Table of Contents
List of figures xiii
List of tables xv
Postulates, statements, theorems, assertions xvii
Conventional notation xix
Preface xxiii
Introduction xxix
1 Quantum logic 1
1.1 Why do we need quantum mechanics? 3
1.1.1 Corpuscular theory of light 3
1.1.2 Wave theory of light 6
1.1.3 Light of low intensity and other experiments 7
1.2 Classical logic 8
1.2.1 Phase space of one classical particle 9
1.2.2 Propositions in phase space 9
1.2.3 Operations with propositions 10
1.2.4 Axioms of logic 11
1.2.5 Phase space from axioms of classical logic 13
1.2.6 Classical observables 14
1.3 Measurements and probabilities 15
1.3.1 Ensembles and measurements 15
1.3.2 States as probability measures 16
1.3.3 Probability distributions and statistical mechanics 17
1.4 Logic of quantum mechanics 18
1.4.1 Partial determinism of quantum mechanics 18
1.4.2 Axioms of quantum logic from probability measures 20
1.4.3 Compatibility of propositions 21
1.4.4 Logic of quantum mechanics 22
1.4.5 Quantum logic and Hilbert space 23
1.4,6 Piron's theorem 23
1.4.7 Should we abandon classical logic? 24
1.5 Physics in Hilbert space 25
1.5.1 Quantum observables 25
1.5.2 States 26
1.5.3 Complete sets of commuting observables 28
1.5.4 Wave functions 29
1.5.5 Expectation values 29
1.5.6 Basic rules of classical and quantum mechanics 30
1.6 Interpretations of quantum mechanics 31
1.6.1 Quantum nonpredictability 31
1.6.2 Collapse of wave function 32
1.6.3 Collapse of classical probability distribution 32
1.6.4 Hidden variables 33
1.6.5 Quantum-logical interpretation 34
1.6.6 Quantum randomness and limits of knowledge 35
2 Poincaré group 37
2.1 Inertial observers 37
2.1.1 Principle of relativity 37
2.1.2 Inertial transformations 38
2.2 Galilei group 39
2.2.1 Composition law in Galilei group 40
2.2.2 Lie algebra of Galilei group 41
2.2.3 Rotations applied to generators 43
2.2.4 Space inversion 45
2.3 Poincaré group 46
2.3.1 Conditions on Poincaré generators 46
2.3.1 Lie algebra of Poincaré group 47
2.3.3 Boosts of translation generators 51
3 Quantum mechanics and relativity 55
3.1 Inertial transformations in quantum mechanics 55
3.1.1 Wigner's theorem 55
3.1.2 Inertial transformations of states 58
3.1.3 Heisenberg and Schrödinger pictures 59
3.2 Unitary representations of Poincaré group 60
3.2.1 Projective representations of groups 60
3.3.2 Generators of projective representation 61
3.2.3 Commutators of projective generators 63
3.2.4 Cancellation of central charges 65
3.2.5 Single-valued and double-valued representations 68
3.2.6 Fundamental statement of relativistic quantum theory 68
3.2.7 Time evolution in moving frame 70
4 Observables 71
4.1 Basic observables 71
4.1.1 Energy, momentum and angular momentum 71
4.1.2 Operator of velocity 73
4.2 Casimir operators 73
4.2.1 4-Vectors 74
4.2.2 Mass operator 74
4.2.3 Pauli-Lubanski 4-vector 75
4.3 Operators of spin and position 77
4.3.1 Physical requirements 77
4.3.2 Spin operator 78
4.3.3 Position operator 79
4.3.4 Commutators of position 81
4.3.5 Alternative set of basic operators 83
4.3.6 Canonical order of operators 84
4.3.7 Power of operator 85
4.3.8 Uniqueness of spin operator 87
4.3.9 Uniqueness of position operator 88
4.3.10 Boost of position 89
5 Elementary particles 91
5.1 Massive particles 93
5.1.1 One-particle Hilbert space 93
5.1.2 Action of rotation subgroup in H0 94
5.1.3 Momentum-spin basis 96
5.1.4 Nonuniqueness of momentum-spin basis 98
5.1.5 Action of translations and rotations on basis vectors 98
5.1.6 Action of boosts on momentum eigenvectors 99
5.1.7 Action of boosts on spin components 100
5.1.8 Wigner angle 101
5.1.9 Irreducibility of representation Ug 102
5.1.10 Method of induced representations 102
5.2 Momentum representation 103
5.2.1 Resolution of identity 103
5.2.2 Boost transformation 104
5.2.3 Wave function in momentum representation 106
5.3 Position representation 107
5.3.1 Basis of localized functions 107
