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.
Highlights
Problem, for which the Hamiltonian formalism is too convenient to dispense with
In relativistic field theories there is an elegant formalism due to Iyer, Lee, Wald, and Zoupas, which, building on earlier ideas from [1,2,3,4], presents Hamiltonian mechanics in a manner that preserves manifest Lorentz or diffeomorphism invariance: the covariant phase space formalism [5,6,7,8,9].1. This method is well-known in the relativity community, where in particular it was used by Wald to derive a generalization of the area formula for black hole entropy to higher-derivative gravity [6], and it has been showing up fairly often in recent discussions of the AdS/CFT correspondence, the asymptotic symmetry structure of gravity in Minkowski space [20, 21], and in attempts to define “near-horizon” symmetries associated to black holes [22,23,24]
In a final section we show that the Poisson bracket in the covariant phase space formalism is generally equivalent to the Peierls bracket, we give a proof of Noether’s theorem for continuous symmetries within the covariant phase space approach, and we comment on some subtleties arising in the application of our results to asymptotic boundaries
Summary
If M has a metric, as it always will for us, we can describe this induced orientation by saying we require that the boundary volume form ∂M is related to the spacetime volume form by. Note in particular that the volume form on ∂Σ is not obtained by viewing ∂Σ as the boundary of its past within ∂M , these differ by a sign. Sometimes we will discuss a Cauchy surface Σ− which is the past boundary of a spacetime M , the most convenient way to maintain our conventions is to say that when this surface appears implicitly as part of ∂M we give it the opposite orientation from when it appears explicitly as Σ−
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