  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.
