QUANTUM-THEORY OF GEOMETRY - III - NON-COMMUTATIVITY OF RIEMANNIAN STRUCTURES

The basic framework for a systematic construction of a quantum theory of Riemannian geometry was introduced recently. The quantum versions o f 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 b etter understand this property and to reconcile it with intuition, we analyse its origin in detail. In particular, a careful study of the un derlying phase space is made and the feature is traced back to the cla ssical theory; there is no anomaly associated with quantization. We al so indicate why the uncertainties associated with this non-commutativi ty become negligible in the semiclassical regime.