Renormalized Path Integral for the Two-Dimensional Delta-Function Interaction

TL;DR

This paper develops a renormalized path integral method for two-dimensional delta-function potentials, addressing their singularities and illustrating quantum anomalies in a scale-invariant quantum mechanics problem.

Contribution

It introduces a nonperturbative, renormalized path integral framework for two-dimensional delta-function interactions, unifying bound-state and scattering analyses.

Findings

Successful regularization of the 2D delta-function potential

Demonstration of quantum anomaly in scale-invariant case

Unified treatment of bound states and scattering

Abstract

A path-integral approach for delta-function potentials is presented. Particular attention is paid to the two-dimensional case, which illustrates the realization of a quantum anomaly for a scale invariant problem in quantum mechanics. Our treatment is based on an infinite summation of perturbation theory that captures the nonperturbative nature of the delta-function bound state. The well-known singular character of the two-dimensional delta-function potential is dealt with by considering the renormalized path integral resulting from a variety of schemes: dimensional, momentum-cutoff, and real-space regularization. Moreover, compatibility of the bound-state and scattering sectors is shown.