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 $L^2(μ)$ space on the circle associated with the oscillator's invariant measure, $ μ$, 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 $L^2(μ)$ case.

Figures (5)

• Figure 1: Commutative diagrams illustrating the correspondence between classical states/observables of the harmonic oscillator and quantum states/observables on the $L^2(\mu)$ space associated with the invariant measure. The diagram on the left shows how classical states (points in the circle $S^1$) embed injectively into Radon probability measures $\mathcal{P}(S^1)$ via the map $\delta$ mapping into Dirac measures, which in turn map injectively into nonzero RKHS functions under the embedding $\mathbb K_\tau$, and then into quantum states and generalized quantum states on $L^2(\mu)$ via the maps $\Pi$ and $\iota$, respectively. The composition $\mathbb K_\tau \circ \delta$ corresponds to the RKHS feature map $F_\tau$ (not shown). These embeddings are all equivariant with the dynamical evolution maps at each level, as depicted in the diagram. The top diagram in the right shows the dynamical equivariance properties of the Banach algebra homomorphism $T$, mapping real-valued continuous functions on $S^1$ to bounded, self-adjoint multiplication operators in $\mathcal{A}(L^2(\mu))$. The middle diagram shows the equivariance properties of $\Omega_\tau : \mathcal{A}(L^2(\mu)) \to C_\mathbb{R}(S^1)$, which is not an algebra homomorphism. The bottom diagram shows the dynamical equivariance properties of the transpose map $T':\mathcal{A}'(L^2(\mu)) \to \mathbb C_\mathbb{R}'(S^1)$.
• Figure 2: As in Fig. \ref{['figCommutL2']}, but for commutative diagrams showing the relationships between the quantum formulations of the circle rotation and the quantum harmonic oscillator on Minkowski space. Note that in the left-hand diagram we abuse notation, using $\iota$ to denote the inclusion maps of both $\mathcal{Q}(L^2(\mu))$ into $\mathcal{A}'(L^2\mu))$ and $\mathcal{Q}(\mathcal{H})$ into $\mathcal{A}'(\mathcal{H})$.
• Figure 3: Wavefunction $\psi^{\sigma_x}_{\theta,\tau}$ associated with the embedding of the classical states (points) on the circle to states of the quantum harmonic oscillator on two-dimensional Minkowski space. Here, the wavefunction is shown as a function of the inertial coordinates $(x^0,x^1)$ of Minkowski space for $\tau = 10^{-3}$ and representative values of $\theta \in S^1$ in the range $[0, \pi]$, using an arbitrary normalization with respect to $L^@(\nu)$ norm.
• Figure 4: As in Fig. \ref{['figCommutL2H']}, but for commutative diagrams showing the relationships between the classical/quantum dynamics associated with the RKHAs $\hat{\mathcal{K}}_\tau$ on the circle and the quantum harmonic oscillator on Minkowski space. In the commutative diagram in the bottom right, $\mathcal{T}(\hat{\mathcal{K}}_\tau)$ denotes the closed subalgebra of $\mathcal{A}(\hat{\mathcal{K}}_\tau)$ consisting of (bounded) multiplication operators by functions in $\hat{\mathcal{K}}_{\tau,\mathbb R}$, and $\mathcal{A}_{\mathcal{T}}(\mathcal{H})$ the image of $\mathcal{T}(\hat{\mathcal{K}}_\tau)$ in $\mathcal{A}(\mathcal{H})$ under the map $\hat{\mathcal{W}}_\tau^+ : A \mapsto \hat{\mathcal{W}}_\tau A \hat{\mathcal{W}}_\tau^*$. Note that such a diagram cannot be drawn in the $L^2(\mu)$ setting of Fig. \ref{['figCommutKH']}.
• Figure 5: As in Fig. \ref{['figPsi']}, but for the wavefunction for the RKHA associated with the fractional diffusion on the circle. Notice the smaller-scale oscillatory features developing due to the linear growth of the eigenvalues of $\mathcal{L}^{1/2}$, as opposed to the quadratic growth of the eigenvalues of $\mathcal{L}$.

Theorems & Definitions (31)

• Theorem 1: Classical-quantum correspondence based on $L^2(\mu)$
• Theorem 2: Canonically commuting operators for the circle rotation
• Theorem 3: Classical-quantum correspondence based on $\hat{\mathcal{K}}_\tau$
• Proposition 4
• Proposition 5
• Proposition 6
• Remark
• Theorem 7
• Lemma 8
• proof