The Quark Model via an hbar expansion of QCD

Abstract

I discuss the possibility that the quark model emerges as the lowest order of an h 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 h (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 h. 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 h and at O (g) in the QCD interaction.