It is shown that the principle of microscopic causality is sufficient and in general also necessary for the existence of dispersion relations for the derivatives of the no-spin-flip amplitude with respect to $cos\vartheta$, taken at zero angle. Physical dispersion relations are derived for these quantities, neglecting the nucleon recoil.

The dispersion formulas for the first derivative reduce to the no-spin-flip parts of Low's $P$-wave equations, if one neglects in addition to the recoil all inelastic processes and contributions from partial waves with $l>\mathrm{}1\mathrm{}$. In combination with corresponding approximate relations, obtained from the exact dispersion formulas for the forward scattering amplitude, one can derive integral equations for $S$-waves only. The inhomogeneous terms of these equations contain the zero-energy scattering lengths.

Finally, it is shown that in general one cannot derive normal dispersion relations for the amplitudes corresponding to individual angular momenta.

• Received 24 October 1955

DOI:https://doi.org/10.1103/PhysRev.102.1174