Can the trace formula describe weak localization?

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Abstract
We attempt to systematically derive perturbative quantum corrections to the Berry diagonal approximation of the two-level correlation function (TLCF) for chaotic systems. To this end, we develop a ‘weak diagonal approximation’ based on a recent description of the first weak localization correction to conductance in terms of the Gutzwiller trace formula. This semiclassical method is tested by using it to derive the weak localization corrections to the TLCF for a semiclassically disordered system. Unfortunately, the method is unable to correctly reproduce the ‘Hikami boxes’ (the relatively small regions where classical paths are glued together by quantum processes). This results in the method failing to reproduce the well known weak localization expansion. It so happens that for the first-order correction it merely produces the wrong prefactor. However, for the second-order correction, it is unable to reproduce certain contributions, and leads to a result which is of a different form to the standard one.
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