of interaction between independent particles of the statistical ensemble is a method

of description of stochastic particles. Different force fields of interaction correspond to

different internal energy of continuous medium of the int-ensemble E [Sst].

In the ideal gas this interaction is described by the collision integral, describing colli-

sions between molecules. This integral describes a real interaction between real molecules.

If there is only one molecule, its motion is deterministic. In the case of stochastic parti-

cles the particle interaction is fictitious in the sense, that the particle motion is stochastic

even in the case of one particle. The reason of sochasticity may be internal reason of a

single particle, but not an interaction with other particles. Nevertheless, the stochasticity

is described as an interaction of a deterministic particle with other deterministic particles

of the int-ensemble. The fact is that the classical dynamics is a dynamics of deterministic

particles. Considering stochastic particles as interacting deterministic particles, one can

describe a motion of stochastic particles by methods of the classical dynamics. Re-

ducing the stochastic particle motion to a motion of interacting deterministic particles,

one uses the form of the κ-field for classification of forms of the particle stochasticity. In

the classical gas dynamics the stochasticity is presented by the internal energy E of the

gas .

At first, we consider the nonrelativistic case of the action (4), (5), when component

κ0 � |κ|. In this case, expanding radical in (4), and setting ξ0 = t, one obtains

AE[Sst] [x ,κ] =

∫t

∫Vξ

(−mc2 + m

2

(dx

dt

)2

+�2

2mκ2 +

�2

2m∇κ − e

cA0 − e

cAdx

dt

)dtd3ξ

(7)

where x = x (t, ξ) = {x1, x2, x3} and κ = κ (t, x) = {κ1, κ2, κ3} , A0 = A0 (t, x), A = A (t, x).

Let us introduce designation

κ (t, x)= − �

mu (t, x) (8)

where u is the mean velocity of a particle in the int-ensemble E [Sst]. Let us set eA0/c =

V (t, x), Aα = 0, α = 1, 2, 3. The action (7) takes the form.

AE[Sst] [x, u ] =∫t

∫Vξ

{m

2

(dx

dt

)2

+m

2u2 − �

2∇u−V

}dtdξ, (9)

The first term of (7) is omitted, because it does not contribute to dynamic equations.

The variable x = x (t, ξ) describes the regular component of the particle velocity. The


variable u = u (t, x) describes the mean value of the stochastic velocity component. The

first term of (9) describes the kinetic energy of regular motion. The second term in (9)

describes the kinetic energy of the stochastic velocity component. The third term de-

scribes interaction between the stochastic component u (t, x) and the regular component

dx/dt. The operator

∇ =

{∂

∂x1,∂

∂x2,∂

∂x3

}(10)

is defined in the space of coordinates x. Dynamic equations for the dynamic system E [Sst]

are obtained as a result of variation of the action (9) with respect to dynamic variables

x and u .

Variation of (9) with respect to u gives

δAE[Sst] [x, u ] =∫ ∫

Vξ

{muδu − �

2∇δu

}dtdξ

=

∫ ∫Vx

{muδu − �

2∇δu

}∂ (ξ1, ξ2, ξ3)

∂ (x1, x2, x3)dtdx

=

∫ ∫Vx

δu

{muρ+

�

2∇ρ

}dtdx−

∫ ∮�

2ρδudtdS (11)

where

ρ =∂ (ξ1, ξ2, ξ3)

∂ (x1, x2, x3)=

(∂ (x1, x2, x3)

∂ (ξ1, ξ2, ξ3)

)−1(12)

We obtain the following dynamic equation

mρu +�

2∇ρ = 0, (13)

Variation of (9) with respect to x gives

md2x

dt2= ∇

(m

2u2 − �

2∇u

)(14)

Here d/dt means the substantial derivative with respect to time t

dF

dt≡ ∂ (F, ξ1, ξ2, ξ3)

∂ (t, ξ1, ξ2, ξ3)(15)

Resolving (13) with respect to u , we obtain the equation

u = − �

2m∇ ln ρ, (16)

which reminds the expression for the mean velocity of the Brownian particle with the

diffusion coefficient D = �/2m.


Eliminating the velocity u from dynamic equations (14) by means of (16), we obtain

the dynamic equations for the mean motion of the stochastic particle Sst

md2x

dt2= −∇V − ∇UB, UB = U

(ρ,∇ρ,∇2ρ

)=

�2

8m

(∇ρ)2

ρ2− �

2

4m

∇2ρ

ρ(17)

Here ρ is considered to be function of t, x, and∇ is the gradient in the space of coordinates

x. The density ρ is defined by (12). UB is so called Bohm potential [2]. Equations (17) are

dynamic equations in the Lagrangian representations, where x = x (t, ξ) , ξ = {ξ1, ξ2, ξ3}To transform dynamic equations to the Euler representation, where dependent dy-

namic variables x ≡ v (t, x) , ξ = ξ (t, x), ρ = ρ (t, x), one should consider the transfor-

mation Jacobian

J = J(ξi,k

)=

∂ (ξ0, ξ1, ξ2, ξ3)

∂ (x0, x1, x2, x3)= det

∣∣∣∣ξi,k∣∣∣∣ , i, k,= 0, 1, 2, 3, ξi,k ≡ ∂ξi∂xk

(18)

After transformation one should set ξ0 = t, x0 = t. One obtains

∂J

∂ξ0,α=∂ (xα, ξ1, ξ2, ξ3)

∂ (x0, x1, x2, x3)=∂ (xα, ξ1, ξ2, ξ3)

∂ (ξ0, ξ1, ξ2, ξ3)

∂ (ξ0, ξ1, ξ2, ξ3)

∂ (x0, x1, x2, x3)(19)

Setting in (19) ξ0 = t, x0 = t, one obtains

∂J

∂ξ0,α=∂ (xα, ξ1, ξ2, ξ3)

∂ (t, ξ1, ξ2, ξ3)

∂ (t, ξ1, ξ2, ξ3)

∂ (t, x1, x2, x3)(20)

According to (15) and (12)

v =dx

dt=∂ (xα, ξ1, ξ2, ξ3)

∂ (t, ξ1, ξ2, ξ3), ρ =

∂ (t, ξ1, ξ2, ξ3)

∂ (t, x1, x2, x3)=∂ (ξ1, ξ2, ξ3)

∂ (x1, x2, x3)(21)

It follows from (20), (21), that According to (12)

∂J

∂ξ0,0= ρ,

∂J

∂ξ0,α= ρvα, α = 1, 2, 3 (22)

Using identity∂

∂xk∂J

∂ξ0,k=

∂

∂x0∂J

∂ξ0,0+

∂

∂xα∂J

∂ξ0,α≡ 0 (23)

and relations (22) one obtains the continuity equation

∂ρ

∂t+∇ (ρv) = 0 (24)

The equation (17) takes the form

dv

dt=∂v

∂t+ (v∇) v = −∇V − ∇UB (25)

Hydrodynamic form (24), (25) of the Schrodinger equation has been obtained by

Madelung [1]. It has been used by Bohm [2] for the case of nonrotational flow, when

v = ∇ϕ, and equation (25) can be written in the form

∂∇ϕ

∂t+∇(∇ϕ)2

2= −∇V − ∇UB (26)

In the case of a rotational flow the equation (25) is not equivalent to the Schrodinger

equation.


4. Relativistic Case

Let us return to the action (4), (5) and show, that int-ensemble E [Sst] can be described in

terms of wave function. We shall consider variables ξ = ξ (x) in (4) as dependent variables

and variables x as independent variables. After manipulations with the transformation

Jacobian (18) and introduction of the wave function (7), (9) one obtains the action (4),

(5) in the form (See details in Appenix)

A [ψ, ψ∗] =∫ {(

i�∂k +e

cAk

)ψ∗

(−i�∂k + e

cAk

)ψ −m2c2ρ− �

2

4(∂lsα)

(∂lsα

)ρ

}d4x

(1)

where

ρ = ψ∗ψ, sα =ψ∗σαψ

ρ, α = 1, 2, 3 (2)

ψ =(ψ1ψ2

), ψ∗ = (ψ∗1, ψ

∗2) , (3)

σα are 2 × 2 Pauli matrices

σ1 =

⎛⎜⎝ 0 1

1 0

⎞⎟⎠ , σ2 =

⎛⎜⎝ 0 −ii 0

⎞⎟⎠ , σ3 =

⎛⎜⎝ 1 0

0 −1

⎞⎟⎠ , (4)

Variations with respect to ψ∗ leads to dynamic equation

(−i�∂k + e

cAk

)(−i�∂k + e

cAk

)ψ −

(m2c2 +

�2

4(∂lsα)

(∂lsα

))ψ

= −�2∂l

(ρ∂lsα

)2ρ

(σα − sα)ψ (5)

It is nonlinear, generally speaking. However, let the wave function be one-component, or

components ψ1 and ψ2 be linear dependent ψ1 = aψ2, a = const. Then s = const and

∂isα = 0. In this case nonlinear terms in (5) vanish, and dynamic equation (5) turns to

the Klein-Gordon equation(−i�∂k + e

cAk

)(−i�∂k + e

cAk

)ψ −m2c2ψ = 0 (6)

5. Discussion

Considering dynamics of continuous medium as a mathematical instrument for description

of stochastic particles, one can found the quantum mechanics and explain the origin of

the wave function as a natural means of a fluid description.

Describing a gas by means of classical gas dynamics, one cannot say anything on

structure and arrangement of the gas, because the gas dynamic equations are simply

conservation laws of the matter and of its energy-momentum. However, adding the

collision integral and describing the gas motion by means of a kinetic equation, one may


determine the distribution function and learn the gas motion mechanism. Analogously, if

one determines the nature of the κ-field, one may to obtain information on arrangement

of elementary particles. Axiomatic conception of quantum mechanics does not admit one

to obtain any information on the elementary particles arrangement. The contemporary

quantum theory describes elementary particles as pointlike objects provided by a set of

quantum numbers. According to quantum theory elementary particles have no internal

structure. When internal structure of hardrons has been discovered experimentally, the

quantum theory explained this fact by existence of pointlike particles (quarks), which

cannot exist singly outside the hardron. However, existence of quarks only inside hardrons

shows that quarks are elements of the hardron structure, but the quantum theory cannot

accept such a supposition. It considers quarks as single particles.

This reminds situation with investigations of chemical elements, where there are two

approaches: (1) empirical approach and (2) structural approach. The empirical approach

is used by chemists. They are not interested in the atom arrangement. They are inter-

ested only in systematization of chemical elements. Chemists ascribe some characteristic

numbers (atomic weight, valency, etc...) to any chemical element and systematize the

chemical elements according to these numbers. Physicists use the structural approach,

which admits one to determine arrangement of atoms (nucleus, electronic envelope, etc.).

Using empirical approach of chemists, one could not create atomic energetics and atomic

weapon.

The quantum theory could explain arrangement of the atom, but it cannot explain

arrangement of elementary particles, because of its empirical approach. For instance, the

quantum mechanics describes the electron by the Dirac equation. In this presentation

the Dirac particle (electron) is a pointlike particle, having the mass m, charge e, spin

�/2, and magnetic moment μ = e�/2mc. These quantum numbers contain the quantum

constant �, even at the classical approach. The world line of a free Dirac particle is a

straight line.

The Dirac equation can be considered as an equation [7], describing some fluid. In

this case the world line of a free Dirac particle is a helix with timelike axis. The particle

rotation along the circles of the helix is a source of spin and of the magnetic moment,

which are not simple quantum numbers now. Such a connection between the quantum

numbers and the structure of the particle world line cannot be obtained in the framework

of the axiomatic conception of quantum theory.

In the fluid dynamics the formula of type of (11) for calculation of mean values

is valid only for some physical quantities. The axiomatic quantum mechanics expands

action of this formula on all physical quantities [6]. As a result one obtains the Neumann’s

theorem on hidden variables, which is not true, because the formula (11) is valid not for

all quantities , as it is supposed in the conditions of the theorem.

Linearity of dynamic equation for a certain kind of fluid is expanded to all dynamic

equations of quantum mechanics without sufficient foundation. As a result one obtains

the linearity principle instead of a special case of the fluid flow.

The force field κ determines properties of the fluid, in terms of which the motion of


stochastic particles is described. One may said, that the κ-field describes properties of

the stochastic particle. Thus, quantum effects are described by the some force field κ, but

not by quantum principles and not by a change of physical quantities by some operators

or matrices.

Changing the effective particle mass (3), the κ-field may make M2 to be negative. It

is a necessary condition of pair production (change of the world line direction in time).

This change of M2 needs a strong κ-field. The κ-field can be strong enough, if it is an

external force field, because the internal κ-field of a particle, which is responsible for

quantum effects, is too weak for such a change of the mass [8]. In the axiomatic quantum

mechanics the internal κ-field is included in the wave function, whereas the external

κ-field is not used. As a result the pair production effect in the quantum field theory

is a corollary of inconsistency of the second quantization procedure in the relativistic

case. [9, 10]. Situation with the second quantization of the nonlinear Klein-Gordon

equation looks as follows. At the conventional second quantization in the relativistic

case the wave function contains both annihilation operators and creation operators. As

a result the Hamiltonian H coincides with the energy E of the system for free particles.

Such a coincidence takes place in the nonrelativistic case (for instance, in the case of

the second quantization of the Schrodinger equation). In the relativistic case such a

coincidence takes place only in the absence of the pair generation. In the case, when the

pair production is possible the conventional method of the second quantization leads to

nonstationary vacuum state. This fact is explained usually in the sense, that vacuum

state does not contain particles, but it contains virtual particles. In reality nonstationary

vacuum is a corollary of inconsistent statement of the the second quantization problem.

At a inconsistent statement of the problem one may obtain any results, which one wants.

It is necessary only to have sufficient ingenuity.

Introducing the distribution function and kinetic equation for it, one obtains a more

detailed information on the gas motion mechanism. In a like way considering a source

of the κ-field, one can obtain a more detailed information on the elementary particle

arrangement [11].

6. Appendix. Transformation of the Action to

Representation in Terms of Wave Function

Let us consider variables ξ = ξ (x) in (4) as dependent variables and variables x as

independent variables. Let the Jacobian (18)

J =∂ (ξ0, ξ1, ξ2, ξ3)

∂ (x0, x1, x2, x3)= det

∣∣∣∣ξi,k∣∣∣∣ , ξi,k ≡ ∂kξi ≡∂ξi∂xk

, i, k = 0, 1, 2, 3 (1)

be considered to be a multilinear function of ξi,k. Then

d4ξ = Jd4x, xi ≡ dxi

dξ0≡ ∂ (xi, ξ1, ξ2, ξ3)

∂ (ξ0, ξ1, ξ2, ξ3)= J−1

∂J

∂ξ0,i(2)


After transformation to dependent variables ξ the action (4) takes the form

A [ξ, κ] =

∫ {−mcK

√gik

∂J

∂ξ0,i

∂J

∂ξ0,k− e

cAk

∂J

∂ξ0,k

}d4x, (3)

K =

√1 + λ2 (κlκl + ∂lκl), λ =

�

mc, (4)

Now variables ξ and κ are considered as functions of independent variables x.

Let us introduce new variables

jk =∂J

∂ξ0,k, k = 0, 1, 2, 3 (5)

by means of Lagrange multipliers pk

A [ξ, κ, j, p] =

∫ {−mcK

√gikjijk − e

cAkj

k + pk

(∂J

∂ξ0,k− jk

)}d4x, (6)

Variation with respect to ξi gives

δAδξi

= −∂l(pk

∂2J

∂ξ0,k∂ξi,l

)= 0, i = 0, 1, 2, 3 (7)

Using identities∂2J

∂ξ0,k∂ξi,l≡ J−1

(∂J

∂ξ0,k

∂J

∂ξi,l− ∂J

∂ξ0,l

∂J

∂ξi,k

)(8)

∂J

∂ξi,lξk,l ≡ Jδik, ∂l

∂J

∂ξi,l≡ 0 ∂l

∂2J

∂ξ0,k∂ξi,l≡ 0 (9)

one can test by direct substitution that the general solution of linear equations (7) has

the form (5)

pk = b0 (∂kϕ+ gα (ξ) ∂kξα) , k = 0, 1, 2, 3 (10)

where b0 �= 0 is a constant, gα (ξ) , α = 1, 2, 3 are arbitrary functions of ξ = {ξ1, ξ2, ξ3},and ϕ is the dynamic variable ξ0, which ceases to be fictitious. Let us substitute (10) in

(6). The term of the form ∂J/∂ξ0,k∂kϕ is reduced to Jacobian and does not contribute

to dynamic equations. The terms of the form ξα,k∂J/∂ξ0,k vanish due to identities (9).

We obtain

A [ϕ, ξ, κ, j] =

∫ {−mcK

√gikjijk − jkπk

}d4x, (11)

where quantities πk are determined by the relations

πk = b0 (∂kϕ+ gα (ξ) ∂kξα) +e

cAk, k = 0, 1, 2, 3 (12)

Integration of (7) in the form (10) is that integration which admits to introduce a

wave function. Note that coefficients in the system of equations (7) at derivatives of pkare constructed of minors of the Jacobian (1). It is the circumstance that admits one to

produce a formal general integration.


Variation of (11) with respect to κl gives

δAδκl

= −λ2mc

√gikjijk

Kκl + ∂l

λ2mc√gikjijk

2K= 0, λ =

�

mc(13)

It can be written in the form

κl = ∂lκ =1

2∂l ln ρ, e2κ =

ρ

ρ0≡

√jsjs

ρ0K, ρ =

√jsjs

K(14)

where the variable κ is potential of the κ-field κi and ρ0 =const is the integration constant.

Substituting (4) in (14), we obtain dynamic equation for κ

�2(∂lκ · ∂lκ+ ∂l∂

lκ)= m2c2

e−4κjsjs

ρ20−m2c2 (15)

Variation of (11) with respect to jk gives

πk = − mcKjk√glsjljs

(16)

or

πkgklπl = m2c2K2 (17)

Substituting√jsjs/K from the second equation (14) in (16), we obtain

jk = − ρ0mc

e2κπk, (18)

Now we eliminate the variables jk from the action (11), using relation (18) and (14).

We obtain

A [ϕ, ξ, κ] =

∫ρ0e

2κ{−m2c2K2 + πkπk

}d4x, (19)

where πk is determined by the relation (12). Using expression (5) for K, the first term

of the action (19) can be transformed as follows.

