Modified commutation relationships from the Berry-Keating program

TL;DR

This paper proposes a new family of modified quantum commutation relations inspired by the Berry-Keating approach to the Riemann hypothesis, leading to a minimal length scale and generalizing previous models.

Contribution

It introduces a novel family of modified position and momentum operators based on the Berry-Keating framework, extending the Bender-Brody-Müller approach.

Findings

Derivation of a family of modified commutators

Establishment of a minimal length scale

Generalization of previous Berry-Keating models

Abstract

Current approaches to quantum gravity suggest there should be a modification of the standard quantum mechanical commutator, $[{\hat x} , {\hat p}] = i \hbar$. Typical modifications are phenomenological and designed to result in a minimal length scale. As a motivating principle for the modification of the position and momentum commutator, we assume the validity of a version of the Bender-Brody-M\"uller variant of the Berry-Keating approach to the Riemann hypothesis. We arrive at a family of modified position and momentum operators, and their associated modified commutator, which lead to a minimal length scale. Additionally, this larger family generalizes the Bender-Brody-M\"uller approach to the Riemann hypothesis.