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Fundamental Theories of Physics 180 Lawrence P. Horwitz Relativistic Quantum Mechanics Fundamental Theories of Physics Volume 180 Series editors Henk van Beijeren Philippe Blanchard Paul Busch Bob Coecke Dennis Dieks Detlef Dürr Roman Frigg Christopher Fuchs Giancarlo Ghirardi Domenico J.W. Giulini Gregg Jaeger Claus Kiefer Nicolaas P. Landsman Christian Maes Hermann Nicolai Vesselin Petkov Alwyn van der Merwe Rainer Verch R.F. Werner Christian Wuthrich The international monograph series “Fundamental Theories of Physics” aims to stretch the boundaries of mainstream physics by clarifying and developing the theoreticalandconceptualframeworkofphysicsandbyapplyingittoawiderange ofinterdisciplinaryscientificfields.Originalcontributionsinwell-establishedfields such as Quantum Physics, Relativity Theory, Cosmology, Quantum Field Theory, Statistical Mechanics and Nonlinear Dynamics are welcome. The series also provides a forum for non-conventional approaches to these fields. Publications should present new and promising ideas, with prospects for their further development, and carefully show how they connect to conventional views of the topic. Although the aim of this series is to go beyond established mainstream physics, a high profile and open-minded Editorial Board will evaluate all contributions carefully to ensure a high scientific standard. More information about this series at http://www.springer.com/series/6001 Lawrence P. Horwitz Relativistic Quantum Mechanics 123 Lawrence P.Horwitz Schoolof Physics RaymondandBeverlySackler Faculty ofExact Sciences TelAviv University Ramat Aviv Israel and Department ofPhysics BarIlan University Ramat Gan Israel and Department ofPhysics Ariel University Samaria Israel FundamentalTheories of Physics ISBN978-94-017-7260-0 ISBN978-94-017-7261-7 (eBook) DOI 10.1007/978-94-017-7261-7 LibraryofCongressControlNumber:2015944499 SpringerDordrechtHeidelbergNewYorkLondon ©SpringerScience+BusinessMediaDordrecht2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerScience+BusinessMediaB.V.DordrechtispartofSpringerScience+BusinessMedia (www.springer.com) Acknowledgments IwouldliketothankConstantinPironforhiscollaborationindevelopingthemain ideasofthetheorydiscussedinthisbookduringvisitstotheUniversityofGeneva, hosted initially by J.M. Jauch, and for many discussions and collaborations which followed.IamgratefultoStephenL.AdlerforhiswarmhospitalityattheInstitute for Advanced Study in Princeton over a period of several visits during which, among other things, much of the research on this subject was done. I would also like to thank William C. Schieve for his collaborations over many years, particu- larlyinthedevelopmentofrelativisticstatistical mechanics,andforthehospitality he and his wife, Florence, offered me at the University of Texas at Austin during manyvisits.IwouldalsoliketothankFritzRohrlichforhishospitalityatSyracuse University and for many discussions on relativistic quantum theory and for his collaboration in our study of the constraint formalism. I am also particularly indebted to Rafael I. Arshansky for his collaboration on developing many of the basic ideas, such as the solution of the two-body bound stateproblem,theLandau-Peierlsrelationandmanyaspectsofscatteringtheory,as well as Ishi Lavie (Z”L), who fell in the first Lebanon-Israel war; he played anessentialroleinthedevelopmentofmanyofthebasicideasinscatteringtheory aswell. IamgratefultoY.Rabinwhoworkedwithmeonthefirstanalysesofthe phenomenon of interference in time in 1976, Ori Oron for his contributions to theapplicationoftheeikonalapproximationstotherelativisticequationsandtothe relativistic generalization of Nelson’s approach to the quantum theory through the use of Brownian motion, and Nadav Shnerb for the development of the second quantized formalism with the help and guidance of Kurt Haller (Z”L), to Igal Aharonovich for his intensive work on the classical electromagnetic properties oftherelativisticallycharged particle andtheassociatedfive-dimensionalfields,to Martin Land for his many invaluable contributions to the theory and many of its applications,toAviGershonforhisinvaluablecontributiontotheapplicationofthe theorytotheTeVeSprogram(ageometricalapproachtothedarkmatterproblem), to Andrew Bennett, who found the way to compute the anomalous magnetic moment of the electron in this framework, as well as many discussions, Meir Zeilig-Hess for his contribution to the black body problem and tensor representa- tions,andtomanyotherstudentsandcolleagueswhohavecontributedtotheeffort to understand the structure and consequences of the theory discussed here. v vi Acknowledgments I also wish to thank Kirsten Theunissen of Springer, for her patience and assistenceduringthepreeparationofthemanuscript,andAlwynvanderMerwefor his encouragement at the beginning of this project. Finally,Iwishtothankmywife,Ruth,forherpatienceandsupportinthemany years during which the research was done in developing and exploring the con- sequencesoftheapproachtorelativisticquantummechanicsthatisdiscussedinthis book. Contents 1 Introduction and Some Problems Encountered in the Construction of a Relativistic Quantum Theory . . . . . . . . . 1 1.1 States in Relativistic Quantum and Classical Mechanics . . . . . . 1 1.2 The Problem of Localization for the Solutions of Relativistic Wave Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Relativistic Classical and Quantum Mechanics. . . . . . . . . . . . . . . 9 2.1 The Einstein Notion of Time . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 The Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 The Newton-Wigner Problem. . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 The Landau-Peierls Problem . . . . . . . . . . . . . . . . . . . . . . . . . 24 Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Spin, Statistics and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1 Relativistic Spin and the Dirac Representation . . . . . . . . . . . . . 