Quantizing G-connections via the tangent groupoid

TL;DR

This paper introduces a novel approach to describing G-connections using the tangent groupoid, linking geometric structures with operator theory, and exploring their role in quantum gravity formulations.

Contribution

It presents a new framework for G-connections via the tangent groupoid, incorporating gauge actions and tetrads as operators on a Hilbert space, advancing quantum gravity models.

Findings

G-connections act as convolution operators away from zero

Gauge actions are characterized within the tangent groupoid framework

Tetrads are formulated as Dirac type operators

Abstract

A description of the space of G-connections using the tangent groupoid is given. As the tangent groupoid parameter is away from zero, the G-connections act as convolution operators on a Hilbert space. The gauge action is examined in the tangent groupoid description of the G-connections. Tetrads are formulated as Dirac type operators. The connection variables and tetrad variables in Ashtekar's gravity are presented as operators on a Hilbert space.