A pseudo-unitary ensemble of random matrices, PT-symmetry and the Riemann Hypothesis

Abstract

An ensemble of 2 x 2 pseudo-Hermitian random matrices is constructed that possesses real eigenvalues with level-spacing distribution exactly as for the Gaussian Unitary Ensemble found by Wigner. By a re-interpretation of Connes' spectral interpretation of the zeros of the Riemann zeta function, we propose to enlarge the scope of search of the Hamiltonian connected with the celebrated Riemann Hypothesis by suggesting that the Hamiltonian could also be PT-symmetric (or pseudo-Hermitian).