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

TL;DR

This paper constructs a rigorous path integral formulation for (1+1) quantum gravity, addressing factor ordering ambiguities and self-adjoint extensions to ensure a well-defined evolution operator.

Contribution

It provides a complete parametrization of self-adjoint extensions for the Hamiltonian and develops a path integral representation incorporating factor ordering ambiguities.

Findings

Identified and parametrized all self-adjoint extensions of the Hamiltonian.

Developed Trotter-type product formulae for path integral construction.

Ensured the Hamiltonian's self-adjointness for a consistent quantum gravity model.

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.