LOCALLY GAUGE-INVARIANT OFF-MASS-SHELL EFFECTIVE POTENTIAL IN QCD(3) AND HOT QCD(4)

We develop a technique whereby we are able to excite the vacuum with e xternal sources that are covariantly conserved even when quantum fluct uations ace considered. This allows us to construct physical, gauge-in variant states and the corresponding gauge-invariant off-mass-shell ef fective potential corresponding to such states. To show a practical ap plication of these ideas, we calculate for Euclidean QCD(3) or ''hot'' QCD(4), to one-loop order, the effective potential of a homogeneous, isotropic state where the gauge fields have a nonzero expectation valu e. The high temperature and possible cosmological consequences of our results are discussed. The definition of physically meaningful Green f unctions of any order, in an arbitrary space-time dimension, is discus sed. In our formalism, in the presence of an external current, it is p ossible to define a BRS symmetry under which the effective Lagrangian is invariant and this allows one to define a physically meaningful eff ective potential.