Pair-interaction approximation in the equations of quantum field theory for a four-body system

It is shown that for the correct formulation of the pair-interaction approximation in the dynamical equations for a system of four and more particles in the various approaches of quantum field theory not only the sum of the pair-interaction potentials but also additional,“two-pair” kernels must be taken into account. Equations of two types with compact kernels that take into account these terms are obtained. The kernels of these equations are expressed in terms of the two- and three-particle scattering amplitudes. The equations of the first type have a structure identical to their nonrelativistic analog, whereas the equations of the second type differ from the nonrelativistic equations even in the number of components. An advantage of these equations is that they can be generalized to the case of arbitrarily many particles.