Perturbative Renormalization in Quantum Mechanics

TL;DR

This paper explores how regularization and renormalization techniques can address ultraviolet divergences in singular quantum mechanical potentials, providing finite, accurate results similar to those in quantum field theories.

Contribution

It demonstrates the application of perturbative renormalization in quantum mechanics using examples like the Dirac delta and Aharonov-Bohm potentials, highlighting their relevance to triviality and anyons.

Findings

Regularization and renormalization yield finite, accurate results for singular potentials.

Examples include Dirac delta potential and Aharonov-Bohm potential.

Methods parallel those used in quantum field theory.

Abstract

Some quantum mechanical potentials, singular at short distances, lead to ultraviolet divergences when used in perturbation theory. Exactly as in quantum field theories, but much simpler, regularization and renormalization lead to finite physical results, which compare correctly to the exact ones. The Dirac delta potential, because of its relevance to triviality, and the Aharonov-Bohm potential, because ot its relevance to anyons, are used as examples here.