Feynman path integral in area tensor Regge calculus and correspondence principle

TL;DR

This paper explores how the quantum measure in area tensor Regge calculus can be aligned with the Feynman path integral and the correspondence principle, using a specific representation of rotation matrices to ensure proper quantization and measure properties.

Contribution

It introduces a representation based on separate selfdual and antiselfdual rotations that allows the quantum measure to satisfy both canonical quantization and the correspondence principle.

Findings

The measure can be modified to satisfy the correspondence principle.

The modification does not affect vacuum expectation values under certain conditions.

The approach provides a consistent way to incorporate the Feynman path integral in Regge calculus.

Abstract

The quantum measure in area tensor Regge calculus can be constructed in such the way that it reduces to the Feynman path integral describing canonical quantisation if the continuous limit along any of the coordinates is taken. This construction does not necessarily mean that Lorentzian (Euclidean) measure satisfies correspondence principle, that is, takes the form proportional to $e^{iS}$ ($e^{-S}$) where $S$ is the action. Requirement to fit this principle means some restriction on the action, or, in the context of representation of the Regge action in terms of independent rotation matrices (connections), restriction on such representation. We show that the representation based on separate treatment of the selfdual and antiselfdual rotations allows to modify the derivation and give sense to the conditionally convergent integrals to implement both the canonical quantisation and correspondence principles. If configurations are considered such that the measure is factorisable into the product of independent measures on the separate areas (thus far it was just the case in our analysis), the considered modification of the measure does not effect the vacuum expectation values.