A Lorentz covariant approach to the bound state problem

Abstract

The relativistic equivalent of the Schr\"odinger equation for a two particle bound state having the total angular momentum $S$ is written in the form of a Lorentz covariant set of equations (p_1^mu+p_2^mu+Omega^mu)Psi(p_1,p_2;P) chi_S(\vec{p}_1,\vec{p}_2)=P^mu Psi(p_1,p_2;P) chi_S(\vec{p}_1,\vec{p}_2) where the operators Omega^mu are the components of a 4-vector quasipotential. The solution of this set is a stationary function representing the distribution of spins and internal momenta in a reference frame where the momentum of the bound system is P^\mu. The contribution of the operators Omega^mu to the bound state momentum is assumed to be the 4-momentum of a vacuum-like effective field entering the bound system as an independent component. It is shown that a state made of free quarks and of the effective field has definite mass and can be normalized like a single particle state. The generalization to the case of three or more particles is immediate.