−m2c2e2κK2 = −m2c2e2κ(1 + λ2

(∂lκ∂

lκ+ ∂l∂lκ))

= −m2c2e2κ + �2e2κ∂lκ∂

lκ− �2

2∂l∂

le2κ

Let us take into account that the last term has the form of divergence. It does not

contribute to dynamic equations and can be omitted. Omitting this term, we obtain

A [ϕ, ξ, κ] =

∫ρ0e

2κ{−m2c2 + �

2∂lκ∂lκ+ πkπk

}d4x, (20)

Here πk is defined by the relation (12), where the integration constant b0 is chosen in the

form b0 = �

πk = � (∂kϕ+ gα (ξ) ∂kξα) +e

cAk, k = 0, 1, 2, 3 (21)


Instead of dynamic variables ϕ, ξ, κ we introduce n-component complex function (7),

(9)

ψ = {ψα} ={√

ρeiϕwα (ξ)}=

{√ρ0e

κ+iϕwα (ξ)}, α = 1, 2, ...n (22)

Here wα are functions of only ξ = {ξ1, ξ2, ξ3}, having the following properties

α=n∑α=1

w∗αwα = 1, − i

2

α=n∑α=1

(w∗α

∂wα

∂ξβ− ∂w∗α∂ξβ

wα

)= gβ (ξ) (23)

where (∗) denotes the complex conjugation. The number n of components of the wave

function ψ depends on the functions gβ (ξ). The number n is chosen in such a way, that

equations (23) have a solution. Then we obtain

ψ∗ψ ≡α=n∑α=1

ψ∗αψα = ρ = ρ0e2κ, ∂lκ =

∂l (ψ∗ψ)

2ψ∗ψ(24)

πk = − i� (ψ∗∂kψ − ∂kψ

∗ · ψ)2ψ∗ψ

+e

cAk, k = 0, 1, 2, 3 (25)

Substituting relations (24), (25) in (20), we obtain the action, written in terms of the

wave function ψ

A [ψ, ψ∗] =∫ {[

i� (ψ∗∂kψ − ∂kψ∗ · ψ)

2ψ∗ψ− e

cAk

] [i�

(ψ∗∂kψ − ∂kψ∗ · ψ)

2ψ∗ψ− e

cAk

]

+ �2∂l (ψ

∗ψ) ∂l (ψ∗ψ)

4 (ψ∗ψ)2−m2c2

}ψ∗ψd4x (26)

Let us use the identity

(ψ∗∂lψ − ∂lψ∗ · ψ) (ψ∗∂lψ − ∂lψ∗ · ψ)

4ψ∗ψ+ ∂lψ

∗∂lψ

≡ ∂l (ψ∗ψ) ∂l (ψ∗ψ)4ψ∗ψ

+gls

2ψ∗ψ

α,β=n∑α,β=1

Q∗αβ,lQαβ,s (27)

where

Qαβ,l =1

ψ∗ψ

∣∣∣∣∣∣∣ψα ψβ

∂lψα ∂lψβ

∣∣∣∣∣∣∣ , Q∗αβ,l =1

ψ∗ψ

∣∣∣∣∣∣∣ψ∗α ψ∗β

∂lψ∗α ∂lψ

∗β

∣∣∣∣∣∣∣ (28)

Then we obtain

A [ψ, ψ∗] =∫ {(

i�∂k +e

cAk

)ψ∗

(−i�∂k + e

cAk

)ψ −m2c2ψ∗ψ

+�2

2

α,β=n∑α,β=1

glsQαβ,lQ∗αβ,sψ

∗ψ

}d4x (29)


Let us consider the case of nonrotational flow, when gα (ξ) = 0. In this case w1 = 1,

w2 = 0, and the function ψ has only one component. It follows from (28), that Qαβ,l = 0.

Then we obtain instead of (29)

A [ψ, ψ∗] =∫ {(

i�∂k +e

cAk

)ψ∗

(−i�∂k + e

cAk

)ψ −m2c2ψ∗ψ

}d4x (30)

Variation of the action (30) with respect to ψ∗ generates the Klein-Gordon equation(−i�∂k + e

cAk

)(−i�∂k + e

cAk

)ψ −m2c2ψ = 0 (31)

Thus, description in terms of the Klein-Gordon equation is a special case of the stochastic

particles description by means of the action (4), (5).

In the case, when the fluid flow is rotational, and the wave function ψ is two-

component, the identity (27) takes the form

(ψ∗∂lψ − ∂lψ∗ · ψ) (ψ∗∂lψ − ∂lψ∗ · ψ)

4ρ− (∂lρ)

(∂lρ

)4ρ

≡ −∂lψ∗∂lψ +1

4(∂lsα)

(∂lsα

)ρ (32)

where 3-vector s = {s1, s2, s3, } is defined by the relations

ρ = ψ∗ψ, sα =ψ∗σαψ

ρ, α = 1, 2, 3 (33)

ψ =(ψ1ψ2

), ψ∗ = (ψ∗1, ψ

∗2) , (34)

and Pauli matrices σ = {σ1, σ2, σ3} have the form (4). Note that 3-vectors s and σ are

vectors in the space Vξ of the Clebsch potentials ξ = {ξ1, ξ2, ξ3}. They transform as

vectors at the transformations

ξα → ξα = ξα (ξ) , α = 1, 2, 3,∂(ξ1, ξ2, ξ3

)∂ (ξ1, ξ2, ξ3)

�= 0 (35)

In general, transformations of Clebsch potentials ξ and those of coordinates x are

independent. However, the action (26) does not contain any reference to the Clebsch

potentials ξ and transformations (35) of ξ. If we consider only linear transformations of

space coordinates x

xα → xα = bα + ωα.βx

β, α = 1, 2, 3 (36)

nothing prevents from accompanying any transformation (36) with the similar transfor-

mation

ξα → ξα = bα + ωα.βξβ, α = 1, 2, 3 (37)

of Clebsch potentials ξ. The formulas for linear transformation of vectors and spinors in

Vx do not contain the coordinates x explicitly, and one can consider vectors and spinors

in Vξ as vectors and spinors in Vx, provided we consider linear transformations (36), (37)

always together.


Using identity (32), we obtain from (26)

A [ψ, ψ∗] =∫ {(

i�∂k +e

cAk

)ψ∗

(−i�∂k + e

cAk

)ψ −m2c2ρ− �

2

4(∂lsα)

(∂lsα

)ρ

}d4x

(38)

Dynamic equation, generated by the action (38), has the form

(−i�∂k + e

cAk

)(−i�∂k + e

cAk

)ψ −

(m2c2 +

�2

4(∂lsα)

(∂lsα

))ψ

= −�2∂l

(ρ∂lsα

)2ρ

(σα − sα)ψ (39)

The gradient of the unit 3-vector s = {s1, s2, s3} describes rotational component of

the fluid flow. If s = const, the dynamic equation (39) turns to the conventional Klein-

Gordon equation (31).

References

[1] E. Madelung, Z.Phys. 40, 322, (1926).

[2] D. Bohm, On possibility of the quantum mechanics interpretation on basis ofrepresentation on hidden variables, Phys.Rev. 85 , 166,(1952), 180,(1952).

[3] Yu.A. Rylov, Spin and wave function as attributes of ideal fluid. Journ. Math. Phys.40 , pp. 256 - 278, (1999).

[4] A. Clebsch, Uber eine allgemaine Transformation der hydrodynamischenGleichungen, J. reine angew. Math. 54 , 293-312 (1857).

[5] A. Clebsch, Ueber die Integration der hydrodynamischen Gleichungen, J. reineangew. Math. 56 , 1-10, (1859).

[6] J. Neumann, Mathematische Grundlagen der Quantenmechanik, Berlin, 1932.

[7] Yu.A. Rylov, Dirac equation in terms of hydrodynamic variables. Advances in AppliedClifford Algebras, 5, pp 1-40, (1995)) See also e-print /1101.5868.

[8] Yu.A.Rylov, Mechanism of pair production in classical dynamics, submitted to EJTP(2015).

[9] Yu.A. Rylov, On connection between the energy-momentum vector and canonicalmomentum in relativistic mechanics. Teoretischeskaya i Matematischeskaya Fizika.2, 333-337.(1970) (in Russian). Theor. and Math. Phys. (USA), 5, 333, (1970)(trnslated from Russian).

[10] Yu.A. Rylov, On quantization of non-linear relativistic field without recourse toperturbation theory. Int. J. Theor. Phys. 6, 181-204, (1972).

[11] Yu. A. Rylov, The way to skeleton conception of elementary particles. Global J.ofScience Frontier Research, vol.14 , iss. 7, ver.1, 43-100, (2014).


Analytic Solution of the Algebraic EquationAssociated to the Ricci Tensor in Extended

Palatini Gravity

Gines R.Perez Teruel∗

Departamento de Fısica Teorica, Universidad de Valencia, Burjassot-46100,Valencia, Spain

Received 1 September 2014, Accepted 1 March 2015, Published 25 August 2015

Abstract: In this work we discuss the exact solution to the algebraic equation associated

to the Ricci tensor in the quadratic f(R,Q) extension of Palatini gravity. We show that an

exact solution always exists, and in the general case it can be found by matrix diagonalization.

Furthermore, the general implications of the solution are analysed in detail, including the

generation of an effective cosmological constant, and the recovery of the f(R) and f(Q) theories

as particular cases in the corresponding limit. In addition, it is proposed a power series expansion

of the solution which is successfully applied to the case of the electromagnetic field. We show

that this power series expansion may be useful to deal perturbatively with some problems in

the context of Palatini gravity.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Extended Palatini Gravity; Ricci Tensor; General Relativity; Analytical Solution

PACS (2010): 02.20.-a; 04.20.-q; 04.40.Nr; 98.80.Es;04.20.-q ;98.80.-k

1. Introduction

Einstein’s theory of general relativity (GR) represents one of the most impressive exer-

cises of human intellect. It implies a huge conceptual jump with respect to Newtonian

gravity in which the idea of gravitational force is reinterpreted in geometrical terms. The

theory has successfully passed numerous precision experimental tests. Its predictions

are in agreement with experiments in scales that range from millimeters to astronomical

units, scales in which weak and strong field phenomena can be observed[1]. The theory is

so successful in those regimes and scales that it is generally accepted that it should also

work at larger and shorter scales, and at weaker and stronger regimes.



This is, however, forcing us today to draw a picture of the universe that is not yet

supported by other independent observations. For instance, to explain the rotation curves

of spiral galaxies, we must accept the existence of vast amounts of unseen matter sur-

rounding those galaxies. A similar situation occurs with the analysis of the light emitted

by distant type-Ia supernovae and some properties of the distribution of matter and ra-

diation at large scales. To make sense of those observations within the framework of GR,

we must accept the existence of yet another source of energy with repulsive gravitational

properties[2]. Together those unseen (or dark) sources of matter and energy are found to

make up to 96% of the total energy of the observable universe! This huge discrepancy

between the gravitationally estimated amounts of matter and energy and the direct mea-

surements via electromagnetic radiation motivates the search for alternative theories of

gravity which can account for the large scale dynamics and structure without the need

for dark matter and/or dark energy.

The extrapolation of the dynamics of GR to the very strong field regime indicates

that the Universe began at a big bang singularity and that the death of a sufficiently

massive star unavoidably leads to the formation of a black hole singularity. Since sin-

gularities generally signal the breakdown of a theory, it is generally accepted that at

high enough energies the dynamics of GR should be replaced by some improved theory.

In this sense, a perturbative approach to quantum gravity indicates that the Einstein-

Hilbert Lagrangian must be supplemented by quadratic curvature terms to render the

theory renormalizable[10, 11]. String theories also regard GR as the low energy limit

of a theory that should pick up increasing corrective terms at high energies[12]. Loop

quantum gravity [13] predicts that the continuum space-time of GR is replaced by a

quantum geometry in which areas and volumes are quantized in bits of an elementary

unit of order the Planck scale. The low energy limit of this theory should also recover

the classical dynamics of GR with corrections signaling the discreteness of the space-time.

The above discussion shows that there are theoretical and phenomenological reasons

to explore the dynamics of alternative theories of gravity, which has led to a burst of activ-

ity in the last years. Among these attempts to go beyond Einstein’s theory, the Palatini

(or metric-affine) approach is particularly promising. By just relaxing the Riemannian

condition on the metric, i.e., by considering that metric and connection are independent

fields, one finds a number of new interesting results and insights, such as new mecha-

nisms to generate an effective cosmological constant or new topological structures in the

interior of black holes. The naturalness of these results contrasts with the difficulties

found within the more standard Riemannian approach of the original formulation of GR,

where the (Levi-Civita) connection is an object derived from the metric and, therefore,

relegated to a secondary role in the structure of the theory.

In the context of the Palatini formalism, there are currently several interesting lines of


research. First we find the so-called f(R) theories, which are constructed using the Ricci

scalar R ≡ Rμνgμν as a basic element of the gravity Lagrangian. These theories have been

thoroughly studied in the literature [18],[20],[22, 23, 24]. The interest on this theories

stems from many good reasons. In the first place, this formalism provides a very elegant

way to derive an effective cosmological constant and, therefore, a novel way to justify the

observed cosmic speedup. Sharing the philosophy of f(R) theories, we also find theories

of the form f(Q), where Q ≡ RμνRμν is the square of the Ricci tensor [21]. A more

general framework is achieved in the context of f(R,Q) theories, whose phenomenology

is much richer than that of the simpler f(R) and f(Q) theories individually. In particular,

quadratic extensions of GR of the f(R,Q) type allow to explore the potential effects that

a minimum length (such as the Planck length) could have on relativistic field theories

[15], produce consistent cosmological models that avoid the big bang singularity by means

of a cosmic bounce [27], and modify the internal structure of black holes in such a way

that their central singularity is replaced by a geometric wormhole structure[31, 32] that

may be free of curvature divergences. Similar properties are also found in the so-called

Born-Infeld-type gravity theory [40, 41, 42] and its extensions [43, 44, 45], which are not

of the f(R,Q) form and contain up to quartic powers of the Ricci tensor.

Indeed, in recent works [35, 40], it has been found that spherically symmetric, electro-

vacuum solutions can be naturally interpreted as geons, i.e., as self-gravitating solutions

of the gravitational-electromagnetic system of equations without sources. This is possible

thanks to the nontrivial topology of the resulting space-time, which through the formation

of a wormhole allows to define electric charges without requiring the explicit existence

of point-like sources of the electric field. In this scenario, massive black holes are almost

identical in their macroscopic properties to those found in GR. However, new relevant

structures arise in the lowest band of the mass and charge spectrum (microscopic regime).

In particular, below a certain critical charge qc = eNc, with Nc =√

2/αem ≈ 16.55, where

αem is the fine structure constant and e the electron charge, one finds a set of solutions

with no event horizon and with smooth curvature invariants everywhere. Moreover, the

mass of these solutions can be exactly identified with the energy stored in the electric

field and their action (evaluated on the solutions) coincides with that of a massive point-

like particle at rest. The topological character of their charge, therefore, makes these

solutions stable against arbitrary perturbations of the metric as long as the topology

does not change. On the other hand, the absence of an event horizon makes these con-

figurations stable against Hawking decay (regular solutions with an event horizon also

exist, though they are unstable). Furthermore, the mass spectrum of the regular geons,

the ones without curvature divergences at the wormhole throat, can be lowered from the

Planck scale down to the Gev scale, which shows that the Planck scale phenomenology

of Palatini gravity can be tested and constrained with currently available experiments [37].

The paper is organized as follows. In sec. II we provide a review of f(R,Q) theories

paying special attention to their algebraic structure. In sec. III we present new methods

to deal with f(R,Q) theories, including the analysis of the general solution of the matrix


equation associated to the Ricci tensor. In particular, we show that this equation always

possesses an exact solution that can be obtained by matrix diagonalization. Furthermore,

we show that not only the solution generates the correct f(R) and f(Q) theories in their

respective limits, but does also provide by a direct computation the effective cosmological

constant Λeff , when it is evaluated in vacuum (T μα=0). In addition, we also develop the

power series representation of the solution, which is applied to some particular models.

This power series expansion may be useful to perform perturbative calculations within

the context of quadratic Palatini gravity.

2. Palatini f(R,Q) Theories

Palatini f(R,Q) theories are defined in terms of the general action

S[g,Γ, ψm] =1

2κ2

∫d4x

√−gf(R,Q) + Sm[g, ψm] , (1)

where gαβ is the space-time metric, Sm[g, ψm] is the matter action, with the matter fields

denoted collectively by ψm, κ2 ≡ 8πG, R = gμνRμν is the Ricci scalar, Q = gμαgνβRμνRαβ

is the Ricci-squared scalar, and Rαβμν = ∂μΓ

ανβ − ∂νΓ

αμβ +Γα

μλΓλνβ −Γα

νλΓλμβ is the Riemann

tensor. The connection Γαβγ has no a priori relation with the metric (Palatini formal-

ism) and must be determined by the theory through the corresponding field equations.

Variation of (1) with respect to metric and connection[14][15], leads to the field equations

fRR(μν) − 1

2fgμν + 2fQR(μα)R

αν = κ2Tμν , (2)

∇α

[√−g(fRgβγ + 2fQR(βγ))

]= 0 , (3)

where fR ≡ ∂Rf , fQ ≡ ∂Qf and R(μν) denotes de symmetric part of the Ricci tensor.

Assuming vanishing torsion, i.e., Γλ[μν] = 0, the Ricci tensor turns out to be symmetric,

i.e., R[μν] = 0. Thus, in what follows symmetry in the indices of Rμν will be implicitly

understood.

2.1 Limit to Einstein’s Theory

Einstein’s GR is automatically recovered from the previous equations when f(R,Q) = R

(Einstein-Hilbert Lagrangian). Indeed, f(R,Q) = R implies fR = 1, fQ = 0. Substituing

these values in equation 2 we obtain the following result

Rμν − 1

2Rgμν = κ2Tμν (4)

which are the famous field equations of GR. On the other hand, equation 3 becomes

∇α

[√−ggβγ ] = 0 , (5)


We can decompose this equation using the product rule

∇α(√−g)gβγ + √−g(∇αg

βγ) = 0 (6)

The explicit expressions for the covariant derivatives of a type (2,0) tensor field gβγ , and

of a tensorial density√−g are the following

∇αgβγ = ∂αg

βγ + Γβραg

ργ + Γγραg

βρ

∇α(√−g) = ∂α(

√−g) − Γλαλ

√−g (7)

Substituing these results in (6) we obtain(∂α(

√−g) − Γλαλ

√−g)gβγ +

√−g(∂αg

βγ + Γβραg

ργ + Γγραg

βρ)= 0 (8)

The derivative of the determinant can be related to the metric tensor by means of the

equation

g−1∂αg = gμν∂αgμν (9)

Using this identity and contracting (8) with gβγ yields

(12gμν∂αgμν − Γλ

λα

)4 + gβγ

(∂αg

βγ + Γβραg

ργ + Γγραg

βρ)= 0 (10)

Where gβγgβγ = 4. The first term can be removed from the equation assuming the relation

12gμν∂αgμν = Γλ

λα. Repeating the remaining equation three times with a convenient

permutation of indices, it is easy to obtain the linear combination that solves the equation.

This relation turns out to be the following

Γβρα =

1

2gβμ (∂αgμρ + ∂ρgμα − ∂μgρα) (11)

which is the Levi-Civita connection of GR. Then, we have proved that equation (5) admits

the symmetric2 metric-compatible Levi-Civita connection as a solution. Therefore, in the

particular case f(R,Q) = R, we consistently recover Einstein’s theory of gravity.

