CLASSICAL AND QUANTUM STRINGS NEAR SPACETIME SINGULARITIES - GRAVITATIONAL PLANE-WAVES WITH ARBITRARY POLARIZATION

We study the classical and quantum dynamics of strings near spacetime singularities. We deal with singular gravitational plane waves of arbitrary polarization and arbitrary profile functions W1(U) and W2(U), whose behaviour for U --> 0 is W1(U) = alpha1/Absolute value of U(beta1), W2(U) = alpha2/Absolute value of U(beta2) (U being a null variable). In these spacetimes, the string dynamics is exactly (and explicitly) solvable even at the spacetime singularities. The string behaviour depends crucially on whether both parameters beta1 and beta2 are smaller or larger than two. When beta1 greater-than-or-equal-to 2 and/or beta2 greater-than-or-equal-to 2, the string does not cross the singularity U = 0 but goes off to infinity in a given direction a depending on the polarization of the gravitational wave. The string time evolution is fully determined by the spacetime geometry, whereas the overall sigma-dependence is fixed by the initial string state. The proper length at fixed tau --> 0 (U --> 0) stretches infinitely. For beta1 < 2 and beta2 < 2, the string passes smoothly through the spacetime singularity and reaches the U > 0 region. In this case, outgoing operators make sense and we find the explicit transformation relating in and out operators. For the quantum string states, this implies spin polarization rotations and particle transmutations. The expectation values of the outgoing mass and mode number operators are computed and the excitation of the string modes after the crossing of the spacetime singularity is analysed.