One non-relativistic particle coupled to a photon field

TL;DR

This paper analyzes the ground state energy of an electron interacting with a photon field, showing how the coupling affects the self-energy and binding energy, with explicit calculations for small coupling constants.

Contribution

It provides a detailed calculation of the electron's self-energy and demonstrates increased binding energy due to photon coupling for small coupling constants.

Findings

Leading order self-energy term derived as a function of coupling and cutoff.

Binding energy increases when electron is in an external potential and coupled to the photon field.

Explicit numerical bounds for coupling constant, cutoff, and radiative correction.

Abstract

We investigate the ground state energy of an electron coupled to a photon field. First, we regard the self-energy of a free electron, which we describe by the Pauli-Fierz Hamiltonian. We show that, in the case of small values of the coupling constant $\alpha$, the leading order term is represented by $2\pi^{-1} \alpha (\Lambda - \ln[1 + \Lambda])$. Next we put the electron in the field of an arbitrary external potential $V$, such that the corresponding Schr\"odinger operator $p^2 + V$ has at least one eigenvalue, and show that by coupling to the radiation field the binding energy increases, at least for small enough values of the coupling constant $\alpha$. Moreover, we provide concrete numbers for $\alpha$, the ultraviolet cut-off $\Lambda$, and the radiative correction for which our procedure works.