A general relationship is established between semiclassical and centroid-based methods for calculating real-time quantum-mechanical correlation functions. It is first shown that the linearized semiclassical initial-value-representation (LSC-IVR) approximation can be obtained via direct linearization of the forward-backward action in the exact path integral expression for the correlation function. A Kubo-transformed two-time correlation function, with the position operator as one of the two operators, is then cast in terms of a carefully crafted exact path integral expression. Linearization of the corresponding forward–backward action, supplemented by the assumption that the dynamics of the centroid is decoupled from that of the higher normal modes, is then shown to lead to the centroid correlation function.
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