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Hamiltonian Mechanics of Gauge Systems

Book Description

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.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. 1. Hamiltonian formalism
    1. 1.1 Hamilton’s principle of stationary action
    2. 1.2 Hamiltonian equations of motion
    3. 1.3 The Poisson bracket
    4. 1.4 Canonical transformations
    5. 1.5 Generating functions of canonical transformations
    6. 1.6 Symmetries and integrals of motion
    7. 1.7 Lagrangian formalism for Grassmann variables
    8. 1.8 Hamiltonian formalism for Grassmann variables
    9. 1.9 Hamiltonian dynamics on supermanifolds
    10. 1.10 Canonical transformations on symplectic supermanifolds
    11. 1.11 Noether’s theorem for systems on supermanifolds
    12. 1.12 Non-canonical transformations
    13. 1.13 Examples of systems with non-canonical symplectic structures
    14. 1.14 Some generalizations of the Hamiltonian dynamics
    15. 1.15 Hamiltonian mechanics. Recent developments
  8. 2. Hamiltonian path integrals
    1. 2.1 Introduction
    2. 2.2 Hamiltonian path integrals in quantum mechanics
    3. 2.3 Non-standard terms and basic equivalence rules
    4. 2.4 Equivalence rules
    5. 2.5 Rules for changing the base point
    6. 2.6 Canonical transformations and Hamiltonian path integrals
    7. 2.7 Problems with non-trivial boundary conditions
    8. 2.8 Quantization by the path integral method
  9. 3. Dynamical systems with constraints
    1. 3.1 Introduction
    2. 3.2 A general analysis of dynamical systems with constraints
    3. 3.3 Physical variables in systems with constraints
    4. 3.4 Nonlinear Poisson brackets and systems with constraints
  10. 4. Quantization of constrained systems
    1. 4.1 The Dirac method
    2. 4.2 The operator ordering problem in constraints
    3. 4.3 Relativistic particle
    4. 4.4 Elimination of non-physical variables. The second-class constraints
  11. 5. Phase space in gauge theories
    1. 5.1 A simple model
    2. 5.2 Harmonic oscillator with a conic phase space
    3. 5.3 The residual discrete gauge group and the choice of physical variables
    4. 5.4 Models with arbitrary simple compact gauge groups
    5. 5.5 Gauge systems with Grassmann variables
    6. 5.6 More general mechanical gauge systems with bosonic variables
    7. 5.7 Systems with Bose and Fermi degrees of freedom
    8. 5.8 Yang–Mills theories
    9. 5.9 Simple effects of the physical phase space structure in quantum theory
  12. 6. Path integrals in gauge theories
    1. 6.1 Preliminary remarks
    2. 6.2 Hamiltonian path integral for gauge systems with conic phase space
    3. 6.3 Models with more complicated structures of the physical phase space
    4. 6.4 Models with Grassmann variables
    5. 6.5 Hamiltonian path integral in an arbitrary gauge
    6. 6.6 Hamiltonian path integrals for gauge systems with bosons and fermions
    7. 6.7 The Kato–Trotter product formula for gauge theories
    8. 6.8 Simple consequences of the modification of the path integral for gauge systems
  13. 7. Confinement
    1. 7.1 Introduction
    2. 7.2 Kinematics. Gauge fields and fiber bundle theory
    3. 7.3 Dynamics. Quantization
    4. 7.4 External fields of charges and static forces. Confinement
  14. 8. Supplementary material
    1. 8.1 A brief survey of the group theory
    2. 8.2 Grassmann variables
    3. 8.3 Gaussian integrals, the Poisson summation formula, kernel Qn, and Van Fleck determinant
    4. 8.4 Elimination of gauge arbitrariness and residual gauge transformations
    5. 8.5 Gauge-invariant representations of the unit operator kernel
  15. References
  16. Index