  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.
