Simplified Calculation of Lamb Shift Using Algebraic Techniques

The Lamb shift in hydrogenlike atoms is treated by algebraic matrix methods using a unitary, infinite-component representation of the group $SO(4, 2)$. The result for the Lamb shift itself, with its dependence upon the Bethe logarithm of the average excitation energy, is identical to previous results, but is obtained more simply. The numerical evaluation of the Bethe logarithm is both simpler and more accurate than by earlier treatments, as demonstrated by a detailed evaluation for the ground state. Numerical values for the Bethe logarithm (obtained without the use of an electronic computer) are $\gamma \mathrm{}\left(1S\right)=2.98412\mathrm{} 85559\left(3\right)\mathrm{}$, $\gamma \mathrm{}\left(2S\right)=2.81176\mathrm{} 98932\left(5\right)\mathrm{}$, and $\gamma \mathrm{}\left(2P\right)=-0.03001\mathrm{} 67089\left(3\right)\mathrm{}$ for the three lowest states, with the figures in parentheses giving the number of units of estimated error in the last decimal place. A series of appendices presents the needed properties of $SO(p, 1)$ and $SO(p, 2)$ representations, the $SO(4, 2)$ formulation of the hydrogen atom, and an alternative treatment of the Bethe logarithm which may also be applied to other operators such as the Coulomb Green's function.