From Local Regge Calculus towards Spin Foam Formalism?

TL;DR

This paper develops an SO(n)-local Regge Calculus framework using a first order formalism on Voronoi complexes, exploring its quantum measure expansion and potential connections to Spin Foam Formalism and Loop Quantum Gravity.

Contribution

It introduces a novel SO(n)-local Regge Calculus formalism with a first order approach on Voronoi complexes, linking it to Spin Foam models and matter coupling.

Findings

Quantum measure expansion in four dimensions resembles Spin Foam Formalism.

Coupling with fermionic matter is straightforward, impacting Loop Quantum Gravity.

The formalism offers new insights into quantum geometry and gravity models.

Abstract

We introduce the basic elements of SO(n)-local theory of Regge Calculus. A first order formalism, in the sense of Palatini, is defined on the metric-dual Voronoi complex of a simplicial complex. The Quantum Measure exhibits an expansion, in four dimensions, in characters of irreducible representation of SO(4) which has close resemblance and differences as well with the Spin Foam Formalism. The coupling with fermionic matter is easily introduced which could have consequences for the Spin Foam Formalism and Loop Quantum Gravity.