  We develop a mathematically rigorous path integral representation of the time
evolution operator for a model of (1+1) quantum gravity that incorporates
factor ordering ambiguity. In obtaining a suitable integral kernel for the
time-evolution operator, one requires that the corresponding Hamiltonian is
self-adjoint; this issue is subtle for a particular category of factor
orderings. We identify and parametrize a complete set of self-adjoint
extensions and provide a canonical description of these extensions in terms of
boundary conditions. Moreover, we use Trotter-type product formulae to
construct path-integral representations of time evolution.
