Quantum Indeterminacy, Polar Duality, and Symplectic Capacities

TL;DR

This paper introduces a new geometric approach to quantum indeterminacy using polar duality and symplectic capacities, linking convex geometry, Fourier analysis, and quantum physics to measure uncertainty.

Contribution

It proposes a novel definition of quantum indeterminacy via hbar-polar quantum pairs and connects this concept to symplectic topology and Hardy's uncertainty principle.

Findings

Quantum indeterminacy can be characterized by hbar-polar pairs.

Symplectic capacities provide a measure of quantum uncertainty.

Polar quantum pairs relate to Fourier transform localization principles.

Abstract

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.