Path integral in area tensor Regge calculus and complex connections

TL;DR

This paper explores a path integral approach in area tensor Regge calculus with complex connections, proposing a well-defined Euclidean quantum measure that is positive and exponentially suppressed at large areas under certain conditions.

Contribution

It introduces a method of integrating over complex connection variables in Regge calculus, leading to a well-defined Euclidean quantum measure on physical configurations.

Findings

The measure can be positive and exponentially suppressed at large areas.

Integration contours in complex connection space can be chosen to define a consistent quantum measure.

The approach excludes degenerate metrics to ensure measure positivity.

Abstract

Euclidean quantum measure in Regge calculus with independent area tensors is considered using example of the Regge manifold of a simple structure. We go over to integrations along certain contours in the hyperplane of complex connection variables. Discrete connection and curvature on classical solutions of the equations of motion are not, strictly speaking, genuine connection and curvature, but more general quantities and, therefore, these do not appear as arguments of a function to be averaged, but are the integration (dummy) variables. We argue that upon integrating out the latter the resulting measure can be well-defined on physical hypersurface (for the area tensors corresponding to certain edge vectors, i.e. to certain metric) as positive and having exponential cutoff at large areas on condition that we confine ourselves to configurations which do not pass through degenerate metrics.