  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 another- this 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.
