Covariant Poisson Brackets in Geometric Field Theory

TL;DR

This paper connects multisymplectic and covariant phase space methods in geometric field theory, showing how the covariant Poisson bracket naturally arises from multisymplectic forms, unifying two approaches.

Contribution

It demonstrates how to derive the covariant phase space symplectic form and Poisson bracket from multisymplectic geometry, clarifying their relationship.

Findings

The covariant Poisson bracket matches the Peierls-DeWitt bracket.

The symplectic form on covariant phase space can be obtained from multisymplectic form.

Unified framework for geometric field theory approaches.

Abstract

We establish a link between the multisymplectic and the covariant phase space approach to geometric field theory by showing how to derive the symplectic form on the latter, as introduced by Crnkovic-Witten and Zuckerman, from the multisymplectic form. The main result is that the Poisson bracket associated with this symplectic structure, according to the standard rules, is precisely the covariant bracket due to Peierls and DeWitt.