Inequivalent representations of canonical commutation and anti-commutation relations

Preface......Page 6
Acknowledgments......Page 10
Contents......Page 11
List of Mathematical Symbols......Page 17
  1.1.1 Definitions and Some Basic Facts......Page 20
  1.1.3 Spectra......Page 27
  1.1.4 Unitary Operators and Related Facts......Page 29
  1.1.5 Symmetric and Self-adjoint Operators......Page 30
  1.1.6 Lowest Energy and Ground State of a Self-adjoint Operator......Page 31
  1.1.7 Spectral Theorem and Some Facts in Functional Calculus......Page 32
  1.1.8 Trace Class and Hilbert–Schmidt Operators......Page 34
  1.1.9 Reduction of Operators......Page 38
  1.1.10 Strongly Continuous One-Parameter Unitary Groups......Page 39
  1.1.11 Strongly Continuous One-Parameter Semi-Groups......Page 41
 1.2 Strong Commutativity of Self-adjoint Operators......Page 43
 1.3 Criteria of Self-adjointness of Symmetric Operators......Page 45
 1.4 Basic Properties of Symmetric Operators......Page 48
 1.5 Tensor Product of Hilbert Spaces......Page 50
 1.6 Symmetric and Anti-symmetric Tensor Product Hilbert Spaces......Page 53
 1.7 Tensor Products of L2-Spaces......Page 55
 1.8 Tensor Products of Operators......Page 57
 1.9 The Natural Isomorphism Between I2(H) and HH......Page 60
 1.10 Convergence of Self-adjoint Operators in Strong Resolvent Sense......Page 62
 1.11 Sesquilinear Form and Representation Theorem......Page 63
 1.12 Fourier Analysis......Page 64
 1.13 Discrete Fourier Transform......Page 67
 1.14 Momentum Operator with Periodic Boundary Condition......Page 69
 1.15 Momentum Operators with Other Boundary Conditions......Page 73
 1.16 Higher-Dimensional Discrete Fourier Transform......Page 76
  2.1.1 Definition of Representation of Canonical Commutation Relations and Remarks......Page 78
  2.1.2 Some General Properties......Page 82
  2.1.3 Unitary Equivalence......Page 85
 2.2 Heisenberg Uncertainty Relation......Page 86
 2.3 Direct Sum Representations......Page 90
 2.4 Tensor Product Representations......Page 91
 2.5 Irreducibility......Page 93
 2.6 Quantum Mechanics as Representations of CCR......Page 100
  2.6.1 Schrödinger Representation......Page 102
  2.6.2 Born–Heisenberg–Jordan Representation......Page 108
  2.6.3 Equivalence of Schrödinger Representation—``Wave Mechanics''—and BHJ Representation—``Matrix Mechanics''......Page 115
   (A) A System with a Periodic Boundary Condition......Page 119
   (B) A System with a Dirichlet Boundary Condition......Page 123
  2.6.5 Quantum System in a Half-Line......Page 125
 2.7 Cyclic Representations......Page 127
  2.8.1 Heuristics......Page 133
  2.8.2 Weyl Relations and CCR......Page 135
  2.8.3 Weyl Representation of CCR......Page 139
  2.8.4 Spectral Properties......Page 140
  2.8.5 Irreducibility......Page 141
  2.8.6 Unitary Invariance......Page 142
  2.8.7 Real Linear Combinations of Representatives......Page 143
  2.8.8 Von Neumann's Uniqueness Theorem and Remarks......Page 146
 2.9 Weak Form of CCR......Page 149
 2.10 Weak Weyl Representations......Page 151
 2.11 Absolute Continuity of Self-adjoint Representatives in Weak Weyl Representations......Page 157
 2.12 Functional Calculus for a Weak Weyl Relation......Page 162
 3.1 Introduction......Page 165
 3.2 Essential Self-adjointness of the Physical Momenta......Page 167
 3.4 Commutation Relations of Exponential Operators......Page 171
 3.5 Remarks on the Case of Constant Magnetic Fields......Page 173
