Unambiguous Phase Spaces for Subregions

TL;DR

This work resolves boundary ambiguities in the covariant phase space for subregions by deriving the subregion symplectic structure from the path integral as a contour integral around a partial Cauchy surface. It defines subregion observables, builds the corresponding Poisson structure, and identifies a subregion phase space whose symplectic form is insensitive to the traditional boundary ambiguities. The results imply that large gauge transformations are non-physical within this framework and highlight edge-mode–driven correlations across entangling surfaces, with potential relevance to holography and black hole information. Overall, the paper provides a covariant, path-integral–based route to well-defined subregion dynamics and clarifies the role of boundaries and gauge structure in covariant 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.

Figures (7)

• Figure 3.1: $\Sigma \subset \mathcal{U}\subset \mathcal{M}$. The action is deformed by a source term at the codimension one surface $\Sigma$.
• Figure 3.2: The support of $\delta\phi$ on $\partial\mathcal{U}$ is contained in $\Sigma^+=J^+(\Sigma)\cap\partial\mathcal{U}$.
• Figure 3.3: A sequence of $\mathcal{U}_n$ such that $\Sigma^+_n\to\Sigma$.
• Figure 3.4: For $\Sigma$ with an asymptotic boundary, the symplectic structure is obtained by integrating over $\partial U_r$, and then taking the limit as $U_r$ grows to contain the entirety of $\Sigma$.
• Figure 4.1: The joint support of $\delta_1\phi,\delta_2\phi$ on $\partial\mathcal{U}$ is contained in $\partial\mathcal{U}\cap J^+(\Sigma_1)\cap J^+(\Sigma_2)$.