- We develop a technique whereby we are able to excite the vacuum with external sources that are covariantly conserved even when quantum fluctuations are considered. This allows us to construct physical, gauge-invariant states and the corresponding gauge-invariant off-mass-shell effective potential corresponding to such states. To show a practical application of these ideas, we calculate for Euclidean QCD3 or ``hot'' QCD4, to one-loop order, the effective potential of a homogeneous, isotropic state where the gauge fields have a nonzero expectation value. The high temperature and possible cosmological consequences of our results are discussed. The definition of physically meaningful Green functions of any order, in an arbitrary space-time dimension, is discussed. In our formalism, in the presence of an external current, it is possible to define a BRS symmetry under which the effective Lagrangian is invariant and this allows one to define a physically meaningful effective potential.