How to construct quantum measure in Regge calculus?

TL;DR

The paper introduces a method to construct a quantum measure in Regge calculus by smoothly transitioning from discrete to continuous dimensions, enabling canonical quantization and finite expectation values.

Contribution

It presents a novel approach to defining quantum measures in Regge calculus that ensures consistency with canonical quantization regardless of the continuous coordinate.

Findings

Constructed a quantum measure that aligns with canonical quantization in 3D Regge calculus.

Achieved finite expectation values for links through averaging with the new measure.

Provided a nearly unique family of measures suitable for the quantization process.

Abstract

We propose the following way of constructing quantum measure in Regge calculus: the full discrete Regge manifold is made continuous in some direction by tending corresponding dimensions of simplices to zero, then functional integral measure corresponding to the canonical quantization (with continuous coordinate playing the role of time) can be constructed. The full discrete measure is chosen so that it would result in canonical quantization one whatever coordinate is made continuous. This strategy is followed in 3D case where full discrete measure is determined in such the way practically uniquely (in fact, family of similar measures is obtained). Averaging with the help of the constructed measure gives finite expectation values for links.