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

Hard thermal loops in the real-time formalism