Abstract
We discuss an approach to characterizing local degrees of freedom of a subregion in diffeomorphism-invariant theories using the extended phase space of Donnelly and Freidel [36]. Such a characterization is important for defining local observables and entanglement entropy in gravitational theories. Traditional phase space constructions for subregions are not invariant with respect to diffeomorphisms that act at the boundary. The extended phase space remedies this problem by introducing edge mode fields at the boundary whose transformations under diffeomorphisms render the extended symplectic structure fully gauge invariant. In this work, we present a general construction for the edge mode symplectic structure. We show that the new fields satisfy a surface symmetry algebra generated by the Noether charges associated with the edge mode fields. For surface-preserving symmetries, the algebra is universal for all diffeomorphism-invariant theories, comprised of diffeomorphisms of the boundary, SL(2, ℝ) transformations of the normal plane, and, in some cases, normal shearing transformations. We also show that if boundary conditions are chosen such that surface translations are symmetries, the algebra acquires a central extension.
Highlights
Impose on the physical Hilbert space prevents such a factorization from occurring.1 One way of handling this nonfactorization is to define the entropy in terms of the algebra of observables for the local subregion [1]
We show that the new fields satisfy a surface symmetry algebra generated by the Noether charges associated with the edge mode fields
The gravitational field equations have been shown to follow from applying the first law of entanglement entropy [10, 11] to subregions, both in holography [12,13,14,15,16] and for more general gravitational theories [17,18,19,20], all of which is predicated on a well-defined notion for entanglement for the local subregion
Summary
The covariant canonical formalism [37,38,39,40] provides a Hamiltonian description of a field theory’s degrees of freedom while maintaining spacetime covariance. The vector field acts on S-functions as a directional derivative, and in particular its action on the functions φx is to give a new function ΦxV ≡ V [φx], which, given a solution, evaluates the linearization Φ of the field φ at the point x. This allows us to define the exterior derivative of the functions φx, denoted δφx. £ξφ must be a solution to the linearized field equations, and the infinitesimal diffeomorphism generated by ξa defines a vector field on S, which we denote ξ, whose action on δφ is.
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