Quantum dynamics of the classical harmonic oscillator

Abstract

A correspondence is established between measure-preserving, ergodic dynamics of a classical harmonic oscillator and a quantum mechanical gauge theory on two-dimensional Minkowski space. This correspondence is realized through an isometric embedding of the $L^2(\mu)$ space on the circle associated with the oscillator's invariant measure, $ \mu $, into a Hilbert space $\mathcal{H}$ of sections of a $\mathbb{C}$-line bundle over Minkowski space. This bundle is equipped with a covariant derivative induced from an SO$^+$(1,1) gauge field (connection 1-form) on the corresponding inertial frame bundle, satisfying the Yang-Mills equations. Under this embedding, the Hamiltonian operator of a Lorentz-invariant quantum system, constructed as a natural Laplace-type operator on bundle sections, pulls back to the generator of the unitary group of Koopman operators governing the evolution of classical observables of the harmonic oscillator, with Koopman eigenfunctions of zero, positive, and negative eigenfrequency corresponding to quantum eigenstates of zero ("vacuum"), positive ("matter"), and negative ("antimatter") energy. The embedding also induces a pair of operators acting on classical observables of the harmonic oscillator, exhibiting canonical position-momentum commutation relationships. These operators have the structure of order-$1/2$ fractional derivatives, and therefore display a form of non-locality. In a second part of this work, we study a quantum mechanical representation of the classical harmonic oscillator using a one-parameter family of reproducing kernel Hilbert algebras associated with the transition kernel of a fractional diffusion on the circle, where a stronger notion of classical-quantum consistency is established than in the $L^2(\mu)$ case.