  Misner space, also known as the Lorentzian orbifold $R^{1,1}/boost$, is the
simplest tree-level solution of string theory with a cosmological singularity.
We compute tree-level scattering amplitudes involving twisted states, using
operator and current algebra techniques. We find that, due to zero-point
quantum fluctuations of the excited modes, twisted strings with a large winding
number $w$ are fuzzy on a scale $\sqrt{\log w}$, which can be much larger than
the string scale. Wave functions are smeared by an operator $\exp(\Delta(\nu)
\partial_+ \partial_-)$ reminiscent of the Moyal-product of non-commutative
geometry, which, since $\Delta(\nu)$ is real, modulates the amplitude rather
than the phase of the wave function, and is purely gravitational in its origin.
We compute the scattering amplitude of two twisted states and one tachyon or
graviton, and find a finite result. The scattering amplitude of two twisted and
two untwisted states is found to diverge, due to the propagation of
intermediate winding strings with vanishing boost momentum. The scattering
amplitude of three twisted fields is computed by analytic continuation from
three-point amplitudes of states with non-zero $p^+$ in the Nappi-Witten plane
wave, and the non-locality of the three-point vertex is found to diverge for
certain kinematical configurations. Our results for the three-point amplitudes
allow in principle to compute, to leading order, the back-reaction on the
metric due to a condensation of coherent winding strings.
