Quantum Mechanics without Spacetime -a Possible Case for Noncommutative Differential Geometry?

The rules of quantum mechanics require a time coordinate for their formulation. However, a notion of time is in general possible only when a classical spacetime geometry exists. Such a geometry is itself produced by classical matter sources. Thus it could be said that the currently known formulation of quantum mechanics pre-assumes the presence of classical matter fields. A more fundamental formulation of quantum mechanics should exist, which avoids having to use a notion of time. In this paper we discuss as to how such a fundamental formulation could be constructed for single particle, non-relativistic quantum mechanics. We argue that there is an underlying non-linear theory of quantum gravity, to which both standard quantum mechanics and classical general relativity are approximations. The timeless formulation of quantum mechanics follows from the underlying theory when the mass of the particle is much smaller than Planck mass. On the other hand, when the particle's mass is much larger than Planck mass, spacetime emerges and the underlying theory should reduce to classical mechanics and general relativity. We also suggest that noncommutative differential geometry is a possible candidate for describing this underlying theory. The rules of quantum mechanics require the concept of time, for their formulation. The time coordinate determines the choice of canonical position and momenta, the normalization of the wavefunction, and of course, the evolution of the quantum system. From the point of view of the special theory of relativity, time is a component of spacetime; furthermore the general theory of relativity endows the spacetime with a pseudo-Riemannean geometry. This geometry however, is determined by the distribution of classical matter-such matter is of course a limiting case of matter obeying the rules of quantum mechanics. Hence, via its dependence on time, quantum mechanics pre-assumes the existence of classical matter, whose very properties it should explain in the first place. A more fundamental formulation of 1