  An exact real time quantum dynamics preaveraged over imaginary time path
integral is formulated for general condensed phase equilibrium ensemble. This
formulation results in the well-known centroid dynamics approach upon filtering
of centroid constraint, and provides a rigorous framework to understand and
analyze a related quantum dynamics approximation method called ring polymer
molecular dynamics. The formulation also serves as the basis for developing new
kinds of quantum dynamics that utilize the cyclic nature of the imaginary time
path integral.
