Hamiltonian Mechanics of Gauge Systems

The principles of gauge symmetry and quantization are fundamental to modern understanding of the laws of electromagnetism, weak and strong subatomic forces and the theory of general relativity. Ideal for graduate students and researchers in theoretical and mathematical physics, this unique book provides a systematic introduction to Hamiltonian mechanics of systems with gauge symmetry. The book reveals how gauge symmetry may lead to a non-trivial geometry of the physical phase space and studies its effect on quantum dynamics by path integral methods. It also covers aspects of Hamiltonian path integral formalism in detail, along with a number of related topics such as the theory of canonical transformations on phase space supermanifolds, non-commutativity of canonical quantization and elimination of non-physical variables. The discussion is accompanied by numerous detailed examples of dynamical models with gauge symmetries, clearly illustrating the key concepts.

Preface	p. xi
Hamiltonian formalism	p. 1
Hamilton's principle of stationary action	p. 1
Hamiltonian equations of motion	p. 7
The Poisson bracket	p. 9
Canonical transformations	p. 10
Generating functions of canonical transformations	p. 12
Symmetries and integrals of motion	p. 15
Lagrangian formalism for Grassmann variables	p. 22
Hamiltonian formalism for Grassmann variables	p. 25
Hamiltonian dynamics on supermanifolds	p. 32
Canonical transformations on symplectic supermanifolds	p. 37
Noether's theorem for systems on supermanifolds	p. 45
Non-canonical transformations	p. 50
Examples of systems with non-canonical symplectic structures	p. 55
Some generalizations of the Hamiltonian dynamics	p. 58
Hamiltonian mechanics. Recent developments	p. 63
Hamiltonian path integrals	p. 68
Introduction	p. 68
Hamiltonian path integrals in quantum mechanics	p. 70
Non-standard terms and basic equivalence rules	p. 75
Equivalence rules	p. 82
Rules for changing the base point	p. 86
Canonical transformations and Hamiltonian path integrals	p. [89
Problems with non-trivial boundary conditions	p. 101
Quantization by the path integral method	p. 130
Dynamical systems with constraints	p. 141
Introduction	p. 141
A general analysis of dynamical systems with constraints	p. 148
Physical variables in systems with constraints	p. 159
Nonlinear Poisson brackets and systems with constraints	p. 165
Quantization of constrained systems	p. 171
The Dirac method	p. 171
The operator ordering problem in constraints	p. 180
Relativistic particle	p. 185
Elimination of non-physical variables. The second-class constraints	p. 189
Phase space in gauge theories	p. 192
A simple model	p. 193
Harmonic oscillator with a conic phase space	p. 205
The residual discrete gauge group and the choice of physical variables	p. 209
Models with arbitrary simple compact gauge groups	p. 222
Gauge systems with Grassmann variables	p. 239
More general mechanical gauge systems with bosonic variables	p. 244
Systems with Bose and Fermi degrees of freedom	p. 256
Yang-Mills theories	p. 265
Simple effects of the physical phase space structure in quantum theory	p. 298
Path integrals in gauge theories	p. 316
Prehminary remarks	p. 316
Hamiltonian path integral for gauge systems with conic phase space	p. 317
Models with more complicated structures of the physical phase space	p. 325
Models with Grassmann variables	p. 336
Hamiltonian path integral in an arbitrary gauge	p. 338
Hamiltonian path integrals for gauge systems with bosons andfermions	p. 349
The Kato-Trotter product formula for gauge theories	p. 358
Simple consequences of the modification of the path integral for gauge systems	p. 380
Confinement	p. 386
Introduction	p. 386
Kinematics. Gauge fields and fiber bundle theory	p. 391
Dynamics. Quantization	p. 403
External fields of charges and static forces. Confinement	p. 405
Supplementary material	p. 419
A brief survey of the group theory	p. 419
Grassmann variables	p. 428
Gaussian integrals, the Poisson summation formula, kernel Q_n, and Van Fleck determinant	p. 436
Elimination of gauge arbitrariness and residual gauge transformations	p. 440
Gauge-invariant representations of the unit operator kernel	p. 448
References	p. 452
Index	p. 463
Table of Contents provided by Ingram. All Rights Reserved.

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Hamiltonian Mechanics of Gauge Systems

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