Quantum Mechanics and Quantum Field Theory: A Mathematical Primer

✍ Scribed by Jonathan Dimock

• Publisher: Cambridge University Press
• Year: 2011
• Tongue: English
• Leaves: 240
• Edition: 1
• Category: Library

✦ Synopsis

Explaining the concepts of quantum mechanics and quantum field theory in a precise mathematical language, this textbook is an ideal introduction for graduate students in mathematics, helping to prepare them for further studies in quantum physics. The textbook covers topics that are central to quantum physics: non-relativistic quantum mechanics, quantum statistical mechanics, relativistic quantum mechanics and quantum field theory. There is also background material on analysis, classical mechanics, relativity and probability. Each topic is explored through a statement of basic principles followed by simple examples. Around 100 problems throughout the textbook help readers develop their understanding.

✦ Table of Contents

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
Introduction......Page 15
Part I Non-relativistic......Page 17
1.1.1 Definitions......Page 19
1.1.2 Sequences......Page 21
1.1.3 Extensions......Page 22
1.1.4 Fourier transform......Page 23
1.2.1 Closed operators......Page 25
1.2.3 Adjoints......Page 27
1.3.1 Definitions......Page 28
1.3.2 Properties......Page 30
1.3.3 Spectral theorem......Page 32
1.3.4 One-parameter groups......Page 34
1.4.1 Properties......Page 36
1.4.2 Hilbert–Schmidt operators......Page 38
1.4.3 Trace class......Page 39
2.1 Hamiltonian mechanics......Page 42
2.2 Examples......Page 43
2.3 Canonical transformations......Page 45
2.4 Symmetries......Page 48
3.1 Principles of quantum mechanics......Page 52
3.2 Canonical quantization......Page 55
3.3 Symmetries......Page 57
3.4 Perspectives and problems......Page 59
4.1 Free particle......Page 61
4.2 Particle in a potential......Page 62
4.3 Spectrum......Page 65
4.4 The harmonic oscillator......Page 67
4.5.1 Wave operators......Page 69
4.5.2 Asymptotic completeness......Page 70
4.5.3 The scattering operator......Page 71
4.6.1 Representations of the rotation group......Page 72
4.6.2 The covering group......Page 73
4.6.3 Spin 1/2 particles......Page 75
5.1.1 A first look......Page 77
5.1.2 Generalization......Page 78
5.1.3 Center ofmass coordinates......Page 79
5.2 Identical particles......Page 80
5.3 n-particles......Page 81
5.4.1 Definitions......Page 84
5.4.2 Fock space over L2......Page 89
6.1 Mixed states......Page 92
6.2.2 Canonical ensemble......Page 93
6.2.4 General problems......Page 95
6.3 Free boson gas......Page 97
6.4 Free fermion gas......Page 100
6.5 Interacting bosons......Page 102
6.6 Further developments......Page 104
Part II Relativistic......Page 107
7.1 Principles of relativity......Page 109
7.2.1 Definitions......Page 110
7.2.2 Free particles......Page 111
7.2.3 Forces......Page 114
7.2.4 Lorentz transformations......Page 115
7.3.1 Scalar fields......Page 117
7.3.2 Charged scalar fields......Page 118
7.3.3 Dirac fields......Page 119
7.3.4 The electromagnetic field......Page 120
7.4.1 The gauge principle......Page 121
7.4.2 Systems......Page 124
7.5 Fundamental solutions......Page 125
8.1.1 Canonical quantization......Page 128
8.1.2 Quantization from Klein-Gordon......Page 129
8.1.3 Covariant quantization......Page 130
8.1.4 Comparison......Page 131
8.2.1 Hamiltonian formulation......Page 132
8.2.2 Canonical quantization......Page 134
8.2.3 Generalization......Page 137
8.2.4 Covariant quantization......Page 138
8.3.1 Hamiltonian formulation......Page 140
8.3.2 Canonical quantization......Page 141
9.1 Spinors......Page 144
9.2.1 Solutions of the Dirac equation......Page 146
9.2.2 Quantum interpretation......Page 147
9.2.3 Translations and rotations......Page 149
9.2.4 A covariant formulation......Page 150
9.2.5 Charge conjugation......Page 151
9.3.1 The problem......Page 153
9.3.2 The field operator......Page 154
9.3.3 Locality......Page 156
9.3.4 A covariant formulation......Page 157
9.4.1 Coulomb gauge......Page 158
9.4.2 A covariant formulation......Page 160
9.5 Electromagnetic field......Page 162
10.1 Lorentzian manifolds......Page 166
10.2 Classical fields on a manifold......Page 168
10.3 Quantum fields on a manifold......Page 169
Part III Probabilistic methods......Page 173
11.1 Probability......Page 175
11.2 Gaussian processes......Page 177
11.3 Brownian motion......Page 179
11.4 The Feynman–Kac formula......Page 182
11.5 Oscillator process......Page 183
11.6 Application: ground states......Page 185
12.1.1 Indexing by an inner product space......Page 188
12.1.2 Wickmonomials......Page 191
12.1.3 Realization onS......Page 193
12.2.1 De.nitions and equivalence......Page 195
12.2.2 The CCR......Page 197
12.3 Path integrals – free fields......Page 198
12.4 Vacuum correlation functions......Page 201
12.5 Thermal correlation functions......Page 203
13.1 The model......Page 206
13.2 Regularization......Page 207
13.3.1 Wightman functions......Page 211
13.3.2 Reconstruction......Page 213
13.4 Path integrals – interacting fields......Page 215
13.5 A reformulation......Page 218
A.1 Banach spaces......Page 222
A.2 Hilbert spaces......Page 223
Appendix B Tensor product......Page 225
Appendix C Distributions......Page 229
References......Page 233
Index......Page 236