The chaotic analytic function

Abstract

In contrast to a stationary Gaussian random function of a real variable which is free to have any correlation function, the closest analogous analytic random function in the complex plane has no true freedom - it is (statistically) unique. Since it has arisen only recently, as an apparently universal feature in the physical context of quantum chaos, I refer to it here as `the chaotic analytic function'. I note that it is implied by the assumption that a quantum chaotic wavefunction has Gaussian randomness and has a constant value for the average of its Wigner function in phase space. Interpreted literally this shows that the chaotic analytic function is the Bargmann function of a pure `white noise' wavefunction. More physically, if `constant' is replaced by `smooth on the scale of a Planck area', these assumptions are the semiclassical ones made by Berry for chaotic eigenstates. The analysis shows that the chaotic analytic function is still obtained semiclassically.