  The Low equation is derived in a functional approach to the reduction of the
S matrix in the canonical formalism. This establishes the vacuum expectation
value of the scattering matrix as the generating functional of non-forward
Green functions, without reference to external currents. The method provides an
alternate derivation of non-perturbative results of field theory, such as the
Low equation, and considerably simplifies their derivation as well as that of
the rules of perturbation theory, the LSZ reduction formula, the
Dyson-Schwinger equations and crossing symmetry. The approach is employed to
further develop the Low equation via reduction of the fermionic sector to
obtain a reduced Dyson-Schwinger equation for boson-fermion scattering.
