Gauge Fixing and Constrained Dynamics

TL;DR

This paper reviews the Dirac formalism for constrained Hamiltonian systems, focusing on gauge fixing, Dirac brackets, and stability, with applications to electrodynamics and future plans for General Relativity.

Contribution

It provides a comprehensive overview of gauge fixing and constrained dynamics using Dirac's formalism, emphasizing stability and well-posedness in gauge theories.

Findings

Dirac brackets offer a consistent method to eliminate gauge freedom.

Fixing all gauge freedom ensures well-posed evolution in gauge theories.

Electrodynamics exemplifies the application of the formalism.

Abstract

We review the Dirac formalism for dealing with constraints in a canonical Hamiltonian formulation and discuss gauge freedom and display constraints for gauge theories in a general context. We introduce the Dirac bracket and show that it provides a consistent method to remove any gauge freedom present. We discuss stability in evolution of gauge theories and show that fixing all gauge freedom is sufficient to ensure well-posedness for a large class of gauge theories. Electrodynamics provides examples of the methods outlined for general gauge theories. Future work will apply the formalism, and results derived here, to General Relativity.