Proof of Dispersion Relations in Quantized Field Theories

The problem of deriving dispersion formulas is reduced to that of the analytic continuation of all functions which are regular in certain domains in the space of several complex variables. The dispersion relations for pion-nucleon scattering are proven for momentum transfers in the center-of-mass system which are smaller than $2\sqrt{2}{M}_{\pi }$. This limit can be improved by further analytic continuation. By using causality and spectral conditions the dispersion formulas for forward nucleon-nucleon scattering could be derived only under the unphysical condition $M_{\pi }>(\sqrt{2}-1)M_N$. We cannot exclude the possibility that this restriction is weakened by taking into account all symmetry properties of the complete four-body Green's function. The situation is similar for the representation of the meson-nucleon vertex function taken on the mass shell of the nucleons. In this case an example of R. Jost shows that the validity of the dispersion formula cannot be guaranteed on the basis of causality, spectrum, and symmetry properties.