Table Of ContentMathematical Physics Studies
Asao Arai
Inequivalent
Representations
of Canonical
Commutation and
Anti-Commutation
Relations
Representation-theoretical Viewpoint
for Quantum Phenomena
Mathematical Physics Studies
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Asao Arai
Inequivalent Representations
of Canonical Commutation
and Anti-Commutation
Relations
Representation-theoretical Viewpoint
for Quantum Phenomena
AsaoArai
DepartmentofMathematics
HokkaidoUniversity
Sapporo
Hokkaido,Japan
ISSN0921-3767 ISSN2352-3905 (electronic)
MathematicalPhysicsStudies
ISBN978-981-15-2179-9 ISBN978-981-15-2180-5 (eBook)
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Preface
As is well known, canonical commutation relations (CCR) and canonical anti-
commutation relations (CAR) are fundamental algebraic relations in quantum
physics. Models in quantum mechanics and quantum field theory are constructed
based on Hilbert space representations of CCR and/or CAR. There are basically
two categories for representations of CCR and CAR, respectively, i.e., reducible
and irreducible, and each of them is divided into two classes: equivalent and
inequivalent.QuantumtheoriesbasedonequivalentrepresentationsofCCRand/or
CARarephysicallyequivalent,beingdifferentonlyintheframeworkofthephysical
picture.1 On the other hand, quantum theories based on inequivalent irreducible
representationsareessentiallydifferentfromeachother,describingnon-comparable
physicalsituations.
Rolesofequivalentrepresentationsandinequivalentirreduciblerepresentations
of CCR and/or CAR are different. Although equivalent representations are phys-
ically equivalent to each other as mentioned above, they may be mathematically
important. For example, there may be mathematical problems which are not so
easy to solve in a representation, but relatively are very easy to solve in other
representations equivalent to the former.2 Concerning inequivalent irreducible
1For example, the Schrödinger representation of the CCR with d degrees of freedom (see
Sect.2.6.1forthedefinition)iswithintheframeworkofthephysicalpictureinwhichtheposition
of a quantum particle is measured in the d-dimensional position space, while the momentum
representation oftheCCRwiththesamedegree, whichisequivalent totheSchrödinger one, is
withinthe framework of thephysical picture in which the momentum of a quantum particle is
measuredinthed-dimensionalmomentumspace.Inthiscase,thed-dimensionalFouriertransform
is the unitary transformation which maps the Schrödinger representation to the momentum
representation,givingtheequivalenceofthem(seeSect.2.6.1formoredetails).
2A typical example is a one-dimensional quantum harmonic oscillator. The spectrum of the
HamiltonianiseasilyfoundintheBorn–Heisenberg–Jordan representationoftheCCRwithone
degree of freedom (see Sect.2.6.2) rather than in the Schrödinger one with the same degree,
whichisequivalenttotheformer.Inquantumfieldtheory,theQ-spacerepresentation(functional
Schrödinger representation), which is equivalent to the Fock representation of the CCR over a
Hilbertspace,isveryuseful(seeSect.8.13).
v
vi Preface
representationsof CCR and/or CAR, the following philosophicalpoint of view is
suggestedbystudiesonmodelsinquantummechanicsandquantumfieldtheory:
The Universe uses inequivalent irreducible representations of CCR and/or
CAR to create or produce “characteristic” quantum phenomena in which
macroscopicobjectsmaybeinvolved.
Examples of such phenomena include: Aharonov–Bohm effect, Casimir effect,
Bose–Einstein condensation, superfluidity, superconductivity, boson masses, and
fermionmasses(seethetextofthisbookforfurtherdetails).
This book is concernedwith mathematicalanalysis on representationsof CCR
andCAR,includingbothfiniteandinfinitedegreesoffreedom.Withaphilosophy
as stated above,the emphasis of the description is put on inequivalentirreducible
representationsofCCRandCARandreadingtheirrolesinthecontextofquantum
physics.
It may be asked why it still is necessary to consider or study representations
of CCR with finite degrees of freedom; is not the von Neumann uniqueness
theoremenough?Theanswerisno,becausethevonNeumannuniquenesstheorem
applies only to Weyl representations of CCR with finite degrees of freedom on
a separable Hilbert space (unfortunately this point seems not to be understood
preciselyingeneralamongnon-expertsofrepresentationtheoryofCCR).Fornon-
Weyl representationsof CCR, even if their degrees of freedom are finite, the von
Neumannuniquenesstheoremisnotvalidanymore.But,interestinglyenough,the
Universemakesuseofthisinvaliditytocreateorproduce“characteristic”quantum
phenomena,givingphysicallyimportantrolestonon-WeylrepresentationsofCCR
withfinitedegreesoffreedomaswell.
