We investigate a case of the Hu-Paz-Zhang master equation of the Caldeira-Leggett model without Lindblad-form obtained in the weak coupling limit up to the second order perturbation. In our study we use Gaussian initial states to be able to employ a sufficient and necessary condition, which can expose positivity violations of the density operator during the time evolution. We demonstrate that the evolution of the non-Markovian master equation has problems when the stationary solution is not a positive operator, i.e., does not have physical interpretation. We also show that solutions always remain physical for small times of evolution. Moreover, we identify a strong anomalous behavior, when the trace of the solution is diverging. We also provide results for the corresponding Markovian master equation and show that positivity violations occur for various types of initial conditions even when the stationary solution is a positive operator. Based on our numerical results we conclude that this non-Markovian master equation is superior to the corresponding Markovian one.

中文翻译：

---

Hu-Paz-Zhang主方程的适用范围

我们研究了在弱耦合极限（直到二阶摄动）中获得的不带Lindblad形式的Caldeira-Leggett模型的Hu-Paz-Zhang主方程的情况。在我们的研究中，我们使用高斯初始状态来满足充分必要的条件，从而可以暴露出时间演化过程中密度算子的正性违背。我们证明了当固定解不是一个正算子时，即没有物理解释时，非马尔可夫主方程的演化存在问题。我们还表明，解决方案在很小的进化时间内始终保持物理状态。此外，当解决方案的痕迹发散时，我们会发现强烈的异常行为。我们还提供了对应的马尔可夫主方程的结果，并表明即使固定解为正算子，各种类型的初始条件也会发生积极性违规。根据我们的数值结果，我们得出结论，该非马尔可夫主方程优于相应的马尔可夫主方程。