  The action for a relativistic free particle of mass m receives a contribution
$-m R(x,y)$ from a path of length R(x,y) connecting the events $x^i$ and $y^i$.
Using this action in a path integral, one can obtain the Feynman propagator for
a spinless particle of mass m in any background spacetime. If one of the
effects of quantizing gravity is to introduce a minimum length scale $L_P$ in
the spacetime, then one would expect the segments of paths with lengths less
than $L_P$ to be suppressed in the path integral. Assuming that the path
integral amplitude is invariant under the `duality' transformation ${\cal R}\to
L_P^2/R$, one can calculate the modified Feynman propagator in an arbitrary
background spacetime. It turns out that the key feature of this modification is
the following: The proper distance $(\Delta x)^2$ between two events, which are
infinitesimally separated, is replaced by $\Delta x^2 + L_P^2$; that is the
spacetime behaves as though it has a `zero-point length' of $L_P$. This
equivalence suggests a deep relationship between introducing a
`zero-point-length' to the spacetime and postulating invariance of path
integral amplitudes under duality transformations. In the Schwinger's proper
time description of the propagator, the weightage for a path with proper time s
becomes $m(s+L_P^2/s)$ rather than as ms. As to be expected, the ultraviolet
behavior of the theory is improved significantly and divergences will disappear
if this modification is taken into account. Implications of this result are
discussed.
