Semiclassical Foundation of Universality in Quantum Chaos

Abstract

Fully chaotic dynamics enjoy ergodicity and thus visit everywhere in the accessible space with uniform likeli- hood, over long periods of time. Even long periodic orbits bring about such uniform coverage. Moreover, classical ergodicity provides quantum chaos with universal char- acteristics. Given chaos, quantum energy levels are correlated within local few-level clusters but become statistically in- dependent as their distance grows much larger than the mean level spacing �. The decay of correlations on the scaleis empirically found system independent, within universality classes distinguished by presence or absence of time-reversal (T ) invariance (1, 2). The corresponding universal long-time characteristics act on the Heisenberg scale TH = 2π¯�, withPlanck's constant. Universal spectral fluctuations were conjectured as a manifestation of quantum chaos two decades ago (3). Now, the semiclassical core of a proof can be given. Based on Gutzwiller's periodic-orbit theory (4), our progress comes with two surprises: one lies in its simplicity, the other in the appearance of interesting mathematics (non- trivial properties of permutations). Moreover, the often disputed intimate relation between periodic orbits and quantum field theory is confirmed for good. We thus expect the underlying ideas to radiate beyond spectral fluctuations, like to transport and localization. Technically speaking, we want to show that each com- pletely hyberbolic classical dynamics has a quantum en- ergy spectrum with the same fluctuations as a random- matrix caricature HRMT of its Hamiltonian, even though that caricature has nothing in common with the Hamilto- nian but symmetry (absence or presence of T invariance). The theory of random matrices (RMT) (1, 2, 5), devel- oped by Wigner and Dyson to account for fluctuations in nuclear spectra yields analytic results for correlators of the level density ρ(E), by averaging over suitable ensem- bles of random matrices. Simplest is the two-point corre- lator ρ(E)ρ(E ' ) −ρ(E) ρ(E ' ), where the overlines denote ensemble average. Its Fourier transform with respect to the energy difference E − E ' , called spectral form factor K(τ), is predicted by RMT for systems without time re- versal invariance (unitary class) and with that symmetry (orthogonal class) as