When Do Measures on the Space of Connections Support the Triad Operators of Loop Quantum Gravity?

TL;DR

This paper examines when measures on the space of generalized connections in loop quantum gravity support the representation of non-Abelian electric flux operators, providing mathematical conditions and no-go results for certain measures.

Contribution

It formulates the problem precisely and proves no-go theorems for the representation of flux operators in the U(1) case, advancing understanding of measure-induced Hilbert spaces in loop quantum gravity.

Findings

No-go theorems for certain measures preventing flux operator representation

Mathematical formulation of the conditions for flux support

Insights into non-Abelian gauge theory quantization

Abstract

In this work we investigate the question, under what conditions Hilbert spaces that are induced by measures on the space of generalized connections carry a representation of certain non-Abelian analogues of the electric flux. We give the problem a precise mathematical formulation and start its investigation. For the technically simple case of U(1) as gauge group, we establish a number of "no-go theorems" asserting that for certain classes of measures, the flux operators can not be represented on the corresponding Hilbert spaces. The flux-observables we consider play an important role in loop quantum gravity since they can be defined without recourse to a background geometry, and they might also be of interest in the general context of quantization of non-Abelian gauge theories.