Canonical-quantization for classical dynamic Neuman-type systems in frames of the Moser spectral approach
Abstract
The classical Neumann type dynamical systems describe the motion of a particles constrained to live on an $N$-sphere $S^N$ in $(N+l)$-dimensional space $\mathbb{R}^{N+1}$ and submitted to quasi-harmonic forces. Following the Moser spectral approach to a connection of the infinite dimensional finite-zoned by Lax dynamical systems with the finite dimensional Neumann type systems on sphere in $\mathbb{R}^{N+1}$, the regular procedure to quantize of them suitably is supposed. The quantum expression of the commuting conserved currents for the quantum Neumann type dynamical systems are determined in a general case via the Dirac canonical quantization procedure.
References
Integrable dynamical systems / Yu. A. Mytropolsky, N. N. Bogoliubov (jr.), A. K. Prykarpatsky, V. G. Samoilenko.— K.: Naukova dumka, 1987.—286 p. (in Russian).
Prykarpatsky A., Mykytiuk I. Algebraic aspects of integrabilily of nonlinear dynamical systems on the manifolds.— K. : Naukova dumka, 1991.— 286 p. (in Russian).
Avan J., Talon M. Alternative Lax structures for the classical and quantum Neumann model // Phys. Lett. B — 1991.— 268, N 2.— P. 209—216.
Moser J. Geometry of quadric and spectral theory // Proc. Chern Symp., 1979.— New York: Springer, 1980.— P. 147—188.
Dirac P. A. M. Principles of quantum mechanics.— Oxford, 1935. 300 p.
Samoilenko V. Gr., Prykarpatsky A. K., Mykytiuk I. V. Abelian integrals, integrable dynamical systems of the Neuman-Rosohatius type and Lax representation// Ukr. Math. J.— 1989.— 41, N 8.— P. 1094—1100 (in Russian).
Hurt N. Geometric quantization in action.— Reidel, 1983.— 336 p.
Copyright (c) 1992 М. М. (jr.), Bogoliubov , I. V. Mykytiuk , В. M. Fil', A. К. Prykarpatsky
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