Unified time-path approach to the generating functional of the Brownian oscillator system: The bilinearly corrected Feynman rule for nonequilibrium processes

We derive fully corrected propagators of a bilinearly interacting Brownian oscillator system by summing up an infinite number of Feynman diagrams. The generating functional of a nonequilibrium system is calculated in terms of the bilinearly corrected propagators thus obtained. The result offers the Feynman rule for systematically studying the effects of both anharmonicity of the potential and nonbilinear system-bath couplings on the Brownian oscillator system. The reduced density matrix, which is useful for investigating the dynamics of the system, is also calculated. The unified time path, which is powerful in calculating propagators, is introduced and used effectively throughout the work.