A new antisymmetric bilinear map for type-I gauge theories

TL;DR

This paper introduces a new antisymmetric bilinear map for gauge theories, demonstrating its Poisson bracket properties in Maxwell theory, and expanding the mathematical tools for analyzing gauge-invariant functionals.

Contribution

It proves that a newly defined antisymmetric bilinear map extends the Peierls map and functions as a Poisson bracket in gauge theories, specifically Maxwell theory.

Findings

The new map is a Poisson bracket in Maxwell theory.

The map belongs to a larger family of antisymmetric bilinear maps.

The construction relies on gauge invariance and geometric structures of the space of histories.

Abstract

In the case of gauge theories, which are ruled by an infinite-dimensional invariance group, various choices of antisymmetric bilinear maps on field functionals are indeed available. This paper proves first that, within this broad framework, the Peierls map (not yet the bracket) is a member of a larger family. At that stage, restriction to gauge-invariant functionals of the fields, with the associated Ward identities and geometric structure of the space of histories, make it possible to prove that the new map is indeed a Poisson bracket in the simple but relevant case of Maxwell theory. The building blocks are available for gauge theories only: vector fields that leave the action functional invariant; the invertible gauge-field operator, and the Green function of the ghost operator.