Semiclassical Foundation of Universality in Quantum Chaos

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.

4 pages, 2 figures; final version published in PRL, minor changes

Cited by in corpus (10)

• Periodic-Orbit Theory of Universality in Quantum Chaos
• Semiclassical Theory of Chaotic Conductors
• Directed Chaotic Transport in Hamiltonian Ratchets
• Quantum-to-classical correspondence in open chaotic systems
• Andreev billiards
• Microscopic Theory for the Quantum to Classical Crossover in Chaotic Transport
• Semiclassical form factor for spectral and matrix element fluctuations of multi-dimensional chaotic systems
• Classical limit of transport in quantum kicked maps
• Quantum decay of an open chaotic system: a semiclassical approach
• Field Theory Approach to Quantum Interference in Chaotic Systems