33 3.2 The Many Body Problem with Spin, and Spin-Statistics . . . . . . 42 3.3 Construction of the Fock Space and Quantum Field Theory. . . . 44 3.4 Induced Representation for Tensor Operators. . . . . . . . . . . . . . 47 Appendix B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4 Gauge Fields and Flavor Oscillations. . . . . . . . . . . . . . . . . . . . . . 51 4.1 Abelian Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 Nonabelian Gauge Fields and Neutrino Oscillations . . . . . . . . . 59 4.3 The Hamiltonian for the Spin 1 Neutrinos . . . . . . . . . . . . . . . . 65 2 4.4 CP and T Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5 The Relativistic Action at a Distance Two Body Problem . . . . . . . 71 5.1 The Two Body Bound State for Scalar Particles. . . . . . . . . . . . 72 5.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3 The Induced Representation. . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4 The Stueckelberg String . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 vii viii Contents 6 Experimental Consequences of Coherence in Time. . . . . . . . . . . . 97 6.1 General Problem of Coherence in Time. . . . . . . . . . . . . . . . . . 97 6.2 The Lindner Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3 Experiment Proposed by Palacios et al.. . . . . . . . . . . . . . . . . . 110 7 Scattering Theory and Resonances. . . . . . . . . . . . . . . . . . . . . . . . 113 7.1 Foundations of Relativistic Scattering Theory. . . . . . . . . . . . . . 114 7.2 The S Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.3 Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.4 Two Body Partial Wave Analysis. . . . . . . . . . . . . . . . . . . . . . 122 7.5 Unitarity and the Levinson Theorem. . . . . . . . . . . . . . . . . . . . 125 7.6 Resonances and Semigroup Evolution. . . . . . . . . . . . . . . . . . . 126 7.7 Lax Phillips Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.8 Relativistic Lee-Friedrichs Model . . . . . . . . . . . . . . . . . . . . . . 137 8 Some Applications: The Electron Anomalous Moment, Invariant Berry Phases and the Spacetime Lattice . . . . . . . . . . . . 143 8.1 The Anomalous Moment of the Electron. . . . . . . . . . . . . . . . . 144 8.2 Invariant Berry Phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.3 The Spacetime Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9 Hamiltonian Map to Conformal Modification of Spacetime Metric: Kaluza-Klein and TeVeS. . . . . . . . . . . . . . . 157 9.1 Dynamics of a Relativistic Geometric Hamiltonian System . . . . 158 9.2 Addition of a Scalar Potential and Conformal Equivalence. . . . . 159 9.3 TeVeS and Kaluza-Klein Theory . . . . . . . . . . . . . . . . . . . . . . 163 9.4 The Bekenstein-Sanders Vector Field as a Gauge Field. . . . . . . 165 9.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 10 Relativistic Classical and Quantum Statistical Mechanics and Covariant Boltzmann Equation. . . . . . . . . . . . . . . . . . . . . . . 173 10.1 A Potential Model for the Many Body System. . . . . . . . . . . . . 174 10.2 The Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . 175 10.3 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 10.4 Grand Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.5 Relativistic Quantum Quantum Statistical Mechanics. . . . . . . . . 187 10.6 Relativistic High Temperature Boson Phase Transition . . . . . . . 191 10.7 Black Body Radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 10.8 Manifestly Covariant Relativistic Boltzmann Equation. . . . . . . . 196 11 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 1 IntroductionandSomeProblems EncounteredintheConstruction ofaRelativisticQuantumTheory 1.1 StatesinRelativisticQuantumandClassicalMechanics Oneofthedeepestandmostdifficultproblemsoftheoreticalphysicsinthepastcen- turyhasbeentheconstructionofasimple,well-definedone-particletheorywhich unites the ideas of quantum mechanics and relativity. Early attempts, such as the constructionoftheKlein-GordonequationandtheDiracequationwereinadequate to provide such a theory since, as shown by Newton and Wigner (1949), they are intrinsicallynon-local,inthesensethatthesolutionsoftheseequationscannotpro- vide a well-defined local probability distribution. This result will be discussed in detailbelow.Relativisticquantumfieldtheories,suchasquantumelectrodynamics, provideamanifestlycovariantframeworkforimportantquestionssuchastheLamb shift and other level shifts, the anomalous moment of the electron and scattering theory,butthediscussionofquantummechanicalinterferencephenomenaandasso- ciatedlocalmanifestationsofthequantumtheoryarenotwithintheirscope;theone particlesectorofsuchtheoriesdisplaythesameproblempointedoutbyNewtonand Wignersincetheysatisfythesameone-particlefieldequations. Ontheotherhand,thenonrelativisticquantumtheorycarriesacompletelylocal interpretationofprobabilitydensity;itcanbeusedasarigorousbasisforthedevelop- mentofnonrelativisticquantumfieldtheory,startingwiththeconstructionoftensor productspacestobuildtheFockspace,andonthatspacetodefineannihilationand creation operators (e.g., Baym 1969). The development of a manifestly covariant singleparticlequantumtheory,withlocalprobabilityinterpretation,couldbeused in the same way to develop a rigorous basis for a relativistic quantum field theory which carries such a local interpretation. A central problem in formulating such a theoryisposedbytherequirementofconstructingadescriptionofthequantumstate ofanelementarysystem(e.g.,a“particle”)asamanifestlycovariantfunctionona manifoldofobservablecoordinateswhichbelongstoaHilbertspace.Theessential propertiesofthequantumtheory,suchasthenotionsofprobability,transitionampli- tudes,linearsuperposition,observablesandtheirexpectationvalues,arerealizedin termsofthestructureofaHilbertspace. ©SpringerScience+BusinessMediaDordrecht2015 1 L.P.Horwitz,RelativisticQuantumMechanics, FundamentalTheoriesofPhysics180,DOI10.1007/978-94-017-7261-7_1