The Generalized Peierls Bracket

TL;DR

The paper develops a covariant framework that extends the Peierls bracket from the solution space to the space of histories and further generalizes it to gauge-dependent quantities using an invariance-breaking construction. It identifies two robust classes of invariance-breaking terms that yield Lie brackets, showing that the generalized Peierls bracket encompasses canonical Dirac, gauge-fixed, and Feynman/Landau gauge algebras, and remains well-defined under pull-backs to subspaces with first-class generalized constraints. This unifies path-integral and algebraic approaches to quantization and clarifies when and how gauge invariance or breaking affects the underlying algebraic structure. The resulting framework provides a covariant, Heisenberg-friendly route to quantization and suggests new quantization paths by treating algebras on histories and their subspaces without relying on a fixed gauge choice.

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.