2D quantum dilaton gravitational Hamiltonian, boundary terms and new definition for total energy

The ADM and Bondi mass for the RST model have been first discussed from Hawking and Horowitzʹs argument. Since there is a nonlocal term in the RST model, the RST lagrangian has to be localized so that Hawking and Horowitzʹs proposal can be carried out. Expressing the localized RST action in terms of the ADM formulation, the RST Hamiltonian can be derived, meanwhile keeping track of all boundary terms. Then the total boundary terms can be taken as the total energy for the RST model. Our result shows that the previous expression for the ADM and Bondi mass actually needs to be modified at quantum level, but at classical level, our mass formula can be reduced to that given by Bilal and Kogan [Phys. Rev. D 47 (1993) 5408] and de Alwis [Phys. Rev. D 49 (1994) 941]. It has been found that there is a new contribution to the ADM and Bondi mass from the RST boundary due to the existence of the hidden dynamical field. The ADM and Bondi mass with and without the RST boundary for the static and dynamical solutions have been discussed respectively in detail, and some new properties have been found. The thunderpop of the RST model has also been encountered in our new Bondi mass formula.