Pointillisme \`a la Signac and Construction of a Quantum Fiber Bundle Over Convex Bodies

TL;DR

This paper constructs a fiber bundle over convex bodies using symplectic and convex geometry, linking geometric quantum states to quantum blobs and Gaussian wavepackets, offering a new geometric perspective on quantum phase space.

Contribution

It introduces a novel fiber bundle model over ellipsoids that connects convex geometry, symplectic geometry, and quantum states, providing a geometric framework for quantum phase space analysis.

Findings

Geometric quantum states correspond to products of convex bodies and their polar duals.

Quantum blobs are characterized as minimal symplectic invariant regions compatible with uncertainty.

Equivalence classes of geometric quantum states are in one-to-one correspondence with Gaussian wavepackets.

Abstract

We use the notion of polar duality from convex geometry and the theory of Lagrangian planes from symplectic geometry to construct a fiber bundle over ellipsoids that can be viewed as a quantum-mechanical substitute for the classical symplectic phase space. The total space of this fiber bundle consists of geometric quantum states, products of convex bodies carried by Lagrangian planes by their polar duals with respect to a second transversal Lagrangian plane.. Using the theory of the John ellipsoid we relate these geometric quantum states to the notion of "quantum blobs" introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets.