From areas to lengths in quantum Regge calculus

TL;DR

This paper explores quantum Regge calculus, focusing on the transition from area-based to length-based formulations, and discusses how different measures affect the finiteness of length expectations in quantum gravity.

Contribution

It introduces a quantisation framework for length-based Regge calculus as a special case of area tensor Regge calculus, with adjustable measures to recover known continuum measures.

Findings

Discrete measures can be tuned to match continuum quantum gravity measures.

Certain measures lead to finite expectation values of lengths.

The quantisation is nearly unique up to a local measure factor.

Abstract

Quantum area tensor Regge calculus is considered, some properties are discussed. The path integral quantisation is defined for the usual length-based Regge calculus considered as a particular case (a kind of a state) of the area tensor Regge calculus. Under natural physical assumptions the quantisation of interest is practically unique up to an additional one-parametric local factor of the type of a power of $\det\|g_{\lambda\mu}\|$ in the measure. In particular, this factor can be adjusted so that in the continuum limit we would have any of the measures usually discussed in the continuum quantum gravity, namely, Misner, DeWitt or Leutwyler measure. It is the latter two cases when the discrete measure turns out to be well-defined at small lengths and lead to finite expectation values of the lengths.