Gosson Maurice A de
University of Vienna Faculty of Mathematics (NuHAG), Wien, Austria.
Found Phys. 2021;51(3):60. doi: 10.1007/s10701-021-00465-6. Epub 2021 May 21.
We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke-Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho-Stark principle from the point of view of quantum polarity.
我们定义并研究了量子极性的概念，它是位置集与动量集之间的一种几何傅里叶变换。扩展我们之前的工作，我们表明量子态的协方差椭球在构型空间和动量空间上的正交投影形成了我们所谓的对偶量子对。此后，我们证明量子极性能够解决高斯波函数的泡利重构问题。量子极性的概念展现了不确定性原理与辛几何和凸几何之间的强烈相互作用，因此我们的方法可能为量子不确定性的几何与拓扑版本铺平道路。我们将我们的结果与布拉施克 - 桑塔洛不等式以及马勒猜想联系起来。我们还从量子极性的角度讨论了哈代不确定性原理和鲜为人知的多诺霍 - 斯塔克原理。
