We propose a model for the propagation of a heavy-quark in a hot plasma, to
be viewed as a first step towards a full description of the dynamics of heavy
quark systems in a quark-gluon plasma, including bound state formation. The
heavy quark is treated as a non relativistic particle interacting with a
fluctuating field, whose correlator is determined by a hard thermal loop
approximation. This approximation, which concerns only the medium in which the
heavy quark propagates, is the only one that is made, and it can be improved.
The dynamics of the heavy quark is given exactly by a quantum mechanical path
integral that is calculated in this paper in the Euclidean space-time using
numerical Monte Carlo techniques. The spectral function of the heavy quark in
the medium is then reconstructed using a Maximum Entropy Method. The path
integral is also evaluated exactly in the case where the mass of the heavy
quark is infinite; one then recovers known results concerning the complex
optical potential that controls the long time behavior of the heavy quark. The
heavy quark correlator and its spectral function is also calculated
semi-analytically at the one-loop order, which allows for a detailed
description of the coupling between the heavy quark and the plasma collective
modes
