Quantum Chaology

This chapter, based on Dodson [66], is somewhat speculative in that it is clear that gamma distributions do not precisely model the analytic systems discussed here, but some features may be useful in studies of qualitative generic properties in applications to data from real systems which manifestly seem to exhibit behaviour reminiscent of near-random processes. Quantum counterparts of certain simple classical systems can exhibit chaotic behaviour through the statistics of their energy levels and the irregular spectra of chaotic systems are modelled by eigenvalues of infinite random matrices. We use known bounds on the distribution function for eigenvalue spacings for the Gaussian orthogonal ensemble (GOE) of infinite random real symmetric matrices and show that gamma distributions, which have the important uniqueness property Theorem 1.1, can yield an approximation to the GOE distribution. This has the advantage that then both chaotic and non chaotic cases fit in the information geometric framework of the manifold of gamma distributions. Additionally, gamma distributions give approximations, to eigenvalue spac-ings for the Gaussian unitary ensemble (GUE) of infinite random hermitian matrices and for the Gaussian symplectic ensemble (GSE) of infinite random hermitian matrices with real quaternionic elements. Interestingly, the spacing distribution between zeros of the Riemann zeta function is approximated by the GUE distribution, and we investigate the stationarity of the coefficient of variation of the numerical data with respect to location and sample size. The review by Deift [52] illustrates how random matrix theory has significant links to a wide range of mathematical problems in the theory of functions as well as to mathematical physics.