The Generalised Born Oscillator and the Berry-Keating Hamiltonian

TL;DR

This paper introduces generalized Born quantum oscillators as deformations of reduced Nambu-Goto systems, explores their relation to $ ext{T}ar{ ext{T}}$ perturbations, and links them to the Riemann-Siegel $ heta$ function, suggesting a novel regularization of the Berry-Keating Hamiltonian.

Contribution

It presents a new family of quantum models deforming Nambu-Goto theory, analyzes their quantization, and uncovers connections to the Riemann-Siegel $ heta$ function and Berry-Keating theory.

Findings

Models resemble $ ext{T}ar{ ext{T}}$-perturbed systems.

Quantization performed up to high orders in $ ext{hbar}$.

Connection established between models and Riemann-Siegel $ heta$ function.

Abstract

In this study, we introduce and investigate a family of quantum mechanical models in 0+1 dimensions, known as generalized Born quantum oscillators. These models represent a one-parameter deformation of a specific system obtained by reducing the Nambu-Goto theory to 0+1 dimensions. Despite these systems showing significant similarities with $\mathrm{T}\overline{\mathrm{T}}$-type perturbations of two-dimensional relativistic models, our analysis reveals their potential as interesting regularizations of the Berry-Keating theory. We quantize these models using the Weyl quantization scheme up to very high orders in $\hbar$. By examining a specific scaling limit, we observe an intriguing connection between the generalized Born quantum oscillators and the Riemann-Siegel $\theta$ function.