Operational quantum theory I: Nonrelativistic structures

423 Pages·2006·3.65 MB·english

by  Saller

OPERATIONAL PHYSICS
OPERATIONAL PHYSICS
Modernphysicsincreasinglyreplacestheobject-basedworld-viewoftheoldphysicswith
an operation-based world-view. This process is supported by a similar one that is taking
place in mathematics at the same time. heTlnaioatrntenIalrnouJfoalicetorheTsicyshP
dedicatestheseries“OperationalPhysics”tothisprocess.Itsgoalistoprovidescientists
andresearcherswithtextbooks,monographs,andreferenceworkstoaddressthegrowing
needforinformation.
Prospective authors are encouraged to correspond with the Editor-in-Chief in advance of
submittingamanuscript.SubmissionofmanuscriptsshouldbemadetotheEditor-in-Chief.
Editor-in-Chief
DavidFinkelstein
GeorgiaInstituteofTechnology
SchoolofPhysics
837StateStreet
Atlanta,GA30332-0430
VolumesPublishedinThisSeries:
OperationalQuantumTheoryI—NonrelativisticStructures
bySaller,H.2006
OperationalQuantumTheoryII—RelativisticStructures
bySaller,H.2006
Heinrich Saller
OperationalQuantum
TheoryI
Nonrelativistic Structures
HeinrichSaller
Werner-HeisenbergInstitut
Max-PlanckInstitutfu¨rPhysik
Mu¨nchen,Germany80805
[email protected]
LibraryofCongressControlNumber:2006920923
ISBN-10:0-387-29199-7
ISBN-13:978-0387-29199-4
Printedonacid-freepaper.
(cid:2)C 2006SpringerScience+BusinessMedia,Inc.
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permissionofthepublisher(SpringerScience+BusinessMedia,Inc.,233SpringStreet,NewYork,
NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use
in connection with any form of information storage and retrieval, electronic adaptation, computer
software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,evenifthey
arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare
subjecttoproprietaryrights.
PrintedintheUnitedStatesofAmerica. (TB/MVY)
9 8 7 6 5 4 3 2 1
springer.com
I would like to thank three people from three generations without whom this
book could not have been written.
First, the late Werner Heisenberg, who implanted in me the conviction
that symmetries with their operations are appropriate basic concepts for un-
derstanding physical interactions and objects.
Second, David Finkelstein, who gave me the feeling, in our fruitful collab-
oration and work over the decades, of not being alone in giving priority to the
operational approach.
Finally, I learned a lot from my first son, Christian, who is a much better
mathematician than I. He taught me that many of the mathematical concepts
denigrated as esoteric and academic by those physicists who have a direct
pipelinetoGodarebasicandexactlytherighttoolsforthephysicalstructures
tobeformalized. Healsohelpedmeverymuchbyknowingandrecommending
the appropriate advanced mathematical literature.
