On positivity of quantum measure and of effective action in area tensor Regge calculus

TL;DR

This paper explores the positivity of the measure and effective action in area tensor Regge calculus, proposing a method to ensure the measure's positivity by complex contour deformation, which may improve the definition of the Euclidean path integral in quantum gravity.

Contribution

It introduces a novel approach to defining a positive measure in area tensor Regge calculus by complex contour deformation, addressing issues of unboundedness in the Euclidean gravitational action.

Findings

Reformulation of the path integral measure with complex contours

Speculation on the measure's positivity across tensor variations

Potential implications for Euclidean quantum gravity

Abstract

Because of unboundedness of the general relativity action, Euclidean version of the path integral in general relativity requires definition. Area tensor Regge calculus is considered in the representation with independent area tensor and finite rotation matrices. Being integrated over rotation matrices the path integral measure in area tensor Regge calculus is rewritten by moving integration contours to complex plain so that it looks as that one with effective action in the exponential with positive real part. We speculate that positivity of the measure can be expected in the most part of range of variation of area tensors.