In the case of CCR with infinite degrees of freedom, even in the category of
irreducible Weyl representations, inequivalent ones appear and some of them in
quantum field theory correspondto physically interesting quantum phenomenaas
mentionedabove.
There may be inequivalentrepresentationsof CCR and CAR whichare related
to spontaneous symmetry breaking.3 In this book, however, we do not discuss
this aspect. To do that, one needs approaches using ∗-algebras (C∗-algebras, von
∗
Neumann algebras, or O -algebras) and their representations. Representations
of CCR and CAR treated in this book are basically the ones that are operator
realizationsofalgebraicrelationsonHilbertspaces.
This book is organized as follows. Chapter 1 is a summary of mathematical
theorieswhichmaybeusedinthefollowingchapters.Itismainlyconcernedwith
operatortheoriesonHilbertspaces. Thosewhoarefamiliarwith thesubjectsmay
skipthischapter.
In Chap.2, we describe a general theory of representationsof CCR with finite
degrees of freedom. For each degree of freedom, three classes of representations
of CCR exist basically: ordinary ones in each of which all the representativesare
3Thisisseen,e.g.,inaBose–EinsteincondensationofaninfinitesystemoffreeBosegasatfinite
temperature.
Preface vii
not bounded, weak Weyl ones in each of which half of the representatives are
unitary,andWeylonesineachofwhicheveryrepresentativeisunitary.Inaddition
to these classes, we consider also weak forms of CCR. We review Schrödinger
representations and Born–Heisenberg–Jordan (BHJ) representations of CCR and
give a direct proof of the equivalence of them with the same degrees of freedom,
which corresponds to the equivalence of Schrödinger’s “wave mechanics” as a
quantum theory4 and “matrix mechanics” by BHJ. Concerning the von Neumann
uniqueness theorem mentioned above, we only state it without proof. A detailed
analysisonweakWeylrepresentationsisgiven.Itisnoteworthythat,inweakWeyl
representations,eachself-adjointrepresentativeisabsolutelycontinuous.
Chapter 3 is devoted to the analysis of a representation of the CCR with two
degreesoffreedomappearinginquantummechanicsofatwo-dimensionalsystem
ofachargedparticlemovingaroundamagneticfieldconcentratedonsomepoints
in the plane. It is well known that, in such a physical situation, a characteristic
phenomenon, called the Aharonov–Bohm (AB) effect (a shift of the interference
patternofelectronbeamsduetotheexistenceofasolenoidalmagneticfield),may
occur. We show that the representation is irreducible and formulate a necessary
and sufficient condition for the representation to be equivalentto the Schrödinger
representationwiththesamedegree.Weshowthatthecasewheretherepresentation
isinequivalenttotheSchrödingeronecorrespondstotheoccurrenceoftheABeffect
inthecontextunderconsideration.Thisisaninterestingcorrespondencetonote.
In Chap.4, we present a general mathematicaltheory of time operators. In the
first stage of cognition, a time operator T is defined to be a canonical conjugate
ofaHamiltonianH,i.e.,itisarepresentative(symmetric,butnotnecessarilyself-
adjoint)inarepresentationoftheCCRwithonedegreeoffreedom([T,H]=iona
suitablesubspaceoftherepresentationspaceunderconsideration).Concerningthis
definition,thefollowingpointmayberemarked.Therehasbeenamisunderstanding
thatthereexistnotimeoperatorsasobservables(self-adjointoperators).Thismay
beduetoPauli’sstatementintheabsenceoftimeoperators(seeSect.4.2formore
details)ormathematicallynon-rigorousargumentswhichdonotcareaboutdomains
of unbounded operators. In fact, Pauli’s statement is false. This is shown by an
explicitconstructionofaself-adjointtimeoperatorforaclassofHamiltonians(see
Sect.4.4) (but, every Hamiltonian does not have a self-adjointtime operator).We
introducefiveclassesoftimeoperators.Arigorousformoftime–energyuncertainty
relation is formulated in terms of time operator. If H is semi-bounded (bounded
frombeloworabove),thentherepresentation(T,H)oftheCCRwithonedegreeof
freedomisobviouslyinequivalenttotheSchrödingerrepresentationwiththesame
degree.Hence,herewehaveclassesofrepresentationsinequivalenttoSchrödinger
representations.We infer that each class of time operatorsplays differentroles in
physics.Forexample,asweshallshow,astrongtimeoperatorcontrolsthedecay-
in-time rate of transition probabilities of state vectors (see Sect.4.5.14). There is
4Inaone-bodyprobleminthree-dimensionalspace,Schrödinger’swavemechanicscanbeviewed
alsoasaclassicalfieldtheory.
viii Preface
a time operatorwhich is related to time delay in a two-bodyscattering processin
non-relativisticquantummechanics(seeRemark4.15).
