  We show from the action integral that in the special environment of a flux
tube, QCD$_4$ in (3+1) dimensional space-time can be approximately compactified
into QCD$_2$ in (1+1) dimensional space-time. In such a process, we find out
how the coupling constant $g_{2D}$ in QCD$_2$ is related to the coupling
constant $g_{4D}$ in QCD$_4$. We show how the quark and the gluon in QCD$_2$
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 QCD$_4$ in the simpler dynamics of
QCD$_2$ where the variation of the gluon fields leads to a bound state.
