New Insights into the Lamb Shift: The Spectral Density of the Shift

In an atom, the interaction of a bound electron with the vacuum fluctuations of the electromagnetic field leads to complex shifts in the energy levels of the electron, with the real part of the shift corresponding to a shift in the energy level and the imaginary part to the width of the energy level. The most celebrated radiative shift is the Lamb shift between the 2s1/2 and the 2p1/2 levels of the hydrogen atom. The measurement of this shift in 1947 by Willis Lamb Jr. proved that the prediction by Dirac theory that the energy levels were degenerate was incorrect. Hans Bethe’s non-relativistic calculation of the shift using second-order perturbation theory demonstrated the renormalization process required to deal with the divergences plaguing the existing theories and led to the understanding that it was essential for theory to include interactions with the zero-point quantum vacuum field. This was the birth of modern quantum electrodynamics (QED). Numerous calculations of the Lamb shift followed including relativistic and covariant calculations, all of which contain a nonrelativistic contribution equal to that computed by Bethe. The semi-quantitative models for the radiative shift of Welton and Power, which were developed in an effort to demonstrate physical mechanisms by which vacuum fluctuations lead to the shift, are also considered here. This paper describes a calculation of the shift using a group theoretical approach which gives the shift as an integral over frequency of a function, which is called the “spectral density of the shift.“ The energy shift computed by group theory is equivalent to that derived by Bethe yet, unlike in other calculations of the non-relativistic radiative shift, no sum over a complete set of states is required. The spectral density, which is obtained by a relatively simple computation, reveals how different frequencies of vacuum fluctuations contribute to the total energy shift. The analysis shows, for example, that half the radiative shift for the ground state 1S level in H comes from virtual photon energies below 9700 eV, and that the expressions of Power and Welton have the correct high-frequency behavior, but not the correct low-frequency behavior, although they do give approximately the correct value for the total shift.