Non-Markovian trajectories involving future in the semi-classical path integral expression

TL;DR

This paper derives a semiclassical path integral for a quantum system coupled to a harmonic bath, revealing a non-Markovian, future-involving integro-differential equation of motion confirmed by numerical tests.

Contribution

It introduces a novel semiclassical path integral formulation showing the system's dynamics depend on both past and future times.

Findings

The system path is non-Markovian and involves future times.

The resulting equations are integro-differential and require root search algorithms.

Numerical tests confirm the necessity of future-involved terms.

Abstract

Semiclassical path integral expression for a quantum system coupled to a harmonic bath is derived based on the stationary phase condition. It is discovered that the system path is non-Markovian. Most strikingly, the system path not only couples to its past (as in the Langevin equation), but also to its future, i.e. the equation of motion for the system is an integro-differential equation that involves all times. Numerical tests are performed to confirm that the future-involved term is indeed necessary. Because of the future-non-Markovian nature of the equation, the numerical solution cannot be obtained by iterative methods. Instead, root search algorithms must be employed.