Perfect and Imperfect Gauge Fixing

TL;DR

This paper explores the distinction between perfect and imperfect gauge fixing in constrained systems, demonstrating how the chain structure influences gauge choice and the number of constants of motion, with examples from physics.

Contribution

It introduces a framework using chain structures to differentiate perfect and imperfect gauge fixings and analyzes their implications in various physical models.

Findings

Perfect gauges determine all gauged degrees of freedom.

Imperfect gauges leave some first class constraints as subsidiary conditions.

The level of gauge fixing affects the number of constants of motion.

Abstract

Gauge fixing may be done in different ways. We show that using the chain structure to describe a constrained system, enables us to use either a perfect gauge, in which all gauged degrees of freedom are determined; or an imperfect gauge, in which some first class constraints remain as subsidiary conditions to be imposed on the solutions of the equations of motion. We also show that the number of constants of motion depends on the level in a constraint chain in which the gauge fixing condition is imposed. The relativistic point particle, electromagnetism and the Polyakov string are discussed as examples and perfect or imperfect gauges are distinguished.