Second quantization for classical nonlinear dynamics

• URL: http://arxiv.org/abs/2501.07419v1
• Date: Mon, 13 Jan 2025 15:36:53 GMT
• Title: Second quantization for classical nonlinear dynamics
• Authors: Dimitrios Giannakis, Mohammad Javad Latifi Jebelli, Michael Montgomerry, Philipp Pfeffer, Jörg Schumacher, Joanna Slawinska,
• Abstract summary: We propose a framework for representing the evolution of observables of measure-preserving ergodic flows through infinite-dimensional rotation systems on tori. We show that their Banach algebra spectra, $sigma(F_w(mathcal H_tau)$, decompose into a family of tori of potentially infinite dimension. Our scheme also employs a procedure for representing observables of the original system by reproducing functions on finite-dimensional tori in $sigma(F_w(mathcal H_tau)$ of arbitrarily large degree.
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• 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 $F_w(\mathcal H_\tau)$ generated by a 1-parameter family of reproducing kernel Hilbert spaces $\mathcal H_\tau$, and endowed with commutative Banach algebra structure under the symmetric tensor product using a subconvolutive weight $w$. We describe the construction of the spaces $F_w(\mathcal H_\tau)$ and show that their Banach algebra spectra, $\sigma(F_w(\mathcal H_\tau))$, decompose into a family of tori of potentially infinite dimension. Spectrally consistent unitary approximations $U^t_\tau$ of the Koopman operator acting on $\mathcal H_\tau$ 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 $\sigma(F_w(\mathcal H_\tau))$ of arbitrarily large degree, with coefficients determined from pointwise products of eigenfunctions of $U^t_\tau$. 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.

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