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Journal of Mathematical Physics

Dynamical Quantization

E. Onofri, Massimo Pauri

Published Apr 1, 1972• 27 citations
Publisher

Abstract

An intrinsic quantization procedure based on higher symmetries of classical dynamical systems and utilizing the techniques of van Hove and Souriau is proposed. The procedure is intrinsically Hamiltonian but not explicitly canonical in that the Heisenberg algebra plays no fundamental role. The proposed method is applied to the n-dimensional harmonic oscillator and to the n-dimensional hydrogen atom. This approach seems to provide the first intrinsic justification of the success of ordinary correspondential quantization for this last system.

DOI
10.1063/1.1666012

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An intrinsic quantization procedure based on higher symmetries of classical dynamical systems and utilizing the techniques of van Hove and Souriau is proposed. The procedure is intrinsically Hamiltonian but not explicitly canonical in that the Heisenberg algebra plays no fundamental role. The proposed method is applied to the n-dimensional harmonic oscillator and to the n-dimensional hydrogen atom. This approach seems to provide the first intrinsic justification of the success of ordinary correspondential quantization for this last system.
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Noncanonical quantization schemes have been proposed in quantum physics long time ago. Recalling some of them I associate these investigations to the quantum mechanical im-plications of the canonically inequivalent Hamilton formula-tions of classical Newtonian dynamics. As the simplest illus-tration of such alternative Hamiltonian descriptions I consider one dimensional system of two particles interacting harmoni-cally. Description of the motion of their center of mass is stan-dard and guarantees the Galilean covariance but the internal motion is described as a noncanonical oscillator for which the Hamiltonian and the total energy are not related by a simple proportionality. Alternative Hamiltonians lead to alternative Poisson structures which may be considered as classical ana-logues of the noncanonical quantum commutation rules. Gen-Concepts of Physics, Vol. I (2004) 11 erated alternative quantizations are consistent with the same Heisenberg equations of motion, i.e., they satisfy the Wigner principle of quantization. For the one dimensional harmonic oscillator the simplest noncanonical algebraic description is given in terms of the Lie algebra so(p, q), p+ q = 3. In such a case the Hamiltonian is still a linear function while the energy is a quadratic polynomial of the main quantum number. This leads to nonstandard termodynamical properties obeyed by such a noncanonical oscillator. The 3-dimensional harmonic oscillator also has a noncanonical description given in terms of the simple Lie algebra, so(p, q), p+ q = 5 in the case under consideration. The most important physical consequence of such a description is the noncommutativity of the coordinate operators, interpreted to be an implication of the noncommu-tative geometry of the underlying spacetime.
We propose a new method of quantization of a wide class of dynamical systems that originates directly from the equations of motion. The method is based on the correspondence between the classical and the quantum Poisson brackets, postulated by Dirac. This correspondence applied to open (non-Hamiltonian) systems allows one to point out the way of transition from the quantum description based on the Lindblad equation to the dynamical description of their classical analogs by the equations of motion and vice versa. As the examples of using of the method we describe the procedure of the quantization of three widely considered dynamical systems: (1) the harmonic oscillator with friction, (2) the oscillator with a nonlinear damping that simulates the process of the emergence of the limit cycle, and (3) the system of two periodic rotators with a weak interaction that synchronizes their oscillations. We discuss a possible application of the method for a description of quantum fluctuations in Josephson junctions with a strong damping and for the quantization of open magnetic systems with a dissipation and a pumping.