Abstract
Einstein-Maxwell theory is not only covariant under diffeomorphisms but also is under U(1) gauge transformations. We introduce a combined transformation constructed out of diffeomorphism and U(1) gauge transformation. We show that symplectic potential, which is defined in covariant phase space method, is not invariant under combined transformations. In order to deal with that problem, following Donnelly and Freidel proposal [19], we introduce new fields. In this way, phase space and consequently symplectic potential will be extended. We show that new fields produce edge modes. We consider surface-preserving symmetries and we show that the group of surface-preserving symmetries is semi-direct sum of 2-dimensional diffeomorphism group on a spacelike codimension two surface with SL(2,R) and U(1). Eventually, we deduce that the Casimir of SL(2,R) is the area element, similar to the pure gravity case [19].
Highlights
The covariant phase space method [22–26] provides a Hamiltonian description so that we can study the generators of infinitesimal gauge transformations without the need to resort to a non-manifestlycovariant decomposition between space and time
We have considered covariant phase space method of obtaining conserved charges in classical field theories, where conserved charges can be extracted from symplectic form which is the exterior derivative of symplectic potential on phase space
We inferred that symplectic potential is not invariant under combined transformation, i.e. under both diffeomorphism and U (1) gauge transformation
Summary
Boundary conditions play crucial role in modern theoretical physics, for example in the holographic principle [1, 2], the AdS/CFT correspondence [3–5], the bulk-boundary correspondence of condensed matter [6–8], or the study of entropy [9–14]. By the introduction edge modes which are compensating fields at the boundary, one can restore gauge invariance fully. Considering these boundary degrees of freedom in quantum theory need to extend Hilbert space [12, 15–18]. By introducing new degrees of freedom on the boundary they have presented a general formalism to associate a gauge-invariant classical phase space to a spatial slice with boundary. In another work [21] Giller has constructed the extended phase space for three-dimensional gravity in first order formalism He has studied the boundary symmetries and the integrability of their generators and has found that the infinite-dimensional algebra of boundary symmetries with first order variables is the same as that with metric variables.
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