V
Contents
INTRODUCTION 1
1 SPACETIME TRANSLATIONS 17
1.1 Time Translations . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2 Position Translations . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.1 Axial Vectors for Rotations . . . . . . . . . . . . . . . . 22
1.2.2 Orientation Manifold of Scalar Products . . . . . . . . . 24
1.3 Spacetime Translations . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.1 Order of Minkowski Space . . . . . . . . . . . . . . . . . 30
1.3.2 Spacetime Topology . . . . . . . . . . . . . . . . . . . . 30
1.3.3 Orientation Manifold of Lorentz Metrics . . . . . . . . . 31
1.4 Decompositions of Spacetime . . . . . . . . . . . . . . . . . . . 32
1.4.1 Decompositions into Time and Position . . . . . . . . . . 33
1.4.2 Decompositions into Position and Light . . . . . . . . . . 37
1.4.3 Lightlike Bases . . . . . . . . . . . . . . . . . . . . . . . 38
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.6 Relations and Mappings . . . . . . . . . . . . . . . . . . . . . . 40
1.7 Equivalence and Order . . . . . . . . . . . . . . . . . . . . . . . 41
1.8 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.9 Monoids and Groups . . . . . . . . . . . . . . . . . . . . . . . . 45
1.9.1 Products of Groups . . . . . . . . . . . . . . . . . . . . . 46
1.10 Vector Space Duality . . . . . . . . . . . . . . . . . . . . . . . . 48
1.11 Bilinearity and Tensor Product . . . . . . . . . . . . . . . . . . 49
1.12 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.12.1 Vector Space Endomorphisms . . . . . . . . . . . . . . . 52
1.12.2 Products of Lie Algebras . . . . . . . . . . . . . . . . . . 55
1.13 Reflections (Conjugations) . . . . . . . . . . . . . . . . . . . . . 56
1.13.1 Inner Products . . . . . . . . . . . . . . . . . . . . . . . 58
1.13.2 Endomorphism Reflections . . . . . . . . . . . . . . . . . 60
1.13.3 Quadratic and Positive Forms (“Squares”) . . . . . . . . 61
1.14 Equivalent Vector Space Bases . . . . . . . . . . . . . . . . . . . 62
1.14.1 Equivalent Inner Products . . . . . . . . . . . . . . . . . 62
1.14.2 Invariance Groups and Lie Algebras of
Inner Products . . . . . . . . . . . . . . . . . . . . . . . 64
1.15 Matrix Diagonalization and Orientation Manifolds . . . . . . . . 66
1.16 Reflections in Orthogonal Groups . . . . . . . . . . . . . . . . . 67
VII
VIII CONTENTS
2 TIME REPRESENTATIONS 71
2.1 The Time Group . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.2 Representations of the Complex Numbers. . . . . . . . . . . . . 73
2.3 Time Representations and Unitarity . . . . . . . . . . . . . . . . 74
2.4 Causal Time Representations . . . . . . . . . . . . . . . . . . . 76
2.5 Nondecomposable Hamiltonians . . . . . . . . . . . . . . . . . . 77
2.6 Time Orbits and Equations of Motion . . . . . . . . . . . . . . . 78
2.7 Self-Dual Time Representations . . . . . . . . . . . . . . . . . . 79
2.8 Compact Time Representations . . . . . . . . . . . . . . . . . . 81
2.9 Noncompact Time Representations . . . . . . . . . . . . . . . . 82
2.10 Invariants and Weights . . . . . . . . . . . . . . . . . . . . . . . 84
2.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.12 Group Realizations and Klein Spaces . . . . . . . . . . . . . . . 86
2.12.1 Self-Realizations of a Group . . . . . . . . . . . . . . . . 88
2.12.2 Fix- and Stabilgroups . . . . . . . . . . . . . . . . . . . . 89
2.12.3 Group Orbits as Irreducible Realizations . . . . . . . . . 90
2.12.4 Left and Right Cosets . . . . . . . . . . . . . . . . . . . 91
2.13 Group and Lie Algebra Representations . . . . . . . . . . . . . . 92
2.13.1 Sum and Product Representations . . . . . . . . . . . . . 94
2.13.2 Scalar and Dual Representations . . . . . . . . . . . . . 94
2.13.3 Representation Monoids . . . . . . . . . . . . . . . . . . 96
2.14 Invariant Inner Products and Self-Dual Representations . . . . . 97
