An Isaac Newton Institute Workshop

Random Matrix Theory and Arithmetic Aspects of Quantum Chaos

Semiclassical foundation of universality in quantum chaos

29th June 2004

Authors: Sebastian Müller (Universität Duisburg-Essen), Stefan Heusler (Universität Duisburg-Essen), Petr Braun (Universität Duisburg-Essen; St Petersburg University), Fritz Haake (Universität Duisburg-Essen), Alexander Altland (Universität zu Köln)

Abstract

We sketch the semiclassical core of a proof of the so-called Bohigas-Giannoni-Schmit conjecture: A dynamical system with full classical chaos has a quantum energy spectrum with universal fluctuations on the scale of the mean level spacing. We show how in the semiclassical limit all system specific properties fade away, leaving only ergodicity, hyperbolicity, and combinatorics as agents determining the contributions of pairs of classical periodic orbits to the quantum spectral form factor. The small-time form factor is thus reproduced semiclassically. Bridges between classical orbits and (the non-linear sigma model of) quantum field theory are built by revealing the contributing orbit pairs as topologically equivalent to Feynman diagrams.

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