标题： 量子不确定性和极对偶性：一种概率方法 显示英文标题

标题： Quantum Indeterminacy and Polar Duality: a Probabilistic Approach

摘要： 我们提出一种概率论证，支持在我们之前的工作中讨论的极对偶性的应用，以表达量子力学的不确定性原理。 我们的方法结合了凸体的马勒体积的性质与调和分析中的Donoho--Stark不确定性原理，该原理描述了函数及其傅里叶变换的集中性。 核心结果表明，位置集中在凸体附近和动量集中在它的极对偶体附近的概率之和等于一，误差项随着自由度数量的增加而迅速减小。 这一结果促使将极对偶性解释为一种几何傅里叶变换。 显示英文摘要

摘要： We present a probabilistic argument supporting the application of polar duality, as discussed in our previous work, to express the indeterminacy principle of quantum mechanics. Our approach combines the properties of the Mahler volume of a convex body with the Donoho--Stark uncertainty principle from harmonic analysis, which characterizes the concentration of a function and its Fourier transform. The central result demonstrates that the sum of the probabilities of position concentration near a convex body and momentum concentration near its polar dual is equal to one, with an error term that diminishes rapidly as the number of degrees of freedom increases. This result motivates the interpretation of polar duality as a kind of geometric Fourier transform.