The Generalized Peierls Bracket

TL;DR

This paper extends the Peierls algebra to gauge dependent functions on the space of histories, providing a unified framework that encompasses various gauge fixing and quantization methods.

Contribution

It introduces a generalized Peierls algebra that unifies gauge invariant and dependent approaches, with a systematic way to choose invariance breaking terms.

Findings

Generalized Peierls algebra can reproduce gauge fixing techniques.

Constraints in the algebra are always first class.

Quantization can be performed using the algebra's properties.

Abstract

We first extend the Peierls algebra of gauge invariant functions from the space ${\cal S}$ of classical solutions to the space ${\cal H}$ of histories used in path integration and some studies of decoherence. We then show that it may be generalized in a number of ways to act on gauge dependent functions on ${\cal H}$. These generalizations (referred to as class I) depend on the choice of an ``invariance breaking term," which must be chosen carefully so that the gauge dependent algebra is a Lie algebra. Another class of invariance breaking terms is also found that leads to an algebra of gauge dependent functions, but only on the space ${\cal S}$ of solutions. By the proper choice of invariance breaking term, we can construct a generalized Peierls algebra that agrees with any gauge dependent algebra constructed through canonical or gauge fixing methods, as well as Feynman and Landau ``gauge." Thus, generalized Peierls algebras present a unified description of these techniques. We study the properties of generalized Peierls algebras and their pull backs to spaces of partial solutions and find that they may posses constraints similar to the canonical case. Such constraints are always first class, and quantization may proceed accordingly.