Spherically symmetric systems of fields and black holes. II. Apparent horizon in canonical formalism

We study the action of a two-dimensional model of gravity found in the preceding paper. We transform the action to the first-order Arnowitt-Deser-Misner form, and work out the generalized momenta and super-Hamiltonians. We propose to foliate the spacetime in such a way that the inside of the apparent horizon will be cut away. In the classical theory, no loss of information for the development of states from ${\mathcal{I}}^{-}$ to ${\mathcal{I}}^{+}$ can result, but in the corresponding quantum theory, some such losses could occur if a black hole evaporates. We study the boundary conditions for the fields at the apparent horizon which are implied by such a foliation, and calculate the corresponding surface correction to the Hamiltonian by the method of Regge and Teitelboim. We generalize the socalled Berger-Chitre-Moncrief-Nutku gauge in such a way that the fields cannot violate the boundary conditions. In this gauge, we perform an explicit total reduction of the canonical formalism so that only the true dynamical variables appear in the Hamiltonian. The reduced Hamiltonian splits into a black hole and a field part.