A Closed Contour of Integration in Regge Calculus

TL;DR

This paper analyzes the analytic structure of the Regge action in simplicial minisuperspace, identifying branch points and constructing a convergent contour integral that yields the wave function of the universe, depending on topology and size.

Contribution

It introduces a closed contour of integration in complex edge length space for Regge calculus, enabling a well-defined wave function for various topologies and dimensions.

Findings

Identifies three finite branch points in the complex plane of edge lengths.

Constructs a closed contour yielding a convergent, oscillating wave function for large universes.

Determines the critical size for transition from exponential to oscillating wave functions across different topologies.

Abstract

The analytic structure of the Regge action on a cone in $d$ dimensions over a boundary of arbitrary topology is determined in simplicial minisuperspace. The minisuperspace is defined by the assignment of a single internal edge length to all 1-simplices emanating from the cone vertex, and a single boundary edge length to all 1-simplices lying on the boundary. The Regge action is analyzed in the space of complex edge lengths, and it is shown that there are three finite branch points in this complex plane. A closed contour of integration encircling the branch points is shown to yield a convergent real wave function. This closed contour can be deformed to a steepest descent contour for all sizes of the bounding universe. In general, the contour yields an oscillating wave function for universes of size greater than a critical value which depends on the topology of the bounding universe. For values less than the critical value the wave function exhibits exponential behaviour. It is shown that the critical value is positive for spherical topology in arbitrary dimensions. In three dimensions we compute the critical value for a boundary universe of arbitrary genus, while in four and five dimensions we study examples of product manifolds and connected sums.