Non-linear $\sigma$-model for long range disorder and quantum chaos

We suggest a new scheme of derivation of a non-linear ballistic $\sigma$-model for a long range disorder and quantum billiards. The derivation is based on writing equations for quasiclassical Green functions for a fixed long range potential and exact representation of their solutions in terms of functional integrals over supermatrices $Q$ with the constraint $Q^2=1$. Averaging over the long range disorder or energy we are able to write a ballistic $\sigma$-model for all distances exceeding the electron wavelength. Neither singling out slow modes nor a saddle-point approximation are used in the derivation. Carrying out a course graining procedure that allows us to get rid off scales in the Lapunov region we come to a reduced $\sigma$-model containing a conventional collision term. For quantum billiards, we demonstrate that, at not very low frequencies, one can reduce the $\sigma$-model to a one-dimensional $\sigma$-model on periodic orbits. Solving the latter model, first approximately and then exactly, we resolve the problem of repetitions.