Quantum Regge Calculus of Einstein-Cartan theory

TL;DR

This paper develops a quantum discretization of Einstein-Cartan theory using Regge calculus, assigning tetrad and spin-connection fields to simplices, and constructs a gauge-invariant quantum action with a bounded partition function.

Contribution

It introduces a novel quantum Regge calculus framework for Einstein-Cartan theory, incorporating torsion-free conditions and gauge invariance on a simplicial complex.

Findings

Constructed a diffeomorphism and gauge-invariant Einstein-Cartan action.

Defined a bounded partition function for quantization.

Derived holonomies of tetrad and spin-connection fields.

Abstract

We study the Quantum Regge Calculus of Einstein-Cartan theory to describe quantum dynamics of Euclidean space-time discretized as a 4-simplices complex. Tetrad field e_\mu(x) and spin-connection field \omega_\mu(x) are assigned to each 1-simplex. Applying the torsion-free Cartan structure equation to each 2-simplex, we discuss parallel transports and construct a diffeomorphism and {\it local} gauge-invariant Einstein-Cartan action. Invariant holonomies of tetrad and spin-connection fields along large loops are also given. Quantization is defined by a bounded partition function with the measure of SO(4)-group valued \omega_\mu(x) fields and Dirac-matrix valued e_\mu(x) fields over 4-simplices complex.