Universal spectral correlations from the ballistic $\sigma$ model

We consider the semiclassical ballistic $\sigma$ model as an effective theory describing the quantum mechanics of classically chaotic systems. Specifically, we elaborate on close analogies to the recently developed semiclassical theory of quantum interference in chaotic systems and show how semiclassical “diagrams” involving near action degenerate sets of periodic orbits emerge in the field theoretical description. We further discuss how the universality phenomenon (i.e., the fact that individual chaotic systems behave according to the prescriptions of random matrix theory) can be understood from the perspective of the field theory.

Figure 1

(Color online) Cartoon of an orbit pair contributing to the first quantum correction to the spectral form factor.

Figure 2

(Color online) Phase space representation of the encounter region. Here the bar in ${\overline{x}}^{\prime },{\overline{y}}^{\prime }$ stands for time reversal, as discussed in the text.

Figure 3

(Color online) Assignment of propagator coordinates (lines terminating in thick points) to Feynman amplitudes (lines continued through the Poincaré section).