Quantum theory of infinite component fields

Abstract

The quantum theory of the infinite component SO(4,2) fields is formulated as a model for relativistic composite objects. We discuss three classes (timelike, lightlike, and spacelike) of physical solutions to a general class of infinite component wave equations. These solutions provide a definite physical interpretation to infinite component wave equations and are obtained by reducing SO(4,2) with respect to its orthogonal, pseudo‐orthogonal, and Euclidean subgroups. The analytic continuations among these solutions are established. In the nonrelativistic limit the timelike physical states exactly reduce to the Schrödinger solution for the hydrogen atom—the simplest composite object. The wave equations for the three classes are studied in two different realizations. In one case the equations describe a three‐dimensional internal Kepler motion with a discrete and a continuous energy spectrum and in the other case the equations describe a four‐dimensional internal oscillatory motion with attractive as well as repulsive potentials. It is found that the Kustaanheimo–Steifel transformation of classical mechanics exactly relates these two internal motions also in the quantum case. Thus a completely relativistic theory of composite systems is established for which the internal dynamics is the generalization of the nonrelativistic two‐body dynamics.