Lecture Notes in Physics

By: AndrÃ© Burnel

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Hardcover | 6 October 2008

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By definition, gauge theories - among the cornerstones of fundamental theoretical physics - involve more degrees of freedom than required by the underlying physics. The unphysical degrees of freedom must be shown not to yield unwarranted effects at every step in the formalism where explicit Lorentz covariance is required.

The present work presents, in a rigorous way, a consistent formulation for the handling of noncovariant gauges in the quantization and renormalization of gauge theories. Though the path integral method is very convenient for the proof of unitarity and renormalizability of gauge theories, the canonical formalism is eventually necessary to to expose the issues in a self-consistent way.

These notes are written as an introduction to postgraduate students, lecturers and researchers in the field and assume prior knowledge of quantum field theory.

Industry Reviews

From the reviews:

"This book develops a consistent formulation for the handling of noncovariant gauges in the quantization process. ... provide the setting of the noncovariant gauge theories on the same levels of consistency as the covariant ones, by means of the canonical formalism. ... the author has also included a detailed and complete description of the quantization theory of constrained systems. ... All in all, this comprehensive ... monograph represents a gratifying addition for graduate students and researchers from high energy physics, gauge theories and related areas." (Vladimir Balan, Zentralblatt MATH, Vol. 1159, 2009)

"In this book the quantization of abelian and non-abelian gauge theories in non-covariant gauges in the canonical formalism is treated in detail. ... great value to anyone involved with the quantization of gauge theories, or even with gauge-fixed gauge theories in general, and provides essential knowledge. ... the book can also be highly recommended to advanced students for an extra insight into the complexities of gauge theories. ... this book is a definite must-read." (Axel Maas, Mathematical Reviews, Issue 2010 a)

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Non-Fiction Science Physics Nuclear Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Mathematics Applied Mathematics Classical Mathematics Wave Mechanics including Vibration & Acoustics Mathematical Physics

