Modification of quantum measure in area tensor Regge calculus and positivity

TL;DR

This paper compares different quantum measures in area tensor Regge calculus, analyzing their properties and positivity conditions, especially in relation to the connection representation and the choice of rotation groups, to better understand quantum gravity discretizations.

Contribution

It introduces and compares two connection-based quantum measures in area tensor Regge calculus, highlighting their positivity properties and conditions for correspondence with canonical quantization.

Findings

SU(2) based measure violates positivity outside physical surface

SO(3) based measure remains positive if area tensors are bounded by Planck scale

Both measures reduce to path integral form in the continuous limit

Abstract

A comparative analysis of the versions of quantum measure in the area tensor Regge calculus is performed on the simplest configurations of the system. The quantum measure is constructed in such the way that it reduces to the Feynman path integral describing canonical quantisation if the continuous limit along any of the coordinates is taken. As we have found earlier, it is possible to implement also the correspondence principle (proportionality of the Lorentzian (Euclidean) measure to $e^{iS}$ ($e^{-S}$), $S$ being the action). For that a certain kind of the connection representation of the Regge action should be used, namely, as a sum of independent contributions of selfdual and antiselfdual sectors (that is, effectively 3-dimensional ones). There are two such representations, the (anti)selfdual connections being SU(2) or SO(3) rotation matrices according to the two ways of decomposing full SO(4) group, as SU(2) $\times$ SU(2) or SO(3) $\times$ SO(3). The measure from SU(2) rotations although positive on physical surface violates positivity outside this surface in the general configuration space of arbitrary independent area tensors. The measure based on SO(3) rotations is expected to be positive in this general configuration space on condition that the scale of area tensors considered as parameters is bounded from above by the value of the order of Plank unit.