2.2 Structure of the Palatini Field Equations

The general algorithm to attack equations (2,3) was described elsewhere [14] (see also

[21] for f(Q) theories), and consists in several steps. First, we need to find a relation

between R(μν) and the matter sources. Rewritting (2), using P νμ = Rμαg

αν we find

2fQPαμ P

να + fRP

νμ − 1

2fδνμ = κ2T ν

μ (12)

This can be seen as a matrix equation, which establishes an algebraic relation P νμ =

P νμ (T

βα ). Once the solution of (12) is known, (3) can be written is terms of gμν and

2 Recall that the symmetry of the coefficients of the connection, Γβρα = Γβ

αρ is equivalent to set the

torsion to zero


the matter, which allows to find a solution for the connection by means of algebraic

manipulations. In particular, the strategy consists of transforming equation (3), into

something similar to (5), which we have proved in the previous subsection that admits

the Levi-Civita connection as a solution. Then, following this reasoning, it seems natural

to propose the ansatz √−g(fRg

βγ + 2fQRβγ

)=

√−hhβγ (13)

with this ansatz, equation (3) acquires the form ∇α[√−hhβγ ] = 0, which is formally

identical to (5), and leads to the Levi-Civita connection for the auxiliary metric hβγ .

In order to find the explicit relation between the auxiliary metric hβγ and the physical

metric gβγ , we need to compute the determimant of the left-and the right hand sides of

(13),which give h = g det(fRI + 2fQP ). Once we know the explicit expression for P we

will be able to compute this determinant. In any case, we have the formal expression [14]

h−1 =g−1Σ√det Σ

(14)

where we have defined the matrix

Σαν = fRδ

αν + 2fQR

αν (15)

Taking the inverse of the matrix (14), we find h = (√

det Σ)Σ−1g.In order to better understand the dynamics of the theory, it is convenient to employ

the relation (14) to obtain a compact form for the field equation (2). Indeed, substituing

2fQRαν = Σα

ν − fRδαν in (2) we find

RμαΣαν =

f

2gμν + κ2Tμν (16)

The contraction of this equation with gνλ gives

RμαΣαν g

νλ =f

2δλμ + κ2T λ

μ (17)

Making use of the nonconformal relation between both metrics (14), it is easy to see

that Σαν g

νλ = hαλ√

det Σ. This provides,

Rμαhαλ =

1√det Σ

(f2δλμ + κ2T λ

μ

)(18)

Finally, taking into account the contraction Rμαhαλ = Rλ

μ(h) we arrive to the compact

expression for the field equations

Rλμ(h) =

1√det Σ

(f2δλμ + κ2T λ

μ

)(19)


3. Analytic Solution for P (T ). Generation of an Effective Cos-

mological Constant

It is worth noting that an explicit analytic solution for Eq. (12) can be found with the

help of linear algebra, in particular matrix algebra. Indeed, given a quadratic matrix

equation

AX2 + BX + C = 0 (20)

A general analytic solution only exists if the matrix coefficients satisfy the conditions:

A = I, [B, C] = 0 and B2 − 4C has a square root. If these conditions are satisfied, the

solution is given by the following expression [17]

X = −1

2B +

1

2

√B2 − 4C (21)

Rewriting Eq. (12) in matrix form we obtain

P 2 +fR2fQ

P − 1

2fQ

(f2I + κ2T

)= 0 (22)

The comparison with Eq.(20) allows us to establish the following identification

A ≡ I B ≡ fR2fQ

I C ≡ − 1

2fQ

(f2I − κ2T

)(23)

Then, the first two conditions are automatically satisfied, and only the third one needs

a detailed analysis. In our case, B2 − 4C = αI + βT , where the coefficients α, β are

functions that depend on the gravity Lagrangian f(R,Q), and are given by

α =1

4

(f 2R + 4fQf

)(24)

β = 2κ2fQ (25)

Therefore, if the matrix αI + βT has a square root, an explicit solution will always

exist. In particular, if T is a diagonal matrix, the linear combination αI + βT will be a

diagonal matrix as well, and the square root of a diagonal matrix can be easily computed.

However, if T is not diagonal, the problem is reduced to the task of diagonalizing the

matrix αI + βT . Note that the matrix αI + βT is symmetric, and we know that a

symmetric matrix is always diagonalizable. We can therefore conclude that the matrix

equation (22) always admits an exact solution, and it is given by (fQ �= 0)

P (T ) = − 1

4fQ

(fRI − 2

√αI + βT

)(26)

On the other hand, using this solution we can write the explicit algebraic equation for Σ

purely in terms of the metric and the matter sources

Σ(T ) ≡ fRI + 2fQP =fR2I +

√αI + βT (27)


It is worth noting that (26) in vacuum (T αβ = 0) boils down to the equations of GR

with the possibility of an effective cosmological constant (depending on the form of theLagrangian). Indeed, to see how this interesting fact emerges directly from our solution,

we only have to set T to zero

P νμ = − 1

4fQ

(fRδ

νμ − 2

√αδνμ + βT ν

μ

)=

− fR4fQ

(1 − 2

fR

√α

)δνμ = − fR

4fQ

(1 −

√1 +

4fQf

f 2R

)δνμ ≡ Λ(R,Q)δνμ (28)

where

Λ(R,Q) = − fR4fQ

(1 −

√1 +

4fQf

f 2R

)(29)

Agrees with the result obtained in Ref. [15]. This equation can be employed to compute

the traces R0 ≡ P μμ |vac= 4Λ(R0, Q0) and Q0 = P β

μPμβ |vac= 4Λ(R0, Q0)

2, which lead

to the standard relation Q0 = R20/4 of de Sitter spacetime. For the quadratic models

f(R,Q) = R + aR2/Rp + Q/Rp, for instance, one can also take the trace of (2) to find

that R0 = 0, from which Q0 = R20/4 = 0 follows. For a generic f(R,Q) model, using

equation (27) in vacuum one finds that Σνμ = (fR/2 +

√α)δνμ, where fR/2 +

√α =

fR

(1 +

√1 +

4fQf

f2R

)/2 ≡ a(R0).

Therefore, in vacuum (19) can be written as Rνμ(h) = Rν

μ(g) = Λeffδνμ, with Λeff =

f(R0, Q0)/2a(R0)2, which shows that the field equations coincide with those of GR with

an effective cosmological constant.

3.1 Limit to f(R) and f(Q) Theories

In order to prove that the analytic solution (26) is consistent with the previous results

obtained in the literature, we should recover the main aspects of f(R) and f(Q) theories

taking the corresponding limit directly from this general solution. For the f(R) case the

consistency conditions for the coefficients are: fQ = 0, β ≡ 2κ2fQ = 0, α ≡ 1/4(f 2R +

4fQf) = f 2R/4. However, a direct substituion in (26) will give a divergent result due

to the fact that the solution is only defined for fQ �= 0. To avoid this problem, in this

particular case it is convenient to make use in first place of the auxiliary matrix Σ defined

in the field equation associated to the independent connection, with the aim of removing

the factors fQ.

Indeed, taking into account the definition of the matrix Σνμ given in (15) we find

Σνμ ≡ fRδ

νμ+2fQP

νμ = fRδ

νμ+2fQ

(− fR4fQ

δνμ +1

2fQ


μ

)= fRδ

νμ−

fR2δνμ+

fR2δνμ = fRδ

νμ

(30)

Therefore, det Σ = det(IfR) = (fR)4, and the relation between the auxiliary metric hμν

and gμν in equation (14), becomes a conformal relation


hαν =gαμΣν

μ√det Σ

=fRg

αν√(fR)4

=gαν

fR(31)

Hence, the full f(R) theory studied in [14] is recovered as a particular case of our analysis.

Regarding the Ricci squared Lagrangians, i.e. f(Q) theories, the consistency condition

is fR = 0, and the coefficients simplify in the form: β = 2κ2fQ, α = fQf .

In these conditions the solution (26) will acquire the following structure, for fQ �= 0

P νμ = − fR

4fQδνμ+

1

2fQ


μ =1

2fQ

√fQfδνμ + 2κ2fQT ν

μ =

√f

4fQδνμ +

κ2

2fQT νμ (32)

This result is in agreement with the theory developed in Ref. [21] for Ricci squared

Lagrangians. In vacuum (T νμ = 0), the latter equation turns out to be

P νμ =

1

2

√f

fQδνμ ≡ Λ(Q)δνμ (33)

Which can also be obtained from the general form of Λ(R,Q) (see Eq.(15)) in the limit

fR → 0. For f(Q) Lagrangians equation (27) collapses to Σνμ =

√αδνμ, with

√α =

√fQf

for fR = 0.

Finally, substituing this result in equation (19) we find that Rμν (h) = Λeffδ

μν , where

Λeff = f/2α = 1/(2fQ) evaluated at Q0. The theory will therefore provide a positive cos-

mological constant (consistent with current astrophysical observations) only when fQ > 0,

a condition that must fulfill all the admissible models.

3.2 Solving for a Diagonal Matrix. The Perfect Fluid

The problem that we have presented here seems to be mathematically well established.

When αI + βT is diagonal the solution (26) will come from a direct computation. It is

easy to understand that αI+βT will be diagonal if T is also a diagonal matrix. However,

if T is a non-diagonal matrix such as the case of the electromagnetic field, we will need

to find a way of diagonalyzing the matrix. For this purpose, an explicit decomposition in

terms of eigenvectors will turn out to be more suitable. In this section, we will illustrate

the utility of the solution given in (26), which in the case of a diagonal T , provides a

compact result in few steps. In a next subsection we will treat in detail the case of the

electromagnetic field from the point of view of their eigenvectors.

The energy-momentum tensor of a perfect fluid can be written as

Tαβ = (p+ ρ)uαuβ + pgαβ (34)

Where p is the pressure of the fluid and ρ its density. Making explicit the matrix


representation

T =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

−ρ 0 0 0

0 p 0 0

0 0 p 0

0 0 0 p

⎞⎟⎟⎟⎟⎟⎟⎟⎠

(35)

αI + βT =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

α− βρ 0 0 0

0 α + βp 0 0

0 0 α + βp 0

0 0 0 α + βp

⎞⎟⎟⎟⎟⎟⎟⎟⎠

(36)

which is a diagonal matrix, and therefore computing its square root will be automatic

√αI + βT =

⎛⎜⎝

√α− βρ

−→0

−→0 (

√α + βp)I3x3

⎞⎟⎠ (37)

In the last expression, it was selected the positive sign of the square roots of the

coefficients in order to be consistent with the limit fQ → 0. These results allow us to

write the matrices P , Σ for the perfect fluid as

P = − 1

4fQ

(fRI − 2

√αI + βT

)=

⎛⎜⎝Ω

−→0

−→0 ωI3x3

⎞⎟⎠ (38)

Σ =fR2I +

√αI + βT =

⎛⎜⎝2ΩfQ + fR

−→0

−→0 (2ωfQ + fR) I3x3

⎞⎟⎠ (39)

where

Ω =2√α− βρ− fR

4fQ(40)

ω =2√α + βp− fR

4fQ(41)

We should point out that these results are in agreement with those obtained elsewhere

[14][27, 29] for the perfect fluid, but the method described here provides a powerful and

direct computation of the matrix P (T ), a calculation that in some particular cases may

be almost automatic.


3.3 An Eigenvector Approach for A General Non-Null Electromagnetic

Field

It is important to investigate what form will acquire the solution for P μα in the case of a

non-diagonal energy-momentum tensor. For instance, this is the case of the electromag-

netic field, whose energy-momentum tensor is given by the standard relation in terms of

the Faraday field strength F μν and the metric gμν

Tμν = FμαFαν − 1

4gμνF

αβFαβ (42)

Nevertheless, it is more convenient to tackle this problem transforming the last expression

into other more adequate, taking an alternative path which will allow us to rewrite the

energy-momentum tensor in terms of more suitable mathematical objects. This will be

provided by the detailed study of the eigenvector problem of the Faraday tensor. For an

eigenvector σμ of the Faraday tensor, we refer to an algebraic object that satisfies the

equation

F ab σ

b = λσa (43)

Due to the skewsymmetric nature of the Faraday tensor, F μν = −F νμ, this automatically

implies that

F abσaσb = λσbσb = 0 (44)

The mathematical objects that satisfy this condition are known as null eigenvectors of

the Faraday tensor. It can be demonstrated that all eigenvector of F ab is an eigenvector

of T ab as well

Tαμ σ

μ =

(FμβF

βα − φ

4δαμ

)σμ =

(λ2 − φ

4

)σα ≡ Ωσα (45)

where φ ≡ F μνFμν . A decomposition of the Faraday tensor in terms of their two real

eigenvectors can be found elsewhere [25][26] and it is given by

Fμν = λ(σμην − σνημ

)− τεμναβσ

αηβ (46)

where λ,τ are real eigenvalues (which depend on φ), of the Faraday tensor.

Although in general a 4x4 matrix will have four independent eigenvectors, there are

only two real eigenvectors ημ, σμ, associated to the type A Faraday tensor discussed in

[25] ( other types yield to a radiative E-M tensor which is not a subject of interest here).

The other two eigenvectors for this type are purely imaginary and it is not necessary

to include them in the decomposition. In order to build an energy-momentum tensor

explicitly in terms of the eigenvectors, one path is the direct substitution of equation (46)

in (42). However, it exists a more economical manner to compute the energy-momentum

tensor. Indeed, the first piece of the energy-momentum tensor is the object FμαFαν , which

for symmetry reasons, in terms of the eigenvectors ημ, σν , can only have the following


structure

FμαFαν = C

(ημσν + σνημ

)(47)

In order to determine the constant C, we compute the trace gμνFμαFαν ≡ φ, and given

the fact [25] that we can choose the eigenvectors satisfying the condition ημσμ = 1, we

obtain C = φ/2. With this result, the final expression for the stress-energy tensor of the

electromagnetic field in terms of the two null real eigenvectors will be

Tμν =φ

2

(ημσν + σμην − 1

2gμν

)(48)

Which agrees with the result obtained in Ref.[25]. It is worth noting that this decom-

position provides gμνTμν = 0, in accordance with the traceless nature of the stress-energy

tensor of the electromagnetic field. At this point, we have collected all the elements re-

quired to investigate the exact form of the solution Pαμ applied to this particular problem.

As in the case of the perfect fluid, we first compute the Matrix αI + βT , of the radicand

of (26)

αδνμ + βT νμ =

(α− βφ

4

)δνμ +

βφ

2

(ημσ

ν + σμην)

≡ M2 (49)

And taking the following ansatz

Mνμ ≡ Aδνμ + B

(ημσ

ν + σμην)

(50)

This will allow us to evaluate the square M2 of this matrix in a transparent way in

order to obtain the relation between the coefficients (A,B) and (α, β)

M2 ≡ MαμM

μβ =

(Aδαμ + B (ημσ

α + σμηα)

) (Aδμβ + B (ηβσμ + σβη

μ))=

A2δαβ +(2AB + B2

) (ηβσ

α + σβηα)

(51)

Comparing this result with equation (49) we can establish the identification

α− βφ

4= A2 (52)

βφ

2= 2AB + B2 (53)

The sum of the two equations provides, B2 + 2AB + A2 −(α +

βφ

4

)= 0, this is a

quadratic equation which possesses the following solutions

B(α, β) = −A(α, β)±√

A(α, β)2 − A(α, β)2 + α +βφ

4= ∓

√α− βφ

4±

√α +

βφ

4(54)

With all these results, the matrix P μν for the electromagnetic field will acquire the compact

expression


P = − 1

4fQ

(fRI − 2

√αI + βT

)= − 1

4fQ

(fRI − 2M

)= − 1

4fQ

[(fR−2A

)δμν−2B

(ηνσ

μ+σνημ)]

(55)

Where the functions A(α, β, φ), B(α, β, φ), are given by the relations deduced in equations

(52-54).

We have therefore been able to compute an exact solution for the P matrix in the

case of the electromagnetic field. It is important to note that this solution is completely

general,neither the particular model of f(R,Q) nor the specific structure of the electro-

magnetic field itself have been specified. It seems reasonable to expect that for specific

scenarios such as spherically symmetric electromagnetic fields the analysis will turn out

to be more economical. In the nex subsection of this work we will explore a power series

expansion of the solution for the particular model f(R,Q) = R + 1Rp(R2 + Q), which is

also studied for the case of the electromagnetic field.

3.4 The power series expansion

In this section we want to explore the perturbative expansion of the solution given in (26).

It is interesting to note that this solution admits a natural power series representation.

Indeed, rewriting equation (26) we find, for α > 0

P (T ) = − fR4fQ

I +

√α

2fQ

(I +

β

αT)1/2

= − fR4fQ

I +

√α

2fQ

∞∑k=0

(1/2

k

) (βα

)k

T k

(56)

It is clear that for β >> α the series will be divergent. However, for β << α the

solution may be approximated by

P (T ) � − fR4fQ

I +

√α

2fQ

(I +

β

2αT)

(57)

The validity of this first order aproximation will depend on the particular modelchosen. If we take for instance the quadratic model f(R,Q) = R + 1

Rp(R2 + Q), where

Rp = 1/l2p with lp ∼ 10−35m the Planck length, one realizes that a priori the power seriesrepresentation does not seem adequate to treat this problem, due to the fact that for thisparticular model, β = 2κ2/Rp ∼ 1/ρp, where ρp ∼ 1091g/cm3, while α = 1/4(f 2

R + 4fQf)in general also includes terms of order 1/Rp. However an interesting feature of this familyof models is that the relation between the traces is R = −κ2T exactly the same expressionas in GR [14].This means that, if we deal with a traceless energy momentum tensor asthe electromagnetic field, the coefficient α will be simplified and some of the problematicterms that avoid the series expansion for this family of models will be removed from theanalysis. For the electromagnetic field, T = 0 implies that R = 0,fR = 1, fQ = 1/Rp. Adirect computation of the coefficients provides, α = 1/4(1 +Q/R2

p), β = 2κ2/Rp = 2/ρp.This means that α does not include terms of order 1/Rp, leaving open the possibility ofperform a series expansion. For this purpose, let us rearranging coefficients in order to


better understand the relevant terms involved

P (T ) = − 1

4fQ

(fRI − 2

√αI + βT

)=

− Rp

4

(I −

√I +

1

ρp

(4Q

κ2Rp

I + 8T

))= −Rp

4

(I −

√I +

1

ρpX

)(58)

where we have defined the matrix

X ≡ 4Q

κ2Rp

I + 8T (59)

Now we can expand the solution in powers of 1/ρp

P = −Rp

4

(I −

(I +

1

2ρpX − 1

8ρ2pX2 + ...

