2013/10/29 by Maurice A. de Gosson, de Gosson, Maurice A.
Computer Science · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1310.7885
openalex publication_date 2013/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The notion of polarity between sets, well-known from convex geometry, is a geometric version of the Fourier transform. We exploit this analogy to propose a new simple definition of quantum indeterminacy, using what we call "hbar-polar quantum pairs", which can be viewed as pairs of position-momentum indeterminacy with minimum spread. The existence of such pairs is guaranteed by the usual uncertainty principle, but is at the same time more general. We use recent advances in symplectic topology to show that this quantum indeterminacy can be measured using a particular symplectic capacity related to action and which reduces to area in the case of one degree of freedom. We show in addition that polar quantum pairs are closely related to Hardy's uncertainty principle about the localization of a function and its Fourier transform.