Physical applications of the resolvent operators on the mathematical formalism of Feynman's theory of the positron

Abstract

The resolvent operators of the theory of linear functional equations are applied to the quantum formalism in general and more specially to the Feynman formulation of the hole theory. A generalization of the resolvent operators is given in order to treat problems with time dependent hamiltonians. It is shown that Feynman's formulation amounts to consider divergent waves for the positive kinetic energies and convergent waves for the negative kinetic energies, in the propagation kernel. Expansions of the propagation kernels are derived from the resolvent, without using Feynman's integral equation which leads to difficulties. A relativistically invariant resolvent is defined in the theory of quantized interacting fields. An operator related to the resolvent describes a new kind of collision which can be used in the theory of the ground state of atomic nuclei.