A version of the connection representation of Regge action

TL;DR

This paper introduces a novel connection-based representation of the Regge action for piecewise flat manifolds, enabling simplified path integral calculations and demonstrating exponential suppression at large scales, which supports the consistency of simplicial spacetime models.

Contribution

It develops a connection and area tensor-based representation of the Regge action for 4-simplices, facilitating path integral analysis in quantum gravity.

Findings

Path integrals over connections factorize over 4-simplices.

Exponential suppression of amplitudes at large areas or lengths.

Representation supports consistent simplicial spacetime quantization.

Abstract

We define for any 4-tetrahedron (4-simplex) the simplest finite closed piecewise flat manifold consisting of this 4-tetrahedron and of the one else 4-tetrahedron identical up to reflection to the present one (call it bisimplex built on the given 4-simplex, or two-sided 4-simplex). We consider arbitrary piecewise flat manifold. Gravity action for it can be expressed in terms of sum of the actions for the bisimplices built on the 4-simplices constituting this manifold. We use representation of each bisimplex action in terms of rotation matrices (connections) and area tensors. This gives some representation of any piecewise flat gravity action in terms of connections. The action is a sum of terms each depending on the connection variables referring to a single 4-tetrahedron. Application of this representation to the path integral formalism is considered. Integrations over connections in the path integral reduce to independent integrations over finite sets of connections on separate 4-simplices. One of the consequences is exponential suppression of the result at large areas or lengths (compared to Plank scale). It is important for the consistency of the simplicial description of spacetime.