New Techniques in the Lamb Shift Calculation

Based on the analogy between the calculations of radiative corrections to electron scattering and the Lamb shift, a new procedure applicable to bound state self-energy problems is developed, wherein the electron propagator is expanded in powers of the external potential. In the resulting sequence, a change in the gauge of the virtual photon field conveniently removes from each term spurious lower order contributions and yields a new and considerably simpler sequence; the expectation value of the first two terms of the latter is shown to account for the major portion of the Bethe logarithm. A simple method is developed to sum all the $\alpha {\left(Z\alpha \right)}^4\mu$ dependence, and the result is the lowest order Lamb shift formula. The case of the calculation, as well as that involved in obtaining the relativistic level shift correction of order $\alpha {\left(Z\alpha \right)}^5\mu$ (not given in the present paper), indicates that the method may find application in the calculation of further higher order effects.