Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics. I. General Theory

TL;DR

This paper explores how dimensional regularization reveals an energy scale in scale-invariant quantum systems, demonstrating the process of dimensional transmutation with the two-dimensional delta-function potential.

Contribution

It introduces a general framework for dimensional transmutation in nonrelativistic quantum mechanics using dimensional regularization, including a renormalization strategy for strong-coupling regimes.

Findings

Identification of scale-invariant potentials and their properties

Demonstration of energy scale emergence in bound and scattering states

Explicit illustration of dimensional transmutation in 2D delta-function potential

Abstract

This is the first in a series of papers addressing the phenomenon of dimensional transmutation in nonrelativistic quantum mechanics within the framework of dimensional regularization. Scale-invariant potentials are identified and their general properties are derived. A strategy for dimensional renormalization of these systems in the strong-coupling regime is presented, and the emergence of an energy scale is shown, both for the bound-state and scattering sectors. Finally, dimensional transmutation is explicitly illustrated for the two-dimensional delta-function potential.