Covariant phase space with boundaries

By D. Harlow, Jie-qiang Wu. Journal of High Energy Physics, 2019

188 citations recorded by the source corpus.

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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3e65b016dcf0cf2e7faba6e5497be1c932404d42

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

In Nuclear and High Energy Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Edge modes of gravity. Part I. Corner potentials and charges (2020, methodology): “ expressed in terms of the boundary Lagrangian (2.6) as L EH= GR+ d EH=GR = EH ; (2.15) showing how the boundary Lagrangian aects the form of the corner symplectic potential, in agreement with [44,49,50]. As expected, this variation is naturally of the form L= E L+ d L, with E Lthe equations of mo”