Quantum Mechanics on a Noncommutative Geometry

Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an underlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. PACS number: 03.65.Ta 1 Why Quantum Mechanics without Classical Spacetime? Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. The new formulation should have the following two properties. Firstly, in the limit in which the quantum system under consideration becomes macroscopic, the ‘quantum spacetime’ should become the standard classical spacetime described by the laws of special and general relativity; and the quantum dynamics should reduce to standard classical dynamics on this spacetime. Secondly, consider the situation in which a dominant part of the quantum system becomes macroscopic and classical, and a sub-dominant part remains microscopic and quantum. By virtue of the first property, the dominant part should look like our classical Universe – classical matter existing in a classical spacetime. Seen from 1310–0157 c © 2006 Heron Press Ltd. 217