论文标题
量子极性二元性和符号骆驼：量化的几何方法
Quantum Polar Duality and the Symplectic Camel: a Geometric Approach to Quantization
论文作者
论文摘要
我们定义和研究量子极性的概念，这是一种几何傅立叶傅立叶变换，位置集和一组动量。为了扩展我们的先前工作，我们表明在配置和动量空间上，量子状态的协方差椭圆形的正交投影形成了我们所谓的双量子对。此后，我们表明量子极性允许解决高斯波施用的Pauli重建问题。量子极性的概念在不确定性原理与符号和凸几何形状之间表现出很强的相互作用，因此我们的方法可以为量子不确定性的几何和拓扑版本铺平道路。我们将结果与Blaschke-Santaló不平等和Mahler猜想联系起来。从量子极性的角度来看，我们还讨论了强壮的不确定性原理和鲜为人知的Donoho原理。
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.
