General covariant xp models and the Riemann zeros

Abstract

We study a general class of models whose classical Hamiltonians are given by H = U(x)p + V(x)/p, where x and p are the position and momentum of a particle moving in one dimension, and U and V are positive functions. This class includes the Hamiltonians HI = x(p + 1/p) and HII = (x + 1/x)(p + 1/p), which have been recently discussed in connection with the nontrivial zeros of the Riemann zeta function. We show that all these models are covariant under general coordinate transformations. This remarkable property becomes explicit in the Lagrangian formulation which describes a relativistic particle moving in a (1+1)-dimensional spacetime whose metric is constructed from the functions U and V. General covariance is maintained by quantization and we find that the spectra are closely related to the geometry of the associated spacetimes. In particular, the Hamiltonian HI corresponds to a flat spacetime, whereas its spectrum approaches the Riemann zeros on average. The latter property also holds for the model HII, whose underlying spacetime is asymptotically flat. These results suggest the existence of a Hamiltonian whose underlying spacetime encodes the prime numbers, and whose spectrum provides the Riemann zeros.