Relativistic quantum mechanics

218 Pages·2015·1.55 MB·English

by  Horwitz

Fundamental Theories of Physics 180
Lawrence P. Horwitz
Relativistic
Quantum
Mechanics
Fundamental Theories of Physics
Volume 180
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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
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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