The Compactification of QCD₄ to QCD₂ in a Flux Tube

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We show from the action integral that in the special environment of a flux tube, QCD₄ in (3+1) dimensional space-time can be approximately compactified into QCD₂ in (1+1) dimensional space-time. In such a process, we find out how the coupling constant g₂D in QCD₂ is related to the coupling constant g₄D in QCD₄. We show how the quark and the gluon in QCD₂ acquire contributions to their masses arising from their confinement within the tube, and how all these quantities depend on the excitation of the partons in the transverse degrees of freedom. The compactification facilitates the investigation of some dynamical problems in QCD₄ in the simpler dynamics of QCD₂ where the variation of the gluon fields leads to a bound state.