Lie algebra of Ashtekar-Barbero connection operators

TL;DR

This paper explores the algebraic structure of Ashtekar-Barbero connection holonomies, linking geometric and algebraic properties, and verifies the consistency of the Hilbert space formulation in loop quantum gravity.

Contribution

It introduces a method to compare geometric and algebraic expansions of holonomies, confirming the reality conditions and gauge invariance in the quantum gravity framework.

Findings

Identification of next-to-the-leading-order terms in holonomy expansions

Verification of the Hilbert space formulation consistency

Confirmation that real connections ensure gauge invariance and Wigner's theorem

Abstract

Holonomies of the Ashtekar-Barbero connection can be considered as abstract elements of a Lie group exponentially mapped from their connections representation. This idea provides a possibility to compare the geometric and algebraic properties of these objects. The result allows to identify the next-to-the-leading-order terms in the geometric and algebraic expansion of a holonomy. This identification leads to the verification of the related Hilbert space formulation. If states are the representations of the holonomy's symmetry group, they preserve gauge transformations according to Wigner's theorem. Thus, the spin network in loop quantum gravity satisfies this theorem. Moreover, the considered identification of the different expansions ensures the reality of the Ashtekar connection. Only the holonomies of real connections lead to the formulation of states that satisfy Wigner's theorem.