Inequivalent representations of canonical commutation and anti-commutation relations PDF
Preview Inequivalent representations of canonical commutation and anti-commutation relations
Mathematical Physics Studies Asao Arai Inequivalent Representations of Canonical Commutation and Anti-Commutation Relations Representation-theoretical Viewpoint for Quantum Phenomena Mathematical Physics Studies SeriesEditors GiuseppeDito,Dijon,France EdwardFrenkel,Berkeley,CA,USA SergeiGukov,Pasadena,CA,USA YasuyukiKawahigashi,Tokyo,Japan MaximKontsevich,Bures-sur-Yvette,France NicolaasP.Landsman,Nijmegen,TheNetherlands BrunoNachtergaele,Davis,CA,USA Theseriespublishesoriginalresearchmonographsoncontemporarymathematical physics.Thefocusis onimportantrecentdevelopmentsattheinterfaceof Mathe- matics,andMathematicalandTheoreticalPhysics.Thesewillinclude,butarenot restricted to: applicationof algebraicgeometry,D-modulesand symplecticgeom- etry,categorytheory,numbertheory,low-dimensionaltopology,mirrorsymmetry, string theory, quantum field theory, noncommutativegeometry, operator algebras, functionalanalysis,spectraltheory,andprobabilitytheory. Moreinformationaboutthisseriesathttp://www.springer.com/series/6316 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) https://doi.org/10.1007/978-981-15-2180-5 ©SpringerNatureSingaporePteLtd.2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore 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
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