Rigorous three-dimensional relativistic equation for quark-antiquark bound states at finite temperature derived from the thermal QCD formulated in the coherent-state representation

A rigorous three-dimensional relativistic equation for quark-antiquark bound states at finite temperature is derived from the thermal QCD generating functional which is formulated in the coherent-state representation. The generating functional is derived newly and given a correct path-integral expression. The perturbative expansion of the generating functional is specifically given by means of the stationary-phase method. Especially, the interaction kernel in the three-dimensional equation is derived by virtue of the equations of motion satisfied by some quark-antiquark Green functions and given in a closed form which is expressed in terms of only a few types of Green functions. This kernel is much suitable to use for exploring the deconfinement of quarks. To demonstrate the applicability of the equation derived, the one-gluon exchange kernel is derived and described in detail.