O(4)Symmetry of the Hydrogen Atom and the Lamb Shift

Abstract

The $O\left(4\right)\mathrm{}$ symmetry of the nonrelativistic hydrogen atom is exploited in conjunction with Green's-function methods as a tool for the solution of Coulomb problems. The Bethe logarithm, the principal contribution to the Lamb shift, is treated by this means, and a rapidly convergent series is obtained. This is used to evaluate the Bethe logarithm of the $1S$, $2S$, and $2P$ states, and a discrepancy with the most precise previously published value for the $1S$ state is indicated. The new value is 2.9841285 with an estimated uncertainty of ±3 in the last figure. The discrepancy is beyond the reach of experimental observation at present.