From Peierls brackets to a generalized Moyal bracket for type-I gauge theories

TL;DR

This paper develops a gauge-invariant Poisson and Moyal bracket framework for type-I gauge theories using the space-of-histories approach, extending Peierls brackets to a generalized quantum setting.

Contribution

It introduces a gauge-invariant Moyal bracket for type-I gauge theories, generalizing Peierls brackets within a covariant formalism, applicable under specific gauge-fixing conditions.

Findings

Peierls bracket is gauge-invariant under certain conditions

A gauge-invariant Moyal bracket reduces to iħ times the Peierls bracket at lowest order

Framework applies to Maxwell, Yang-Mills, and gravitational fields

Abstract

In the space-of-histories approach to gauge fields and their quantization, the Maxwell, Yang--Mills and gravitational field are well known to share the property of being type-I theories, i.e. Lie brackets of the vector fields which leave the action functional invariant are linear combinations of such vector fields, with coefficients of linear combination given by structure constants. The corresponding gauge-field operator in the functional integral for the in-out amplitude is an invertible second-order differential operator. For such an operator, we consider advanced and retarded Green functions giving rise to a Peierls bracket among group-invariant functionals. Our Peierls bracket is a Poisson bracket on the space of all group-invariant functionals in two cases only: either the gauge-fixing is arbitrary but the gauge fields lie on the dynamical sub-space; or the gauge-fixing is a linear functional of gauge fields, which are generic points of the space of histories. In both cases, the resulting Peierls bracket is proved to be gauge-invariant by exploiting the manifestly covariant formalism. Moreover, on quantization, a gauge-invariant Moyal bracket is defined that reduces to i hbar times the Peierls bracket to lowest order in hbar.