Chapter 5 is a brief description of part of representation theory of CAR with
finite degreesof freedom.Representationsof CAR with finite degreesof freedom
arephysicallyrelatedtointernaldegreesoffreedomlikespin.
In Chap.6 (resp. Chap.7), we describe elements of the theory of boson (resp.
fermion)Fockspace,whichisusedtoconstructBose(resp.Fermi)fieldsinquantum
fieldtheory.Onlyrequisitesforthefollowingchaptersaregiven.
InChap.8(resp.Chap.9),weanalyzerepresentationsofCCR(resp.CAR)with
infinitedegreesoffreedominanabstractframework.
Thelastchapterisdevotedtodescriptionsofvariousinequivalentrepresentations
of CCR or CAR with infinite degrees of freedom and physical correspondences
of them, including scaling of quantum fields, infinite volume theories, Bose–
Einstein condensation, temperatures, infrared and ultraviolet renormalizations,
bosonmasses,andmassesofDiracparticles.
For the reader’sconvenience,fourappendicesare addedin thisbook.Theyare
concernedwithbasicmathematicalsubjects.
This book is written as an introductory text to theories of representations of
CCR and CAR in relation to quantum physics with special attention given to
their inequivalent representations, being directed to general audiences including
graduate students and researchers in mathematics and physics who are interested
in mathematical structures of quantum mechanics and quantum field theory. For
this reason, in the main text (Chaps.2–10) of the book, the author tried to make
mathematicaldescriptionsasdetailedaspossible.Nocompletenessisintendedfor
thebibliographyinthisbook,andreferencesinthebibliographyarerestrictedmore
or less to those which are directly related to the subjects discussed in the text.
Articles and books written in Japanese also are included in the bibliography for
convenienceofreaderswhocanreadJapanese.
It would be the author’s great pleasure if this book could demonstrate the
above-mentioned philosophical point of view to a certain extent and suggest the
unfathomablerichnessanddepthoftheworldofrepresentationsofCCRandCAR
asfundamentalconceptsunderlyingquantumphenomena.
Sapporo,Japan AsaoArai
October2019
Acknowledgments
Theory of representations of CCR has been one of my research subjects since
I started my research career in 1978. During past years, I have benefited from
conversations or discussions with many researchers on the subject. First of all,
I would like to express my hearty thanks to Professor Hiroshi Ezawa (Emeritus
Professor, Gakushuin University) who inspired me with the importance of the
representation-theoreticalpointofviewforquantummechanicsandquantumfield
theory,inparticular,theimportanceofinequivalentrepresentationsofCCR.Ihave
learnedalotfromhisexcellentbookStructuresofQuantumMechanics(Iwanami-
Shoten, 1978, in Japanese) and from his seminars on mathematical physics at
Gakushuin University. Professor Ezawa gave me also the opportunityto deliver a
series of lectures on representations of CCR and physics in summer schools on
mathematicalphysicsofwhichhewasoneoftheorganizers.Iwouldliketothank
also Professor Shige-Toshi Kuroda (Emeritus Professor, Gakushuin University),
who was my teacher when I was a graduate student at the University of Tokyo,
andProfessorHuzihiroAraki(EmeritusProfessor,KyotoUniversity)fortheirkind
interestinmyworkonrepresentationsofCCR.Mythanksgoalsotothefollowing
researcherswhogavemetheopportunitytodeliveraseriesoflecturesoraseminar
on representations of CCR at their institutions (honorific titles omitted): Daisuke
Fujiwara, Harald Grosse, Masao Hirokawa, Fumio Hiroshima, Takashi Ichinose,
Atsushi Inoue, Hiroshi Isozaki, Tadashi Kawanago, Hideki Kurose, Itaru Mitoma
(deceased), Nobuaki Obata, Masanori Ohya (deceased), Izumi Ojima, Tomohiro
Sasamoto,HerbertSpohn,KenjiYajima.
ix
