Bi-local field equation out of Bethe-Salpeter equation

The bi-local (BL) field equations offer a useful phenomenological approach to two-body bound state problems by means of relativistic potentials, altho ugh their field theoretical basis is obscure. On the other hand, the Bethe- Salpeter (BS) equation for two-body bound states is obtained under an appro ximation within the framework of field theory. In some cases, the BS equati ons are known to be reduced to the BL field equations, since the order of t he BS equations is higher than that of the BL field equations as differenti al equations. In this paper, we attempt to find a systematic method of redu ction by regarding those equations as constraints in the homogeneous canoni cal formalism (HCF). It is shown that if the interaction kernel contains a delta function representing an instantaneous interaction, then reduction is possible even for the BS equation for two-body scalar fields. Discussion i s also given on the relation between the normalization of the BS amplitude and that of the reduced BL field.