How classical is the quantum universe?

TL;DR

This paper explores the classical foundations of quantum mechanics, demonstrating that key quantum concepts like the Schrödinger equation and uncertainty principle can be derived or formulated using classical physics and topology.

Contribution

It shows that the Schrödinger equation can be be derived from classical Hamiltonian mechanics and that the uncertainty principle can be expressed through classical symplectic topology.

Findings

Schrödinger equation derivable from Hamilton's equations

Uncertainty principle expressed via symplectic capacity

Quantum concepts have classical underpinnings

Abstract

We discuss two topics that are usually considered to be exclusively "quantum": the Schroedinger equation, and the uncertainty principle. We show (or rather recall) that the Schroedinger equation can be derived from Hamilton's equations using the metaplectic representation. We also show that the uncertainty principle, stated in the form of the Robertson-Schroedinger-Heisenberg inequalities can be formulated in perfectly classical terms using the topological notion of symplectic capacity.