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.
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BibTeX
@misc{gosson2026quantum,
title = {{On Quantum Indeterminacy and the Uncertainty Principle}},
author = {Maurice de Gosson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.01103},
eprint = {2605.01103},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.01103},
}
RIS
TY - UNPB
TI - On Quantum Indeterminacy and the Uncertainty Principle
AU - Maurice de Gosson
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2605.01103
UR - https://arxiv.org/abs/2605.01103
AB - 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.
ER -