Quantization of Non-standard Hamiltonians and the Riemann Zeros

Abstract

Relations between number theory, quantum mechanics and statistical mechanics are of interest to mathematicians and physicists since it was suggested that the zeros of the Riemann zeta function might be related to the spectrum of a self-adjoint quantum mechanical operator related to a one-dimensional Hamiltonian $$ H = xp $$ known as Berry–Keating–Connes Hamiltonian. However, this type of Hamiltonian is integrable and the classical trajectories of particles are not closed leading to a continuum spectrum. Recently, Sierra and Rodriguez-Laguna conjectured that the Hamiltonian $$ H = x(p \,+\, {\xi \mathord{\left/ {\vphantom {\xi p}} \right. \kern-0pt} p}) $$ where $$ \xi $$ is a coupling constant with dimensions of momentum square is characterized by a quantum spectrum which coincides with the average Riemann zeros and contains closed periodic orbits. In this paper, we show first that the Sierra–Rodriguez-Laguna Hamiltonian may be obtained by means of non-standard singular Lagrangians and besides the Hamiltonians $$ H = x(p + {\xi \mathord{\left/ {\vphantom {\xi p}} \right. \kern-0pt} p}) $$ and $$ H(x,p) = px $$ are not the only semiclassical Hamiltonians connected to the average Riemann zeros. We show the presence of a new Hamiltonian where its quantization revealed a number of interesting properties, in particular, the sign of a trace of the Riemann zeros.