论文标题
响应场和量子机械路径积分的鞍点
The Response Field and the Saddle Points of Quantum Mechanical Path Integrals
论文作者
论文摘要
在量子统计力学中,Moyal方程控制了Wigner函数的时间演变和代表任意混合状态的密度矩阵的更通用的Weyl符号的时间演变。马里诺夫的路径整体给出了一种正式的Moyal方程解决方案。在本文中,我们证明了该路径积分可以被视为量子力学中几种概念,几何和动力学问题之间的自然联系。通过强调响应字段(马里诺夫积分中的集成变量之一)甚至对纯状态发挥着关键作用来实现统一的观点。讨论的重点是积分的半经典近似与其严格的经典限制之间的关系。与Feynman型路径积分不同,后者在Marinov案中的定义很好。涵盖的主题包括基于“通风平均”的概念,关于描述隧道过程的阳性竞争性智能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。分析了电子和Bohm-Aharonov效应的双缝实验,作为说明性示例。此外,在分析持续的(“ wick旋转”)响应场上,马里诺夫路径的激数之间存在令人惊讶的关系,发现了Feynman-type积分的复杂Instantons。后者在最近的工作中对Picard-Lefschetz理论的最重要作用起着重要的作用,适用于振荡路径积分和复兴计划。
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.