  We derive an effective classical theory for real-time SU($N$) gauge theories
at high temperature. By separating off and integrating out quantum fluctuations
we obtain a 3D classical path integral over the initial fields and conjugate
momenta. The leading hard mode contribution is incorporated in the equations of
motion for the classical fields. This yields the gauge invariant hard thermal
loop (HTL) effective equation of motion. No gauge-variant terms are generated
as in treatments with an intermediate momentum cut-off. Quantum corrections to
classical correlation functions can be calculated perturbatively. The 4D
renormalizability of the theory ensures that the 4D counterterms are sufficient
to render the theory finite. The HTL contributions of the quantum fluctuations
provide the counterterms for the linear divergences in the classical theory.