))(60)

Our main interest focuses on the first order contribution, so neglecting terms that go

as O(

1

ρ2p

), we find

P

Rp

� 1

8

X

ρp� 1

8

(4Q

κ4ρ2pI +

8

ρpT

)� T

ρp(61)

which leads to Σ ≡ fRI + 2fQP � I +2T

ρp. With these results, the relation between gμν

and the auxiliary metric hμν , for the electromagnetic field will be

hμν � 1

Ω

(gμν +

2

ρpT μν

)(62)

Where we have defined

Ω ≡√det

(I +

2

ρpT

)(63)

The results of equations (62-63) clearly show how the local densities of energy and mo-

mentum of the electromagnetic field perturb the metric. If ρEM/ρp � 1, then Ω � 1,

which implies hμν � gμν , the connection turns out to be the Levi-Civita connection and

therefore the geometry is essentially the same as in GR. However, when ρEM/ρp is not

so small, we expect a significant departure from GR. The peturbation of the metric due

to the local densities of energy and momentum of very intense light beams in the early

universe, may lead to quantum gravity effects that will be explored elsewhere.

4. Summary and Conclusions

In this work we have introduced new methods to deal with Palatini f(R,Q) theories. In

particular, these new methods include the study of the general analytic solution associ-

ated to the Ricci tensor, a general algorithm to compute the square root of the matrix


αI + βT in the nondiagonal case, and a power series representation of the solution that

can be useful to deal perturbatively with some problems in the context of Palatini grav-

ity. Indeed, we have applied the method successfully to the electromagnetic field, and

we have shown that when ρEM/ρp << 1, the independent connection turns out to be

the Levi-Civita connection and therefore the geometry is essentially the same as in GR.

Nevertheless, if ρEM/ρp is not so small, the local densities of energy and momentum per-

turb the metric,the connection can no longer be expressed as the Levi-Civita connection

and therefore we expect a significant departure from GR. Then, at very high energies,

the structure of the theory may include quantum gravity effects that will be studied in

future works.

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.


Testing the Everett Interpretation of QuantumMechanics with Cosmology

Aurelien Barrau∗

Laboratoire de Physique Subatomique et de Cosmologie, Universite Grenoble-Alpes,CNRS-IN2P3

53,avenue des Martyrs, 38026 Grenoble cedex, France

Received 2 June 2015, Accepted 1 July 2015, Published 25 August 2015

Abstract: In this brief note, we argue that contrarily to what is still often stated, the Everett

many-worlds interpretation of quantum mechanics is not in principle impossible to test. It is

actually not more difficult (but not easier either) to test than most other kinds of multiverse

theories. We also remind why multiverse scenarios can be falsified.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Interpretation of Quantum Mechanics; Quantum Cosmology; Everett many-worlds

interpretation of quantum mechanics

PACS (2010): 03.65.-w; 98.80.Qc; 98.80.Bp

1. The Everett Many World Interpretation

This brief article aims at contradicting a naive –and in our view incorrect– belief that

seems to be dominant in the physics community: the unfalsifiability of the Everett ”many-

worlds” interpretation of quantum mechanics. Although the Everett interpretation is

taken more and more seriously, the reason for this choice seems to be mostly aesthetic as

most articles about the many-worlds view mention its fundamental untestability.

At the intuitive level, the Everett interpretation of quantum mechanics (see [1, 2] for

historical articles) states that all possible alternate histories of a quantum system are

actually real. Each of them takes place in a different universe, there is no collapse of

the wave function. All that could have happened in our past but didn’t happen here,

did indeed happen in another world. All that can happen in the future will happen in

different universes. Every possible quantum outcome is real in one universe. How strange



it might look, this vision is now considered as one of the mainstream interpretation of

quantum mechanics. Its obvious advantage is somehow to take quantum mechanics ”lit-

erally” without adding by hands a non-unitary evolution mysteriously triggered by the

measurement process.

Slightly more rigorously, the Everett interpretation could be said to rely on two hy-

pothesis. Firstly that the wavefunction has an observer-independent objective existence

and actually is the fundamental object. For a non-relativistic N-particle system the

wavefunction is a complex-valued field in a 3N dimensional space. Secondly that the

wavefunction obeys the standard linear deterministic wave equations at all times. The

observer plays no special role in the theory and, consequently, there is no collapse of the

wavefunction. For non-relativistic systems, the Schrodinger wave equation describes the

evolution.

2. The Multiverse and Experiments

Apart from the Everett interpretation of quantum mechanics, many kinds of ”multi-

verses” are now discussed in cosmology. One can cite for example the infinite space that

would arise when k = 0 or k = −1 (if the topology is simple) and contains infinitely many

Hubble volumes, the interior of charged or rotating black holes if one takes seriously an-

alytically extended Penrose-Carter diagrams, the inflationary bubbles possibly filled by

different laws of physics, etc. Different laws of physics can in particular be obtained

by compactification schemes and generalized magnetic fluxes in string theory or, more

generically, by any kind of theory that has several local minima or vacua. We will, in the

following of this section focus on such models. In each of the previously given examples,

the word ”universe” has a different meaning depending on the case.

Are those multiverses testable ? The answer is : in principle, yes. There are at

least two different arguments in this direction. The first one is to emphasize (see, e.g.,

[3] in particular arguments by M. Tegmark) that the multiverse is not a hypothesis but

a consequence. It would indeed be an artificial and ad hoc hypothesis. But it actually

appears as an output of theories that can be tested locally and that are built for completely

other purposes, for example particle physics or quantum gravity. It is not an input. If

those theories were to be falsified, all their predictions would disappear, including the

multiverse. The other way round, if those theories were to be part of our main paradigm

it would be inconsistant to discard the multiverse they produce just because one does

not like the idea. So, basically, the point is that we should not test the multiverse, but

rather the theory that predicts it. And this is certainly in principle possible. Exactly as

one uses general relativity (GR) to describe the internal structure of black holes even if

this specific prediction cannot be tested, just because we are confident enough in GR to

trust this.


The second argument is more closely related with the multiverse in itself. There is –in

those approaches– a large (possibly infinite) number of universes. And we see only a single

one. Can something be said about the whole ensemble ? Obviously yes ! Let us imagine

than instead of billions of particle collisions we were to see only one at the LHC. Would

we have discovered the Higgs ? Of course not. But would be have been able to disproof

a crazy model of particle physic? Most probably yes. A single sample contains less

information than the full set but it does contain some information. It makes a statistical

test possible, as usual in physics (either for quantum of classical reasons, all confirmations

or exclusions are anyway statistical).To make precise predictions in the multiverse and

compare them with our own universe, it is necessary but not sufficient to know the shape

of the landscape. It is also necessary to account for a possible bias in the sample we

observe (this is also very common in science). This precaution might be referred to as

the anthropic principle. Let us make it very clear that there is absolutely no link at

all between this principle (which is not actually a principle but rather a reminder) and

intelligent design, anthropocentrism, theology or teleology. Just the other way round, the

point is only to take care of the fact that as our planet is obviously not at all representative

of our full universe, our universe might not be representative of the full multiverse. There

is nothing mysterious or mystic here, this is just a bias that should be accounted for: we

live in a place which is favorable to complexity, just because we are ourselves complex

structure, of course we do not live in empty space or at the center of a neutron star.

Similarly, our universe might be more complexity-friendly than most others and when

trying to test the compatibility of a given theory with our observations we have to take

care of that.

Let us assume that a model predicts a billion of universes, all of them but one being

empty. Let us take a random sample. If this sample is full of stars, one can conclude that

the theory is excluded at a high confidence level. But if we now consider a random sample

selected from within by an observer and take into account the selection bias due to the

fact that we are ourselves not made of vacuum, the model will not anymore be disfavored.

So, basically, when comparing the structure of the landscape (of solutions of the con-

sidered theory) with the sample universe we live in, the important quantity is the product

of the probability of a given solution Si, P (Si), by the number of observer N(Si) in this

solution. If one considers a random human being wondering where he lives, there are

more chances that he is Indian rather than Australian, even though Australia is larger

than India, just because there are more Indian observers. This has to be accounted for

when trying to reconstruct the landscapes of the Earth from local descriptions. It is well

known that this formulation is obviously too vague and ill defined. We don’t know how to

define properly an observer in a general enough way. In addition, there are many severe

limits associated with general problems in constructing a proper and unique measure.

These are, however, mostly technical problems that can, in principle, be solved. Then,

the compatibility between our world and this observer-weighted probability distribution

over the landscape can be evaluated and the model excluded or confirmed at a given


confidence level.

This view is not sterile. Even if one takes a narrow-minded non-ontological definition

of physics (assumed to be only useful in making local predictions), this has consequences.

As an example, one can consider the status of the inflationary model, in view of the

Planck mission results. Depending on whether one takes into account the multiverse

–that is a generic prediction of eternal inflation– or ignore it, the compatibility of the

model with data might change completely (see, e.g., [4]). Actually, our universe might

even be exponentially disfavored in an inflationary multiverse [5], which is not at all the

case in a single universe.

3. Basic Idea for Testing the Everett Hypothesis

Following [6], we would like to argue that the Everett interpretation of quantum mechan-

ics can be tested in the very same way.

The question of testing the Everett vision has been raised long ago. Three possible

”tests” that we won’t describe in details here (see [7]) have been considered. The first one

is related with linearity. In principle, because of linearity, it is possible to detect the pres-

ence of other nearby worlds, through the existence of interference effects. This has been

pushed forward in [8] where an observer splits into two copies making different observa-

tions. They remember having made observations but not the results and can therefore

be rejoined coherently into a single copy. This is very clever but nearly impossible to be

made experimentally.

The second one is that many-worlds actually requires that gravity be quantized, in

contrast to other interpretations which are silent about the role of gravity. The reason is

that if gravity was to remain non-quantized, all the universes that the Everett interpre-

tation predicts should be easily detectable by their gravitational presence –they would

all share the same background metric with our co-existing quantum worlds. Of course,

gravity could be quantized and, still, the Everett interpretation could be wrong.

The third one, related with the first one and still using linearity (see [9]) is based on

reversible quantum computers. It requires currently beyond of reach artificial intelligence

and reversible nanoelectronics.

What we want to emphasize here, as D. Page somehow did more than a decade ago,

is that in the framework of quantum cosmology, testing the Everett interpretation of

quantum mechanics is not different from testing any other multiverse proposal.

In a single universe interpretation of quantum mechanics, only the relative probabil-

ity for different universes matters. There is only one universe, so only the probabilities

between different outcomes are important. If the probability for a x-universe is much

higher than the probability for a y-universe, we should be in the x-universe, whatever the


number of observers it generates. However, in the Everett vision, all possible universes

do actually exist. So, if the number of observers are different, say Nx and Ny respectively

in the x-universe and the y-universe, the relevant probabilities are now the ones weighted

by the number of observers. If the ratio Ny/Nx is higher that P (x)/P (y), the prediction

is now the opposite. We should be in the y-universe. The point is that the situation is

basically the same as in any multiverse situation. If there in only one World we should

compute probability for this World, if there are many worlds the observer-weighted prob-

ability is the correct distribution. In principle, observations can select which one is true.

The methodology is very close to now standard approaches to testing the multiverse.

Instead of just testing a model the point here is to test two interpretations of quantum

mechanics. As long as the two interpretations lead to different final probability distribu-

tions, which is the case, the proposal is testable.

4. Digression on What Science is?

It is often argued in physics articles that ”science is what Popper says: something that

can be falsified, and we should not step away from that”. First, we would like to under-

line that the multiverse is not outside this definition. It is not a theory, it is part of the

predictions of theories that can in principle be falsified. As we explained before, many

researches are devoted to the internal structure of black holes in GR. This is considered as

usual science whereas nobody can go there and come back telling us if that is true. Once

a model, in this exemple GR, is well enough tested to be part of the dominant paradigm,

it is perfectly legitimate to use it even where it cannot be tested. It has hopefully never

been necessary to check all the predictions of a theory to consider it as reliable and to

use it.

Beyond that, we believe that we should not take Popper (or, more precisely this

caricatural simplification of Popper’s claims) too seriously. First, because it is only one

epistemology among many others, say, for example (considering only the the XXth cen-

tury), Carnap, Bachelard, Canguilhem, Cavailles, Bourdieu, Feyerabend, Kuhn, Koyre,

Lakatos, Quine, Latour, Hacking, Merton, etc. All those epistemologies lead to consistent

views on science that are completely different from the one of Popper whose popularity

might just be due to its simplicity. But every scientist knows that falsifiability does not

work in practice : all theories have troubles and depending on the situation we might

assume invisible objects (dark matter for exemple) or a more limited domain of appli-

cation of the theory to save it anyway. No serious theory has ever been refuted by a

single experiment. Not to mention that most new theories were initially believed to be

impossible to test/falsify (remember Auguste Comte believing, just before the discovery

of spectroscopy, that discussing the composition of stars was not science as no one will

ever be able to go there and analyse them). Popper’s epistemology comes together with

(or is deeply rooted in) a highly contestable firm belief that science will solve and has al-


ready solved most problems of the humanity (”today, misery has essentially disappeared

from occidental Europe; excessive social disparity do not exist anymore”, said he before

criticizing ”intellectuals” that do not agree with this obvious remark !).

Anyway, even if Popper was right in describing what science was –and we don’t

believe so–, say in the 1950’s, why should we take this as a correct definition forever ?

In essence, science is a dynamical process. Playing with the rules if obviously part of the

game. Hopefully, the rules are not the same now as what they were for Galileo. Science

is about deconstruction: nothing can be taken for granted, everything can –in principle–

be revised and changed. If one wants certainty and immuable rules, one should better

consider theology. Of course, we all expect science to make predictions and to describe

the World(s) more precisely. Everything is for sure not allowed. But one cannot know

what are the correct revisions before exploring them and considering the consequences.

So, even if the multiverse was –and we don’t think it is the case– to impose a change

of the rules of the game, we think it would be more dangerous to a priori forbid the

idea than to seriously study the new paradigm it might generate. All fields get redefined

from within. This is obviously the case in art where contemporary musical composition,

poetical writing and pictorial creation do not obey the aesthetics as it was defined initially

by Baumgarten. Why wouldn’t this evolution of the rules be allowed for science which

is, probably more than any other, the cognitive field where changes are essential ?

5. Specific Possible Tests of the Everett Interpretation

In [6], the particular example of the Hartle-Hawking no-boundary proposal was consid-

ered as a consistent proposal for a quantum state of the Universe. The Hartle-Hawking

proposal basically does away with the initial three geometry, i.e. only includes four di-

mensional geometries that match onto the final three geometry. The path integral is

interpreted as giving the probability of a universe with certain properties (i.e. those

of the boundary three geometry) being created from essentially nothing. In a S3 min-

isuperspace approach with a single φ2 potential scalar field, the no-boundary proposal

instanton leads to a set of FLRW universes with various number of inflationnary e-folds

and therefore various properties, with an amplitude given by:

A ∝ eD2

,

D being the linear spatial size of the Euclidean 4-dimensionnal hemisphere where the

solution nucleates [10]. Actually, D ∝ 1/Φ0, where Φ0 is the initial value of the massive

scalar field, and N ∝ exp(Φ0), where N is the number of e-folds of inflation. Taking

the volume of the resulting universe as a rough mesure proportional to the number of

observers, one can compare the probability for our Universe being what it is in a single

universe history on the one hand and, on the other hand, with the observer-weighted

probability that should be chosen in the Everett many-worlds view where all universes

do actually exist. The result strongly favors the Everett view. Obviously the conclu-


sion should not be taken too seriously. Firstly, because it could very well be that the

Hartle-Hawking proposal is not correct. Secondly, because even in this framework, the

preliminary result from Page relies on many controversial assumptions. But this shows

that testing the Everett proposal is possible.

This approach can be generalized to basically any model of quantum cosmology. The

most delicate point is to define the measure for the number of observers for each out-

come. This is quite straightforward in the case of closed universes (that are anyway

often better defined in quantum cosmology) but very difficult in general. In the frame-

work of Loop Quantum Cosmology [11], for example, the situation is more complicated.

All quantum states can be shown to undergo a bounce in the sense that the expec-

tation value of the volume operator has a non-zero lower bound in any state. Basi-

cally, the Wheel-DeWitt equation is replaced by a difference equation : ∂2ϕΨ(ν, ϕ) =3πG4λ2 ν

[(ν + 2λ)Ψ(ν + 4λ) − 2νΨ(ν) + (ν − 2λ)Ψ(ν − 4λ)

]where Ψ is the wave function

of the Universe and λ is the square root of the minimum area gap. The situation is tricky

because at the semi-classical order, the duration of inflation is here determined by the

phase of the inflaton field in the remote past, that is in the classical contracting branch.

Although the most probable value can be determined [12], this remains a random classical

process. However, all quantum trajectories are still possible around the semi-classical ap-

proximation of highly peaked states. One has in principle all the required ingredients to

calculate the weighted and unweighted probabilities for different universes and compare

with ours.

Clearly, the procedure is still mostly out of reach just because we don’t yet have

an established theory of quantum cosmology (not to mention that some physicists even

question the fact that quantum mechanics should apply to the Universe as whole [13])

and it would be difficult to establish this theory if the interpretation of quantum me-

chanics if not fixed. The most promising avenue would be to have empirical evidence for

a given quantum gravity proposal (e.g. by measuring Lorentz invariance violations, by

detecting quantum gravity effects in black holes, or by any other means) and then apply

this otherwise confirmed model to the Universe.

This proposal should be taken with care. It could probably be pushed ad absurdum.

There is usually a non-vanishing quantum probability for basically everything. For ex-

ample, there is a non-vanishing quantum probability for me to survive any kind of illness

or aging process. In the Everett view, I will somehow be ”immortal”. Not for my friends

and family, of course, seing me dying in most worlds. So I might argue that the simple

fact that I am writing this article at an age which is not much above (and actually even

smaller) that the mean life expectancy is already an argument disfavoring the model (the

probability than I am less than 80 years-old if I am allowed to be eternal is very small).

This would however forgets that the number of copies of myself decreases very fast as

time goes on and this might counterbalance the naive argument.


The main point we wanted to make is that testing the Everett many-worlds is not

fundamentally different from testing any multiverse model and is in principle possible.

References

[1] H. Everett, Reviews of modern physics, 29, 454 , 1957.

[2] B.S. DeWitt, R.N. Graham, eds, The Many-Worlds Interpretation of QuantumMechanics, Princeton Series in Physics, Princeton University Press (1973).

[3] B. Carr, ed., Universe or Multiverse, Cambridge, Cambridge University Press (2007).

[4] A. Ijjas, P.J. Steinhardt, A. Loeb, Phys. Lett. B 723, 261, 2013.

[5] A. Ijjas, P.J. Steinhardt, A. Loeb, arXiv:1402.6980.

[6] D. Page, arXiv:quant-ph/9904004, D. Page, AIP Conf.Proc. 493 (1999) 225.

[7] M.C. Clive, The Everett FAQ, 1995.

[8] D. Deutsch, Int. J. Theor. Phys., 24, 1, 1995.

[9] D. Deutsch, ”Three connections between Everett’s interpretation and experimentQuantum Concepts of Space and Time”, eds R. Penrose and C. Isham, OxfordUniversity Press, 1986.

[10] J. B. Hartle & S. W. Hawking, Phys. Rev. D 28, 2960, 1983;S. W. Hawking, Nucl. Phys. B 239, 257, 1984.

[11] A. Ashtekar & P. Singh, Class. Quantum Grav. 28, 213001, 2011.