2.15 Characters of Groups . . . . . . . . . . . . . . . . . . . . . . . 99
2.16 Representations of Ordered Monoids . . . . . . . . . . . . . . . 100
2.17 Minimal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 101
2.17.1 Algebraic Elements . . . . . . . . . . . . . . . . . . . . . 103
2.17.2 Projectors and Nilpotents; Jordan Bases . . . . . . . . . 103
2.17.3 Exponential and Logarithm . . . . . . . . . . . . . . . . 105
2.18 The HausdorffProduct . . . . . . . . . . . . . . . . . . . . . . . 107
2.19 (Semi)Simple and Decomposable Endomorphisms . . . . . . . . 107
2.20 Representations of Compact (Finite)
Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.21 Algebra Representations and Modules . . . . . . . . . . . . . . . 110
2.21.1 Group Algebra . . . . . . . . . . . . . . . . . . . . . . . 113
2.21.2 (Semi)Simple Associative Algebras . . . . . . . . . . . . 114
2.22 Characteristic and Minimal Polynomial . . . . . . . . . . . . . . 116
2.22.1 Triagonalization and Diagonalization . . . . . . . . . . . 117
2.22.2 Eigenspaces and Eigenvalues . . . . . . . . . . . . . . . . 118
2.22.3 Principal Spaces. . . . . . . . . . . . . . . . . . . . . . . 120
3 SPIN, ROTATIONS, AND POSITION 125
3.1 Linear Operations on the Alternative . . . . . . . . . . . . . . . 126
3.2 Pauli Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.3 Spin Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.4 Spinor Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.5 Spin Representations . . . . . . . . . . . . . . . . . . . . . . . . 131
3.6 Position Translations from Adjoint Spin Structures . . . . . . . 134
CONTENTS IX
3.7 Polynomials with Spin Group Action . . . . . . . . . . . . . . . 136
3.7.1 Spinor Polynomials . . . . . . . . . . . . . . . . . . . . . 136
3.7.2 Harmonic Polynomials and Spherical Harmonics . . . . . 136
3.8 Spin Representation Matrix Elements . . . . . . . . . . . . . . . 139
3.9 Spin Invariants and Weights . . . . . . . . . . . . . . . . . . . . 141
3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
3.11 Derivations of Algebras . . . . . . . . . . . . . . . . . . . . . . . 143
3.11.1 Inner Derivations . . . . . . . . . . . . . . . . . . . . . . 143
3.11.2 Adjoint Affine Lie Algebra . . . . . . . . . . . . . . . . . 144
3.11.3 Adjoint Affine Lie Group . . . . . . . . . . . . . . . . . . 145
3.12 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . 146
3.13 Exponential and Logarithmic Mappings . . . . . . . . . . . . . . 147
3.13.1 Lie Algebra-Lie Group Relations . . . . . . . . . . . . . 148
3.13.2 Lie-Jacobi Transformation . . . . . . . . . . . . . . . . . 150
3.14 (Semi)Simple Lie Algebras . . . . . . . . . . . . . . . . . . . . . 151
3.15 Lie Algebra Inner Products . . . . . . . . . . . . . . . . . . . . 152
3.16 Lie Algebra Decompositions . . . . . . . . . . . . . . . . . . . . 154
3.17 Multilinearity and Tensor Algebra . . . . . . . . . . . . . . . . . 154
3.17.1 Grassmann and Polynomial Algebra. . . . . . . . . . . . 156
3.17.2 Volume Elements and Axial Vectors . . . . . . . . . . . . 158
3.17.3 Derivations of Tensor Algebras . . . . . . . . . . . . . . 159
3.18 Enveloping Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 161
3.18.1 Lie Algebra Invariants . . . . . . . . . . . . . . . . . . . 162
4 ANTISTRUCTURES: The Real in the Complex 167
4.1 Anticonjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.2 The Complex Quartet . . . . . . . . . . . . . . . . . . . . . . . 170
4.2.1 Canonical Real Substructures . . . . . . . . . . . . . . . 171
4.2.2 Isomorphisms in the Complex Quartet . . . . . . . . . . 171
4.3 Antidoubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.3.1 The Anticonjugation Invariance Group . . . . . . . . . . 173
4.4 Dual and Antirepresentations . . . . . . . . . . . . . . . . . . . 174
4.5 Particles and Antiparticles . . . . . . . . . . . . . . . . . . . . . 176
