Eigenvalue Problem in Quantum Electrodynamics

The eigenvalue problem in quantum electrodynamics is discussed from the point of view of the Fredholm theory of integral equations. Starting with positron theory—the theory of a quantized Dirac field interacting with an external field only—the external potentials are replaced by bare photon fields. To insure causality the photon operators are ordered in time. Certain integral equations for the Fredholm minors constructed on the Feynman kernel are taken to be the equations for the wave functions of $n$ particle systems. An expansion in interaction patterns rather than the coupling constant is indicated. The one particle problem is treated in the first pattern approximation. Procedures proposed by Snyder and Snow and the mass renormalization scheme are discussed in this connection. Finally a purely formal derivation of the Bethe-Salpeter equation for the two-body problem in the lowest pattern approximation is given.