[12] L. Linsefors & A. Barrau, Phys. Rev. D 87, 12, 123509, 2013.

[13] G.E.F. Ellis, private communication.


Elliott Formula for Particle-hole Pair of Dirac Cone

Lyubov E. Lokot∗

Institute of Semiconductor Physics, NAS of Ukraine, 41, Nauky Ave., Kyiv 03028,Ukraine


Abstract: In the paper a theoretical study the both the quantized energies of excitonic states

and their wave functions in graphene is presented. An integral two-dimensional Schrodinger

equation of the electron-hole pairing for a particles with electron-hole symmetry of reflection

is exactly solved. The solutions of Schrodinger equation in momentum space in graphene by

projection the two-dimensional space of momentum on the three-dimensional sphere are found

exactly. We analytically solve an integral two-dimensional Schrodinger equation of the electron-

hole pairing for particles with electron-hole symmetry of reflection. In single-layer graphene

(SLG) the electron-hole pairing leads to the exciton insulator states. Quantized spectral series

and light absorption rates of the excitonic states which distribute in valence cone are found

exactly. If the electron and hole are separated, their energy is higher than if they are paired.

The particle-hole symmetry of Dirac equation of layered materials allows perfect pairing between

electron Fermi sphere and hole Fermi sphere in the valence cone and conduction cone and hence

driving the Cooper instability.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: graphene; Electron-Hole Pair;Dirac Cone; Elliott Formula; Honeycomb

Lattice;Schrodinger Equation

PACS (2010): 81.05.ue; 81.05.U-; 71.30.+h; 71.10.-w

1. Introduction

The graphene [1, 2, 3] presents a new state of matter of layered materials. The energy

bands for graphite was found using ”tight-binding” approximation by P.R. Wallace [4].

In the low-energy limit the single-particle spectrum is Dirac cone similarly to the light

cone in relativistic physics, where the light velocity is substituted by the Fermi velocity

vF and describes by the massless Dirac equation.

In the paper we present a theoretical investigation of excitonic states as well as their

∗ E-mail: [email protected].


Fig. 1 (Color online) Single-particle spectrum of graphene for massless Dirac fermions (Majoranafermions).

wave functions in graphene. An integral form of the two-dimensional Schrodinger equa-

tion of Kepler problem in momentum space is solved exactly by projection the two-

dimensional space of momentum on the three-dimensional sphere in the paper [6].

The integral Schrodinger equation was analytically solved by the projection the three-

dimensional momentum space onto the surface of a four-dimensional unit sphere by Fock

in 1935 [5].

We consider the pairing between oppositely charged particles with complex dispersion.

The Coulomb interaction leads to the electron-hole bound states scrutiny study of which

acquire significant attention in the explanations of superconductivity.

If the exciton binding energy is greater than the flat band gap in narrow-gap semi-

conductor or semimetal then at sufficiently low temperature the insulator ground state is

instable with respect to the exciton formation [7, 8]. And excitons may be spontaneously

created. In a system undergo a phase transition into a exciton insulator phase similarly

to Bardeen-Cooper-Schrieffer (BCS) superconductor. In a single-layer graphene (SLG)

the electron-hole pairing leads to the exciton insulator states [9].

In the paper an integral two-dimensional Schrodinger equation of the electron-hole

pairing for particles with complex dispersion is analytically solved. A complex dispersions

lead to fundamental difference in exciton insulator states and their wave functions.

We analytically solve an integral two-dimensional Schrodinger equation of the electron-

hole pairing for particles with electron-hole symmetry of reflection.

For graphene in vacuum the effective fine structure parameter αG = e2

vF �κ√π= 1.23.

For graphene in substrate αG = 0.77, when the permittivity of graphene in substrate is

estimated to be κ = 1.6 [10]. It means the prominent Coulomb effects [11].

It is known that the Coulomb interaction leads to the semimetal-exciton insulator

transition, where gap is opened by electron-electron exchange interaction [8, 12, 13, 14].

The perfect host combines a small gap and a large exciton binding energy [8, 7].

In graphene the existing of bound pair states are still subject matter of researches

[15, 16, 17, 18, 19].


It is known [20] in the weak-coupling limit [21], exciton condensation is a consequence

of the Cooper instability of materials with electron-hole symmetry of reflection inside

identical Fermi surface. The identical Fermi surfaces is a consequence of the particle-hole

symmetry of massless Dirac equation for Majorana fermions.

2. Theoretical study

2.1 Quantized spectral series of the excitonic states of valence Dirac cone.

In the honeycomb lattice of graphene with two carbon atoms per unit cell the space group

is D13h [22]:

D13h {E|0} {C(+,−)

3 |0} {C′(A,B,C)2 |0} {σh|τ} {S(−,+)

3 |τ} {σ(A,B,C)v |τ}

K+3 2 -1 0 2 -1 0

g2 {E|0} {C(+,−)3 |0} {E|0} {E|0} {S(−,+)

3 |τ} {E|0}

χ2(g) 4 1 0 4 1 0 K+1 +K+

2 +K+3

χ(g2) 2 -1 2 2 -1 2

12[χ2(g)+χ(g2)] 3 0 1 3 0 1 K+

1 +K+3

12{χ2(g)−χ(g2)} 1 1 -1 1 1 -1 K+

2

The direct production of two irreducible presentations of wave function and wave

vector of difference κ−K or κ−K ′ expansion is K+3 × K+�

3 and can be expanded on

pα : τψ × τk = (K+1 +K+

2 +K+3 ) × K+

3 = K+3 × K+

3 . (1)

In the low-energy limit the single-particle spectrum is Dirac cone describes of the

massless Dirac equation for a massless Dirac fermions (Majorana fermions). The Hamil-

tonian of graphene for a massless Dirac fermions [4]

H = vF (τ qxσx + qyσy), (2)

where qx, qy are Cartesian components of a wave vector, τ = ± 1 is the valley index,

vF = 106 m/s is the graphene Fermi velocity, σx, σy are Pauli matrices (here we assume

that � = 1).

The dispersion of energy bands may be found in the form [4]

ε± = ± vF q, (3)

where q =√q2x + q2y .

The Schrodinger equation for the calculating of exciton states can be written in the

general form

(ε(q) + q20)Φ(q) =1

π

∫Φ(q ′)

|q − q ′|dq′, (4)


where q20 = −ε, ε is a quantized energy. We look for the bound states and hence the

energy will be negative.

For the single layer graphene

ε(q)+q20q2+q20

= ± vF2q0

sin (θ) + 1−cos θ2

. (5)

An integral form of the two-dimensional Schrodinger equation in momentum space

for the graphene is solved exactly by projection the two-dimensional space of momentum

on the three-dimensional sphere.

When an each point on sphere is defined of two spherical angles θ, φ, which are knitted

with a momentum q [5, 6]. A space angle Ω may be found as surface element on sphere

dΩ = sin(θ)dθ dφ = ( 2q0q2+q20

)2dq [5, 6]. A spherical angle θ and a momentum q are shown

[5, 6] to be knitted as

cos θ =q2 − q20q2 + q20

, sin θ =2qq0q2 + q20

, q2 = q20(1 + cos θ

1 − cos θ). (6)

Using spherical symmetry the solution of integral Schrodinger equation can look for

in the form

Φ(q) =√q0(

2q0q2 + q20

)3/2∞∑l=0

AlY0l (θ, φ), (7)

where

Y 0l (θ, φ) =

√2l+14πP 0l (cos θ). (8)

Since [6]

(q2+q20)1/2(q′2+q20)

1/2

2q01

|q−q′| =∑∞

λ=0

∑λμ=−λ

4π2λ+1

Y μλ (θ, φ)Y

μ,∗λ (θ′, φ′), (9)

then substituting (7), (9) in (4), can find equation

ε(q)+q20q2+q20

∑∞l=0AlY

0l (θ, φ) =

2q0

∑∞l=0

∑∞λ=0

∑λμ=−λ

∫1

2λ+1Y μλ (θ, φ)Y

μ,∗λ (θ′, φ′)Y 0

l (θ′, φ′)Al(

2q0q′2+q20

)2dq ′.(10)

The integral equations for SLG based on Eq. (5) may be found in the form

∫(± vF

2q0sin (θ) + 1−cos θ

2)∑∞

l=0AlY0l (θ, φ)Y

n,∗k (θ, φ)dΩ =

= 2q0

∫ ∑∞λ=0

∑λμ=−λ

∑∞l′=0

12λ+1

Y μλ (θ, φ)Y

μ,∗λ (θ′, φ′)Y 0

l′ (θ′, φ′)Y n,∗

k (θ, φ)dΩ dΩ′Al′ .(11)

Since [23]

cos θPml (cos θ) =

√l2−m2√4l2−1 P

ml−1(cos θ) +

√(l+1)2−m2√4(l+1)2−1 P

ml+1(cos θ), (12)

sin θPml (cos θ) =

√(l−m)(l−m−1)√

4l2−1 Pm+1l−1 (cos θ) +

√(l+m+1)(l+m+2)√

4(l+1)2−1 Pm+1l+1 (cos θ), (13)


ε0 ε1 ε2 ε3 Ry

1107.94 122.47 39.59 17.97 87.37

Table 1. Quantized spectral series of the excitonic states which distribute in valence cone εn,

n = 0, 1, 2, 3, ... in meV, exciton Rydberg Ry in meV.

then solutions of the integral equation (10) for the energies and wave functions corre-

spondingly can be found analytically with taken into account the normalization condition

( 12π)2

∫ q2+q202q20

|Φ(q)|2dq = 1.

From equation (11) one can obtain the eigenvalue and eigenfunction problem one can

find recurrence relation

1

2(l +

1

2)Al +

1

q0Al +

1

2Al−1(l +

1

2)al +

1

2Al+1(l +

1

2)bl = 0. (14)

The solutions of the quantized series in excitonic Rydbergs where Ry=87.37 meV,

and wave functions of the integral equation (11) one can find in the form

ε0 = − 1

(14+ 1

2(1 + 1

2)a1)2

, (15)

ε1 = − 1

(12(1 + 1

2) + 1

4b0 +

12(2 + 1

2)a2)2

, (16)

ε2 = − 1

(12(2 + 1

2) + 1

2(1 + 1

2)b1 +

12(3 + 1

2)a3)2

, (17)

ε3 = − 1

(12(3 + 1

2) + 1

2(2 + 1

2)b2 +

12(4 + 1

2)a4)2

, (18)

Φl(cos θ) =√

2π(q0l)3

∑∞n=0(1 − cos θ)3/2P 0

n(cos θ), (19)

where q20l = −εl, l = 0, 1, 2, 3, 4, ....,

al =1

2π

√2(l − 1) + 1

2

√2

2l + 1

l√4l2 + 1

, (20)

bl =1

4π

√2(l + 1) + 1

√2l + 1

l + 1√4(l + 1)2 − 1

. (21)

Quantized spectral series of the excitonic states distribute in valence Dirac cone. The

energies of bound states are shown to be found as negative, i. e. below of Fermi level.

Thus if the electron and hole are separated, their energy is higher than if they are paired.

2.2 Elliott formula and light absorption rates of the excitonic states of

valence Dirac cone.

The intervalley transitions probability caused intervalley photoexcitations taken into ac-

count Coulomb interaction of electron-hole pair one can obtain from Fermi golden rule

in the form


α0 α1

7.67*1022 1.14*1025

Table 2. Light absorption rate of quantized spectral series of the excitonic states which

distribute in valence cone αn, n = 0, 1, ... in cm−1.

P = 2π�( evFEω

�ω)2

∑n(

∑q |〈 ± 1, q|σx,y| ∓ 1, q〉|×

×Φn(q2−q20q2+q20

))2[δγ(εn − �ω) + δγ(εn + �ω)].(22)

Considering the case of relatively weak excitation the total rate of increase of the

number of photons in the fixed mode one can obtain in the form

R = 2π�( evFEω

�ω)2

∑n(

∑q |〈 ± 1, q|σx,y| ∓ 1, q〉|×

×Φn(q2−q20q2+q20

))2[δγ(εn − �ω) + δγ(εn + �ω)].(23)

The change in the energy density of electromagnetic waves can be presented in the

form

dW

dt=

�ω

SR. (24)

Under ac electric field Eωe cos (qz − ω t) the energy density of electromagnetic waves

one can obtain in the form W = 18πκE2

ω. Light absorption rate one can obtain in the

form α(ω) = 1W

dWdz

. Since dWdz

= dWdt

√κc

then light absorption rate with taken into account

|〈 ± 1, q|σx,y| ∓ 1, q〉|2 = 1/2 can be rewritten in the form

&(α(ω)) = 16√κc�2ω

(evF )2∑

n(∫dq×

×Φn(q2−q20q2+q20

))2[δγ(εn − �ω) + δγ(εn + �ω)],(25)

where∑

q → S(2π)2

∫dq in a formula (23).

3. Results and discussions

The integral Schrodinger equation for a parabolic bands was analytically solved by the

projection the three-dimensional momentum space onto the surface of a four-dimensional

unit sphere by Fock in 1935 [5].

In the paper an integral two-dimensional Schrodinger equation of the electron-hole

pairing for particles with complex dispersion is analytically solved. A complex dispersion

leads to fundamental difference in the energy of exciton insulator states and their wave

functions.


hole pairing for particles with electron-hole symmetry of reflection.


It is known that the Coulomb interaction leads to the semimetal-exciton insulator

transition, where gap is opened by electron-electron exchange interaction [8, 12, 13, 14].

The perfect host combines a small gap and a large exciton binding energy [8, 7].

We consider the pairing between oppositely charged particles in graphene. The

Coulomb interaction leads to the electron-hole bound states scrutiny study of which

acquire significant attention in the explanations of superconductivity.

It is known [7, 8] if the exciton binding energy is greater than the flat band gap in

narrow-gap semiconductor or semimetal then at sufficiently low temperature the insulator

ground state is instable concerning to the exciton formation with follow up spontaneous

production of excitons. In a system undergo a phase transition into a exciton insulator

phase similarly to BCS superconductor. In a SLG the electron-hole pairing leads to the

exciton insulator states.

The particle-hole symmetry of Dirac equation of layered materials allows perfect pair-

ing between electron Fermi sphere and hole Fermi sphere in the valence band and con-

duction band and hence driving the Cooper instability. In the weak-coupling limit in

graphene with the occupied conduction-band states and empty valence-band states in-

side identical Fermi surfaces in band structure, the exciton condensation is a consequence

of the Cooper instability.

4. Conclusions

In this paper we found the solution the integral Schrodinger equation in a momentum

space of two interacting via a Coulomb potential Dirac particles that form the exciton in

graphene.

In low-energy limit this problem is solved analytically. We obtained the energy dis-

persion and wave function of the exciton in graphene. The excitons were considered as a

system of two oppositely charge Dirac particles interacting via a Coulomb potential.

We solve this problem in a momentum space because on the whole the center-of-mass

and the relative motion of the two Dirac particles can not be separated.


hole pairing for particles with electron-hole symmetry of reflection. An integral form of

the two-dimensional Schrodinger equation in momentum space for graphene is solved ex-

actly by projection the two-dimensional space of momentum on the three-dimensional

sphere.

Quantized spectral series of the excitonic states distribute in valence Dirac cone. The

energies of bound states are shown to be found as negative, i. e. below of Fermi level.

Thus if the electron and hole are separated, their energy is higher than if they are paired.

In the SLG the electron-hole pairing leads to the exciton insulator states.

5. Appendix A

Table 3. The irreducible representational of D13h [24].


D13h {E|0} {C(+,−)

3 |0} {C′(A,B,C)2 |0} {σh|τ} {S(−,+)

3 |τ} {σ(A,B,C)v |τ}

K+1 1 1 1 1 1 1 x2+y2, z2

K+2 1 1 -1 1 1 -1 Jz

K+3 2 -1 0 2 -1 0 (x, y)

K−1 1 1 1 -1 -1 -1

K−2 1 1 -1 -1 -1 1 z

K−3 2 -1 0 -2 1 0 (x2−y2,xy), (Jx,Jy)

6. Appendix B

From a trigonometric calculation one can find a following recurrence relations

cot θPm+1l (cos θ) =

Pm+2l (cos θ)+[l(l+1)−m(m+1)]Pm

l (cos θ)

2(m+1), (26)

1sin θ

Pml−1(cos θ) =

(2l+1)Pm+1l (cos θ)+(l−m)(l−m+1)(2l+1)Pm−1

l (cos θ)

((l+m)(l+m+1)−(l−m)(l−m+1)), (27)

cot θPml (cos 2θ) = ( 1

sin 2θ+ cot 2θ)Pm

l (cos 2θ), (28)

(12(3 + 4(cot θ)2) − 1

2− 1

sin θ)Pm

l (cos 2θ) = cot θPml (cos 2θ), (29)

where

Pml (x) = 1

2m(l+m)!

(l−m)!m!(1 − x2)m/2F (m− l,m+ l + 1,m+ 1, 1−x

2), (30)

F (α, β, γ, z) = − 12π i

Γ(1−α)Γ(γ)Γ(γ−α)

∮(−t)α−1(1 − t)γ−α−1(1 − tz)−βdt. (31)

In order to find a light absorption rates necessarily to solve the integral

J =∫ 1

−1 dxF (−l, l + 1, 1, 1−x2). (32)

Substituting (31) into (32) we obtain the integral in the form

J = − 12π i

Γ(1+l)Γ(1)Γ(1+l)

∫ 1

−1 dx∮(−t)−l−1(1 − t)l(1 − t

2+ tx

2)−l−1dt, (33)

which can be rewritten as follows

J = − 12π i

Γ(1+l)Γ(1)Γ(1+l)

∫ 1

−1 dx∮(−t)−l−1(1 − t)l(1 − t

2)−l−1(1 + tx

2−t)−l−1dt. (34)

The solution the following integral


J =∫ 1

−1(1 +tx2−t)

−l−1dx, (35)

may be found by substitution

y = tx2−t . (36)

We find the solution of the integral

J =∫ t

2−t

− t2−t

(1 + y)−l−1dy = − 2−l

l+2(2 − t)l + 2−l

l+2(1 − t)−l(2 − t)l. (37)

Then substituting (37) into (34) we obtain the integral in the form

J = − 12π i

Γ(1+l)Γ(1)Γ(1+l)

2−l

l+2(∮(−t)−l−1(1 − t)l(1 − t

2)−1dt− ∮

(−t)−l−1(1 − t)0(1 − t2)−1dt)2l,

(38)

which can be expressed via a hypergeometric functions as follows

J = Γ(1)l+2

(F (−l, 1, 1, 12) 1Γ(1)

− F (−l, 1,−l, 12) Γ(1)Γ(1+l)Γ(−l)). (39)

In a similar form can be calculated the integral

J =∫ 1

−1 dxPml (x). (40)

Substituting (30) into (40) we obtain the integral in the form

J =∫ 1

−1 dx12m

(l+m)!(l−m)!m!

