Non-linear σ-model for long range disorder and quantum chaos

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We suggest a new scheme of derivation of a non-linear ballistic σ-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 Q²=1. Averaging over the long range disorder or energy we are able to write a ballistic σ-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 σ-model containing a conventional collision term. For quantum billiards, we demonstrate that, at not very low frequencies, one can reduce the σ-model to a one-dimensional σ-model on periodic orbits. Solving the latter model, first approximately and then exactly, we resolve the problem of repetitions.