Secondary Calculus and the Covariant Phase Space

TL;DR

This paper rigorously develops the covariant phase space in the context of secondary calculus, clarifying its structure and gauge symmetries for potential covariant quantization of Lagrangian field theories.

Contribution

It provides a fully rigorous mathematical framework for the covariant phase space, including the degeneracy distribution and gauge invariants, using secondary calculus.

Findings

Describes the degeneracy distribution of the presymplectic form

Reestablishes the Lie bracket among gauge invariant functions

Offers a rigorous approach to covariant phase space in secondary calculus

Abstract

The covariant phase space of a Lagrangian field theory is the solution space of the associated Euler-Lagrange equations. It is, in principle, a nice environment for covariant quantization of a Lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) "presymplectic structure" w (as first noticed by Zuckerman in 1986) whose degeneracy (functional) distribution is naturally interpreted as the Lie algebra of gauge transformations. We propose a fully rigorous approach to the covariant phase space in the framework of secondary calculus. In particular we describe the degeneracy distribution of w. As a byproduct we rederive the existence of a Lie bracket among gauge invariant functions on the covariant phase space.