 3.6 Flatness of Vector Potentials and Existence of Representations of CCR......Page 175
 3.7 Inequivalent Representations......Page 179
 3.8 Correspondence to Aharonov–Bohm Effect......Page 181
 3.9 Representations for Different Vector Potentials......Page 183
 3.10 Charges and Inequivalent Irreducible Representations of CCR......Page 187
 3.11 Notes......Page 188
 4.1 Introduction......Page 189
 4.2 An Abstract Definition of Time Operator and Time-Energy Uncertainty Relation......Page 194
 4.3 A Structure Generating Pairs of a Hamiltonian and a Time Operator......Page 196
 4.4 Time Operators of Hamiltonians with Purely Discrete Spectrum......Page 197
  4.4.1 An Operator Associated with a Self-adjoint Operator with Purely Discrete Spectrum......Page 198
  4.4.2 A Time Operator of H......Page 200
  4.4.3 A Sufficient Condition for Tmax to be Bounded......Page 203
  4.4.4 Unboundedness of TG......Page 205
  4.4.5 Concluding Remarks......Page 206
  4.5.1 Definition and Basic Properties......Page 207
  4.5.2 Self-adjoint Strong Time Operator and Weyl Representation of CCR......Page 210
  4.5.3 Non-Self-adjointness of Strong Time Operators......Page 211
  4.5.4 A Perturbation Theorem......Page 212
  4.5.5 Spectral Properties of Strong Time Operators......Page 213
  4.5.6 Quasi-Weyl Relation......Page 215
  4.5.7 A Remark on Uniqueness of Weak Weyl Representations......Page 218
  4.5.8 Strong Time Operator of HF......Page 219
  4.5.9 Strong Time Operator of an N-body System......Page 221
  4.5.10 Aharonov–Bohm Time Operators......Page 225
  4.5.11 Strong Time Operators of Free Relativistic Schrödinger Operators......Page 228
  4.5.12 Strong Time Operators of a Free Dirac Operator......Page 231
  4.5.13 A Structure Generating Pairs of a Hamiltonian and a Strong Time Operator......Page 236
  4.5.14 Decay-in-Time of Transition Probabilities......Page 238
  4.5.15 Existence of Strong Time Operators......Page 241
  4.5.16 Construction of Strong Time Operators of a Self-adjoint Operator from Those of Another Self-adjoint Operator......Page 245
 4.6 Other Classes of Time Operators......Page 247
 4.7 Generalized Time Operators......Page 250
 5.1 Introduction......Page 252
 5.2 Representations of the CAR with One Degree of Freedom......Page 258
 5.3 Representations of the CAR with N Degrees of Freedom......Page 261
 6.1 The Boson Fock Space Over a Hilbert Space......Page 264
 6.2 Boson Second Quantization Operators......Page 265
 6.3 Boson Γ-Operators......Page 269
 6.4 Creation and Annihilation Operators......Page 273
 6.5 Commutation Relations......Page 279
 6.7 Relative Boundedness of Creation and Annihilation Operators......Page 284
 6.9 Representations of Boson Second Quantization Operators in Terms of A(·)#......Page 291
 6.10 Commutation Relations Between A(f)# and db(T)......Page 298
 6.11 Segal Field Operators......Page 300
  6.11.1 Basic Properties......Page 301
  6.11.2 Self-Adjointness of the Segal Field Operator......Page 304
  6.11.3 Vacuum Expectation Values......Page 307
  6.11.4 Irreducibility of Segal Field Operators......Page 308
  6.11.5 Some Formulae and Spectrum of ΦS(f)......Page 309
  6.11.6 Properties of Weyl Operators......Page 311
 6.12 Canonical Free Bose Field and Canonical Conjugate Momentum......Page 314
 6.13 Symplectic Spaces and Generalization of Segal Field Operators......Page 315
 6.14 Quadratic Operators......Page 317
