Abstract
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.
Highlights
The notion of duality is omnipresent in science and philosophy, and in human thinking [38]
In this article we introduce a new kind of duality in quantum mechanics, having its roots in convex geometry
This example suggests that the uncertainty principle (UP) can be expressed using polar duality, which is a tool from convex geometry
Summary
The notion of duality is omnipresent in science and philosophy, and in human thinking [38]. Duality in science is usually implemented using a transformation which serves as a dictionary for translating between two different representations of an object. In quantum mechanics this role is played by the Fourier transform which allows one to switch from the position representation to the momentum representation. In this article we introduce a new kind of duality in quantum mechanics, having its roots in convex geometry. While the Fourier transform turns a function in x-space into a function in p-space our duality turns a set of positions into a set of momenta: it is a kind of proto-Fourier transform operating between sets, and
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