Path integral in the simplest Regge calculus model

TL;DR

This paper analyzes a basic (3+1)D Regge calculus model with discrete space and continuous time, deriving its path integral measure and exploring the effects of selfdual splitting on the integral's complexity.

Contribution

It derives the path integral measure for the simplest Regge calculus model and investigates the simplifications from selfdual-antiselfdual splitting of variables.

Findings

Path integral measure for the model is explicitly obtained.

Selfdual-antiselfdual splitting simplifies the integral.

Complete separation of (anti-)selfdual sectors is not achieved.

Abstract

The simplest (3+1)D Regge calculus model (with three-dimensional discrete space and continuous time) is considered which describes evolution of the simplest closed two-tetrahedron piecewise flat manifold in the continuous time. The measure in the path integral which describes canonical quantisation of the model in terms of area bivectors and connections as independent variables is found. It is shown that selfdual-antiselfdual splitting of the variables simplifies the integral although does not admit complete separation of (anti-)selfdual sector.