Local subsystems in gauge theory and gravity (1601.04744v2)

Abstract: We consider the problem of defining localized subsystems in gauge theory and gravity. Such systems are associated to spacelike hypersurfaces with boundaries and provide the natural setting for studying entanglement entropy of regions of space. We present a general formalism to associate a gauge-invariant classical phase space to a spatial slice with boundary by introducing new degrees of freedom on the boundary. In Yang-Mills theory the new degrees of freedom are a choice of gauge on the boundary, transformations of which are generated by the normal component of the nonabelian electric field. In general relativity the new degrees of freedom are the location of a codimension-2 surface and a choice of conformal normal frame. These degrees of freedom transform under a group of surface symmetries, consisting of diffeomorphisms of the codimension-2 boundary, and position-dependent linear deformations of its normal plane. We find the observables which generate these symmetries, consisting of the conformal normal metric and curvature of the normal connection. We discuss the implications for the problem of defining entanglement entropy in quantum gravity. Our work suggests that the Bekenstein-Hawking entropy may arise from the different ways of gluing together two partial Cauchy surfaces at a cross-section of the horizon.

• The paper proposes an extended phase space that incorporates boundary degrees of freedom to preserve gauge invariance in localized subsystems.
• It distinguishes between pure gauge transformations and nontrivial surface symmetries, revealing an algebra with local SL(D-2, ℝ) structure.
• The framework suggests an extended Hilbert space approach, offering a refined method to define entanglement entropy in quantum gravity.

Local Subsystems in Gauge Theory and Gravity

The paper "Local Subsystems in Gauge Theory and Gravity" by William Donnelly and Laurent Freidel addresses the challenge of defining localized subsystems in gauge theories and general relativity, aiming to provide a framework for understanding entanglement entropy in these contexts. Entanglement entropy, which relates closely to black hole thermodynamics, is thought to play a significant role in quantum gravity, but its definition in the presence of gauge symmetries and general covariance is not straightforward.

Main Contributions

1. Extended Phase Space: The authors introduce a formalism that extends the classical phase space associated with a region in space. This extension is crucial for accommodating gauge invariance when the region has a boundary. In gauge theories, the new phase space variables include additional boundary degrees of freedom, which transform under gauge groups.
2. Gauge Invariance: The authors provide a method to maintain gauge invariance by adding boundary terms to the symplectic potential. In Yang-Mills theory, this involves introducing a gauge degree of freedom at the boundary, while in gravity, it involves a choice of coordinates describing the boundary's location.
3. Symmetry and Dynamics: They identify two distinct types of transformations: gauge transformations, which are pure gauge and have vanishing Hamiltonians, and surface symmetries, which are genuine symmetries of the phase space with nontrivial generators. In gravity, surface symmetries consist of more complex transformations that include boosts and surface diffeomorphisms.
4. Algebra of Surface Symmetries: A significant finding is the algebra satisfied by generators of surface-preserving transformations in gravity. This algebra includes surface boosts and diffeomorphisms, and highlights the role of a local $SL(D-2, \mathbb{R})$ symmetry that acts on normal planes to the surface.
5. Extended Hilbert Space: At the quantum level, the construction points towards an extended Hilbert space that carries representations of boundary symmetry groups, suggesting that entanglement entropy can be defined via an embedding into this extended Hilbert space.

Implications and Future Directions

The implications of this research are both conceptual and practical. Conceptually, it suggests a novel way of looking at subsystems in quantum gravity, moving beyond the conventional tensor product structure that is problematic in the presence of gauge symmetries. Practically, it lays the groundwork for a new approach to calculating entanglement entropy which aligns it more closely with horizon entropy.

The introduction of a local symmetry group at the boundary could potentially lead to a richer understanding of quantum gravity and black hole entropy. If these symmetries can be precisely related to the microstates counted by the Bekenstein-Hawking entropy, they might provide a more detailed picture of what these microstates represent at the quantum level.

Future directions could involve rigorous quantization of the constructed classical phase space, which would require understanding the representation theory of the surface symmetry algebra identified. Another interesting aspect is to see how these concepts apply to dynamical boundaries, which are relevant for cosmological spacetimes and evolving horizons, perhaps shedding light on the relation between quantum entanglement and spacetime geometry.