Modified Commutation Relationships from the Berry-Keating Program

This work is a study of the introduction of gravity to quantum theory. The presence of the Riemann zeta function in string theoretic arguments suggests a connection between the zeta function and quantum theory. Current approaches to quantum gravity suggest there should be a modification of the standard quantum mechanical commutator, [x,p] = i~. 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-Mu¨ller variant of the Berry-Keating approach to the Riemann hypothesis. Through our research we arrive at a family of modified position and momentum operators, and their associated modified commutator, which leads to a minimal length scale. Additionally, this larger family generalizes the Bender-Brody-Mu¨ller approach to the Riemann hypothesis.