Area expectation values in quantum area Regge calculus

TL;DR

This paper develops a quantum measure for area tensor Regge calculus, enabling finite expectation values for areas and connecting discrete and continuous formulations through a well-defined path integral approach.

Contribution

It introduces a quantum measure for area tensor Regge calculus that remains consistent in the continuous time limit and allows for finite area expectations.

Findings

Derived the continuous time limit of area tensor Regge calculus.

Constructed a quantum measure compatible with canonical quantisation.

Obtained finite expectation values for areas using the new measure.

Abstract

The Regge calculus generalised to independent area tensor variables is considered. The continuous time limit is found and formal Feynman path integral measure corresponding to the canonical quantisation is written out. The quantum measure in the completely discrete theory is found which possesses the property to lead to the Feynman path integral in the continuous time limit whatever coordinate is chosen as time. This measure can be well defined by passing to the integration over imaginary field variables (area tensors). Averaging with the help of this measure gives finite expectation values for areas.