# Unambiguous Phase Spaces for Subregions

## TL;DR

This paper resolves boundary ambiguities in the covariant phase space formalism for subregions by deriving a contour integral expression for the symplectic structure, impacting gauge symmetry and entanglement studies.

## Contribution

It introduces a method to compute the symplectic structure as a contour integral, clarifying boundary issues in covariant phase space applications to subregions.

## Key findings

- Symplectic structure expressed as a contour integral around a partial Cauchy surface.
- Resolution of boundary ambiguities in covariant phase space for subregions.
- Implications for gauge symmetry and entanglement in field theories.

## Abstract

The covariant phase space technique is a powerful formalism for understanding the Hamiltonian description of covariant field theories. However, applications of this technique to problems involving subregions, such as the exterior of a black hole, have heretofore been plagued by boundary ambiguities. We provide a resolution of these ambiguities by directly computing the symplectic structure from the path integral, showing that it may be written as a contour integral around a partial Cauchy surface. This result has implications for gauge symmetry and entanglement.