Journal of Mathematical Physics, Volume 62, Issue 4, April 2021. 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 L2(μ) space on the circle associated with the oscillator’s invariant measure, μ, into a Hilbert space [math] of sections of a [math]-line bundle over Minkowski space. This bundle is equipped with a covariant derivative induced from an SO+(1, 1) gauge field 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 geometrical 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 eigenfrequencies corresponding to quantum eigenstates of zero (“vacuum”), positive (“matter”), and negative (“antimatter”) energies. 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 spaces, [math], associated with the time-τ transition kernel of a fractional diffusion on the circle. As shown in recent work, these spaces are unital Banach *-algebras of functions. It is found that the evolution of classical observables in these spaces takes place via a strongly continuous, unitary Koopman evolution group, which exhibits a stronger form of classical–quantum consistency than the L2(μ) case. Specifically, for every real-valued classical observable in [math], there exists a quantum mechanical observable, whose expectation value is consistent with classical function evaluation. This allows for a description of classical state space dynamics, classical statistics, and quantum statistics of the harmonic oscillator within a unified framework.

## Authors

## Departments

## Libraries

## Recent Articles

- David G. Seely
- Key chemical reaction pathways in a helium-nitrogen atmospheric glow discharge plasma based on a global model coupled with the genetic algorithm and dynamic programming
- Virtual probe stimulated tip-enhanced Raman spectroscopy: The extreme field enhancement in virtual-real probe dimer
- Improving cold-atom sensors with quantum entanglement: Prospects and challenges
- Prospects for application of ferroelectric manganites with controlled vortex density
- Interface engineering for high-efficiency perovskite solar cells
- Applications of chip-scale semiconductor metamaterials based on plasmon-induced transparency in modulation and sensing
- Auger recombination in narrow gap HgCdTe/CdHgTe quantum well heterostructures
- Quantum dynamics of the classical harmonic oscillator
- Integrated density of states: From the finite range to the periodic Airy–Schrödinger operator
- Quantum computation of dominant products in lithium–sulfur batteries
- Stochastic evaluation of fourth-order many-body perturbation energies
- New concepts for hydrogen fueling stations take form
- GRIN metamaterial generalized Luneburg lens for ultra-long acoustic jet
- A planar microwave resonator with odd resonance for calibration in permanent moisture sensing applications
- Electronic and magnetic properties of two-dimensional of FeX (X = S, Se, Te) monolayers crystallize in the orthorhombic structures
- Strain-modulated magnetization precession in skyrmion-based spin transfer nano-oscillator
- Structure evidence of Pna21 phase and field-induced transition of Pna21/R3c in Bi1−xSmxFe0.99Ti0.01O3 ceramics
- Equilibrium approach for modeling erosional failure of granular dams
- Highly solar-blind ultraviolet selective metal-semiconductor-metal photodetector based on back-illuminated AlGaN heterostructure with integrated photonic crystal filter