Partial differential equations with respect to coupling constants: electromagnetic mass difference of hadrons

The partial differential equations with respect to strong and electromagnetic coupling constants are established for the S matrix, Heisenberg operators, and "out" field operators. The macrocausality relations-vanishing of derivatives with respect to coupling constants for free "in" field operators-are important in the formalism. As a consequence of macrocausality relations, observable masses do not depend on strong and electromagnetic coupling constants. This makes the observable n-p mass difference uncomputable. Thus, our approach goes along the lines of divergent (but renormalizable) field theory. We are assuming (although not discussing) the existence of nonelectromagnetic interactions which make $m_n-m_p\ne 0$ when e=0. The partial differential equations with respect to strong and electromagnetic coupling constants are derived for bare n and p masses. In order to integrate these differential equations, the observable n and p must be known a priori. After integrating these differential equations, a formal expression for the observable n-p mass difference is obtained. This expression, compared with the usual expression from the literature, besides containing the difference of n and p mass shifts due to the electromagnetic interactions ("renormalized" by strong interactions), also contains the bare n-p mass difference and the difference of n and p mass shifts due to strong interactions. One cannot "recover" the usual expression for the n-p mass difference since our expression, as far as the observable n-p mass difference is concerned, is an identity and not the relation from which it can be computed.