(1 − x2)m/2F (m− l,m+ l + 1,m+ 1, 1−x2). (41)

Using the formula (31) the integral (41) one can transform into the integral

J = − 12π i

Γ(1−m+l)Γ(m+1)Γ(1+l)

12m

(l+m)!(l−m)!m!

∫ 1

−1 dx(1 − x2)m/2∮(−t)m−l−1(1 − t)l(1 − t1−x

2)−m−l−1dt,

(42)

which can be rewritten in the form

J = − 1

2π i

Γ(1 −m+ l)Γ(m+ 1)

Γ(1 + l)

1

2m(l +m)!

(l −m)!m!×∫ 1

−1dx(1 − x2)m/2

∮(−t)m−l−1(1 − t)l(1 − t

2)−m−l−1(1 +

tx

2 − t)−m−l−1dt. (43)

In order to find the solution of the integral (43) it is necessarily to consider the integral

of form:

J =∫ 1

−1(1 − x2)m/2(1 + tx2−t)

−m−l−1dx, (44)

which can be transformed into the integral of form:


J = ( t2−t)

−m−l−1 ∫ 1

−1(1 − x2)m/2(2−tt

+ x)−m−l−1dx. (45)

The solution the integral (45) one can find using the binomial theorem and following

replacements

J = ( t2−t)

−m−l−1 ∫ 1

−1∑γ

k=0γ!

k!(γ−k)!(1)γ−k(−1)kx2k(2−t

t+ x)−m−l−1dx, (46)

γ = m/2,

u = (2−tt

+x

x)−1/(m+l+1). (47)

So integral (46) may be rewritten as follows

J = (2−tt)2k−m−l−1

∫u2(m+l+1)(k+1)−(m+l+1)2−(m+l+1)(1 − u(m+l+1))−2(k+1)+m+l+1du. (48)

The solution of the integral (48) one can find by replacement

u = (y)1/(m+l+1). (49)

We obtain the following expression for the looking for integral:

J = 1(m+l+1)

(2−tt)2k−m−l−1

∫ t/2

−t/(2−2t) y2(k+1)−m−l−3+ 1

m+l+1 (1 − y)−2(k+1)+m+l+1dy =

= 1(m+l+1)

(2−tt)2k−m−l−1

∫ t/2

−t/(2−2t) y2(k+1)−m−l−3+ 1

m+l+1×× ∑−2k+m+l−1

n=0(−2k+m+l−1)!

(n)!(−2k+m+l−1−n)!(1)(−2k+m+l−1−n)(−y)ndy.

(50)

The solution the integral (50) one can find using the binomial theorem

(1 − y)−2(k+1)+m+l+1 =∑−2k+m+l−1

n=0(−2k+m+l−1)!

(n)!(−2k+m+l−1−n)!(1)(−2k+m+l−1−n)(−y)n. (51)

Substituting equation (51) in the integral (50) one can obtain the looking for integral

in the form

J = 1(m+l+1)

(2−tt)2k−m−l−1

∑−2k+m+l−1n=0

(−2k+m+l−1)!(n)!(−2k+m+l−1−n)!(1)

(−2k+m+l−1−n)(−1)n×× ∫ t/2

−t/(2−2t) y2(k+1)−m−l−3+ 1

m+l+1+ndy.

(52)

We find the solution of the integral (52) in the form:

J = 1(m+l+1)

(2−tt)2k−m−l−1

∑−2k+m+l−1n=0

(−2k+m+l−1)!(n)!(−2k+m+l−1−n)!(1)

(−2k+m+l−1−n)(−1)n×× y

2k−m−l+ 1m+l+1

+n

(2k−m−l+ 1m+l+1

+n)|t/2−t/(2−2t).

(53)


Substituting (53) in (46) one can rewrite the integral (46) in the form:

J = ( t2−t)

−m−l−1 ∑γk=0

γ!k!(γ−k)!(1)

γ−k(−1)k×× 1

(m+l+1)(2−t

t)2k−m−l−1

∑−2k+m+l−1n=0

(−2k+m+l−1)!(n)!(−2k+m+l−1−n)!(1)

(−2k+m+l−1−n)(−1)n×× y

2k−m−l+ 1m+l+1

+n

(2k−m−l+ 1m+l+1

+n)|t/2−t/(2−2t).

(54)

Substituting (54) in the looking for integral (43) one can rewrite the integral (43) as

follows:

J = − 12π i

Γ(1−m+l)Γ(m+1)Γ(1+l)

12m

(l+m)!(l−m)!m!

1(m+l+1)

∑γk=0

γ!k!(γ−k)!(1)

γ−k(−1)k××∑−2k+m+l−1

n=0(−2k+m+l−1)!

(n)!(−2k+m+l−1−n)!(1)(−2k+m+l−1−n)(−1)n×

× ∑2ks=0

(2k)!(s)!(2k−s)!(−1)s×

× ∮(−t)m−l−1(1 − t)l(1 − t

2)−m−l−1×

× (2t)(2k−s) y

2k−m−l+ 1m+l+1

+n

(2k−m−l+ 1m+l+1

+n)|t/2−t/(2−2t)dt,

(55)

which can be rewritten in the form

J = − 12π i

Γ(1−m+l)Γ(m+1)Γ(1+l)

12m

(l+m)!(l−m)!m!

1(m+l+1)

∑γk=0

γ!k!(γ−k)!(1)

γ−k(−1)k××∑−2k+m+l−1

n=0(−2k+m+l−1)!

(n)!(−2k+m+l−1−n)!(1)(−2k+m+l−1−n)(−1)n×

× ∑2ks=0

(2k)!(s)!(2k−s)!(−1)s×

× 1(2k−m−l+ 1

m+l+1+n)

∮(−t)m−l−1(1 − t)l(1 − t

2)−m−l−1×

× (2t)(2k−s)((t/2)2k−m−l+

1m+l+1

+n − (−t/(2 − 2t))2k−m−l+1

m+l+1+n)dt,

(56)

or as follows

J = − 12π i

Γ(1−m+l)Γ(m+1)Γ(1+l)

12m

(l+m)!(l−m)!m!

1(m+l+1)

∑γk=0

γ!k!(γ−k)!(1)

γ−k(−1)k××∑−2k+m+l−1

n=0(−2k+m+l−1)!

(n)!(−2k+m+l−1−n)!(1)(−2k+m+l−1−n)(−1)n×

× ∑2ks=0

(2k)!(s)!(2k−s)!(−1)s×

× 2m+l− 1

m+l+1−n−s

(2k−m−l+ 1m+l+1

+n)

∮(−t)m−l−1(1 − t)l(1 − t

2)−m−l−1×

× (t)s−m−l+1

m+l+1+n(1 − (−1)2k−m−l+

1m+l+1

+n(1 − t)−2k+m+l− 1m+l+1

−n)dt.

(57)

We find the solution of the looking for integral as follows:


J =Γ(1 −m+ l)Γ(m+ 1)

Γ(1 + l)

1

2m(l +m)!

(l −m)!m!

1

(m+ l + 1)

γ∑k=0

γ!

k!(γ − k)!(1)γ−k(−1)k ×

−2k+m+l−1∑n=0

(−2k +m+ l − 1)!

(n)!(−2k +m+ l − 1 − n)!(1)(−2k+m+l−1−n)(−1)n ×

2k∑s=0

(2k)!

(s)!(2k − s)!(−1)s × 2m+l− 1

m+l+1−n−s

(2k −m− l + 1m+l+1

+ n)(−1)−s+m+l− 1

m+l+1−n ×

(F (s− 2l +1

m+ l + 1+ n,m+ l + 1, s− l +

1

m+ l + 1+ n+ 1, 1/2) ×

Γ(l + 1)

Γ(1 − s+ 2l − 1m+l+1

− n)Γ(s− l + 1m+l+1

+ n+ 1)−

−(−1)2k−m−l+1

m+l+1+nF (s− 2l +

1

m+ l + 1+ n,m+ l + 1, s+ 1 − 2k +m, 1/2) ×

Γ(1 − 2k +m+ 2l − 1m+l+1

− n)

Γ(1 − s+ 2l − 1m+l+1

− n)Γ(s+ 1 − 2k +m)). (58)

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Hamilton-Jacobi Formulation of Supermembrane

M. Kh. Srour1∗, M. Alwer2 and N. I.Farahat2†

1Physics Department, Al Aqsa University , P.O. Box 4051 Gaza, Palestine.2Physics Department, Islamic University of Gaza, P.O. Box 108 Gaza, Palestine.


Abstract: The Hamilton-Jacobi formalism of constrained systems is applied to Supermembrane

system. The equations of motion for a singular system are obtained as total differential equations

in many variables. These equations of motion are in exact agreement with those obtained by

Dirac’s method.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Supermembrane System; Hamilton-Jacobi formalism; Singular Lagrangian Systems;

Dirac Method; Classical Field Theory

PACS (2010): 03.50.-z; 11.10.Cd; 03.70.+k; 11.10.Ef; 45.50.Dd; 45.20.Jj; 47.10.Df

1. Introduction

The Hamiltonian Formulation of constrained systems was initiated by Dirac[1], he ob-

tained the equations of motion of singular Lagrangian systems by using the consistency

conditions. He also showed that the number of degrees of freedom of the mechanical

system can be reduced, this formalism has a wide application in field theory [2,3]. An

alternative Hamilton-Jacobi scheme for constrained systems was proposed by Guler[4,5].

Guler used the Hamilton-Jacobi formulation to obtain the equations of motion as total

differential equations. This approach based on Caratheodory’s equivalent Lagrangian

method [6]. This method has been applied to very few physical examples [7-21]. A bet-

ter understanding of its features, in the study of constrained systems when compared

to Dirac’s Hamiltonian formalism is still necessary. In this paper we wish to discuss

the classical mechanics of the Supermembrane system using the general procedure of

Hamilton-Jacobi formulation .

Now let us make a brief review on the Hamilton-Jacobi formulation of singular system

∗ E-mail: [email protected].† E-mail: [email protected].


(canonical method). The Lagrangian function of any physical system with n degrees of

freedom is a function of n generalized coordinates, n generalized velocities and parameter

t .i.e L ≡ L(qi, qi, t). The Hess matrix is defined as

Aij =∂2L(qi, qi, t)

∂qi∂qji, j = 1, 2, . . . , n. (1)

If the rank of this matrix is n, then the Lagrangian is called regular, otherwise it is called

singular. Moreover, systems which have singular Lagrangian are called singular systems

or constrained system.

One starts from the singular Lagrangian L(qi, qi, t) with Hess matrix of rank (n − p),

p < n. The generalized momenta pi corresponding to the generalized coordinates qi are

defined as

pa =∂L

∂qa, a = 1, 2, . . . , n− p, (2)

pμ =∂L

∂qμ, μ = n− p+ 1, . . . , n. (3)

Since the rank of the Hess matrix is (n− p), one may solve eqn. (2) for qa as

qa = qa(qi, qμ, pa; t) ≡ ωa (4)

Substituting eqn. (4), into eqn. (3), we get

pμ =∂L

∂qμ

∣∣∣∣qa=ωa

≡ −Hμ(qi, qμ, pa; t). (5)

The canonical Hamiltonian Ho is defined as

H0 = −L(qi, qν , qa; t) + paqa + pμqμ∣∣pν=−Hν

. (6)

The set of Hamilton-Jacobi partial differential equations (HJPDE) is expressed as

H ′α

(τ, qν , qa, Pi =

∂S

∂qi, P0 =

∂S

∂τ

)= 0, (7)

where

H ′0 = p0 +H0 , (8)

H ′μ = pμ +Hμ. (9)

We define P0 =∂S

∂τ, and Pi =

∂S

∂qi, with q0 = t, and S being the action.

The equations of motion are obtained as total differential equations and take the form

dqa =∂H ′

α

∂padtα, (10)

dpr = −∂H′α

∂qrdtα, r = 0, 1, . . . , n. (11)


dZ = (−Hα + pa∂H ′

α

∂pa)dtα. (12)

where Z = S(tα, qa). These equations are integrable if and only if

dH ′0 = 0, (13)

dH ′μ = 0, μ = n− r + 1, . . . , n. (14)

If conditions (13) and (14) are not satisfied identically, one may considered them as a new

constraints and again test the integrability conditions, then repeating this procedure, a

set of conditions may be obtained.

In the last works [13,14], the superparticles and superstring were studied using Hamilton-

Jacobi approach. In the recent work we will use the above formalism to discuss the clas-

sical mechanics of the supermembrane.

2. Hamilton-Jacobi Formulation of Subermembrane in Four Di-

mensions

Consider the following supermembrane action in four dimensions,[22]

I =

∫d3ξ[

1

2

√γN−1Πμ

0Π0μ − √γNaN−1Πμ

0Πaμ

− 1

2

√γ(γab −NaN bN−1)Πμ

aΠbμ +1

2

√γN + 3εabΠA

0 ΠBa Π

Cb BCBA].

(15)

The Lagrangian density is

L =1

2

√γN−1Πμ

0Π0μ − √γNaN−1Πμ

0Πaμ

− 1

2

√γ(γab −NaN bN−1)Πμ

aΠbμ +1

2

√γN + 3εabΠA

0 ΠBa Π

Cb BCBA,

(16)

where

ε12 = −ε21 = 1,Πμi = ∂iX

μ − iψΓμ∂iψ,Πμi = ∂iψ

α, ζ i = (τ, σ, ρ)(i = 1, 2, 3) are the

coordinates, ψα is a 32-components Majorana spinor, and (Xμ, ψα) are the coordinates

of the eleven-dimensional superspace. Using the definitions (2) and (3), the canonical

momenta (Pμ, Pα,Π,Πa,Πab) conjugated to canonical variables (Xμ, ψα, N,Na, γab) take

the forms

Pμ =∂L

∂(∂0Xμ)=

√γN

−1

Π0μ − √γNaN−1Πaμ + Sμ, (17)

Pα =∂L

∂(∂0ψα)= i(ΨΓμ)αPμ + Sα = −Hα, (18)

Π =∂L

∂(∂0N)= 0 = −HΠ, (19)

Πa =∂L

∂(∂0Na)= 0 = −Ha, (20)


Πab =∂L

∂(∂0γab)= 0 = −Hab, (21)

We can solve (17) forXμ in terms of Pμ and other coordinates as

Xμ ≡ ∂0Xμ =2Pμ√γN−1 + iψΓμψ + 2NaΠaμ − 2Sμ√

γN−1 (22)

Now , we introduce the Hamiltonian density H0 as

H0 =Pμ(∂0Xμ) + Pα(∂

0Ψα) + Π(∂0N) + Πa(∂0Na) + Πab(∂

0Nab) − L=

N

2√γ(Pμ − Sμ)(P

μ − Sμ) +1

2N

√γγabΠμ

aΠbμ − 1

2N

√γ

+NaΠμa(Pμ − Sμ),

(23)

and the canonical Hamiltonian may be written as

H0 =

∫dσdρH0

=

∫dσdρ

[N


μ − Sμ) +1

2N

√γγabΠμ

aΠbμ − 1

2N

√γ

+NaΠμa(Pμ − Sμ)

].

(24)

The set of HJPDEs defined in (8) and (9) reads as

H′0 = P0 +H0 =P0 +

N


μ − Sμ) +1

2N

√γγabΠμ

aΠab − 1

2N

√γ

+NaΠμa(Pμ − Sμ),

(25)

H′α = Pα +Hα = Pα + i(ΨΓμ)αPμ + Sα = 0, (26)

H′Π = Π = 0, (27)

and

H′ab = Πab = 0 (28)

Using (10) and (11), the set of HJPDE (25)-(29) leads to the following total differential

equations :

dXμ = (2Pμ√γN−1 + iψΓμψ + 2NaΠaμ − 2Sμ√

γN−1 )dτ (29)

dPμ = 0, (30)

dPα = 0, (31)

dΠ = (1


μ − Sμ) +1

2

√γγabΠμ

aΠbμ − 1

2

√γ)dτ, (32)

dΠa = Πμa(Pμ − Sμ)dτ, (33)

and

dΠab =1

2NΠμ

aΠbμdτ (34)

The set of total differential equations (29) to (34) are integrable if they satisfy the

integrability conditions (13) and (14). In fact the variations of H′0 and H

′α are not

identically satisfied, therefore the system is not integrable.


3. Conclusion

In this work the supermembrane in four dimensions is studied as a dynamical constrained

Hamiltonian system. The Hamilton -Jacobi method is examine to obtain the equations of

motion as total differential equations. In Hamilton-Jacobi treatment one obtains the con-

straints in phase space directly in forming Hamilton-Jacobi partial differential equations.

It is no need to classify the constraints as primary or secondary constraints, first class or

second class as in Dirac’s method. Furthermore, in Dirac’s method we must reduce any

constrained system to one with first class constraints only, which require to introduce an

arbitrary variables, therefore we have made a gauge fixing. This is not necessary at all

in Hamilton-Jacobi treatment. The final point in this conclusion is that the equations of

motion are not consequence, since the integrability conditions are not identically satisfied,

and the theory needs to add more constraints to the system.

References

[1] Dirac P.A. M., Lectures on Quantum Mechanics (Belfer Graduate School of Science,Yeshiva University, New York, N.Y.)1964.

[2] Sundermeyer K., Constrained Dynamics, Lect. Notes Phys., 169 (Springer,Verlag)1982.

[3] Gitman D.M. and Tyutin I.V., Quantization of Fields with Constraints(Springer,Verlag)1990.

[4] Guler Y., Nuovo Cimento,B 107 (1992) 1389.

[5] Guler Y.,Nuovo Cimento, B 107 (1992) 1143.

[6] CARATHEODORY C.Calculus of Variations and Partial Differential Equations ofFirst Order, Part II (Holden- Day) 1967,pp.205.

[7] W. I. Eshraim, and N.I. Farahat, Electronic J. of Theoretical Phys, 17, (2008), 65.

[8] N.I.Farahat, and Guler Y.,Nuovo Cimento B 111 (1996) 513.

[9] S.Muslih, N.I.Farahat, and M.Heles, Nuovo Cimento B119 ,(2004)531.

[10] N.I.Farahat, and Z.M.Nassar, Hadronic J.25 ,(2002)239.

[11] W. I. Eshraim, and N.I. Farahat, Hadronic Journal, 29 , (2006), 553.

[12] N.I. Farahat, and H. A. Elegla, Tur. J. Phys, 30 , (2006), 473.

[13] W. I. Eshraim, and N.I.Farahat, Electronic journal of theotetical Physics, 14, (2007).

[14] W. I. Eshraim, and N.I. Farahat, Islamic University Journal, 15 , (2007).

[15] N. I. Farahat, and H. A. Elegla, Modern physics Letters A, 25, No.2,(2010), 135.

[16] N. I. Farahat, and H. A. Elegla, Turkish J. of Phys, 32 , (2008), 1.

[17] W. I. Eshraim, and N.I. Farahat, Romanian J. of Phys, 53 , (2008), 437.

[18] N. I. Farahat, and H. A. Elegla, Electronic J of Theor. Phys, 19 , (2008), 1.


[19] N. I. Farahat, and H. A. Elegla, Int J. Theor. Phys, 49 , (2010), 384.

