The Classical Lamb Shift: Why Jackson Is Wrong!

Abstract

At the Edirne summer school, a discussion developed on how it seemed to be possible to use classical methods to calculate atomic spontaneous emission rates, but not the Lamb shift. This point of view has particularly been emphasized in Jackson’s book, Classical Electrodynamics, in which he flat-out states that a classical argument based on radiation-reaction theory and the Abraham-Lorentz equation will give the correct Einstein A coefficient for spontaneous decay, but gives the wrong value for the Lamb shift¹. This bit of folklore is often touted as some sort of proof that classical methods can not reproduce elements of quantum electrodynamics, even in a limiting sense. Usually, as Jackson does, the blame is placed on electromagnetic vacuum fluctuations — a manifestly nonclassical phenomenon — that is claimed to be the “true” origin of the Lamb shift. But people claim that vacuum fluctuations are also the cause of atomic spontaneous emission — how come classical radiation-reaction theory is right there? “Just good luck,” is the usual reply. This seems an unsatisfactory state of affairs, for several reasons. To me, the correspondence principle presents the biggest objection to the notion that there is no classical analog to the Lamb shift. In the limit of large quantum numbers, one would expect the quantum electrodynamical Lamb shift to go over into a classical frequency pulling that an oscillating macroscopic charge experiences by interacting with its self field. An additional disturbing fact is that classical theory is quite often used to compute cavity corrections to the Lamb shift², and at least for excited atoms the results agree exactly with full QED calculations³. Why then can the classical approach get the cavity-induced Lamb shift right, but not the free-space shift? As we shall see, there is nothing wrong with the classical approach to the free-space Lamb shift; it is Jackson who is wrong!