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.7 Twin Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.8 Complexification of Real Vector Spaces . . . . . . . . . . . . . . 180
5 SIMPLE LIE OPERATIONS 183
5.1 Diagonalization of Operations . . . . . . . . . . . . . . . . . . . 184
5.1.1 Eigenspaces and Eigenforms (Weights) . . . . . . . . . . 184
5.1.2 Projectors and Nilpotents . . . . . . . . . . . . . . . . . 186
5.2 Abelian, Nilpotent, and Solvable . . . . . . . . . . . . . . . . . . 187
5.2.1 Solvable and Nilpotent Algebras . . . . . . . . . . . . . . 188
5.2.2 Theorems of Engel and Lie . . . . . . . . . . . . . . . . . 189
5.3 The Basic Lie Operations . . . . . . . . . . . . . . . . . . . . . 191
5.3.1 The Lie Algebras with Dimensions up to Three . . . . . 191
5.3.2 Heisenberg Lie Algebras . . . . . . . . . . . . . . . . . . 194
X CONTENTS
5.4 Spectral Decompositions of Lie Algebras . . . . . . . . . . . . . 194
5.4.1 Spectral Decompositions for Nilpotent Lie Algebras . . . 195
5.4.2 Cartan Subalgebras . . . . . . . . . . . . . . . . . . . . . 196
5.4.3 Spectral Decomposition of Simple Lie Algebras . . . . . 198
5.5 “Spin” Structure of Simple Lie Algebras . . . . . . . . . . . . . 199
5.5.1 Canonical Triplet Generators . . . . . . . . . . . . . . . 201
5.5.2 A and Ac : The Lie Algebras
n−1 n−1
of SL(Cn) and SU(n) . . . . . . . . . . . . . . . . . . . 203
5.6 Roots and Weights . . . . . . . . . . . . . . . . . . . . . . . . . 206
5.6.1 Root Systems with Reflections . . . . . . . . . . . . . . . 206
5.6.2 Fundamental Roots and Weights . . . . . . . . . . . . . 208
5.6.3 Weight Modules and Weight Cones . . . . . . . . . . . . 210
5.6.4 Euclidean Structure for Weights . . . . . . . . . . . . . . 211
5.7 Classification of Complex Simple Lie Algebras and
Dynkin Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 214
5.8 Simple Complex and Compact Lie Groups . . . . . . . . . . . . 216
5.9 Simple Root Systems . . . . . . . . . . . . . . . . . . . . . . . . 217
5.9.1 The Root Systems A . . . . . . . . . . . . . . . . . . . . 217
r
5.9.2 The Root Systems C . . . . . . . . . . . . . . . . . . . . 218
r
5.9.3 The Root Systems B . . . . . . . . . . . . . . . . . . . 219
r
5.9.4 The Root Systems D . . . . . . . . . . . . . . . . . . . 220
r
5.9.5 The Root System G . . . . . . . . . . . . . . . . . . . . 221
2
5.9.6 The Root System F . . . . . . . . . . . . . . . . . . . . 222
4
5.9.7 The Exceptional Lie Algebras E . . . . . . . . . . . . 223
6,7,8
5.10 Real Simple Lie Algebras . . . . . . . . . . . . . . . . . . . . . . 223
5.10.1 The Normal and Compact Forms of a Simple
Complex Lie Algebra . . . . . . . . . . . . . . . . . . . . 224
5.10.2 Reflections of Compact Lie Algebras . . . . . . . . . . . 225
5.10.3 Classification of Real Simple Lie Algebras . . . . . . . . 226
6 RATIONAL QUANTUM NUMBERS 231
6.1 Simple Representations of Simple Lie Symmetries . . . . . . . . 232
6.2 Representation Invariants and Weights of Simple
Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
6.2.1 Weight Module and Representation Cone . . . . . . . . . 236
6.2.2 Dual Representations and Invariant Forms . . . . . . . . 237
6.2.3 Permutation Symmetry of Product Representations . . . 238
6.3 Representations of Simple Lie Algebras . . . . . . . . . . . . . . 239
6.3.1 Representations of the Lie Algebras A . . . . . . . . . . 239
r
6.3.2 Representations of the Lie Algebras C . . . . . . . . . . 244
r
6.3.3 Representations of the Lie Algebras B . . . . . . . . . . 245
r
6.3.4 Representations of the Lie Algebras D . . . . . . . . . . 246
r
6.3.5 Representations of the Exceptional Lie Algebras . . . . . 247
6.4 Centrality of Representations . . . . . . . . . . . . . . . . . . . 249
6.4.1 Broken Winding Numbers as Lepton and Quark
Hypercharges . . . . . . . . . . . . . . . . . . . . . . . . 251