Regularization of ultraviolet divergence for a particle interacting with a scalar quantum field

TL;DR

This paper introduces a new calculation scheme for a particle interacting with a scalar quantum field that avoids ultraviolet divergence and dependence on arbitrary cutoffs, ensuring finite and continuous observable quantities.

Contribution

It proposes a nonasymptotic state approach that eliminates ultraviolet divergence and the need for a momentum cutoff in perturbation theory.

Findings

Observable quantities remain finite and cutoff-independent.

Traditional perturbation series diverge logarithmically at small coupling.

The new method provides a continuous dependence on the coupling constant.

Abstract

When a nonrelativistic particle interacts with a scalar quantum field, the standard perturbation theory leads to a dependence of the energy of its ground state on an undefined parameter---"momentum cutoff"---due to the ultraviolet divergence. We show that the use of nonasymptotic states of the system results in a calculation scheme in which all observable quantities remain finite and continuously depend on the coupling constant without any additional parameters. It is furthermore demonstrated that the divergence of traditional perturbation series is caused by the energy being a function with a logarithmic singularity for small values of the coupling constant.