Second quantization for classical nonlinear dynamics

TL;DR

This work develops a rigorous second-quantization framework for classical, measure-preserving ergodic flows by embedding the Koopman evolution into a weighted Fock space $F_w(\mathcal{H}_\tau)$ built from a family of RKHSs $\mathcal{H}_\tau$. Regularized, diagonalizable generators $W_\tau$ yield unitary evolutions $U^t_\tau$ that lift to rotation systems on spectral tori $\sigma(F_w(\mathcal{H}_\tau))$, providing asymptotically consistent topological models of Koopman dynamics. Observables are embedded into the Fock space and represented on finite-dimensional tori via polynomial (Fourier) expansions, enabling data-driven, kernel-based approximations with positivity preservation. The approach is demonstrated numerically on Stepanoff flows and Lorenz 63, showing improved predictive skill over traditional Koopman-eigenfunction methods while maintaining a provable convergence framework in the appropriate limits. This framework offers a flexible bridge between ergodic theory, RKHS methods, and quantum-inspired representations, with potential pathways to quantum computing implementations and broader applicability to nonlinear dynamical systems. All mathematics are presented with explicit operators and spectra, and the asymptotic consistency results support reliable approximation of continuous spectra in practice.

Abstract

Using techniques from many-body quantum theory, we propose a framework for representing the evolution of observables of measure-preserving ergodic flows through infinite-dimensional rotation systems on tori. This approach is based on a class of weighted Fock spaces $F_w(\mathcal H_τ)$ generated by a 1-parameter family of reproducing kernel Hilbert spaces $\mathcal H_τ$, and endowed with commutative Banach algebra structure under the symmetric tensor product using a subconvolutive weight $w$. We describe the construction of the spaces $F_w(\mathcal H_τ)$ and show that their Banach algebra spectra, $σ(F_w(\mathcal H_τ))$, decompose into a family of tori of potentially infinite dimension. Spectrally consistent unitary approximations $U^t_τ$ of the Koopman operator acting on $\mathcal H_τ$ are then lifted to rotation systems on these tori akin to the topological models of ergodic systems with pure point spectra in the Halmos--von Neumann theorem. Our scheme also employs a procedure for representing observables of the original system by polynomial functions on finite-dimensional tori in $σ(F_w(\mathcal H_τ))$ of arbitrarily large degree, with coefficients determined from pointwise products of eigenfunctions of $U^t_τ$. This leads to models for the Koopman evolution of observables on $L^2$ built from tensor products of finite collections of approximate Koopman eigenfunctions. Numerically, the scheme is amenable to consistent data-driven implementation using kernel methods. We illustrate it with applications to Stepanoff flows on the 2-torus and the Lorenz 63 system. Connections with quantum computing are also discussed.

Figures (9)

• Figure 1: Generating vector field \ref{['eq:vec_stepanoff']} of the Stepanoff flow for $\alpha = \sqrt{20}$.
• Figure 2: Time evolution of the von Mises density function from \ref{['eq:von_mises']} with $\gamma = 1$ under the Stepanoff flow ($U^{t}f$; left column), the classical approximation based on $2d+1$ eigenfunctions ($f^{(t)}_\text{cl}$; center column), and the Fock space approximation for torus dimension $d = 50$ and degree $m = 4$ ($f^{(t)}_\text{Fock}$; right column). Rows from top to bottom show snapshots at the evolution times $t=0, 0.5, 1, 2, 4$, respectively.
• Figure 3: Spectrum of the regularized generator $W_{\tau,l,N}$ for the Stepanoff flow (left) and L63 system (right). Eigenfrequencies $\omega_{j,\tau,l,N}$ are plotted versus their corresponding Dirichlet energies from the RKHS $\mathcal{H}_\tau$. The plotted points are colored by the logarithms of the expansion coefficient amplitudes $\lvert \langle f, \xi_{j,\tau,l,N}\rangle_{H_N}\rvert$ of the prediction observable $f$ in the eigenbasis $\xi_{j,\tau,l,N}$.
• Figure 4: Real and imaginary parts of representative eigenfunctions $\zeta_{j,\tau,l,N}$ for the Stepanoff flow. The eigenfunctions shown have index $j=1, 8, 97$ with respect to the Dirichlet energy ordering, and are members of the complex-conjugate (nonconstant) eigenfunction pairs with the 7th, 1st, 5th largest projection amplitudes $\lvert \langle \xi_{j,\tau,l,N}, f \rangle_{H_N}\rvert$, respectively.
• Figure 5: Error in the classical and Fock space approximations from \ref{['fig:evo_stepanoff']} (center and right columns, respectively) relative to the true Stepanoff evolution. The true evolution is plotted in the left column for reference.

Theorems & Definitions (26)

• Theorem 1
• Definition 2
• Theorem 3
• Definition 4
• Remark 5
• Theorem 6
• proof
• Proposition 7
• proof
• Proposition 8