Covariant phase space with boundaries
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Springer
Received: July 6, Revised: August 21, Accepted: September 20, Published: October 22,
2020 2020 2020 2020
Covariant phase space with boundaries
Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.
E-mail: [email protected], [email protected] 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. Keywords: Classical Theories of Gravity, AdS-CFT Correspondence ArXiv ePrint: 1906.08616
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)146
JHEP10(2020)146
Daniel Harlow and Jie-qiang Wu
Contents 1 Introduction 1.1 Notation
1 4 5 5 9 13 16
3 Examples 3.1 Particle mechanics 3.2 Two-derivative scalar field 3.3 Maxwell theory 3.4 Higher-derivative scalar 3.5 General relativity 3.6 Brown-York stress tensor 3.7 Jackiw-Teitelboim gravity
20 20 21 22 23 24 28 30
4 Discussion 4.1 Meaning of the Poisson bracket 4.2 Noether’s theorem 4.3 Asymptotic boundaries 4.4 Black hole entropy
36 36 39 41 43
A Non-covariant Hamiltonian analysis of general relativity
45
1
Introduction
The most basic problem in physics is the initial-value problem: given the state of a system at some initial time, in what state do we find it at a later time? This problem is most naturally discussed within the Hamiltonian formulation of classical/quantum mechanics. In relativistic theories howev
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