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

TL;DR

This work analyzes when measures on the space of generalized connections admit representations of flux-like observables in loop quantum gravity, focusing on the $U(1)$ gauge group. Using projective techniques, the authors characterize measures by families $\{f_\gamma\}$ and derive a necessary and sufficient admissibility condition ensuring symmetric representations of the flux operators, encoded in the divergences $\Delta_{S,f}$. In the $U(1)$ case they prove no-diff-invariant-measure exists beyond the Ashtekar-Lewandowski measure that supports flux representations; Euclidean-invariant, factorizing measures are drawn toward AL, and Varadarajan-type measures fail admissibility due to Bochner-Minlos-type arguments. The results suggest strong constraints on background-dependent representations and raise questions about extending to non-Abelian groups, with implications for semiclassical regimes and the consistency of flux observables in quantum gravity. Overall, AL measure emerges as uniquely compatible with a flux representation under diffeomorphism invariance, highlighting the delicate interplay between measure, geometry, and operator realization in LQG.

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.

Figures (5)

• Figure 1: Operation $\mathop{\mathrm{\bf sub}}\nolimits_{e,v_*}$ subdividing an edge $e$ of a graph
• Figure 2: Operation $\mathop{\mathrm{\bf add}}\nolimits_e$, adding an edge $e$ to a graph (note that $e$ does not necessarily have to begin and end in vertices of $\gamma$)
• Figure 3: An example for $e$ and $\phi(e)$
• Figure 4: The tube $T$
• Figure 5: Surface and graph considered in the proof of Proposition \ref{['pr2']}

Theorems & Definitions (16)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Proposition 3.1
• Proposition 3.2
• proof
• Proposition 4.1
• Lemma 4.2
• proof
• Theorem 4.3: Theorem 1.4 of huebsch