Quantum Theory of Non-Local Fields. Part II. Irreducible Fields and their Interaction

Abstract

General properties of non-local operators are considered in connection with the problem of invariance with respect to the group of inhomogeneous Lorentz transformations. It is shown that irreducible fields can be classified by the eigenvalues of four invariant quantities. Three of these quantities can be interpreted, respectively, as the mass, radius, and magnitude of the internal angular momentum of the particles associated with the quantized non-local field in question. Further, space-time displacement operators are introduced as a particular kind of non-local operator. As a tentative method of dealing with the interaction of non-local fields, an invariant matrix is defined by the space-time integral of a certain invariant operator, which is a sum of products of non-local field operators and displacement operators. It is shown that the matrix thus constructed satisfies the requirements that it be unitary and invariant and that the matrix elements are different from zero only if the initial and final states had the same energy and momentum. However, the remaining conditions of correspondence and convergence cannot be fulfilled simultaneously, in general, by the $S$-matrix for the non-local fields. It is yet to be investigated whether all of these requirements are satisfied by an appropriate change in the definition of the $S$-matrix.