# Covariant phase space with boundaries

## TL;DR

This paper refines the covariant phase space formalism for field theories with boundaries, providing a systematic algorithm that clarifies boundary contributions to the Hamiltonian and aligns with previous results, with implications for black hole entropy.

## Contribution

It introduces an algorithmic procedure to incorporate boundary terms into the covariant phase space formalism, clarifying the physical interpretation of boundary quantities and extending the framework to general boundary conditions.

## Key findings

- Derived a natural emergence of boundary term B in the Hamiltonian.
- Identified an additional boundary term present in general relativity.
- Confirmed the covariant phase space Poisson bracket equals the Peierls bracket.

## Abstract

The covariant phase space method of Iyer, Lee, Wald, and Zoupas gives an elegant way to understand the Hamiltonian dynamics of Lagrangian field theories without breaking covariance. The original literature however does not systematically treat total derivatives and boundary terms, which has led to some confusion about how exactly to apply the formalism in the presence of boundaries. In particular the original construction of the canonical Hamiltonian relies on the assumed existence of a certain boundary quantity "$B$", whose physical interpretation has not been clear. We here give an algorithmic procedure for applying the covariant phase space formalism to field theories with spatial boundaries, from which the term in the Hamiltonian involving $B$ emerges naturally. Our procedure also produces an additional boundary term, which was not present in the original literature and which so far has only appeared implicitly in specific examples, and which is already nonvanishing even in general relativity with sufficiently permissive boundary conditions. The only requirement we impose is that at solutions of the equations of motion the action is stationary modulo future/past boundary terms under arbitrary variations obeying the spatial boundary conditions; from this the symplectic structure and the Hamiltonian for any diffeomorphism that preserves the theory are unambiguously constructed. We show in examples that the Hamiltonian so constructed agrees with previous results. We also show that the Poisson bracket on covariant phase space directly coincides with the Peierls bracket, without any need for non-covariant intermediate steps, and we discuss possible implications for the entropy of dynamical black hole horizons.