[20] W. I. Eshraim, and N.I. Farahat, Electronic Journal of Theoretical Physics, 22 ,(2009) 189.

[21] N. I. Farahat, and H. A. Elegla, J of App. Math. And Phys., 1, (2013)105.

[22] E. Bergshoeff, E. Sezgin, and Y. Tanii, Nuclear Phys. B298 , (1988)187.


Fractional Spin Through Quantum Affine AlgebraA(n) and Quantum Affine Super-algebra A(n,m)

Mostafa Mansour1∗ and Mohammed Daoud2

1 Departement de Physique, Faculte Polydisciplinaire de Beni Mellal.Universite Sultan Moulay Slimane. Beni Mellal. Morocco.

2 Departement de Physique, Faculte des Sciences. B. P. 28/SUniversite Ibnou Zohr. Agadir. Morocco.

Received 18 January 2015, Accepted 1 May 2015, Published 25 August 2015

Abstract: Using the splitting of a Q-deformed boson, in the Q → q = e2πik limit, the fractional

decomposition of the quantum affine algebra A(n) and the quantum affine super-algebra A(n,m)

are found. This decomposition is based on the oscillator representation and can be related to

the fractional supersymmetry and k-fermionic spin. We establish also the equivalence between

the quantum affine algebra A(n) and the classical one in the fermionic realization.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Quantum Algebra; Lie Algebra; Fractional Spin; Quantum Affine Superalgebra

PACS (2010): 03.65.Fd; 02.20.Sv; 02.20.Hj; 02.20.Uw;03.65.-w; 02.20.Tw; 75.10.Jm;04.60.Pp

1. Introduction

The concept of quantum algebra [1, 2], has extensively entered mathematical and phys-

ical literatures. From a mathematical point of view, quantum algebras can be regarded

as deformations of the universal enveloping algebras of semi-simple Lie algebras. The

quantum analogues of Lie superalgebras has been constructed in [3, 4]. Quantized en-

veloping algebras associated to affine algebras and superalgebras are given in [1, 5]. Many

properties of quantum algebras are quit similar to or richer than ones of the usual Lie

groups and algebras in connection with the representations theory. It is well known that

boson realization method is a very powerful and elegant method for the study of quantum

algebras representations. Based on this method, the representation theory of quantum

affine algebras has been an object of intensive studies; Available are the results for the

oscillator representations of affine algebra. There are obtained [6, 7, 8] through consistent

∗ email: [email protected].


realizations involving deformed Bose and Fermi operators [9, 10].

A geometric interpretation of fractional supersymmetry has been developed in [11,

12, 13, 14, 15]. In these works, the authors show that the one-dimensional superspace is

isomorphic to the braided line when the deformation parameter goes to a root of unity.

Similar techniques are used, in the reference [16], to show how internal spin arises natu-

rally in certain limit of the Q-deformed angular momentum algebra UQ(sl(2)). Indeed,

using Q-Schwinger realization, it is shown that the decomposition of the UQ(sl(2)) into

a direct product of not deformed U(sl(2)) and Uq(sl(2)) which is the same version of

UQ(sl(2)) at Q = q. The property of splitting of quantum algebras An, Bn, Cn and Dn

and quantum superalgebra A(m,n), B(m,n), C(n + 1) and D(n,m) in the Q → q limit

is investigated in [17]. The case of deformed Virasoro algebra and some other particular

quantum (Super)-algebra is given in [18] and the property of splitting of quantum (su-

per) Virasoro algebras is given in [19]. the case of quantum affine algebras with vanishing

central charge is developed in [20].

The aim of this paper is to investigate the property of decomposition of the

quantum affine algebra A(n) and the quantum affine superalgebra A(n,m) in the Q → q

limit. As a first step we wish to present in the next section (section 2) a some results

concerning k-fermions. In section 3, we discuss the property of Q-boson decomposition

in the Q → q limit. We introduce the way in which one obtains two independent objects

( an ordinary boson and a k-fermion ) from one Q-deformed boson when Q goes to

a root of unity. We establish also the equivalence between a Q-deformed fermion and

a conventional (ordinary ) one. Using these results, we analyse the Q → q limit of the

quantum affine algebra UQ(A(n)) (section 4) and the quantum affine superalgebra A(m,n)

(section 5). We note that Q-oscillator realization is crucial in this paper. Therefore, the

results obtained in this work are valid for the oscillator representations. In the last section

(section 6) we shall give some concluding remarks .

2. Introducing k-Fermionic Algebra.

The usual starting commutation relations of q-deformed bosonic algebra∑

q are

a−a+ − qa+a− = q−N

a−a+ − q−1a+a− = qN

qNa+q−N = qa+

qNa−q−N = q−1a−

qNq−N = q−NqN = 1

(1)

where the deformation parameter

q = exp(2πi

l) l ∈ N − {0, 1} (2)


is a root of unity. The annihilation operator a− is hermitic conjugated to creation operator

a+. the number operator N is hermitic. From equation (1), it is not difficult to obtain

the following relations

a−(a+)n = [[n]]q−N(a+)n−1 + qN(a+)na−

(a−)na+ = [[n]](a+)n−1q−N + qNa+(a−)n(3)

Where the symbol [[]] is defined by:

[[n]] =1 − q2n

1 − q2(4)

The cases of odd and even values of � have to be treated in slightly different ways. Hence,

we introduce a new variable k defined by

k = l for odd values of l

k = l2

for even values of l(5)

such that for odd l (rep. even l), we have qk = 1 (resp.,qk = −1). In the particular case

n = k, equations (3) are amenable to the form

a−(a+)k = ± (a+)ka−

(a−)ka+ = ± a+(a−)k(6)

In addition, the equation (1) yield

qN(a+)k = (a+)kqN

qN(a−)k = (a−)kqN(7)

We point out that the elements (a+)k and (a−)k are elements of the centre of∑

q

algebra (odd l). The irreducible representations are k-dimensional. Due to the fact that

the elements (a+)k and (a−)k are central, if one deals with a k-dimensional representation,

we have

(a+)k = α I, (a−)k = β I (8)

The extra possibilities parameterised by

(i) α = 0 β �= 0

(ii) α �= 0 β = 0

(iii) α �= 0 β �= 0


are not relevant for the considerations of this paper. The case (iii) correspond to

the periodic representation and in the cases (i) and (ii) we have the so-called semiperi-

odic(semicyclic) representation. In what follows, we shall deal with a representation of

the algebra∑

q such that

(a+)k = 0 , (a−)k = 0 (9)

are satisfied. We note that the algebra∑−1 obtained for k = 2, correspond to ordinary

fermion operators with (a+)2 = 0 and (a−)2 = 0 which reflects the Pauli exclusion

principle. In the limit case where k → ∞, we have the algebra∑

1, which correspond

to the ordinary boson operators. For k arbitrary, the algebra∑

q correspond to the

k−fermions(or anyons with fractional spin in the sense of Majid [21, 22]) operators that

interpolate between fermion and boson operators.

3. Fractional Spin through Q-boson.

In the previous section, we have been working with q a root of unity. When ql = 1,

quantum oscillator(k−fermionic) algebra exhibit rich representation behaviour with very

special properties different from the generic case. In the first case the Hilbert space is

finite dimensional, wile in the generic case the Fock space is infinite dimensional Hilbert

space. Now, let us consider, in order to investigate the decomposition of a Q−deformed

boson (q ∈ C) in the Q → exp(2π ik), the Q−deformed algebra ΔQ. The algebra ΔQ is

generated by an annihilation operator B−, a creation operator B+ and a number operator

NB with the relations

B−B+ −QB+B− = Q−NB

B−B+ −Q−1B+B− = QNB

QNBB+Q−NB = QB+

QNBB−Q−NB = Q−1B−

QNBQ−NB = Q−NBQNB = 1

(10)

From equation (10), we obtain

[Q−NBB−, [Q−NBB−, [. . . [Q−NBB−, (B+)k]Q2k . . . ]Q4 ]Q2 ] = Qk(k−1)

2 [k]! (11)

Where the Q−deformed factorial is given by

[k]! = [k][k − 1][k − 2] . . . [1]

[0]! = 1

with

[k] =Qk −Q−k

Q1 −Q−1


The Q−commutator, in (11), of two operators A and B is defined by

[A,B]Q = AB −QBA

The aim of this section is to determine the limit of the ΔQ algebra when Q goes to the

root of unity q see (2). The starting point is the limit Q → q of equation (11)

limQ→q1k[Q−NBB−, [Q−NBB−, [. . . [Q−NBB−, B+k

]Q2k . . . ]Q4 ]Q2 ]

= limQ→qQ

k(k−1)2

[k]![Q−NB(B−)k, (B+)k]

= qk(k−1)

2 .

(12)

The equation (12) can be reduced to :

limQ→q[Q

kNB2 (B−)k

([k]!)12

,(B+)

kQ

kNB2

([k]!)12

] = 1 (13)

We note that since q is a root of unity, it is possible to change the sign on the exponent

of qkNB

2 terms in the above and in the following definitions (when Q → q).

Following the work[18], we define the operators

b− = limQ→qQ±kNB

2 (B−)k

([k]!)12

b+ = limQ→q(B+)

kQ±kNB

2

([k]!)12

(14)

then we obtain

[b−, b+] = 1 (15)

Which are nothing but the commutation relation of an ordinary boson. The number

operator of this new bosonic oscillator is defined, in the usual ways, as Nb = b+b−.This type of reasoning, concerning the Q

toq limit of Q−boson, has been invoked for the first time in the references [13-16,18] in

order to investigate the fractional supersymmetry. (In these references, the authors show

that there is an isomorphism between the braided line and one dimensional superspace.)

Now, we are in a position to discuss the splitting of Q−deformed boson in the Q → q

limit. Let us introduce the new set of generators given by:

A− = B−q−kNb

2

A+ = B+q−kNb

2

NA = NB − kNb

(16)

which satisfies the following commutation relations

[A−, A+]q−1 = qNA

[A−, A+]q = q−NA

[NA, A±] = ±A±

(17)


and then define a k−fermion. The two algebras generated by the set of operators

{b+, b−, Nb} and {A+, A−, NA} are mutually commutative. We thus conclude that in the

Q

toq limit, the Q−deformed bosonic algebra oscillator decomposes into two independents

oscillators, an ordinary boson and k−fermion.

There is also a natural question which emerges: is it possible to find similar splitting

property for Q−deformed fermionic operators when the deformation parameter Q reduce

to a root of unity q ? To answer to this question, we consider the Q−deformed fermionic

algebra generated by the operators F−, F+ and NF satisfying the following relations

F−F+ + QF+F− = QN

F−F+ + Q−1F+F− = Q−N

QNFF+Q−NF = QF+

QNFF−Q−NF = Q−1F−

QNFQ−NF = Q−NFQNF = 1

(F+)2 = 0, (F−)2 = 0

(18)

We define the new operators

f− = Q−NF

2 F−

f+ = F+Q−NF

2

(19)

We obtain by a direct calculation the following anti-commutation relation

{f−, f+} = 1 (20)

Moreover, we have the nilpotency conditions

(f−)2 = 0

(f+)2 = 0(21)

Thus, we see that theQ−deformed fermion reproduces the conventional(ordinary)fermion.

4. Quantum Affine Algebra UQ(A(n)) at Q a Root of Unity

We use now the above results to derive the property of decomposition of quantum affine

algebras UQ(A(n)) in the Q → q limit. Recall that the UQ(A(n)) algebra is generated by

the set of generators {ei, fi, k±i = Q±hii = Qdi ±hi , 0 ≤ i ≤ n} satisfying the following

relations:

[ei, fj] = δijki−k−1

i

Qi−Q−1i

kiejk−1i = Q

aiji ej, kifjk

−1i = Q

−aiji fj

kik−1i = k−1i ki = 1, kikj = kjki

(22)


and the quantum Serre relations described by the expressions:

∑0≤p≤1−aij(−1)p

⎡⎢⎣ 1 − aij

p

⎤⎥⎦

Qi

e1−aij−pi eje

pi = 0

∑0≤p≤1−aij(−1)p

⎡⎢⎣ 1 − aij

p

⎤⎥⎦

Qi

f1−aij−pi fjf

pi = 0

(23)

In equations (22) ,(23) aij is the ij-element of n× n generalised Cartan matrix:

An =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

2 −1 0 · · · · · · −1

−1 2 −1 0

0 −1. . . . . . . . .

...

... 0. . . . . . . . . 0

...

2 −1

−1 0 · · · 0 −1 2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

and (di) are the non zero integers such that diaij = aijdi. The quantity

⎡⎢⎣ m

n

⎤⎥⎦Qi

in

equation (22) is defined by:

⎡⎢⎣ m

n

⎤⎥⎦Qi

=[m]Qi

!

[m− n]Qi! [n]Qi

!(24)

with

[Hi]Qi=QHi

i −Q−Hii

Qi −Q−1i

The quantum affine algebra UQ(An) admits two Q-oscillator representations: bosonic

and fermionic. in the bosonic realization, the generators of UQ(An) algebra can be con-


structed, by introducing (n +1) Q-deformed bosons as follows.

ei = B−i B+i+1, 1 ≤ i ≤ n

fi = B+i B

−i+1, 1 ≤ i ≤ n

ki = Q−Ni+Ni+1 , 1 ≤ i ≤ n

e0 = B−n+1B+1 ,

f0 = B−1 B+n+1,

k0 = QN1−Nn+1 ,

(25)

The fermionic realization is given by:

ei = F+i F

−i+1, 1 ≤ i ≤ n

fi = F−i F+i+1, 1 ≤ i ≤ n

ki = QNi−Ni+1 , 1 ≤ i ≤ n

e0 = F+n+1F

−1 ,

f0 = F+1 F

−n+1,

k0 = Q−N1+Nn+1 ,

(26)

At this stage we investigate the limit Q → q of the quantum affine algebra UQ(An). As

already mentioned in the introduction, our analysis is based on the Q-oscillator represen-

tation. Therefore all results obtained are specific to the use of Q-Schwinger realization. In

the Q → q, the splitting of Q-deformed bosons leads to classical bosons {b+i , b−i , Nbi , (1 ≤i ≤ n)} given by the equation (14)and k-fermionic operators {A+

i , A−i , NAi

, (1 ≤ i ≤ n)}defined by equations (16). From the classical bosons, we define the operators

ei = b−i b+i+1,

fi = b+i b−i+1,

hi = −Nbi +Nbi+1,

(27)

for i = 1, ..., n and

e0 = b−1 b+n+1,

f0 = b+1 b−n+1,

h0 = Nb1 −Nbn+1 ,

(28)

The set {ei, fi, ki, 0 ≤ i ≤ n} generate the classical algebra U(A(n)). From the

remaining operators {A+i , A

−i , NAi

, (1 ≤ i ≤ n + 1)}, one can realize the Uq(A(n))


algebra. Indeed, the generators defined by

Ei = A−i A+i+1, (1 ≤ i ≤ n)

Fi = A+i A

−i+1, (1 ≤ i ≤ n)

Ki = q−NAi+NAi+1 , (1 ≤ i ≤ n)

E0 = A−1 A+n+1,

F0 = A+1 A

−n+1,

K0 = qNA1−NAi+1 ,

(29)

generate the Uq(A(n)) algebra which is the same version of UQ(An) obtained by simply

setting Q = q, rather than by taking the limit as above.

Due to the commutativity of elements of Uq(An) and U(An), we obtain the following

decomposition of the quantum affine algebra Uq(An):

limQ→q

UQ(An) = Uq(An) ⊗ U(An).

in the bosonic realization.

To end this section, we discuss the equivalence between UQ(A(n)) and U(A(n)) algebras

in the fermionic construction. Indeed, We have discussed in the second section how one

can identify the conventional fermions with Q-deformed fermions. There have an equiv-

alence between these two objects. Consequently, due to this equivalence, it is possible

to construct Q-deformed affine algebras UQ(An) using ordinary fermions. It is also pos-

sible to construct the affine algebra An by considering Q-deformed fermions. So, in the

fermionic realization we have equivalence between U(An) and UQ(An). To be more clear,

we consider the UQ(An) in the Q-fermionic representation. The generators are given by:

ei = F−i F+i+1, 1 ≤ i ≤ n

fi = F+i F

−i+1, 1 ≤ i ≤ n

ki = QNFi−NFi+1 , 1 ≤ i ≤ n

e0 = F+n+1F

−1 ,

f0 = F+1 F

−n+1,

k0 = Q−NF1+NFn+1 ,

(30)

due to equivalence fermion - Q-fermion, the operators f−i , f−i are defined as a constant

multiple of conventional fermion operators, i.e,

f−i = Q−NFi2 F−i

f+i = F+

i Q−NFi

2

(31)


from which we can realize the generators:

Ei = f−i f+i+1, 1 ≤ i ≤ n

Fi = f+i f

−i+1, 1 ≤ i ≤ n

Hi = Nfi −Nfi+1, 1 ≤ i ≤ n

E0 = f+n+1f

−1 ,

F0 = f+1 f

−n+1,

H0 = −Nf1 +Nfn+1 ,

(32)

The set {Ei, Fi, Hi , 0 ≤ i ≤ n} generate the classical affine algebra U(An).

5. Quantum Affine Superalgebra UQ(A(m,n)) at Q a Root of

Unity

Let Q ∈ C − {0} be the deformation parameter. we shall use also Qi = Qdi with di are

numbers , that symmetries the Cartan matrix (aij). The quantum affine superalgebra

UQ(A(m,n)) is described in the Serre-Chevalley basis in terms of the simple root ei, fiand Cartan generators hi where i = 1, ....,m + n − 1 which satisfy the following super-

commutations relations

[ei, fj] = δijQdihi−Q−dihi

Qi−Q−1i

[hi, hj] = 0

[hi, ej] = aijej, [hi, fj] = −aijfj[ei, ei] = [fi, fi] = 0, if aii = 0

(33)

with the bracket [, ] is the Z2-graded one

[X, Y ] = XY − (−1)deg(X)deg(Y )Y X.


In the equation (33), (aij) is the element of the following Cartan matrix:

A(m,n) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 · · · · · · · · · · · · 0 −1

−1 2 −1 0 0

0 −1. . . . . . . . .

...

... 0. . . . . . 0

...

.... . . . . . . . . −1

. . .

0 −1 2 −1. . .

. . . −1 0 1. . .

. . . −1 2 −1 0...

.... . . −1

. . . . . . . . ....

... 0. . . . . . 0

0. . . . . . . . . −1

−1 0 · · · · · · · · · · · · 0 · · · 0 −1 2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

It is convenient to introduce the quantities ki = Qdihii in terms of which the defining

relations ( 33) become

[ei, fj] = δijki−k−1

i

Qi−Q−1i

kiejk−1i = Q

aiji ej, kifjk

−1i = Q

−aiji fj

kik−1i = k−1i ki = 1, kikj = kjki

[ei, ei] = [fi, fi] = 0, if aii = 0

(34)

Further the quantum affine superalgebra UQ(A(m,n)) generators obey to generalised

Serre relations .The latter’s are most simply presented in terms of the following rescaled

generators [25]:

ξi = eik− 1

2i , ζi = fik

− 12

i

they then take the form

(adQξi)1−aijξj = 0, i �= j,

(adQζi)1−aijζj = 0, i �= j.

