The Interaction of the Bi-Local Field with the External Field

Abstract

The wave equation of the hi-local field interacting with the external field is introduced on the basis of the hi-local field equation proposed previously. The subsidiary conditions are compatible with the wave equation in the interacting case. As examples of the present formalism, the form factor and the scattering amplitudes are studied. The form factor is decreasing exponentially as t--7-oo. The scattering amplitude arising from the exchange of the hi-local field in t-channel shows Regge behaviour. § I. Introduction Recently, the multi-local fields have been extensively studied as models of hadrons. From the viewpoint of the relativistic quark model, the hi-local and tri-local fields with Hooke potential have been used to investigate the. mass spectrum and the form factors of hadrons.!l On the other hand, by following the extended particle pictures, the mechanical models of the multi-local fields have been proposed by several authors!) From the latter point of view, however, it seems very diffi­ cult to introduce the interactions consistently even in the case of those with external local fields. This difficulty comes from the subsidiary conditions which suppress the relative time freedom. In this paper, we shall discuss the interaction of the hi-local field with an external scalar field based on the wave equation of the hi-local field proposed recently by one of the authors!) In our formalism, the geometrical center of two mass points is composed of two parts one of which is a part of Zitterbewegung and another is that of a free motion. The mean center is defined by the coordinate representing a free motion and it commutes with the subsidiary conditions which are characteristic in our formalism. Using the mean center, we can obtain the wave equation of the hi-local field in the external local field. In § 2, we shall summarize our formalism and introduce the mean center X, which is commutable with the subsidiary condition. The wave equation in the interacting case will be given in this section. The general form of the wave functions satisfying the free wave equation and the subsidiary conditions are also discussed. As a simple appli­ cation of the interacting hi-local field, we shall study the form factor of the hi-local field in the ground state in § 3. The form factor obtained here is decreasing exponentially as the transferred momentum - t goes very large. In § 4, we shall