On Quantum Indeterminacy and the Uncertainty Principle

Abstract:We propose a geometric formulation of quantum indeterminacy based on polar duality between convex bodies representing position and momentum data. A pair (X,P) of centrally symmetric convex bodies is said to satisfy the indeterminacy condition when the h-polar of X is included in P. We show that this condition naturally arises from the geometry of quantum blobs and is closely related to Hardy's uncertainty principle and the Donoho--Stark inequalities. Using symplectic capacities and John ellipsoids, we associate a canonical covariance ellipsoid with every quantum polar pair and prove that it satisfies the quantum condition. The Robertson--Schrödinger inequalities follow as a consequence. This provides a non-statistical formulation of quantum indeterminacy from which the usual uncertainty relations emerge. which the usual uncertainty relations emerge.

Final revision; to appear in Proc. Royal Soc. A

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Geometric Topology (math.GT); Symplectic Geometry (math.SG)

arXiv:2605.01103 [quant-ph]

(or arXiv:2605.01103v4 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2605.01103 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.