Quantization of nonlinear non-Hamiltonian systems

Abstract:
Several important dynamical systems are in ℝ2, 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 Schrödinger 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 have shown, constructively, that every polynomial system admits a physical generator of time evolution in the form of a Lindbladian [1]. In this talk I will explain the problem of quantization, briefly discuss some of its history, and illustrate our solution to the problem with two well-known examples from nonlinear dynamics—the saddle-node bifurcation, and the van der Pol oscillator.

[1] Quantization of nonlinear non-Hamiltonian systems, Phys. Rev. E 112, 054206
https://journals.aps.org/pre/abstract/10.1103/l54l-sff5