(35)

where aij matrix is obtained from the non-symmetric Cartan matrix aij by substitut-

ing -1 for the strictly positive elements in the rows with 0 on the diagonal entry. The

quantum adjoint action adQ can explicitly written in terms of the coproduct and the

antipode as :

(adQX)Y = (−1)deg(X(2)).deg(Y )X(1)Y S(X(2))


with Δ(X) = X(1) ⊗X(2). and some supplementary relations for i such that aii = 0.

[[ei−1, ei]Q, [ei, ei+1]Q] = 0

[[fi−1, fi]Q, [fi, fi+1]Q] = 0

The universal quantum affine Lie superalgebra UQ(A(m,n)) is endowed with a Hopf

superalgebra structure with coproduct:

Δ(ki) = ki ⊗ ki

Δ(ei) = ei ⊗ k12i + k

− 12

i ⊗ ei

Δ(fi) = fi ⊗ k12i + k

− 12

i ⊗ fi

counit:

ε(ki) = 1, ε(ei) = ε(fi) = 0

and antipode :

S(ki) = −ki, S(ei) = −Qaii2

i ei, S(fi) = −Q−aii

2i fi

We shall give now the Q-oscillator representation of the quantum affine superalgebra

UQ(A(m,n)). We shall provide explicit expressions for corresponding generators as linear

and bilinear in Q-deformed bosonic and fermionic oscillators. The quantum affine super-

algera UQ(A(m,n)) can be realized simply by (m+1)Q-deformed fermions and (n+1)

Q-bosons. Explicitly the generators of AQ(m,n) are given by:

ei = F+i F

−i+1, 1 ≤ i ≤ m

fi = F−i F+i+1, 1 ≤ i ≤ m

ki = Q(NFi−MFi+1

), 1 ≤ i ≤ m

em+1 = F+m+1B

−1 , fm+1 = F+

m+1B+1 , km+1 = Q(NFm+1

+NB1)

em+j = B+j−1B

−j , 2 ≤ j ≤ n+ 1

fm+j = B−j−1B+j , 2 ≤ j ≤ n+ 1

km+j = Q(NBj−1−NBj

), 2 ≤ j ≤ n+ 1

e0 = B+n+1F

−1 ,

f0 = F+1 B

+n+1,

k0 = Q(NFn+1+NB1

)

(36)

Due to the property of Q-boson decomposition in the Q → q limit, each Q-boson

{B−i , B+i , NBi

} reproduce an ordinary boson {b−i , b+i , Nbi} and a k-fermion operator {A−i , A+i , NAi

}.


In the limit the Q-fermions become q-fermions which are objects equivalents to conven-

tional fermions {f−i , f+i , Nfi}. From the classical bosons {b−i , b+i , Nbi} and conventional

fermions {f−i , f+i , Nfi}, one can realize the classical affine algebra U(A(m,n)):

Ei = f+i f

−i+1, 1 ≤ i ≤ m

Fi = f−i f+i+1, 1 ≤ i ≤ m

Hi = Nfi −Nfi+1, 1 ≤ i ≤ m

Em+1 = f+m+1b

−1 , Fm+1 = f+

m+1b+1 , Hm+1 = Nfm+1 +Nb1

Em+j = b+j−1b−j , 2 ≤ j ≤ n

Fm+j = b−j−1b+j , 2 ≤ j ≤ n

Hm+j = Nbj−1−Nbj , 2 ≤ j ≤ n

E0 = b+n+1f−1 ,

F0 = f+1 b

+n+1,

H0 = Nfn+1 +Nb1

(37)

From the operators {A−i , A−i , NAi} we construct the generators

ei = A−i A+i+1, 1 ≤ i ≤ n+ 1

fi = A+i A

−i+1, 1 ≤ i ≤ n+ 1

ki = q−NAi+NAi+1 , 1 ≤ i ≤ n+ 1

e0 = A−n+1A+1 ,

f0 = A+n+1A

−1 ,

k0 = qNAn+1+NA1

(38)

for 1 ≤ i ≤ n+ 1, which generates the algebra Uq(An). It is easy to verify that Uq(A(n))

and A(m,n) are mutually commutative. As results, we have the following decomposition

of quantum superalgebra AQ(m,n) in the Q → q limit

limQ→qUQ(A(m,n)) = U(A(m,n)) ⊗ Uq(A(n)).

6. Conclusion

We have presented the general method leading to the investigation the Q → q = e2πik

limit of the quantum affine algebra UQ(An) and quantum affine superalgebra UQ(A(m,n))

based on the decomposition of Q-bosons in this limit. We note that Q-oscillator realiza-

tion is crucial in this manner of spitting in this paper. We believe that the techniques


and formulae, used here, will be useful foundation to extend this study to all quantum

affine algebras, quantum affine superalgebras and Q-deformed exceptional Lie algebras

and superalgebras.

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Nobel Lecture: Spontaneous Symmetry Breaking InParticle Physics: A Case of Cross Fertilization∗

Yoichiro Nambu

University of Chicago, The Enrico Fermi Institute, Chicago, Illinois 60637, USA

Abstract: We reproduce here the Nobel lecture of Yoichiro Nambu on the Spontaneous

Symmetry Breaking In Particle Physics: A Case of Cross Fertilization.

Reprinted with permission from the Nobel Foundation and Reviews Of Modern Physics,

Volume 81, JulySeptember 2009.

Ignazio Licata and Ammar Sakajic© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Elementary Particle; Spontaneous Symmetry Breaking; Cross Fertilization

PACS (2010): 11.30.Qc; 03.65.-w; 03.70.+k; 11.15.-q; 11.30.-j; 11.30.Rd; 12.10.-g; 12.15.-y;

11.30.Hv, 11.30.Ly

I will begin by a short story about my background. I studied physics at the University

of Tokyo. I was attracted to particle physics because of the three famous names, Nishina,

Tomonaga, and Yukawa, who were the founders of particle physics in Japan. But these

people were at different institutions than mine. On the other hand, condensed matter

physics was pretty good at Tokyo. I got into particle physics only when I came back

to Tokyo after the war. In hindsight, though, I must say that my early exposure to

condensed matter physics has been quite beneficial to me.

Particle physics is an outgrowth of nuclear physics which began in the early 1930s with

the discovery of the neutron by Chadwick, the invention of the cyclotron by Lawrence,

and the ınvention’ of meson theory by Yukawa (Nambu, 2007). The appearance of an

ever-increasing array of new particles in the subsequent decades, and the advances in

quantum Field theory gradu-ally led to our understanding of the basic laws of nature,

culminating in the present standard model.

When we faced those new particles, our first attempts were to make sense out of them

by finding some regularities in their properties. They invoked the symmetry principle to

∗ The 2008 Nobel Prize for Physics was shared by Yoichiro Nambu, Makoto Kobayashi, and Toshihide

Maskawa. This paper is the text of the address given in conjunction with the award.


classify them. Symmetry in physics leads to a conservation law. Some conservation laws

are exact, like energy and electric charge, but these attempts were based on approximate

similarities of masses and interactions.

Nevertheless, seeing similarities is a natural and very useful trait of the human mind.

The near equality of proton and neutron masses and their interactions led to the concept

of isospin SU(2) symmetry (Heisenberg, 1932). On the other hand, one could also go

in the opposite direction, and elevate symmetry to a more elaborate gauged symmetry.

Then symmetry will determine the dynamics as well, a most attractive possibility. Thus

the beautiful properties of electromagnetism was ex-tended to the SU(2) non-Abelian

gauge Field (Yang and Mills, 1954). But strong interactions are short range. Giving a

mass to a gauge Field destroys gauge invariance.

Spontaneous symmetry breaking SSB , which is the main subject of my talk, is a

phenomenon where a symmetry in the basic laws of physics appears to be broken. In

fact, it is a very familiar one in our daily life, although the name SSB is not the name is

due to (Baker and Glashow, 1962). For example, consider a elastic straight rod standing

vertically. It has a rotational symmetry; it looks the same from any horizontal direction.

But if one applies increasing pressure to squeeze it, it will bend in some direction, and the

symmetry is lost. The bending can occur in principle in any direction since all directions

are equivalent. But you do not see it unless you repeat the experiment many times. This

is SSB.

The SSB in quantum mechanics occurs typically in a uniform medium consisting of

a large number of elements. It is a dynamical effect. Symmetry allows some freedom of

action to each of them but the interaction among them forces them, figuratively speaking,

to line up like a crowd of people looking in the same direction. Then it is not easy to

change the direction wholesale even if it is allowed by the symmetry and hence does not

take energy, because the action is not local operator. So the symmetry appears to be

lost. It is still possible to recover the lost symmetry by a global operation, but it would

amount to a kind of phase transition. Some of the examples are

Physical system Broken symmetry

Ferromagnets Rotational invariance with respect to spin

Crystals Translational and rotational invariance (modulo discrete values)

Superconductors Local gauge invariance (particle number)

SSB in a medium then has the following characteristic properties:

(1) The ground state has a huge degeneracy. A symmetry operation takes one ground

state to another.

(2) Only one of the ground states and a whole spectrum of excited states built on it are

realized in a given situation.

(3) SSB is, in general, lost at sufficiently high temperatures.

In relativistic quantum Field theory, this phenomenon becomes also possible for the

entire space-time, for the ’vacuum’ is not void, but has many intrinsic degrees of freedom.


In this context, it may play an important role in cosmology. As the universe expands

and cools down, it may undergo one or more SSB phase transitions from states of higher

symmetries to lower ones, which change the governing laws of physics.

I will now recall the chain of events which led me to the idea of SSB and its application

to particle physics. One day in 1956 R. Schrieffer gave us a seminar on what would

come to be called the BCS theory (Bardeen et al., 1957) of superconductivity. I was

impressed by the bold-ness of their ansatz for the state vector, but at the same time I

became worried about the fact that it did not appear to respect gauge invariance. Soon

thereafter (Bogoliubov 1958) this is a fermionic version of transformation which he First

introduced in a description of super fluidity (1947) and (Valatin 1958) independently

introduced the concept of quasiparticles as fermionic ex-citations in the BCS medium.

The quasiparticles did not carry a definite charge as they were a superposition of electron

and hole, with their proportion depending on the momentum. How can one then trust

the BCS theory for discussing the electromagnetic properties like the Meissner effect? It

actually took two years for me to resolve the problem to my satisfaction. There were a

number of people who also addressed the same problem, but I wanted to understand it

in my own way. Essentially it is the presence of a massless collective mode, now known

by the generic name of Nambu-Goldstone NG boson, that saves charge conservation or

gauge in-variance.

The Bogoliubov-Valatin BV quasiparticles are de-scribed by the equations (Nambu,1960a),

Eψp,+ = εpψp,+ +Δψ†−p,−,

Eψ†−p,− = −εpψ†−p,− +Δψp,+,

E =√ε2p +Δ2 (1)

Here ψp,+ and ψ†−p,− are the wave functions for an electron and a hole of momentum

p and spin + or − , εp is the kinetic energy relative to the Fermi energy, and 2Δ is

the energy gap. In terms of spinlike matrices τi, the corresponding Hamiltonian and the

charge current are

H0 = εpΨ†τ3Ψ+ΔΨ†τ1Ψ,

ρ0 = Ψ†τ3Ψ, j0 = Ψ†(p/m)Ψ (2)

The BV ground state is Ψ†p = 0 for all p . The charge does not commute with H0, and the

continuity equation does not hold, which is the problem. But it has turned out that the

same interaction that led to the BCS-BV ground state also leads to collective excitations

f , which contributes to the charge current and restores the continuity equation. The

correct expression is

ρ � ρ0 +1

α2∂tf,

j � j0 − ∇f(∇2 − 1

α2∂t)f � −2ΔΨ†τ2Ψ (3)


f represents the NG mode. Physically, it corresponds to excitations that tend to restore

the lost symmetry. Its energy goes to zero in the long wavelength limit as it corresponds

to the global symmetry operation. It also happens that the above NG mode also mixes

with the Coulomb interaction among the electrons because of their common long-range

nature, and turns into the well-known plasmons with

ω2 = e2n/m (4)

where e, n, and m are, respectively, the charge, density, and mass of the electron.

The formal similarity of the BV equation to the Dirac equation naturally led me to

transport the BCS theory to particle physics (Nambu, 1960b). The gap Δ goes over

to the mass M , which breaks chirality γ5 rather than the ordinary charge 1. The axial

current is the analog of the electromagnetic vector current in the BCS theory. If chirality

is a broken symmetry, the matrix elements of the axial current between nucleon states of

four-momentum p and p′ should have the form

Γμ5(p′, p) = (γμγ5 − 2Mγ5qμ/q

2)F (q2), qμ = p′μ − pμ (5)

So chiral symmetry is compatible with a finite nucleon mass M provided that there

exists a massless pseudo-scalar NG boson. In reality, there is a pseudoscalar pion, and

the vector and axial vector interactions that appear in weak decays of the nucleons and

the pion had the properties

gV � gA, gπ �√2MgA/G, (6)

where gV and gA are vector and axial vector couplings of the nucleon, gπ is axial coupling

for the pion, G is the pion-nucleon coupling, and M is the nucleon mass. The second

equation was known as the Goldberger-Treiman relation (Goldberger and Treiman, 1958),

and it implies that the matrix element of the axial vector part of nucleon decay is

ΓμA= (γμγ5 − 2Mγ5qμ)/(q

2 −m2π), (7)

which differs from Eq.(5) by the presence of pion mass. In view of the smallness of

mπ compared to M, I made the hypothesis that the axial current is an approximately

conserved quantity, the nucleon mass is generated by an SSB of chirality, and the pion

is the corresponding NG boson which should become massless in the limit of ex-act

conservation. (Proton and neutron masses should also become equal).

The model system [Nambu and Jona-Lasinio (1961a); the first presentation of the

model was presented by Nambu and Jona-Lasinio (1961b)] I worked out subsequently

with Jona-Lasinio is a concrete realization of the proposed SSB. It has the form similar

to the BCS model

L = −ψγμ∂μψ + g[(ψψ)2 − (ψγ5ψ)], (8)

which is invariant against the particle number and chiral transformations,

ψ → exp(iα)ψ, ψ → ψ exp(−iα),ψ → exp(iγ5α)ψ, ψ → ψ exp(iγ5α). (9)


After SSB, the ’nucleon’ acquires a mass M ∼ 2g < ψψ > . Although the model is

nonrenormalizable, it is easy to demonstrate the SSB mechanism. The generated mass

M is determined by the ’gap equation’

2π2

gΛ2= 1 − M2

Λ2ln(1 +

Λ2

M2), (10)

where Λ is a cutoff. Bound states of nucleon-antinucleon meson and nucleon-nucleon

dibaryon pairs of spin 0 and 1 were also found. In particular, the masses of 0−(∼ ψγ5ψ)

and 0+(∼ ψψ) mesons are found to be 0 and 2M , respectively. A more realistic two-flavor

model was also considered by generalizing Eq. (8) to

L = −ψγμ∂μψ + g[(ψψ)2 − Σi(ψγ5τiψ)(ψγ5τiψ)], (11)

with a similar gap equation. We get an isovector 0 pion and a isoscalar 0+. The actual

pion mass was generated by a small explicit bare mass in the Lagrangian of the order of

5 MeV. This also induced a change in axial cou-pling constant gA in the right direction.

Other examples of the BCS-type SSB are 3He super-fluidity and nucleon pairing in

nuclei (Arima and Iachello, 1975, 1976). In general, there exist simple mass relations

among the fermion and the boson in BCS-type theories (Nambu, 1985).

The BCS theory also accounts for the generation of the London mass for the elec-

tromagnetic Field. This problem is made simple in terms of the Higgs scalar Field

(Anderson, 1963; Englert and Brout, 1964; Higgs, 1964). The relativistic analog of the

London relation is, in mo-mentum space,

jμ(q) = Kμν(q)Λk,

Kμν = (δμν − qμqν/q2)K(q2),

K(q2) � q2/(q2 −m2). (12)

The third relation shows the massless NG boson turn-ing into a massive ’plasmon,’ a

process corresponding to Eq. (4) . This was successfully applied to weak gauge Field in

the Weinberg-Salam WS theory (Weinberg, 1967; Salam, 1968) of electroweak unification.

The fermion masses are also generated and break chiral invariance. These so-called

current masses for the up and down quarks play the role of the bare mass in the Nambu-

Jona-Lasinio model.

In the current standard model of particle physics, the NJL model may be regarded

as an effective theory for the QCD with respect to generation of the so-called constituent

masses. One is interested in the low energy degrees of freedom on a scale smaller than

some cutoff Λ ∼ 1 GeV. The short distance dynamics above Λ as well as the confinement

may be treated as a perturbation The problem has been extensively studied by many

people. The Lagrangian adopted by Hatsuda and Kunihiro (1994) is

L = LQCD + LNJL + LKMT + δL (13)

LNJL is for the quarks, and contains ’current mass’ terms. LKMT refers to the Kabayashi-

Maskawa-’t Hooft chiral anomaly

LKMT = gD{det[qi(1 − γ5)qj] +H.c.}. (14)


Both of them contribute to the explicit breaking of chiral invariance. (δL contains the

effects of confinement and one gluon exchange). The WS theory resembles the Ginzburg-

Landau (1950) description of superconductivity which was shown to follow from the BCS

theory by Gor’kov (1959). In the same way the NJL model goes over to the model of

Gell-Mann and Levy (1960) . If this analogy turns out real, the Higgs Field might be an

effective description of underlying dynamics.

Finally, I will end this lecture with a comment on the mass hierarchy problem. Hi-

erarchical structure is an outstanding feature of the universe. The masses of known

fundamental fermions also make a hierarchy stretching 11 orders of magnitude. Mass is

not quantized in a simple regular manner like charge and spin. Mass is a dynamical quan-

tity since it receives contributions from interactions. But we do not see yet a pattern like

those in the hydrogen atom which led to quantum mechanics, or the Regge trajectories

which led to the dual string picture.

The BCS mechanism seems relevant to the problem, as was remarked earlier. It

generates a mass gap for fermions, plus the Goldstone and Higgs modes as low-lying

bosons. The bosons may act in turn as an agent for further SSB, leading to the possibility

of hierarchical SSB or ’tumbling’ (Raby et al., 1980). In fact we already have examples

of it (Nambu, 2004):

(1) The chain atoms-crystal-phonon-superconductivity. The NG mode for crystal for-

mation is the phonon which induces the Cooper pairing of electrons to cause super-

conductivity.

(2) The chain QCD-chiral SSB of quarks and baryons−π, σ, and other mesons-nuclei for-

mation and nucleon pairing-nuclear collective modes. No further elaboration would

be required.

I am greatly thankful to G. Jona-Lasinio for his help in the planning of the lecture.

References

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