Exact Determination of the Coulomb Amplitude byS-Matrix Methods

Abstract

An extension of the $S$-matrix methods to the case of the exchange of zero-mass particles allows us to determine the Coulomb scattering amplitude and the levels of the hydrogen atom completely. We start from the special analytic properties in $t$ due to the zero mass of the photon, and write the amplitude as an entire function of $\ln t$. Two of our main assumptions are: (1) The amplitude is of exponential type with respect to $\ln t$; (2) the partial-wave amplitude has only Regge poles in $l$. We find several classes of solutions. One of them is the Schrödinger solution. The others lead to ghosts or to bound states in the $e^{+}p$ system and are therefore discarded. The same method can be extended to the nonrelativistic problem with spin, and leads to the same result as in the case without spin. In the relativistic case, we get results which disagree with the Dirac and Klein-Gordon solutions.