The Quark Model via an hbar expansion of QCD

I discuss the possibility that the quark model emerges as the lowest order of an h̄ expansion of QCD bound states. In a hamiltonian approach the instantaneous A0 potential is determined by the field equations separately for each Fock component. These equations allow also a linear potential as a homogeneous solution. Stationarity of the action sets the direction of the ensuing constant electric field to be along the fermion pair separation. States bound by this non-perturbative, linear A0 potential are analogous to the Born term of standard perturbative expansions of scattering amplitudes, in that they represent the dominant contribution at lowest order in h̄ (no loops) and at O (g) in the potential. The Dirac equation for relativistic fermions bound by an external A0 potential is most easily derived using retarded boundary conditions. I demonstrate why this boundary condition does not affect the bound state energies at lowest order in h̄. Translated to physical Feynman boundary conditions the Dirac bound states are a superposition of Fock states with any number of fermionantifermion pairs. Applying this approach to relativistic quark-antiquark states in QCD results in a bound state equation which was previously proposed without derivation and shown to provide a reasonable description of the meson spectrum, including linear Regge trajectories. The equal-time wave functions have unique Lorentz transformation properties, which ensure the correct dependence of the bound state energy on the center-of-mass momentum. This indicates that the solution is exact at the Born level, i.e., at lowest order in h̄ and at O (g) in the QCD interaction.