TTP06-29 Equations for Relativistic Particles, Binaries and Ternaries

Relativistic differential equations for bound states of $n$ Dirac particles ($n=1,2,3$) are put into forms that depend only on the square of the total cms energy. Instead of coupling $4^n$ Dirac components, they decompose into two equivalent equations for $4^n/2$ components $\psi$ and $\chi$ of total chiralities $+1$ and $-1$, respectively. Their time-dependent versions are of the Klein-Gordon type; they reproduce the relativistic kinematics for the emission of photons or pions. Although the equations are presently extended to mesons and baryons within perturbative QCD only, the necessity of free Klein-Gordon equations for closed systems implies $E^2$-spectra not only for mesons, but also for baryons. For binaries, there exists an intricate transformation which turns the equation for $\psi$ into an effective one-body Dirac equation. The corresponding transformation for $\chi$ is derived here.