  We establish the path integral approach for the time-dependent heat exchange
of an externally driven quantum system coupled to a thermal reservoir. We
derive the relevant influence functional and present an exact formal expression
for the moment generating functional which carries all statistical properties
of the heat exchange process for general linear dissipation. The general method
is applied to the time-dependent average heat transfer in the dissipative
two-state system. We show that the heat can be written as a convolution
integral which involves the population and coherence correlation functions of
the two-state system and additional correlations due to a polarization of the
reservoir. The corresponding expression can be solved in the weak-damping limit
both for white noise and for quantum mechanical coloured noise. The
implications of pure quantum effects are discussed. Altogether a complete
description of the dynamics of the average heat transfer ranging from the
classical regime down to zero temperature is achieved.
