  We study the Hawking radiation in two dimensional dilaton black hole by means
of quantum gravity holding near the apparent horizon. First of all, we
construct the canonical formalism of the dilaton gravity in two dimensions.
Then the Vaidya metric corresponding to the dilaton black hole is established
where it is shown that the dilaton field takes a form of the linear dilaton.
Based on the canonical formalism and the Vaidya metric, we proceed to analyze
quantum properties of a dynamical black hole. It is found that the mass loss
rate of the Hawking radiation is independent of the black hole mass and at the
same time the apparent horizon recedes to the singularity as shown in other
studies of two dimensional gravity. It is interesting that one can construct
quantum gravity even near the curvature singularity and draw the same
conclusion with respect to the Hawking radiation as the above-mentioned
picture. Unfortunately, the present formalism seems to be ignorant of the
contributions from the functional measures over the gravitational field, the
dilaton and the ghosts.
