Polymeric quantum mechanics and the zeros of the Riemann zeta function

TL;DR

This paper explores polymeric quantum models related to the Berry-Keating and Sierra-Rodríguez-Laguna Hamiltonians, demonstrating how polymer quantization reproduces the Riemann-von Mangoldt formula's smooth part and models Riemann zero fluctuations.

Contribution

It introduces a polymer quantization approach to these models, deriving Hamiltonians, wave functions, and spectra that connect to the distribution of Riemann zeros.

Findings

Polymeric quantization yields Hamiltonians with energy spectra dependent on a scale parameter.

The model reproduces the smooth part of the Riemann-von Mangoldt formula.

A correction term depending on energy and scale parameter models Riemann zero fluctuations.

Abstract

We analize the Berry-Keating model and the Sierra and Rodr\'iguez-Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding stationary wave functions. The self-adjointness condition provide a proper domain for the Hamiltonian operator and the energy spectrum, which turned out to be dependent on an introduced scale parameter. By performing a counting of semiclassical states, we prove that the polymer representation reproduces the smooth part of the Riemann-von Mangoldt formula, and introduces a correction depending on the energy and the scale parameter, which resembles the fluctuation behavior of the Riemann zeros.