Three-Dimensional Formulation of the Relativistic Two-Body Problem and Infinite-Component Wave Equations

A relativistic quasipotential equation is derived from the conventional Hamiltonian formalism and old-fashioned "noncovariant" off-energy-shell perturbation theory in a similar way to that by which the four-dimensional Bathe-Salpeter equation is obtained from the off-mass-shell Feynman rules. The three-dimensional equation for the (off-energy-shell) scattering amplitude appears as a straightforward generalization of the nonrelativistic Lippmann-Schwinger equation. The corresponding homogeneous equation for the bound-state wave function and the normalization condition for its solutions are derived from the equation for the complete four-point Green's function. In order to obtain a solvable model, we consider a simplified version of the quasipotential equation which still reproduces correctly the on-shell scattering amplitude and is consistent with the elastic unitarity condition. It involves a "local" approximation to the potential $V(p-q)\mathrm{}$ which defines the kernel of our integral equation (the integration being carried over a two-sheeted hyperboloid in the energy-momentum space). It is shown that for the scalar Coulomb potential $V(p-q)=\frac{\alpha }{{\mathrm{}(p-q)\mathrm{}}^2}$, our model equation is equivalent to a simple infinite-component wave equation of the type considered by Nambu, Barut, and Fronsdal. The energy eigenvalues for the bound-state problem are calculated explicitly in this case and are found to be $O\left(4\right)\mathrm{}$ degenerate (just as in the nonrelativistic Coulomb problem and in Wick and Cutkosky's treatment of the Bethe-Salpeter equation in the same approximation).