can be determined for quarkonium systems interacting with a heat bath. In Section II, we review the reduced density matrix, apply the model of Caldeira and Leggett for quantum Brownian motion to imaginary time, and then describe briefly path-integral Monte Carlo techniques for calculating this reduced density matrix. In Section III, we show how the imaginary-time propagator for charmonium may be calculated to high accuracy with these techniques, for any potential or drag coefficient, and we show the results for G ( τ, T ) for the vector channel at temperatures just above the deconfinement transition. Not many before have considered the effect of the medium on heavy quark and quarko- nium Euclidean correlators calculated on the lattice. Current work by Beraudo et al. [15] does consider the medium effect on heavy quark correlators. Starting from the QCD La- grangian, they make the HTL approximation for the heavy quark’s interaction with the light degrees of freedom at finite temperature, and then they integrate these degrees of freedom out. The focus of our work is on a heavy particle interacting quite strongly with the medium, as suggested by calculations taking advantage of gauge-gravity duality [10]. We show how the Lagrangian for a particle interacting strongly with a heat bath can be renormalized so that functional integration is well-defined, and where analytic results are possible. For an excellent review of the functional integral approach to quantum Brownian motion, in both imaginary and real time, see [16]. For a briefer discussion of these ideas and how they apply to quarkonium, see [17]. These results are discussed in generality, and shown to arise from the Schwinger-Keldysh contour integral for a heavy quark’s real-time partition function, by Son and Teaney [18]. These systems can be studied with approaches besides the path-integral formulation, and the study of quantum Brownian motion, in terms of a partial differential equation for the density matrix, has been studied by Hu, Paz, and Zhang [19, 20]. II. THE PATH INTEGRAL FORM FOR THE REDUCED DENSITY MATRIX OF A DISSIPATIVE SYSTEM 3