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Mathematical Aspects of Quantum Field Theory : Cambridge Studies in Advanced Mathematics (Hardcover) - Edson de Faria

Mathematical Aspects of Quantum Field Theory

Cambridge Studies in Advanced Mathematics (Hardcover)


Published: 12th August 2010
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Over the last century quantum field theory has made a significant impact on the formulation and solution of mathematical problems and inspired powerful advances in pure mathematics. However, most accounts are written by physicists, and mathematicians struggle to find clear definitions and statements of the concepts involved. This graduate-level introduction presents the basic ideas and tools from quantum field theory to a mathematical audience. Topics include classical and quantum mechanics, classical field theory, quantization of classical fields, perturbative quantum field theory, renormalization, and the standard model. The material is also accessible to physicists seeking a better understanding of the mathematical background, providing the necessary tools from differential geometry on such topics as connections and gauge fields, vector and spinor bundles, symmetries and group representations.

'... a brief, but careful, reasonably balanced, and representative introduction for mathematicians to QFT as it is used in physics.' The Mathematical Intelligencer

Forewordp. ix
Prefacep. xi
Classical mechanicsp. 1
Newtonian mechanicsp. 1
Lagrangian mechanicsp. 4
Hamiltonian mechanicsp. 7
Poisson brackets and Lie algebra structure of observablesp. 10
Symmetry and conservation laws: Noether's theoremp. 11
Quantum mechanicsp. 14
The birth of quantum theoryp. 14
The basic principles of quantum mechanicsp. 16
Canonical quantizationp. 21
From classical to quantum mechanics: the C* algebra approachp. 24
The Weyl C* algebrap. 26
The quantum harmonic oscillatorp. 29
Angular momentum quantization and spinp. 35
Path integral quantizationp. 40
Deformation quantizationp. 49
Relativity, the Lorentz group, and Dirac's equationp. 51
Relativity and the Lorentz groupp. 51
Relativistic kinematicsp. 56
Relativistic dynamicsp. 57
The relativistic Lagrangianp. 58
Dirac's equationp. 60
Fiber bundles, connections, and representationsp. 65
Fiber bundles and cocyclesp. 65
Principal bundlesp. 68
Connectionsp. 71
The gauge groupp. 73
The Hodge * operatorp. 75
Clifford algebras and spinor bundlesp. 77
Representationsp. 82
Classical field theoryp. 93
Introductionp. 93
Electromagnetic fieldp. 94
Conservation laws in field theoryp. 99
The Dirac fieldp. 103
Scalar fieldsp. 108
Yang-Mills fieldsp. 110
Gravitational fieldsp. 111
Quantization of classical fieldsp. 117
Quantization of free fields: general schemep. 117
Axiomatic field theoryp. 118
Quantization of bosonic free fieldsp. 122
Quantization of fermionic fieldsp. 128
Quantization of the free electromagnetic fieldp. 140
Wick rotations and axioms for Euclidean QFTp. 141
The CPT theoremp. 142
Scattering processes and LSZ reductionp. 144
Perturbative quantum field theoryp. 153
Discretization of functional integralsp. 153
Gaussian measures and Wick's theoremp. 154
Discretization of Euclidean scalar fieldsp. 159
Perturbative quantum field theoryp. 164
Perturbative Yang-Mills theoryp. 182
Renormalizationp. 192
Renormalization in perturbative QFTp. 192
Constructive field theoryp. 201
The Standard Modelp. 204
Particles and fieldsp. 205
Particles and their quantum numbersp. 206
The quark modelp. 207
Non-abelian gauge theoriesp. 209
Lagrangian formulation of the standard modelp. 213
The intrinsic formulation of the Lagrangianp. 225
Hilbert spaces and operatorsp. 232
Hubert spacesp. 232
Linear operatorsp. 233
Spectral theorem for compact operatorsp. 235
Spectral theorem for normal operatorsp. 237
Spectral theorem for unbounded operatorsp. 238
Functional calculusp. 245
Essential self-adjointnessp. 247
A note on the spectrump. 249
Stone's theoremp. 250
The Kato-Rellich theoremp. 253
C* algebras and spectral theoryp. 258
Banach algebrasp. 258
C* algebrasp. 262
The spectral theoremp. 268
States and GNS representationp. 271
Representations and spectral resolutionsp. 276
Algebraic quantum field theoryp. 282
Bibliographyp. 289
Indexp. 293
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521115773
ISBN-10: 0521115779
Series: Cambridge Studies in Advanced Mathematics (Hardcover)
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 314
Published: 12th August 2010
Country of Publication: GB
Dimensions (cm): 22.86 x 15.49  x 2.03
Weight (kg): 0.57
Edition Number: 1