Classical and Quantum Dynamics of Constrained Hamiltonian Systems

The following sections are included:

• Preface

• Notation

• Contents

Elementary particle physicists are well familiar with Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD), the theories describing the electromagnetic and strong interactions. These theories are based on Lagrangians possessing local (gauge) symmetries. The weak interactions are also based on a gauge theory, but whose gauge invariance is spontaneously broken. In fact, all theories of the fundamental interactions are gauge theories. This makes the study of systems possessing local symmetries so important. As is well known, the quantization of such systems is endowed with special problems, since their classical action is invariant along gauge orbits. On the level of the equations of motion this implies the existence of an infinite number of solutions connected by local (gauge) transformations, which are physically completely equivalent. Several methods of quantization, such as the canonical Gupta-Bleuler formalism, or the functional path integral approach, which makes use of the so-called Faddeev-Popov trick, are well known to theoretical elementary particle physicists. These methods are described in any textbook on Quantum Field Theory…

The following sections are included:

• Introduction

• Singular Lagrangians

• Algorithm for detecting local symmetries on Lagrangian level

• Examples

• Generator of gauge transformations and Noether identities

The following sections are included:

• Introduction

• Primary constraints

• The Hamilton equations of motion
  • Streamlining the Hamilton equations of motion
  • Alternative derivation of the Hamilton equations
  • Examples

• Iterative procedure for generating the constraints
  • Particular algorithm for generating the constraints

• First and second class constraints. Dirac brackets

The following sections are included:

• Introduction

• The case f_ab singular
  • Example: particle on a hypersphere

• Interpretation of W^(L) and F

• The Faddeev-Jackiw reduction

The following sections are included:

• Introduction

• Local symmetries and canonical transformations

• Local symmetries of the Hamilton equations of motion

• Local symmetries of the total and extended action

• Local symmetries of the Lagrangian action

• Solution of the recursive relations

• Reparametrization invariant approach

The following sections are included:

• Introduction

• Gauge identities and Dirac's conjecture

• General system with two primaries and one secondary constraint

• Counterexamples to Dirac's conjecture?

The following sections are included:

• Introduction

• Summary of the BFT-procedure

• The BFT construction

• Examples of BFT embedding
  • The multidimensional rotator
  • The Abelian self-dual model
  • Abelian self-dual model and Maxwell-Chern-Simons theory
  • The non-abelian SD model

The following sections are included:

• Introduction
  • Carathéodory's integrability conditions
  • Characteristic curves of the HJ-equations

• HJ equations for first class systems

• HJ equations for second class systems
  • HPF for reduced second class systems
  • Examples
  • HJ equations for second class systems via BFT embedding
  • Examples

The following sections are included:

• Introduction

• Systems with only second class constraints

• Systems with first and second class constraints
  • Example: the free Maxwell field in the Coulomb gauge
  • Concluding remark

The following sections are included:

• Introduction

• Partition function for second class systems

The following sections are included:

• Introduction

• Grassmann variables

• BFV quantization of a quantum mechanical model
  • The gauge-fixed effective Lagrangian
  • The conserved BRST charge in configuration space
  • The gauge fixed effective Hamiltonian
  • The BRST charge in phase space

• Quantization of Yang-Mills theory in the Lorentz gauge

• Axiomatic BRST approach
  • The BRST charge and Hamiltonian for rank one theories
  • FV Principal Theorem
  • A large class of gauges
  • Connecting Z_Ψ with the quantum partition function in a physical gauge. The SU(N) Yang-Mills theory

• Equivalence of the SD and MCS models

• The physical Hilbert space. Some remarks

The following sections are included:

• Introduction

• Axiomatic field-antifield formalism

• Constructive proof of the field-antifield formalism for a restricted class of theories
  • From the FV-phase-space action to the Hamiltonian master equation
  • Transition to configuration space

• The Lagrangian master equation

• The quantum master equation
  • An alternative derivation of the quantum master equation
  • Gauge invariant correlation functions

• Anomalous gauge theories. The chiral Schwinger model
  • Quantum Master equation and the anomaly

The following sections are included:

• Appendix A Local Symmetries and Singular Lagrangians
  • Local symmetry transformations
  • Bianchi identities and singular Lagrangians

• Appendix B The BRST Charge of Rank One

• Appendix C BRST Hamiltonian of Rank One

• Appendix D The FV Principal Theorem

• Appendix E BRST Quantization of SU(3) Yang-Mills Theory in α-gauges

• Bibliography

• Index