A Pseudo-Quantum Triad: Schrödinger's Equation, the Uncertainty Principle, and the Heisenberg Group

We show that the paradigmatic quantum triad "Schrödinger equation–Uncertainty principle–Heisenberg group" emerges mathematically from classical mechanics. In the case of the Schrödinger equation, this is done by extending the metaplectic representation of linear Hamiltonian flows to arbitrary flows; for the Heisenberg group this follows from a careful analysis of the notion of phase of a Lagrangian manifold, and for the uncertainty principle it suffices to use tools from multivariate statistics together with the theory of John's minimum volume ellipsoid. Thus, the mathematical structure needed to make quantum mechanics emerge already exists in classical mechanics.