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arXiv:2412.10314

Quantum Indeterminacy and Polar Duality: a Probabilistic Approach

Maurice A. de Gosson

Published Dec 13, 2024• 0 citations
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Abstract

We present a probabilistic argument supporting the application of polar duality, as discussed in our previous work, to express the indeterminacy principle of quantum mechanics. Our approach combines the properties of the Mahler volume of a convex body with the Donoho--Stark uncertainty principle from harmonic analysis, which characterizes the concentration of a function and its Fourier transform. The central result demonstrates that the sum of the probabilities of position concentration near a convex body and momentum concentration near its polar dual is equal to one, with an error term that diminishes rapidly as the number of degrees of freedom increases. This result motivates the interpretation of polar duality as a kind of geometric Fourier transform.

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We present a probabilistic argument supporting the application of polar duality, as discussed in our previous work, to express the indeterminacy principle of quantum mechanics. Our approach combines the properties of the Mahler volume of a convex body with the Donoho--Stark uncertainty principle from harmonic analysis, which characterizes the concentration of a function and its Fourier transform. The central result demonstrates that the sum of the probabilities of position concentration near a convex body and momentum concentration near its polar dual is equal to one, with an error term that diminishes rapidly as the number of degrees of freedom increases. This result motivates the interpretation of polar duality as a kind of geometric Fourier transform.
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.
We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke-Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho--Stark principle from the point of view of quantum polarity.
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.
We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke-Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho-Stark principle from the point of view of quantum polarity.