Bulg. J. Phys. vol.33 no.3 (2006), pp. 217-229

Quantum Mechanics on a Noncommutative Geometry

T.P. Singh
Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
Abstract. 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.

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