Regge calculus from discontinuous metrics

TL;DR

This paper explores a generalized form of Regge calculus using discontinuous metrics, proposing a quantum measure on this superspace and deriving a nearly unique phase factor for the quantum Regge measure that respects key physical properties.

Contribution

It introduces a framework for quantum Regge calculus based on discontinuous metrics and derives a specific phase factor for the quantum measure that maintains physical consistency.

Findings

Quantum measure on discontinuous metrics is constructed.

A nearly unique phase factor for the quantum Regge measure is derived.

The approach ensures positivity, continuum limit, and minimal lattice artifacts.

Abstract

Regge calculus is considered as a particular case of the more general system where the linklengths of any two neighbouring 4-tetrahedra do not necessarily coincide on their common face. This system is treated as that one described by metric discontinuous on the faces. In the superspace of all discontinuous metrics the Regge calculus metrics form some hypersurface. Quantum theory of the discontinuous metric system is assumed to be fixed somehow in the form of quantum measure on (the space of functionals on) the superspace. The problem of reducing this measure to the Regge hypersurface is addressed. The quantum Regge calculus measure is defined from a discontinuous metric measure by inserting the $\delta$-function-like phase factor. The requirement that this reduction would respect natural physical properties (positivity, well-defined continuum limit, absence of lattice artefacts) put rather severe restrictions and allows to define practically uniquely this phase factor.