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

Abhay Ashtekar·Alejandro Corichi·Jose. A. Zapata
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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.

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Abhay Ashtekar, Alejandro Corichi, Jose. A. Zapata. 1998-08-12. Quantum Theory of Geometry III: Non-commutativity of Riemannian Structures. https://doi.org/10.1088/0264-9381%2F15%2F10%2F006

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