Although it is generally accepted that the spectrum of clean chaotic billiards is well described by Wigner-Dyson random matrix theory, an analytical confirmation of this fact is not easy. The situation differs from the one for disordered billiards where the equivalence to the random matrix theory has been established long ago using a supermatrix $sigma $-model. Attempts to construct a “ballistic” $sigma $-model are reviewed and it is shown how one can construct the most general type of a $sigma $-model that has a form of a model of non-commutative quantum mechanics. Nevertheless, simplifying this general model to a semiclassical limit is not simple. It is suggested to use for explicit calculations a Balian-Bloch representation for Green functions in the billiard. Proceeding in this way a ballistic $sigma $-model with a “regularizer” mixing classical orbits is suggested. It is argued that in contrast to previous hypotheses the regularizer can exist only on the boundary of the billiard. The existence of the regularizer makes the zero-dimensional version of the $sigma $-model suitable for description of the low energy limit, which should lead to the Wigner-Dyson statistics.

Ruhr Univ. Bochum ; L.D. Landau Inst. for Theoretical Physics Moscow