On the Faddeev-Popov determinant in Regge calculus

TL;DR

This paper investigates the Faddeev-Popov determinant in 4D Regge calculus, highlighting issues with the measure due to singularities and proposing discretisation of the gauge field as a potential solution.

Contribution

It identifies the ill-defined nature of the Faddeev-Popov factor in Regge calculus and suggests a discretisation approach to resolve the singularities.

Findings

Faddeev-Popov factor is generally ill-defined due to conical singularities.

Discretising the gauge field can potentially resolve measure singularities.

Singularities occur near the flat background where physical and gauge degrees of freedom mix.

Abstract

The functional integral measure in the 4D Regge calculus normalised w.r.t. the DeWitt supermetric on the space of metrics is considered. The Faddeev-Popov factor in the measure is shown according to the previous author's work on the continuous fields in Regge calculus to be generally ill-defined due to the conical singularities. Possible resolution of this problem is discretisation of the gravity ghost (gauge) field by, e.g., confining ourselves to the affine transformations of the affine frames in the simplices. This results in the singularity of the functional measure in the vicinity of the flat background, where part of the physical degrees of freedom connected with linklengths become gauge ones.