Related papers: Quantization of nonlinear non-Hamiltonian systems

Quantization of nonlinear non-Hamiltonian systems

URL: http://arxiv.org/abs/2503.06939v2
Date: Thu, 03 Apr 2025 07:25:41 GMT
Title: Quantization of nonlinear non-Hamiltonian systems
Authors: Andy Chia, Wai-Keong Mok, Leong-Chuan Kwek, Changsuk Noh,
Abstract summary: In developing quantum theory, Dirac and others realized that classical Hamiltonian systems can be mapped to their quantum counterparts via canonical quantization.<n>Here we resolve whether non-Hamiltonian systems can be quantized systematically while respecting the same physical requirements.<n>By leveraging open-systems theory, we prove constructively that every system admits a physical generator of time evolution in the form of a Lindbladian.
Score: 0.5249805590164902
License: http://creativecommons.org/licenses/by/4.0/
Abstract: Several important dynamical systems are in $\mathbb{R}^2$, defined by the pair of differential equations $(x',y')=(f(x,y),g(x,y))$. A question of fundamental importance is how such systems might behave quantum mechanically. In developing quantum theory, Dirac and others realized that classical Hamiltonian systems can be mapped to their quantum counterparts via canonical quantization. The resulting quantum dynamics is always physical, characterized by completely-positive and trace-preserving evolutions in the Schr\"{o}dinger picture. However, whether non-Hamiltonian systems can be quantized systematically while respecting the same physical requirements has remained a long-standing problem. Here we resolve this question when $f(x,y)$ and $g(x,y)$ are arbitrary polynomials. By leveraging open-systems theory, we prove constructively that every polynomial system admits a physical generator of time evolution in the form of a Lindbladian. We call our method cascade quantization, and demonstrate its power by analyzing several paradigmatic examples of nonlinear dynamics such as bifurcations, noise-activated spiking, and Li\'{e}nard systems. In effect, our method can quantize any classical system whose $f(x,y)$ and $g(x,y)$ are analytic with arbitrary precision. More importantly, cascade quantization is exact. This means restrictive system properties usually assumed in the literature to facilitate quantization, such as weak nonlinearity, rotational symmetry, or semiclassical dynamics, can all be dispensed with by cascade quantization. We also highlight the advantages of cascade quantization over existing proposals, by weighing it against examples from the variational paradigm using Lagrangians, as well as non-variational approaches.

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