  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. This mass-dependent non-linearity is
significant when particle masses are comparable to Planck mass, and negligible
otherwise. Such a non-linearity is in principle detectable through experimental
tests of quantum mechanics for mesoscopic systems, and is a valuable empirical
probe of theories of quantum gravity. We also briefly remark on the possible
connection our approach could have with loop quantum gravity and string theory.
