Mathematical Aspects of Quantum Field Theory

, by Edson de Faria , Welington de Melo

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ISBN: 9780521115773 | 0521115779
Cover: Hardcover
Copyright: 9/27/2010

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Presents some of the basic ideas and tools of quantum field theory to a mathematical audience.

Foreword	p. ix
Preface	p. xi
Classical mechanics	p. 1
Newtonian mechanics	p. 1
Lagrangian mechanics	p. 4
Hamiltonian mechanics	p. 7
Poisson brackets and Lie algebra structure of observables	p. 10
Symmetry and conservation laws: Noether's theorem	p. 11
Quantum mechanics	p. 14
The birth of quantum theory	p. 14
The basic principles of quantum mechanics	p. 16
Canonical quantization	p. 21
From classical to quantum mechanics: the C^* algebra approach	p. 24
The Weyl C^* algebra	p. 26
The quantum harmonic oscillator	p. 29
Angular momentum quantization and spin	p. 35
Path integral quantization	p. 40
Deformation quantization	p. 49
Relativity, the Lorentz group, and Dirac's equation	p. 51
Relativity and the Lorentz group	p. 51
Relativistic kinematics	p. 56
Relativistic dynamics	p. 57
The relativistic Lagrangian	p. 58
Dirac's equation	p. 60
Fiber bundles, connections, and representations	p. 65
Fiber bundles and cocycles	p. 65
Principal bundles	p. 68
Connections	p. 71
The gauge group	p. 73
The Hodge * operator	p. 75
Clifford algebras and spinor bundles	p. 77
Representations	p. 82
Classical field theory	p. 93
Introduction	p. 93
Electromagnetic field	p. 94
Conservation laws in field theory	p. 99
The Dirac field	p. 103
Scalar fields	p. 108
Yang-Mills fields	p. 110
Gravitational fields	p. 111
Quantization of classical fields	p. 117
Quantization of free fields: general scheme	p. 117
Axiomatic field theory	p. 118
Quantization of bosonic free fields	p. 122
Quantization of fermionic fields	p. 128
Quantization of the free electromagnetic field	p. 140
Wick rotations and axioms for Euclidean QFT	p. 141
The CPT theorem	p. 142
Scattering processes and LSZ reduction	p. 144
Perturbative quantum field theory	p. 153
Discretization of functional integrals	p. 153
Gaussian measures and Wick's theorem	p. 154
Discretization of Euclidean scalar fields	p. 159
Perturbative quantum field theory	p. 164
Perturbative Yang-Mills theory	p. 182
Renormalization	p. 192
Renormalization in perturbative QFT	p. 192
Constructive field theory	p. 201
The Standard Model	p. 204
Particles and fields	p. 205
Particles and their quantum numbers	p. 206
The quark model	p. 207
Non-abelian gauge theories	p. 209
Lagrangian formulation of the standard model	p. 213
The intrinsic formulation of the Lagrangian	p. 225
Hilbert spaces and operators	p. 232
Hubert spaces	p. 232
Linear operators	p. 233
Spectral theorem for compact operators	p. 235
Spectral theorem for normal operators	p. 237
Spectral theorem for unbounded operators	p. 238
Functional calculus	p. 245
Essential self-adjointness	p. 247
A note on the spectrum	p. 249
Stone's theorem	p. 250
The Kato-Rellich theorem	p. 253
C^* algebras and spectral theory	p. 258
Banach algebras	p. 258
C^* algebras	p. 262
The spectral theorem	p. 268
States and GNS representation	p. 271
Representations and spectral resolutions	p. 276
Algebraic quantum field theory	p. 282
Bibliography	p. 289
Index	p. 293
Table of Contents provided by Ingram. All Rights Reserved.

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Mathematical Aspects of Quantum Field Theory

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Mathematical Aspects of Quantum Field Theory

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