On Quantum Indeterminacy and the Uncertainty Principle

Maurice de Gosson

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arXiv · 2605.01103

On Quantum Indeterminacy and the Uncertainty Principle

Maurice de Gosson

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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--Schr\"odinger 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.

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Maurice de Gosson. 2026-05-01. On Quantum Indeterminacy and the Uncertainty Principle. https://arxiv.org/abs/2605.01103

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