Misner space, also known as the Lorentzian orbifold , is the simplest tree-level solution of string theory with a cosmological singularity. Wecompute tree-level scattering amplitudes involving twisted states, using operatorand current algebra techniques. We find that, due to zero-point quantumfluctuations of the excited modes, twisted strings with a large winding numberw are fuzzy on a scale , which can be much larger than the string scale. Wavefunctions are smeared by anoperator reminiscent of the Moyal product of non-commutative geometry, which, sinceΔ(ν) is real, modulates the amplitude rather than the phase of the wavefunction, and is purelygravitational in its origin. We compute the scattering amplitude of two twisted states and onetachyon or graviton, and find a finite result. The scattering amplitude of two twisted andtwo untwisted states is found to diverge, due to the propagation of intermediate windingstrings with vanishing boost momentum. The scattering amplitude of three twisted fields iscomputed by analytic continuation from three-point amplitudes of states with non-zerop+ in the Nappi–Witten plane wave, and the non-locality of the three-point vertexis found to diverge for certain kinematical configurations. Our results for thethree-point amplitudes allow us in principle to compute, to leading order, theback-reaction on the metric due to a condensation of coherent winding strings.