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Starting from a total Lagrangian describing an oscillator-bath system, anovel derivation of exact quantum propagator is presented. Having the quantumpropagator, the exact density matrix, reduced density matrix of the mainoscillator and thermal equilibrium fixed point are obtained. The problem isgeneralised to the cases where the main oscillator is under the influence of aclassical external force. By introducing generalised auxiliary classicalfields, the generalised quantum propagator or generating functional of positioncorrelation functions is obtained.

Exact Density matrix of an oscillator-bath system: Alternative derivation