### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 039, 19 pages      arXiv:1612.05622      https://doi.org/10.3842/SIGMA.2017.039

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

John Haga and Rachel Lash Maitra
Department of Applied Mathematics, Wentworth Institute of Technology, 550 Huntington Ave., Boston MA 02115, USA

Received December 19, 2016, in final form June 01, 2017; Published online June 07, 2017

Abstract
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.

Key words: factor ordering in quantum gravity; path integrals in quantum gravity; singularity avoidance in quantum gravity; quantization on a half-line.

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