In quantum statistical mechanics, Moyal’s equation governs the time evolution of Wigner functions and of more general Weyl symbols that represent the density matrix of arbitrary mixed states. A formal solution to Moyal’s equation is given by Marinov’s path integral. In this paper we demonstrate that this path integral can be regarded as the natural link between several conceptual, geometric, and dynamical issues in quantum mechanics. A unifying perspective is achieved by highlighting the pivotal role which the response field, one of the integration variables in Marinov’s integral, plays for pure states even. The discussion focuses on how the integral’s semiclassical approximation relates to its strictly classical limit; unlike for Feynman type path integrals, the latter is well defined in the Marinov case. The topics covered include a random force representation of Marinov’s integral based upon the concept of “Airy averaging”, a related discussion of positivity-violating Wigner functions describing tunneling processes, and the role of the response field in maintaining quantum coherence and enabling interference phenomena. The double slit experiment for electrons and the Bohm–Aharonov effect are analyzed as illustrative examples. Furthermore, a surprising relationship between the instantons of the Marinov path integral over an analytically continued (“Wick rotated”) response field, and the complex instantons of Feynman-type integrals is found. The latter play a prominent role in recent work towards a Picard–Lefschetz theory applicable to oscillatory path integrals and the resurgence program.

在量子统计力学中，Moyal方程控制着Wigner函数以及代表任意混合态密度矩阵的更通用的Weyl符号的时间演化。马里诺夫的路径积分给出了Moyal方程的形式化解。在本文中，我们证明了该路径积分可以视为量子力学中几个概念，几何和动力学问题之间的自然联系。通过强调响应场（即Marinov积分中的积分变量之一）甚至对纯态所起的关键作用，可以实现统一的观点。讨论着重于积分的半经典逼近如何与其严格的经典极限有关。与费曼型路径积分不同，后者在马里诺夫案例中有很好的定义。涵盖的主题包括基于“艾里平均”概念的Marinov积分的随机力表示，描述隧道过程的违反正定性Wigner函数的相关讨论，以及响应场在维持量子相干性和启用干扰现象中的作用。作为示例，分析了电子的双缝实验和Bohm–Aharonov效应。此外，在分析连续（“维克旋转”）响应场上的Marinov路径积分的实例与Feynman型积分的复杂实例之间发现了令人惊讶的关系。后者在最近的皮卡德-莱夫切茨理论的研究中起着重要作用，该理论适用于振荡路径积分和回潮程序。