The Born Oscillator

TL;DR

This paper explores a quantum oscillator derived from Born's nonlinear electrodynamics, proposing it as a regularization method for a Hamiltonian linked to the Riemann zeta function's zeros.

Contribution

It introduces a quantized version of Born's oscillator, connecting nonlinear electrodynamics with number theory and providing a potential regularization for a Hamiltonian related to the Riemann zeros.

Findings

Quantization of Born's oscillator offers a new approach to regularize Hamiltonians.

The oscillator's properties relate to the nonlinear theory of electrodynamics.

Potential implications for understanding the zeros of the Riemann zeta function.

Abstract

The paper studies the properties of an oscillator whose Hamiltonian is $[(1+q^2)(1+p^2)]^{1/2}-1$. It can be deduced from the nonlinear theory of electrodynamics originally proposed by Max Born in 1934. The quantization of such oscillator represents a possible regularization of the Barry and Keating's Hamiltonian, which has been proposed in the framework of the theory of non-trivial zeros of the Riemann's $\zeta$ function.