Covariant phase space with boundaries (1906.08616v4)

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.

• The paper provides a systematic algorithm for deriving Hamiltonians in theories with spatial boundaries, resolving ambiguities in traditional treatments.
• It introduces a crucial additional boundary term into the covariant phase space formalism, ensuring consistency with symplectic structures.
• The study demonstrates that the Poisson bracket naturally coincides with the Peierls bracket, enhancing its applicability to general relativity and black hole entropy.

Covariant Phase Space with Boundaries

The research paper "Covariant Phase Space with Boundaries" by Daniel Harlow and Jie-qiang Wu addresses crucial gaps in the application of the covariant phase space formalism to field theories with spatial boundaries. The methodology primarily originates from the work of Iyer, Lee, Wald, and Zoupas, which has been instrumental in presenting Hamiltonian dynamics of Lagrangian field theories without compromising covariance. Despite its elegance, the original formalism has historically struggled with the systematic treatment of boundary terms and total derivatives, leading to ambiguities that this paper seeks to resolve.

Summary

This paper proposes an enhanced algorithmic framework for applying the covariant phase space formalism to contexts with spatial boundaries. Particularly, the authors revisit the derivation of the canonical Hamiltonian, emphasizing the natural emergence of the boundary term denoted as "$B$". This term has been previously introduced without a clear physical interpretation. The paper underscores the importance of an additional boundary term, which unveils itself conspicuously even within general relativity, provided sufficiently permissive boundary conditions are imposed.

The authors impose a foundational criterion on field theories: the stationarity of the action should be maintained modulo future/past boundary terms under arbitrary variations conforming to spatial boundary conditions. Through this, both the symplectic structure and the Hamiltonian for any diffeomorphism preserving the theory are meticulously constructed. The results are verified across several examples, aligning the newly constructed Hamiltonian with those found in prior studies.

A particularly notable outcome of this analysis is the demonstration that the Poisson bracket on covariant phase space inherently coincides with the Peierls bracket, thereby eliminating non-covariant intermediate processes that have previously obfuscated consistent interpretations. Additionally, the findings have intriguing potential implications for understanding the entropy of dynamical black hole horizons.

Key Results

1. Algorithmic Derivation of Hamiltonians: The paper provides a systematic approach to derive the Hamiltonian in theories with spatial boundaries, resolving the ambiguity associated with existing treatments.
2. Boundary Term Inclusion: Unlike previous literature where boundary terms were not systematically incorporated, this paper derives an additional boundary term essential for appropriately capturing the dynamics within the covariant framework.
3. Symplectic Current and Structure: The researchers define a pre-symplectic current through the pullback of the symplectic potential to the pre-phase space that respects boundary conditions, thus ensuring independence from the choice of Cauchy surface.
4. Peierls Bracket Equivalence: Establishing equivalence with the Peierls bracket reinforces the internal consistency of the covariant phase space formalism and its applicability to broader contexts.

Theoretical Implications

General Relativity and Beyond: The paper significantly advances the theoretical landscape in understanding boundary contributions in Hamiltonian formulations, which is pivotal not only for general relativity but also for theories involving higher derivative actions.

Black Hole Entropy: The connection to black hole entropy opens potential avenues for reevaluating entropy concepts in non-stationary regimes, potentially providing insights that align with holographic principles and quantum gravity.

Future Directions

The research sets the stage for several promising directions. First, expanding this formalism to accommodate different types of boundaries, such as null boundaries, would be an invaluable extension. Second, connecting the findings with quantum gravity research and AdS/CFT correspondence could provide new insights into the interfacial dynamics between quantum field theories and general relativity. Lastly, the potential link between the additional boundary term and holographic prescriptions for black hole entropy should be investigated more thoroughly, particularly in the context of non-equilibrium thermodynamics.

In conclusion, "Covariant Phase Space with Boundaries" represents a meticulous reexamination of a foundational aspect of theoretical physics, offering novel insights and tools for researchers exploring the complexities at the intersection of geometry and dynamics in field theories.