Some Remarks on Non-Local Field Theory

Abstract

The $S$-matrix formalism introduced by Yukawa for non-local field theory is considered. For suitable types of non-local fields and interactions it is shown that the $S$-matrix is convergent through the second order of interaction. In the limiting case in which the non-local fields become loca, it is found that the $S$-matrix formalism yields results inconsistent with the usual formalism unless a certain limiting process is introduced. The limiting process brings agreement between the two formalisms only through the second order of interaction; and the higher orders will, in general, disagree. Unfortunately, the limiting process also destroys the convergence of the $S$-matrix in the general case of non-local fields. These results suggest that the $S$-matrix formalism will need to be revised, but no definite recommendations for doing this are made here. An internal angular momentum operator for non-local fields is introduced; this operator aids in the decomposition of the field into irreducible parts of different spins.