Exact solution for a quantum field with $\delta$-like interaction

TL;DR

This paper derives exact solutions for a quantum field with a delta-like potential, providing explicit formulas for resolvent and heat kernel, and explores renormalization and non-perturbative effects across different dimensions.

Contribution

It offers the first exact expressions for the resolvent and heat kernel of a quantum field with delta-like interactions in various dimensions.

Findings

Exact heat kernel and Green's functions derived

Renormalization of coupling constant observed

Non-perturbative behavior of effective action identified

Abstract

A quantum field described by the field operator $\Delta_{a}=\Delta+ a\delta_\Sigma$ involving a $\delta$-like potential is considered. Mathematically, the treatment of the $\delta$-potential is based on the theory of self-adjoint extension of the unperturbed operator $\Delta$. We give the general expressions for the resolvent and the heat kernel of the perturbed operator $\Delta_{a}$. The main attention is payed to $d=2$ $\delta$-potential though $d=1$ and $d=3$ cases are considered in some detail. We calculate exactly the heat kernel, Green's functions and the effective action for the operator $\Delta_{a}$ in diverse dimensions and for various spaces $\Sigma$. The renormalization phenomenon for the coupling constant $a$ of $d=2$ and $d=3$ $\delta$-potentials is observed. We find the non-perturbative behavior of the effective action with respect to the renormalized coupling $a_{ren}$.