构造宏观可观测量的Ogata定理的近似交换矩阵
Constructing Nearby Commuting Matrices for Ogata's Theorem on Macroscopic Observables
AI总结：
本文提出一种构造近似交换矩阵的方法，为位置维度$d=2$的Ogata定理提供了构造性证明，并确保实宏观可观测量渐近成为近似交换的实可观测量。
摘要：
为解决von Neumann猜想，arXiv:1111.5933中的Ogata定理证明了一个高度非平凡的结论：当$N \to \infty$时，对应于具有$N$个位置和固定位置维度$d$的宏观可观测量的任意多个矩阵，均渐近成为近似交换的可观测量。我们开发了一种方法，为$su(2)$的归一化高度可约表示构造近似交换矩阵，其中不可约子表示的重数表现出特定的单调递减行为。随后，我们针对位置维度$d=2$的情况提供了Ogata定理的构造性证明，并给出了近似可观测量接近程度的显式估计。此外，受arXiv:1012.3494中探讨的时间反演对称性应用的启发，我们的构造具有以下性质：实宏观可观测量渐近成为近似交换的实可观测量。本文还包含理解所获结果证明所需的预备矩阵分析介绍、文献中关于近似交换算子的$C^\ast$-代数泛函分析入门、近似交换矩阵问题的历史与主要结果的详细讨论、有限维量子力学系统中可观测量的测量基础回顾，以及非交换有界算子不确定性原理的讨论，并给出了具有近似交换逼近的可观测量的相关结果。
英文摘要：
Resolving a conjecture of von Neumann, Ogata's theorem in arXiv:1111.5933 showed the highly nontrivial result that arbitrarily many matrices corresponding to macroscopic observables with $N$ sites and a fixed site dimension $d$ are asymptotically nearby commuting observables as $N \to \infty$. We develop a method to construct nearby commuting matrices for normalized highly reducible representations of $su(2)$ whose multiplicities of irreducible subrepresentations exhibit a certain monotonically decreasing behavior. We then provide a constructive proof of Ogata's theorem for site dimension $d=2$ with explicit estimates for how close the nearby observables are. Moreover, motivated by the application to time-reversal symmetry explored in arXiv:1012.3494, our construction has the property that real macroscopic observables are asymptotically nearby real commuting observables. This thesis also contains an introduction to the prerequisite matrix analysis for understanding the proof of the results gotten, a primer on the functional analysis of $C^\ast$-algebras frequently used in the literature of almost commuting operators, a detailed discussion of the history and main results in the problem of almost commuting matrices, a review of the basics of measurement of observables in finite dimensional quantum mechanical systems, and a discussion of the uncertainty principle for non-commuting bounded operators with some results obtained for observables with nearby commuting approximants.
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