Path integral measure in Regge calculus from the functional Fourier transform

TL;DR

This paper investigates the measure in the path integral for Regge calculus by comparing functional Fourier transforms to the continuum GR measure, deriving explicit expressions that suggest a proper continuum limit reproduces general relativity.

Contribution

It introduces a rigorous method to fix the Regge measure by comparing functional Fourier transforms, providing explicit formulas for simple cases and linking to the continuum GR measure.

Findings

Derived explicit measures for simple Regge manifolds

Showed the Regge measure can approximate the continuum GR measure

Indicated a proper continuum limit reproduces general relativity

Abstract

The problem of fixing measure in the path integral for the Regge-discretised gravity is considered from the viewpoint of it's "best approximation" to the already known formal continuum general relativity (GR) measure. A rigorous formulation may consist in comparing functional Fourier transforms of the measures, i.e. characteristic or generating functionals, and requiring these to coincide on some dense set in the functional space. The possibility for such set to exist is due to the Regge manifold being a particular case of general Riemannian one (Regge calculus is a minisuperspace theory). The two versions of the measure are obtained depending on what metric tensor, covariant or contravariant one, is taken as fundamental field variable. The closed expressions for the measure are obtained in the two simple cases of Regge manifold. These turn out to be quite reasonable one of them indicating that appropriately defined continuum limit of the Regge measure would reproduce the original continuum GR measure.