On the Calculation of Self-Energies in Quantum Theory by Analytic Continuation

Riesz's method of solving hyperbolic differential equations by analytic continuation has been used by Gustafson to eliminate infinities in quantum theory. Treating the one-electron case, he found finite values of the self-energy integrals in the second approximation, also for those integrals for which the $\lambda$-limiting process fails (without the further assumption of negative-energy photons). In the present paper it is shown that the general result of Gustafson's procedure is to remove all divergences normally appearing in self-energy expressions, except logarithmic divergences. Thus the total self-energy of the electron, to the second approximation, is found to be zero on the one-electron theory if calculated by this method, whereas in the hole theory the logarithmically divergent expression of Weisskopf is retained. A proposal by Pauli to alter the commutation rules in a certain way gives essentially the same results.