Exact Green's functions for delta-function potentials and renormalization in quantum mechanics

TL;DR

This paper provides a straightforward method to compute Green's functions for Hamiltonians with delta-function potentials, incorporating renormalization techniques to handle infinities in higher dimensions.

Contribution

It introduces a simple recipe for constructing Green's functions with delta potentials and naturally incorporates renormalization concepts in quantum mechanics.

Findings

Explicit Green's functions for delta potentials derived

Renormalization techniques applied to higher-dimensional cases

Connection between quantum mechanics and field theory concepts established

Abstract

We present a simple recipe to construct the Green's function associated with a Hamiltonian of the form H=H_0+V, where H_0 is a Hamiltonian for which the associated Green's function is known and V is a delta-function potential. We apply this result to the case in which H_0 is the Hamiltonian of a free particle in D dimensions. Field theoretic concepts such as regularization, renormalization, dimensional transmutation and triviality are introduced naturally in order to deal with an infinity which shows up in the formal expression of the Green's function for D>1.