Polar Duality and Quantum Mechanics

Mathematical points do not have any physical meaning: they are abstractions belonging to the Platonic realm of geometry. Still, In classical physics, mathematical points serve as fundamental elements, representing precise locations in space and time. These points facilitate a continuous description of physical phenomena through analysis and geometry, rooted in Newtonian principles. However, These abstract entities present challenges when transitioning to the quantum realm due to Heisenbergs principle of indeterminacy: in quantum mechanics, particles cannot be precisely localized, making the concept of points obsolete. To address this challenge, we propose a new approach involving the replacement of the ordinary position space with a covering of convex bodies; these sets represent the available knowledge about the position of a system, while their polar duals represent the best possible knowledge about the systems momenta. This view introduces a pointillism-like perspective, reminiscent of the painter Paul Signacs technique of using small, distinct dots of color to form an image. (Technically, our approach implies a more general geometric principle of indeterminacy than the usual expression using Heisenbergs uncertainty principle). To summarize, we aim to construct a substitute for a quantum phase space, extending previous work of ours to arbitrary convex subsets carried by Lagrangian manifolds. This extension requires advanced techniques from convex and symplectic geometry, as well as harmonic analysis.