  The calculation of quantum canonical time correlation functions is considered
in this paper. Transport properties, such as diffusion and reaction rate
coefficients, can be determined from time integrals of these correlation
functions. Approximate, quantum-classical expressions for correlation
functions, which are amenable to simulation, are derived. These expressions
incorporate the full quantum equilibrium structure of the system but
approximate the dynamics by quantum-classical evolution where a quantum
subsystem is coupled to a classical environment. The main feature of the
formulation is the use of a mapping basis where the subsystem quantum states
are represented by fictitious harmonic oscillator states. This leads to a full
phase space representation of the dynamics that can be simulated without appeal
to surface-hopping methods. The results in this paper form the basis for new
simulation algorithms for the computation of quantum transport properties of
large many-body systems.
