On Yukawa's Theory of Non-local Field, I: The Case of Free Field

Abstract

The relation of Yukawa's non·local field theory1) to the ordinary local field theory is investigated. In § 3, it is shown that the equations of motion and the commutation relations of the non·local field are derived from those of the local field by a canonical transformation, which means that both fields are equivalent at least in the case of no interaction. § 4 is devoted to some remarks on the feature of the non.local field as a mixed field composed of those with various spins. The considerations in this paper are all restricted to the free field. § 1. Introduction and summary Recently, Yukawa discussed a generalization of the field concept by introducing a non-local field, which is free from the restriction that a field is determined as a point function in the ordinary coordinate space. This attempt is noteworthy at present, when the difficulty to remove various divergences in the present field theory within the frame­ work of the local field theory has become clearer, as giving a new scope to the future theories. In his theory a finite radius of the elementary particle can be introduced with­ out contradiction to the condition of Lorentz-invariance. In view of this promising nature of the non-local field, it would be interesting to investigate to what extent the theory of the non-local field will succeed in resolving the difficulties of the present field theory and in elucidating the structure of the elementary particles. In trying such a generalization, however, we have no guiding principle generally acceptable, and the Yukawa theory which was proposed under the guiding principle of Lorentz-invariance and reciprocity, could not avoid to include some speculative nature. Then, in order to make the character of the non-local field clearer, it may be desirable to investigate its relation to the local field. This may also serve to give a preparation for a further development of the non· local field theory. The first step toward this line was made by Fierz2). He anticipated the equivalence of the Yukawa model to the local field, at least in the case of no interaction, by pointing out that the Yukawa model could be regarded as a superposition of local fields of various spins. We investigate this situation furthermore by using the method of canonical trans­ formation. In § 2, the equations of motion of the non-local field are derived by a canonical transformation from those of the local field. In the course of the procedure, a represent­ ation, in which K, L and .11£ in (2.3) are diagonal, is used to make the correspondence