General covariant xp models and the Riemann zeros

TL;DR

This paper explores a class of covariant Hamiltonian models linked to the Riemann zeros, revealing geometric structures that relate their spectra to the zeros and suggesting a spacetime encoding of prime numbers.

Contribution

It introduces a covariant framework for Hamiltonian models connected to the Riemann zeros, unifying classical and quantum descriptions through spacetime geometry.

Findings

Models are covariant under coordinate transformations.

Spectra relate to the geometry of associated spacetimes.

H_I and H_II models approximate Riemann zeros in their spectra.

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 H_I =x (p+1/p) and H_II=(x+ 1/x)(p+ 1/p), which have been recently discussed in connection with the non trivial 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 H_I corresponds to a flat spacetime, whereas its spectrum approaches the Riemann zeros in average. The latter property also holds for the model H_II, 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.