5.3.2 Operators of observables in position representation 109
5.3.3 Inertial transformations of observables and states 110
5.3.4 Time translations of observables and states 113
5.4 Massless particles 113
5.4.1 Spectra of momentum, energy and velocity 113
5.4.2 Representations of small groups 114
5.4.3 Basis in Hilbert space of massless particle 117
5.4.4 Massless representations of Poincaré group 118
5.4.5 Doppler effect and aberration 120
6 Interaction 123
6.1 Hilbert space of multiparticle system 123
6.1.1 Tensor product theorem 123
6.1.2 Particle observables in multiparticle systems 125
6.1.3 Statistics 126
6.2 Relativistic Hamiltonian dynamics 127
6.2.1 Noninteracting representation of Poincaré group 128
6.2.2 Dirac's Forms of dynamics 129
6.2.3 Total observables in multiparticle systems 131
6.3 Instant form of dynamics 131
6.3.1 General instant-form interaction 131
6.3.2 Bakamjian-Thomas construction 132
6.3.3 Example: two-particle system 133
6.3.4 Other variants of instant-form dynamics 135
6.4 Cluster separability 136
6.4.1 Definition of cluster separability 136
6.4.2 Examples of interaction potentials 137
6.4.3 Smooth potentials 139
6.4.4 Nonseparability of Bakamjian-Thomas dynamics 139
6.4.5 Cluster-separable three-particle interaction 140
6.4.6 Relativistic addition of interactions 141
6.5 Bound states and time evolution 144
6.5.1 Spectra of mass and energy operators 144
6.5.2 Perturbation theory 146
6.5.3 Once again about the Doppler effect 146
6.5.4 Time evolution 148
6.6 Classical Hamiltonian dynamics 150
6.6.1 Quasiclassical states 151
6.6.2 Heisenberg uncertainty relation 152
6.6.3 Spreading of quasiclassical wave packets 153
6.6.4 Phase space 154
6.6.5 Poisson bracket 154
6.6.6 Time evolution of wave packets 157
6.6.7 Once again about experiments with two holes 159
7 Scattering 161
7.1 Scattering operators 161
7.1.1 Physical meaning of S-operator 161
7.1.2 S-operator in perturbation theory 164
7.1.3 Convenient notation for t-integrals 165
7.1.4 Adiabatic switching of interaction 167
7.1.5 T-matrix 169
7.1.6 S-matrix and bound states 170
7.2 Scattering equivalence 171
7.2.1 Equivalent Hamiltonians 172
7.2.2 Bakamjian construction of point-form dynamics 173
7.2.3 Unitary link between point and instant forms of dynamics 175
7.2.4 Scattering equivalence of forms of dynamics 176
A Delta function 179
B Orthocomplemented lattices 181
B.1 Derivation of quantum axioms 181
B.2 Some lemmas and theorems 182
C Groups and vector spaces 187
C.1 Groups 187
C.2 Vector spaces 187
D Group of rotations 191
D.1 Basics of 3D space 191
D.2 Scalars and vectors 192
D.3 Orthogonal matrices 193
D.4 Invariant tensors 195
D.5 Vector parametrization of rotations 197
D.6 Group properties of rotations 200
D.7 Generators of rotations 201
E Lie groups and Lie algebras 203
E.1 Lie groups 203
E.2 Lie algebras 205
E.3 One-parameter subgroups of Lie groups 207
E.4 Baker-Campbell-Hausdorff formula 208
F Hilbert space 209
F.1 Internal product 209
F.2 Orthonormal bases 209
F.3 Bra and ket vectors 210
F.4 Tensor product of Hilbert spaces 212
G Operators 213
G.1 Linear operators 213
G.2 Matrices and operators 214
G.3 Functions of operators 216
G.4 Hermitian and unitary operators 218
G.5 Linear operators in different orthonormal bases 219
G.6 Diagonalization of Hermitian and unitary matrices 220
H Subspaces and projections 225
H.1 Projections 225
H.2 Commuting operators 226
I Representations of groups and algebras 231
I.1 Unitary representations of groups 231
I.2 Stone theorem 232
I.3 Heisenberg algebra 232
I.4 Double-valued representations of rotation group 233
I.5 Unitary irreducible representations of rotation group 235
J Pseudo-orthogonal representation of Lorentz group 237
J.1 Minkowski space-time 237
J.2 General properties of representation 238
J.3 Matrices of pseudo-orthogonal representation 239
J.4 Representation of Lorentz Lie algebra 240
Bibliography 243
Index 247
