冲绳熵论讲义
The Okinawa Lectures on Entropy
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中文总结 AI 辅助
本讲义在介绍熵论历史后，基于概率论、模理论等数学工具，系统阐述经典与量子熵的构造、相关定理及应用，涵盖多种熵的定义与性质，兼具数学严谨性与理论物理适用性。
中文摘要 AI 辅助
在一段历史介绍后，本讲义将介绍经典与量子概率论框架下最重要的经典熵与量子熵的构造。经典熵从大偏差理论角度展开研究，涵盖Sanov定理、Cramér定理、Gärtner-Ellis定理与Varadhan定理，并举例说明其在玻尔兹曼统计物理与吉布斯统计物理中的若干应用。量子熵在公式层面表面上与经典熵相关联，而在更深层面则通过熵在统计假设检验中的关键作用建立联系。经典（相对）Kullback-Leibler熵、Umegaki提出的其量子对应形式，以及Renyi提出的它们的变形形式，均自然地契合于这一语境。量子熵面临着为冯·诺依曼代数形式化的子系统上的一对态定义和计算相对熵的新问题，这需要模（又称Tomita-Takesaki）理论，该理论为Araki与Uhlmann提出的量子相对熵提供了框架（这些熵包含Kullback-Leibler熵与Umegaki熵作为特例）。为了用密度算子与迹来（重新）定义和计算这些熵，还需要进一步的构造，即Haagerup的非交换L^p空间。冯·诺依曼代数也为Connes-Stormer-Narnhofer-Thirring熵提供了设定，该熵是动力系统与遍历理论中Kolmogorov-Sinai熵的量子版本，本讲义也将对其进行介绍。本课程最初受黑洞热力学启发，且应与其相关，不过我们未讨论该应用及热力学第二定律。本课程力求数学上严谨，同时也能让理论物理学家感兴趣。先修要求为本科概率论、泛函分析与量子理论。
英文摘要
After a historical introduction, the most important classical and quantum entropies are introduced as constructions in classical and quantum probability theory. Classical entropies are studied from large deviation theory, including theorems of Sanov, Cramér, Gärtner-Ellis, and Varadhan, and are illustrated in some applications to both Boltzmannian and Gibbsian statistical physics. Quantum entropies superficially connect to classical entropies at the formula level, but more deeply do so via the crucial role of entropy in statistical hypothesis testing. The classical (relative) Kullback-Leibler entropy, its quantum counterpart introduced by Umegaki, as well as their deformations proposed by Renyi all fit naturally in this context. Quantum entropy faces the new problem of defining and computing the relative entropy of a pair of states on a subsystem, here formalized as a von Neumann algebra. This requires modular (aka Tomita-Takesaki) theory, which provides the framework for the relative quantum entropies introduced by Araki and Uhlmann (these encompass both the Kullback-Leibler and Umegaki entropies as special cases). To (re)define and compute these entropies in terms of density operators and traces, further constructions are needed, namely Haagerup's noncommutative L^p spaces. Von Neumann algebras also provide the setting for the Connes-Stormer-Narnhofer-Thirring entropy, which is a quantum version of the Kolmogorov-Sinai entropy in dynamical systems and ergodic theory, to which we also provide an introduction. This course was originally inspired by, and should be relevant to, black hole thermodynamics, although we discuss neither this application nor the second law. The course tries to be both mathematically rigorous and interesting to theoretical physicists. Prerequisites are undergraduate probability theory, functional analysis, and quantum theory.
