Letter: Quantum Mechanics Without Spacetime II: Noncommutative Geometry and the Free Point Particle
Abstract
In a recent paper we have suggested that a formulation of quantum mechanics should exist, which does not require the concept of time, and that the appropriate mathematical language for such a formulation is noncommutative differential geometry. In the present paper we discuss this formulation for the free point particle, by introducing a commutation relation for a set of noncommuting coordinates. The sought for background independent quantum mechanics is derived from this commutation relation for the coordinates. We propose that the basic equations are invariant under automorphisms which map one set of coordinates to anotherthis is a natural generalization of diffeomorphism invariance when one makes a transition to noncommutative geometry. The background independent description becomes equivalent to standard quantum mechanics if a spacetime manifold exists, because of the proposed automorphism invariance. The suggested basic equations also give a quantum gravitational description of the free particle.
 Publication:

General Relativity and Gravitation
 Pub Date:
 May 2003
 DOI:
 10.1023/A:1022999205139
 arXiv:
 arXiv:grqc/0205056
 Bibcode:
 2003GReGr..35..869S
 Keywords:

 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Quantum Physics
 EPrint:
 8 pages