  If there exists a formulation of quantum mechanics which does not refer to a
background classical spacetime manifold, it then follows as a consequence,
(upon making one plausible assumption), that a quantum description of gravity
should be necessarily non-linear. This is true independent of the mathematical
structure used for describing such a formulation of quantum mechanics. A
specific model which exhibits this non-linearity is constructed, using the
language of noncommutative geometry. We derive a non-linear Schrodinger
equation for the quantum dynamics of a particle; this equation reduces to the
standard linear Schrodinger equation when the mass of the particle is much
smaller than Planck mass. It turns out that the non-linear equation found by us
is very similar to a non-linear Schrodinger equation found by Doebner and
Goldin in 1992 from considerations of unitary representaions of the
infinite-dimensional group of diffeomorphisms in three spatial dimensions. Our
analysis suggests that the diffusion constant introduced by Doebner and Goldin
depends on the mass of the particle, and that this constant tends to zero in
the limit in which the particle mass is much smaller than Planck mass, so that
in this limit the non-linear theory reduces to standard linear quantum
mechanics. A similar effective non-linear Schrodinger equation was also found
for the quantum dynamics of a system of D0-branes, by Mavromatos and Szabo.