 6.15 The Boson Fock Space Over a Direct Sum Hilbert Space......Page 323
 7.1 Definitions and Basic Properties......Page 328
 7.2 Fermion Second Quantization Operators......Page 329
 7.3 Fermion Γ-Operators......Page 331
 7.4 Fermion Annihilation and Creation Operators......Page 333
 7.5 Commutation Relations Between B(·)# and dΓf(·)......Page 337
 7.6 Tranformations of B(·)# by Γf(·)......Page 338
 7.7 Representations of Fermion Second Quantization Operators in Terms of B(·)#......Page 339
 7.8 Uniform Differentiability of Basic Operator-Valued Functions......Page 340
 7.9 The Fermion Fock Space Over a Direct Sum Hilbert Space......Page 342
 8.1 Representation of CCR......Page 345
 8.2 Cyclic Representations......Page 349
 8.3 Second Quantization Operator Associated with a Representation of CCR......Page 351
  8.3.1 A General Fact on a Sesquilinear Form......Page 352
  8.3.2 A Second Quantization Operator Associated with a Representation of CCR......Page 354
 8.4 Diagonalization of HC(T)......Page 361
 8.5 Analysis of Bogoliubov Translations......Page 363
 8.6 Bogoliubov Transformations......Page 364
 8.7 Second Quantization Operator Associated with {B(f)|f H}......Page 372
 8.8 Representations of Heisenberg CCR......Page 376
 8.9 Weyl Representations of CCR Over Real Inner Product Spaces......Page 381
 8.10 Translations of Fock Representation of Heisenberg CCR......Page 384
 8.11 A Class of Irreducible Weyl Representations of CCR (I)......Page 386
 8.12 A Class of Irreducible Weyl Representations of CCR (II)......Page 389
 8.13 Functional Schrödinger Representation......Page 396
 9.1 Definitions......Page 400
 9.2 Fermionic Bogoliubov Transformations......Page 402
 9.3 A Class of Representations of CAR......Page 405
 10.1 Quantum Field Models in Hamiltonian Formalism......Page 410
 10.2 Scale Transformations of Time-Zero Fields......Page 416
  10.3.1 Finite Volume Case......Page 417
  10.3.2 Infinite Volume Case......Page 418
 10.4 Translations of Heisenberg CCR and BEC......Page 419
 10.5 Improper Bogoliubov Transformation and Renormalization......Page 421
  10.5.1 A Model Equivalent to Mqd......Page 422
  10.5.2 A Renormalized Model......Page 423
  10.6.1 Finite Volume Theory......Page 425
  10.6.2 Infinite Volume Theory......Page 432
  10.7.1 Definition......Page 438
  10.7.2 Boson Masses as Indices of a Family of Mutually Inequivalent Representations of CCR......Page 442
 10.8 Quantum Fields at Finite Temperatures......Page 443
 10.9 Van Hove Model......Page 446
  10.9.1 The Infrared Regular Case......Page 449
  10.9.2 The Infrared Singular Case......Page 452
  10.9.3 Infrared Divergence......Page 453
  10.9.4 Infrared Renormalization......Page 455
  10.9.5 Representations Indexed by Sources......Page 458
  10.9.6 Ultraviolet Renormalization......Page 459
  10.10.1 Eigenvectors of hD(k) and Some Operators......Page 465
  10.10.2 Construction of a Free Quantum Dirac Field......Page 468
  10.10.3 Inequivalence of Free Quantum Dirac Fields of Different Masses......Page 471
A Multiplication Operators......Page 474
B Spectral Measures and Functional Calculus......Page 477
 C.1 Finite Direct Sums of Hilbert Spaces and Operators......Page 483
 C.2 Infinite Direct Sums of Hilbert Spaces and Operators......Page 485
 C.3 A Theorem on Essential Self-adjointness......Page 487
D Spectra of a Self-adjoint Operator......Page 489
Bibliography......Page 494
Index......Page 501