Quantum Indeterminacy and Polar Duality: a Probabilistic Approach

TL;DR

This paper introduces a probabilistic framework linking quantum indeterminacy with polar duality, utilizing convex geometry and harmonic analysis to quantify the uncertainty principle in quantum mechanics.

Contribution

It presents a novel probabilistic argument connecting polar duality with quantum uncertainty, incorporating Mahler volume and the Donoho--Stark principle.

Findings

Sum of probabilities near a convex body and its polar dual approaches one

Error term diminishes rapidly with increasing degrees of freedom

Supports interpreting polar duality as a geometric Fourier transform

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.