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arXiv:quant-ph/0511068 · Journal of Physics A Mathematical and General

Variational approach to dequantization

Ricardo Antonio Mosna, Ian P. Hamilton, Luigi Delle Site

Published Mar 22, 2006• 12 citations
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Abstract

We present a dequantization procedure based on a variational approach whereby quantum fluctuations latent in the quantum momentum are suppressed. This is done by adding generic local deformations to the quantum momentum operator which give rise to a deformed kinetic term quantifying the amount of ``fuzzyness'' caused by such fluctuations. Considered as a functional of such deformations, the deformed kinetic term is shown to possess a unique minimum which is seen to be the classical kinetic energy. Furthermore, we show that extremization of the associated deformed action functional introduces an essential nonlinearity to the resulting field equations which are seen to be the classical Hamilton-Jacobi and continuity equations. Thus, a variational procedure determines the particular deformation that has the effect of suppressing the quantum fluctuations, resulting in dequantization of the system.

DOI
10.1088/0305-4470/39/14/l03

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A Nonlinear qqq-Deformed Schrödinger Equation
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We study the connection between dissipation and reality in macroscopic quantum systems. We present the following scenario; if we consider the dynamics of a `partial' wave function, the dissipation is represented as a nonlocal term and it causes destructive interference to suppress the quantum fluctuation. Using the variational method, we confirm that this dissipation term is a reasonable extension of the standard (Schrödinger) description for isolated systems, from which we also derive the classical action. Consequently, in macroscopic systems, the states whose time-integrated dissipation takes an extreme value come true. This description, which is consistent with our sense of reality, coexists with the usual linear-time-dependent description.
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The dynamical variational approach based on geometrical quantization is demonstrated to be capable in describing the most important quantum mechanical quantities. In particular, the consistent prescription for calculating the transition probabilities is presented. For several reasons, the method is expected to be better under control than the semiclassical methods in treating the systems whose classical counterparts are chaotic. The formal considerations are illustrated using an exactly solvable SU(3)-spin system
Abstract We develop a deformation quantization approach to quantum mechanics that exhibits a nonzero minimal uncertainty in position through the modification of commutation relations for the operators of position and momentum. Deformation quantization is a method of quantizing a classical Hamiltonian system by suitably deforming the Poisson algebra of the system. The developed theory of deformation quantization is non-formal. An appropriate integral formula for the star-product is introduced, along with a suitable space of functions on which the star-product is well defined. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, an example of states of maximal localization is given in terms of generalized Wigner functions.