The Born approximation for bound states

TL;DR

This paper explores a novel approach to bound state calculations in QED and QCD by using the Born approximation, based on classical potentials, as a fundamental starting point for perturbation theory.

Contribution

It introduces a Born approximation framework for bound states, linking classical gauge fields to quantum bound state descriptions and proposing a perturbative expansion in ar for these systems.

Findings

Born states correspond to classical potential solutions in QED.

Perturbation theory around bound states involves interacting in/out states.

Classical boundary conditions can generate QCD scale mbda_{QCD}.

Abstract

Bound states are stationary in time and interact continuously. Even a first approximation of atomic wave functions in QED requires contributions of all orders in \alpha. Bound state perturbation theory depends on the choice of this first approximation, just as the Taylor expansion of an ordinary function depends on the expansion point. Considering the expansion to be not in $\alpha$ but in $\hbar$, i.e., in the number of loops, defines the perturbative expansion uniquely also for bound states. I show how the Schr\"odinger equation for Positronium with the classical potential $V(r)=-\alpha/r$ corresponds to the Born, $O(\hbar^0)$ bound state approximation in QED. Standard perturbation theory is based on an expansion around $O(\alpha^0)$ free states that have no overlap with bound states. Perturbing around bound states requires using interacting $in$ and $out$ states. For Born states the binding potential arises from a classical gauge field. In the absence of loops the QCD scale $\Lambda_{QCD}$ can originate from a boundary condition imposed on the solution of the classical gluon field equations. A perturbative expansion may be relevant even for hadrons, if their non-perturbative features such as confinement and chiral symmetry breaking are present already in the Born term.