Proof of the equivalence of the symplectic forms derived from the canonical and the covariant phase space formalisms

TL;DR

This paper demonstrates that the symplectic forms obtained from the covariant and canonical phase space formalisms are mathematically equivalent for theories defined on space-times with boundaries.

Contribution

It provides a rigorous proof of the equivalence between covariant and canonical symplectic forms in boundary-including space-time theories.

Findings

Symplectic forms from both formalisms are equivalent.

The proof applies to theories with space-time boundaries.

Establishes consistency between two fundamental phase space approaches.

Abstract

We prove that, for any theory defined over a space-time with boundary, the symplectic form derived in the covariant phase space is equivalent to the one derived from the canonical formalism.