Random matrix theory: From Riemann zeros to quantum chaos

Abstract

Random matrices have applications in many fields of physics, such as nuclear physics and quantum chaos, but they also have connections to number theory. In this thesis, we study the similarities between eigenvalues of random matrices, zeros of the Riemann zeta function and energy levels of quantum chaotic systems. We give an introduction to random matrix theory and study the eigenvalue statistics of the Gaussian ensembles. The n−level correlation function of the non-trivial zeros of the Riemann zeta functions are studied, which Montgomery conjectured to be the same as for the Gaussian unitary ensemble. The Riemann zeta function is part of the Dirichlet L-functions, and their zeros all exhibit the same statistics. Chaotic quantum systems also show the same statistics in their energy levels. Using quantum billiards as an example, we show that chaotic motion in the billiard leads to Gaussian level spacing. The boundary roughness and effect of time-reversal symmetry breaking are also discussed.