Canonical Quantization for Constrained Systems	p. 1
Canonical Quantization of Mechanical Unconstrained Systems	p. 1
Lagrangian Formalism	p. 1
Hamiltonian	p. 1
Poisson Brackets	p. 2
Quantization	p. 2
Field Theory	p. 3
Lagrangian and Lagrangian Density	p. 3
Hamiltonian	p. 3
Poisson Brackets	p. 4
Quantization	p. 4
The Free Scalar Field	p. 5
Solution of the Klein-Gordon Equation	p. 5
The Electromagnetic Field	p. 6
Primary and Secondary Constraints	p. 8
Primary Constraints	p. 8
Secondary Constraints	p. 8
First and Second Class Constraints	p. 9
Quantization in the Presence of Second-Class Constraints	p. 10
Quantization in Presence of First-Class Constraints	p. 11
BRST Quantization	p. 13
Summary	p. 14
Back to the Free Massless Vector Field	p. 14
Lagrangian and Constraints	p. 14
Gauge Fixing	p. 16
Covariant Gauge for the Free Massless Vector Field	p. 16
Commutation Relations for Any Time	p. 18
Creation and Annihilation Operators	p. 21
The Gupta-Bleuler Formalism	p. 22
References	p. 23
Quantization of the Free Electromagnetic Field in General Class III Linear Gauges	p. 25
Introduction	p. 25
Lagrangian and Field Equations	p. 26
The Lagrangian	p. 26
Euler-Lagrange Equations	p. 26
Derived Field Equations	p. 28
Sketching the Solution of the Cauchy Problem	p. 28
Particular Cases of Interest	p. 30
Planar-Type Gauges	p. 31
Constraint Analysis	p. 32
Canonical Momenta and Primary Constraints	p. 32
The Hamiltonian	p. 33
Constraint Chains	p. 33
Singular Frames	p. 33
Commutation Relations for Any Time	p. 34
Canonical Commutation Relations	p. 34
Commutation Relations Involving the S Field	p. 34
Commutation Relations Involving the S' Field	p. 36
Commutation Relations Involving B = [partial differential] [middle dot] A	p. 38
Commutation Relations Between A[subscript mu]'s	p. 42
Summary of the Commutation Relations for Any Time	p. 44
Creation and Annihilation Operators	p. 45
Momentum Space Expansion of the Fields	p. 45
Commutation Relations Between Creation and Annihilation Operators	p. 50
Summary of the Algebra of Creation and Annihilation Operators	p. 55
The Gupta-Bleuler Formalism	p. 55
Covariance Problems	p. 56
Translation Invariance	p. 56
Lorentz Transformations	p. 58
References	p. 60
Quantization of the Free Electromagnetic Field in Class II Axial Gauges	p. 61
Introduction	p. 61
Lagrangian and Field Equations	p. 61
The Lagrangian	p. 61
Euler-Lagrange Equations	p. 62
Derived Field Equations	p. 63
Summary of the Field Equations	p. 63
Constraint Analysis and Effective Hamiltonian	p. 63
Solution of Field Equations	p. 65
Solution of n [middle dot] [partial differential]S = 0	p. 65
Solution of (n [middle dot] [partial differential] + [kappa])S = 0	p. 66
Solution of n [middle dot] [partial differential]B = (n[superscript 2] - a)S	p. 67
Elementary Solution of n [middle dot] [partial differential square]A = 0	p. 67
Solution of the Cauchy Problem for [square]A(x - z) = L[subscript x]D[subscript n](x - z)	p. 70
Elementary Solution of (n [middle dot] [partial differential superscript 2 square]A = 0	p. 70
Commutation Relations	p. 71
Equal Time Commutators	p. 71
Commutation Relations Involving the S-Field	p. 72
Commutation Relations Involving the S'-Field	p. 72
Commutation Relations Involving [partial differential] [middle dot] A	p. 73
Commutation Relations [A subscript mu](x), A[subscript v](z)	p. 74
Association of a Feynman Propagator with the Operator n [middle dot] [partial differential]	p. 76
Associating Creation and Annihilation Operators with Fields	p. 77
The S-Field	p. 78
Other Fields	p. 78
Correct Momentum Space Expansion of the Fields	p. 80
The S-Field	p. 80
The B-Field	p. 80
The S'-Field	p. 82
The A[subscript mu]-Field	p. 83
A Toy Model for the Unphysical S and B Fields	p. 84
Conclusions	p. 85
Interpolating Between Axial and Relativistic Gauges	p. 85
Summary	p. 86
References	p. 86
Gauge Fields in Interaction	p. 87
Introduction	p. 87
General Formalism of Gauge Invariance	p. 88
General Transformations	p. 88
Infinitesimal Transformations	p. 89
Remarks	p. 90
Class III Gauges in Yang-Mills Theory	p. 91
Building the Lagrangian	p. 91
Field Equations	p. 94
Canonical Momenta and Constraints	p. 94
The Hamiltonian	p. 95
The BRST Charge	p. 95
Ghost Number	p. 97
Global Internal Symmetry	p. 97
Physical States	p. 98
Problems of Covariance	p. 98
References	p. 99
Perturbation Theory: Renormalization and All That	p. 101
Introduction	p. 101
The S-Matrix	p. 101
Definition and Properties	p. 101
Perturbative Expansion	p. 103
Expansion and Consequences of the Unitarity Condition	p. 103
Consequences of the Causality Condition	p. 104
The First Term T[subscript 1]	p. 106
Nonunicity of the T[subscript n]	p. 108
The Fixed Part of T[subscript n]	p. 108
Wick's Theorem	p. 109
Feynman Rules	p. 111
Feynman Rules in Momentum Space	p. 113
