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.

中文翻译：

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半经典路径积分表达式中涉及未来的非马尔可夫轨迹

耦合到谐波浴的量子系统的半经典路径积分表达式是基于固定相条件导出的。发现系统路径是非马尔可夫的。最引人注目的是，系统路径不仅与它的过去（如朗之万方程）有关，而且与它的未来有关，即系统的运动方程是一个涉及所有时间的积分微分方程。进行数值测试以确认涉及未来的项确实是必要的。由于方程的未来非马尔可夫性质，数值解无法通过迭代方法获得。相反，必须采用根搜索算法。