Covariant phase space with boundaries

TL;DR

This work provides a systematic, fully covariant procedure to form the covariant phase space for field theories with spatial boundaries, explicitly accounting for boundary terms through a boundary form C and an extended boundary Lagrangian ℓ. It derives a general, unambiguous expression for the diffeomorphism generators H_ξ that includes a natural boundary contribution beyond the original B-term, recovers known results (e.g., Brown–York tensor in GR), and shows the covariant Poisson bracket coincides with the Peierls bracket. By illustrating with particle mechanics, scalar and Maxwell fields, GR, and JT gravity, the paper demonstrates how boundary conditions and boundary terms shape the physical phase space and charges, including asymptotic boundary charges and their relation to entropy and holography. The framework clarifies Noether's theorem, asymptotic symmetries, and potential extensions to dynamical horizons, providing a cohesive, covariant foundation for boundary-aware Hamiltonian dynamics in diffeomorphism-invariant theories.

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.

Figures (2)

• Figure 1: The natural dynamical region for JT gravity: two asymptotic boundaries connected by a wormhole. The dashed lines indicate the horizons of this wormhole, which cross at the extremal point where $\Phi=\Phi_e$. The dotted lines show where the spatial boundaries are located at finite $r_c$; these boundaries are parametrized by $t\in (-\infty,\infty)$.
• Figure 2: The various solutions used in computing the Peierls bracket, in the special case where $g$ is a local operator. The last one gives the direction in pre-phase space in which evolution by $g$ moves $\phi_0$.