The Born approximation for bound states

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.