If , the 3-manifold on which the ADM canonical variables , are defined, has a boundary , then the usual vacuum constraints21. However, as Balachandran, Chandar, and Momen [32, 31] and Carlip  realized, the boundary conditions for the smearing fields can be relaxed by adding appropriate boundary terms to the constraints. Namely, for any vector field and function they define and are well-defined 2-surface observables corresponding to the momentum and the Hamiltonian constraint, respectively. Moreover, the observables form an infinite-dimensional Lie algebra with respect to the Poisson bracket. In this Lie algebra the momentum constraints form an ideal, and the quotient of the algebra of the observables and the constraint ideal is still infinite-dimensional. (In the asymptotically flat case, cf. [47, 364].) In fact, defines a homomorphism of the Lie algebra of the 2-surface vector fields into the quotient algebra of these observables modulo constraints [32, 31, 99]. To understand the meaning of these observables, recall that any vector field on generates a diffeomorphism, which is an exact (gauge) symmetry of general relativity, and the role of the momentum constraint is just to generate this gauge symmetry in the phase space. However, the boundary breaks the diffeomorphism invariance of the system, and hence on the boundary the diffeomorphism gauge motions yield the observables and the gauge degrees of freedom give raise to physical degrees of freedom, making it possible to introduce the so-called edge states [32, 31, 99]. Evaluating and on the constraint surface we obtain
Analogous investigations were done by Husain and Major in . Using Ashtekar’s complex variables  they determined all the local boundary conditions for the canonical variables , and for the lapse , the shift , and the internal gauge generator on that ensure the functional differentiability of the Gauss, the diffeomorphism, and the Hamiltonian constraints. Although there are several possibilities, they discussed the two most significant cases. In the first case the generators , , and are vanishing on , whenever there are infinitely many 2-surface observables both from the diffeomorphism and the Gauss constraints, but no observable from the Hamiltonian constraint. The structure of these observables is similar to that of those coming from the ADM diffeomorphism constraint above. The other case considered is when the canonical momentum (and hence, in particular, the 3-metric) is fixed on the 2-boundary. Then the quasi-local energy could be an observable, as in the ADM analysis above.
All of the papers [32, 31, 99, 210] discuss the analogous phenomenon of how the gauge freedoms are getting to be true physical degrees of freedom in the presence of 2-surfaces on the 2-surfaces themselves in the Chern-Simons and BF theories. Weakening the boundary conditions further (allowing certain boundary terms in the variation of the constraints) a more general algebra of ‘observables’ can be obtained [101, 296]: They form the Virasoro algebra with a central charge. (In fact, Carlip’s analysis in  is based on the so-called covariant Noether charge formalism below.) Since this algebra is well known in conformal field theories, this approach might be a basis of understanding the microscopic origin of the black hole entropy [100, 101, 102, 296, 103]. However, this quantum issue is beyond the scope of the present review.
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