Boson-Fermion Scattering in the Heisenberg Representation

Abstract

It is shown that the $S$-matrix for boson-fermion scattering can be simply expressed in the Heisenberg representation. By performing a time integration one obtains the $S$-matrix in the Schr\"odinger representation, which has the same form as the conventional perturbation theory sum over states. Suitably limiting the nature of the intermediate states entering into this sum leads to integral equations for certain matrix elements which are equal to the $S$-matrix elements on the energy shell. These equations appear in a completely renormalized form. For example, in the fixed source limit, the four pion-nucleon scattering states satisfy the same equation (with different numerical coefficients). The equations are nonlinear, but involve only the scattering phase shifts. The equivalent equation for photopion production is linear, and in the fixed source limit can be written down from a knowledge of the experimental scattering phase shifts. The zero-pion-mass theorems of Gell-Mann and Goldberger (concerning the isotopic spin independence of the zero-energy $S$-wave scattering) and of Kroll and Ruderman, [Phys. Rev. 93, 233 (1954)] follow simply from the formalism.