Spectral statistics in disordered metals: A trajectories approach

We show that the perturbative expansion of the two-level correlation function, R(ω), in disordered conductors can be understood semiclassically in terms of self-intersecting particle trajectories. This requires the extension of the standard diagonal approximation to include pairs of paths that are nonidentical but have almost identical action. The number of diagrams thus produced is much smaller than in a standard field-theoretical approach. We show that such a simplification occurs because R(ω) has a natural representation as the second derivative of free energy F(ω). We calculate R(ω) to three-loop order, and verify a one-parameter scaling hypothesis for it in two dimensions. We discuss the possibility of applying our ``weak diagonal approximation'' to generic chaotic systems.