Variational Method for Few-Body Bound States in QFT

We discuss a variational method for deriving relativistic few-body equations in QFT. The method is illustrated on two, three and four body systems in a strongly coupled model, namely the Scalar Yukawa (Wick-Cutkosky) model, and on QED (the muonium, positronium and muonic hydrogen systems). A reformulation of the models is employed, in which covariant Green functions are used to solve the field equations partially, so as to express the mediating field in terms of the particle fields. The resulting reformulated Hamiltonian of the quantized system thereby contains an interaction term in which the mediating-field Green function is sandwiched between the particle currents. Numerical solutions of two, three and four body equations are presented for the scalar Yukawa model for various strengths of the coupling. The results are compared, where possible, to Bethe-Salpeter based and other calculations. Perturbative solutions of the two-fermion equations are presented and compared to other calculations as well as to experimental results for some states of muonium.