Divergences, Power Counting and Renormalizability	p. 118
Example	p. 118
Power Counting and Superficial Degree of Divergence	p. 118
Renormalizability by Power Counting	p. 120
Dimensional Regularization of Covariant Divergent Integrals	p. 120
A General One-Loop Integral	p. 120
Euclidean Space	p. 121
Use of the Feynman Formula	p. 121
Elimination of the Denominators	p. 122
Complex Dimension	p. 122
Tensor Integrals	p. 123
Extension to General Covariant or Noncovariant Integrals in a Preferred Frame	p. 124
The General One-Loop Integral	p. 124
Euclidean Space	p. 125
Elimination of the Denominators	p. 126
Calculation of the Derivatives	p. 127
Introduction of the Feynman Variables	p. 128
Integration Over [lambda]	p. 129
Regularization of Ultraviolet Divergences	p. 129
Consequences of the Nonsingularity of the B[superscript -1] Matrix	p. 130
Computation of the Ghost Loop With Two Legs	p. 131
Renormalization and Counter-Terms	p. 132
Various Renormalization Schemes	p. 133
Multiplicative Renormalization	p. 134
Summary	p. 135
References	p. 136
Slavnov-Taylor Identities for Yang-Mills Theory	p. 137
Introduction	p. 137
The Reduction Formula	p. 137
One-Particle Irreducible Vertex Functions	p. 138
Yang-Mills Theory in a General Class III Gauge	p. 139
The Lagrangian and Superficially Divergent Processes	p. 139
BRST Symmetry, Field Equations and Canonical Commutation Relations	p. 141
Commuting Derivatives and Time-Ordered Products	p. 143
The Ward-Takahashi-Slavnov-Taylor Identity for the Gluon Self-Energy	p. 143
Covariant Gauges	p. 143
General Gauges	p. 146
Renormalization	p. 148
Identity for the Three-Gluon Vertex Function	p. 149
Derivation of the Identity	p. 150
Renormalization	p. 150
Ghost Propagator	p. 151
Ghost-Ghost-Gluon Vertex	p. 153
Identity in Coordinate Space	p. 153
Momentum Space	p. 153
Remark	p. 154
Multiplicative Renormalization	p. 154
Summary	p. 156
References	p. 156
Field Theory Without Infinities	p. 157
Introduction	p. 157
Iterative Construction of the S-Matrix Without Time-Ordering	p. 158
Splitting of Causal Distributions into Advanced and Retarded Parts	p. 159
Distribution and Fourier Transform	p. 159
Order of Singularity of a Distribution	p. 160
Splitting of Distribution with Negative Order of Singularity	p. 161
Nonnegative Singularity Order	p. 162
Application to Yang-Mills Theory	p. 164
First Order	p. 164
Example of a One-Loop Process: The Gluon Self-energy	p. 179
Calculation of I[subscript 1]([xi];[kappa])	p. 186
Calculation of I[subscript 2](k;[kappa subscript 1], [kappa subscript 2])	p. 187
Calculation of the Tensor Distributions	p. 188
The Final Result for the Gluon Self-energy	p. 190
Summary	p. 190
References	p. 191
Gauges with a Singular C Matrix	p. 193
Introduction	p. 193
The Ghost Loop Contribution	p. 194
The Leibbrandt Gauges	p. 194
Interpolating Between Leibbrandt and Relativistic Gauges	p. 197
Singularities Generated by the Lack of Power Counting in Ultra-Violet Divergent Perturbative Theory	p. 199
The Loop Integration in a General Gauge	p. 200
Restriction to Loops with Two External Particles	p. 201
Further Restriction for m [greater than or equal right] 6	p. 201
Interpolating Between Leibbrandt and Relativistic Gauges	p. 205
Cancellation of Divergences at [alpha] = 0 in the Self-Energy	p. 207
References	p. 208
Conclusion	p. 209
Notations	p. 211
A Useful Fourier Transform	p. 213
Generalized Functions	p. 215
Elementary Solutions of the Klein-Gordon Equation	p. 215
The [Delta] Function	p. 215
The [Delta superscript F] Function	p. 216
The E Function	p. 216
Definition	p. 216
Derivation of the Elementary Solution	p. 216
Zero-Time Properties	p. 217
Integration Over k[subscript 0]	p. 218
The Cauchy Problem Associated with [square superscript 2]	p. 218
Generalized Functions Associated with the [square subscript C] Operator	p. 219
Elementary Solution D[subscript C superscript (1)]	p. 220
Positive Frequency Part	p. 221
Negative Frequency Part	p. 222
Elementary Solution with Causal Support	p. 222
Generalization with a Mass Parameter	p. 223
Analogous of the Feynman Propagator	p. 224
Solution of the Cauchy Problem	p. 225
Remarks	p. 226
Generalized Functions Associated with [square subscript C superscript 2]	p. 226
The Elementary Function with Causal Support	p. 226
Zero Time Properties	p. 227
Integration Over k[subscript 0]	p. 227
The Cauchy Problem	p. 228
Generalized Functions Associated with [square square subscript C]	p. 228
Elementary Solution with Causal Support	p. 228
Zero-Time Properties of F[subscript C] and its Time Derivatives	p. 230
Preferred Frame	p. 231
The Cauchy Problem	p. 231
The G-Functions	p. 232
Index	p. 235
Table of Contents provided by Ingram. All Rights Reserved.

Lecture Notes in Physics

At a Glance

Hardcover

Industry Reviews

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