Inducing Barbero-Immirzi Connections along SU(2)-reductions of Bundles on Spacetime

TL;DR

This paper introduces a novel mathematical technique for inducing connections along bundle reductions on spacetime, providing a more general and flexible approach to defining Barbero-Immirzi connections in loop quantum gravity.

Contribution

It presents a new method for inducing connections along bundle reductions that generalizes the standard restriction approach, allowing greater control over transformation laws and holonomy constraints.

Findings

The new method clarifies and generalizes the definition of BI connections.

Induced connections have no holonomy constraints once a bundle reduction is given.

The approach offers a mathematically rigorous way to handle spacetime connections in LQG.

Abstract

We shall present here a general apt technique to induce connections along bundle reductions which is different from the standard restriction. This clarifies and generalizes the standard procedure to define Barbero-Immirzi (BI) connection, though on spacetime. The standard spacial BI connection used in LQG is then obtained by its spacetime version by standard restriction. The general prescription to define such a reduced connection is interesting from a mathematical viewpoint and it allows a general and direct control on transformation laws of the induced object. Moreover, unlike what happens by using standard restriction, we shall show that once a bundle reduction is given, then any connection induces a reduced connection with no constraint on the original holonomy as it happens when connections are simply restricted.