Factor Ordering and Path Integral Measure for Quantum Gravity in (1+1) Dimensions

[摘要] 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.

[关键词] factor ordering in quantum gravity;path integrals in quantum gravity;singularity avoidance in quantum gravity;quantization on a half-line