The Symplectic Egg

TL;DR

This paper explores the transition from classical to quantum mechanics through symplectic geometry, highlighting key theorems and proposing a symplectic perspective on the uncertainty principle.

Contribution

It introduces a symplectic formulation of the quantum uncertainty principle, connecting classical symplectic geometry to quantum mechanics insights.

Findings

Derivation of Gromov's non-squeezing theorem and its dynamical interpretation

A novel symplectic formulation of the quantum uncertainty principle

Potential for new discoveries in quantum mechanics using symplectic methods

Abstract

We invite the reader (presumably an upper level undergraduate student) to a journey leading from the continent of Classical Mechanics to the new territories of Quantum Mechanics. We'll be riding the symplectic camel and have William of Occam as travel companion, so no excess baggage is allowed. The first part of our trip takes us from the symplectic egg to Gromov's non-squeezing theorem and its dynamical interpretation. The second part leads us to a symplectic formulation of the quantum uncertainty principle, which opens the way to new discoveries.