  We present a systematic discussion of Braaten and Pisarski's hard thermal
loop (HTL) effective theory within the framework of the real-time
(Schwinger-Keldysh) formalism. As is well known, the standard imaginary-time
HTL amplitudes for hot gauge theory express the polarization of a medium made
out of nonabelian charged point-particles; we show that the complete real-time
HTL theory includes, in addition, a second set of amplitudes which account for
Gaussian fluctuations in the charge distributions, but nothing else. We give a
concise set of graphical rules which generate both set of functions, and
discuss its relation to classical plasma physics.
