Quantization of Hamiltonian and non-Hamiltonian systems

Sergey A. Rashkovskiy; Sergey A. Rashkovskiy

doi:10.3934/cam.2023014

Communications in Analysis and Mechanics

2023, Volume 15, Issue 2: 267-288. doi: 10.3934/cam.2023014

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Research article

Quantization of Hamiltonian and non-Hamiltonian systems

Sergey A. Rashkovskiy ^,

Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Vernadskogo Ave., 101/1, Moscow, 119526, Russia

Received: 15 April 2023 Revised: 05 June 2023 Accepted: 05 June 2023 Published: 16 June 2023
81Sxx, 81P10, 81Q80

The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.
- Hamiltonian and non-Hamiltonian systems,
- quantization,
- Hamilton-Jacobi theory,
- Kostin equation,
- non-linear Dirac equation
Citation: Sergey A. Rashkovskiy. Quantization of Hamiltonian and non-Hamiltonian systems[J]. Communications in Analysis and Mechanics, 2023, 15(2): 267-288. doi: 10.3934/cam.2023014

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Abstract

The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.

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