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

Quantum Physics