Quantum Theory of Geometry III: Non-commutativity of Riemannian Structures

TL;DR

The paper analyzes why a background-independent quantum theory of Riemannian geometry—built from holonomies and surface-smeared triads—naturally yields non-commuting geometric operators. It shows that naive Poisson brackets fail the Jacobi identity and that a consistent quantization arises from a Lie algebra built on Cyl and derivations $X_{S,f}$, leveraging the cotangent-bundle structure. The non-commutativity has a classical origin in the singular smearings and is not a quantum anomaly, yet it is negligible in the semiclassical regime. The results reinforce that the apparent lack of a triad (metric) representation is a direct consequence of the chosen gauge- and diffeomorphism-invariant variables, with physical implications tempered in semiclassical states.

Abstract

The basic framework for a systematic construction of a quantum theory of Riemannian geometry was introduced recently. The quantum versions of Riemannian structures --such as triad and area operators-- exhibit a non-commutativity. At first sight, this feature is surprising because it implies that the framework does not admit a triad representation. To better understand this property and to reconcile it with intuition, we analyze its origin in detail. In particular, a careful study of the underlying phase space is made and the feature is traced back to the classical theory; there is no anomaly associated with quantization. We also indicate why the uncertainties associated with this non-commutativity become negligible in the semi-classical regime.