Ashtekar-Barbero holonomy on the hyperboloid: Immirzi parameter as a Cut-off for Quantum Gravity

TL;DR

This paper investigates the Ashtekar-Barbero holonomies on hyperboloids in Loop Quantum Gravity, revealing the Immirzi parameter's role as a quantum gravity cut-off and highlighting the connection's dependence on spacetime embedding.

Contribution

It demonstrates the non-space-time nature of the Ashtekar-Barbero connection and uncovers the Immirzi parameter's periodicity as a quantum cut-off in gravity.

Findings

Holonomies depend on spacetime embedding.

Immirzi parameter acts as a quantum cut-off.

Limitations in reconstructing extrinsic curvature.

Abstract

In the context of the geometrical interpretation of the spin network states of Loop Quantum Gravity, we look at the holonomies of the Ashtekar-Barbero connection on loops embedded in space-like hyperboloids. We use this simple setting to illustrate two points. First, the Ashtekar-Barbero connection is not a space-time connection, its holonomies depend on the spacetime embedding of the canonical hypersurface. This fact is usually interpreted as an inconvenience, but we use it to extract the extrinsic curvature from the holonomy and separate it from the 3d intrinsic curvature. Second, we show the limitations of this reconstruction procedure, due to a periodicity of the holonomy in the Immirzi parameter, which underlines the role of a real Immirzi parameter as a cut-off for general relativity at the quantum level in contrast with its role of a mere coupling constant at the classical level.