Reflection positivity in simplicial gravity

TL;DR

This paper demonstrates reflection positivity in simplicial gravity, enabling the construction of a quantum Hilbert space and operators in a non-perturbative framework based on Regge calculus.

Contribution

It establishes reflection positivity within simplicial gravity and constructs a quantum Hilbert space without relying on perturbation theory.

Findings

Reflection positivity is proven for the Regge action.

A Hilbert space for quantum gravity is constructed.

Field operators and observables are defined in this framework.

Abstract

Within the context of piecewise linear manifolds we establish reflection positivity with a Hilbert action given in terms of the Regge curvature and a cosmological term. Using this positivity a Hilbert space for a quantum theory is constructed and some field operators and observables are given. The set-up allows to introduce time reversal though no time exists. All constructions are non-perturbative.