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

TL;DR

This paper explores the non-commutative nature of quantum Riemannian structures, clarifying its classical origin and showing that uncertainties diminish in the semi-classical limit, advancing the understanding of quantum geometry.

Contribution

It provides a detailed analysis of the non-commutativity in quantum Riemannian geometry and clarifies its classical roots and semi-classical behavior.

Findings

Non-commutativity arises from classical phase space structure.

No anomaly is introduced during quantization.

Uncertainties become negligible in the semi-classical limit.

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.