On the self-energies and cross-sections of orthodox quantum mechanics

Abstract

In this paper are introduced several novel techniques having as their objects, first, the gain of a new understanding of the divergence difficulties of orthodox quantum theory, and secondly, the systematic development of the quantum mechanics of fields in terms of the density matrix. There is included a new presentation of a form of perturbation theory which has the outstanding advantages over the usual one that it is much more quickly convergent, and leaves the density matrix normalized at every stage so that the diagonal elements representing occupational probabilities cannot diverge. Exact general formulae are derived for self-energies and cross-sections for the purpose of examining their convergence properties. The general theory of fields is developed ab initio , and it is shown that the use of the density matrix in place of the wave vector illuminates and simplifies the customary theory. The second quantization of the density matrix which follows throws unexpected light on the existence of particles with negative expectation values. Finally, the whole theory is applied to the crucial example of the electron or positron in an electromagnetic field. It is confirmed that the self-energy of the electron in the orthodox theory cannot be made finite without the introduction of negative energy photons.