Correlations in the actions of periodic orbits derived from quantum chaos

We discuss two-point correlations of the actions of classical periodic orbits in chaotic systems. For systems where the semiclassical trace formula is exact and the spectral statistics follow random matrix theory, there exist nontrivial correlations between actions, which we express in a universal form. We illustrate this result with the analogous problem of the pair correlations between prime numbers. We also report on numerical studies of three chaotic systems where the semiclassical trace formula is only approximate, but nevertheless these unexpected action correlations are observed.