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arXiv:hep-th/9212121 · International Journal of Modern Physics A

PHYSICAL NONLINEAR ASPECTS OF CLASSICAL AND QUANTUM q-OSCILLATORS

Vladimir I. Man’ko, Giuseppe Marmo, S. Solimeno, F. Zaccaria

Published Aug 10, 1993• 128 citations
arXivHTMLPDF

Abstract

The classical limit of quantum q-oscillators suggests an interpretation of the deformation as a way to introduce nonlinearity. Guided by this idea, we considered q fields, the partition function, and compute a consequence on the specific heat and second order correlation function of the q-oscillator which may serve for experimental checks for the nonlinearity.

DOI
10.1142/s0217751x93001454

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The classical limit of quantum q-oscillators suggests an interpretation of the deformation as a way to introduce nonlinearity. Guided by this idea, we considered q fields, the partition function, and compute a consequence on the specific heat and second order correlation function of the q-oscillator which may serve for experimental checks for the nonlinearity.
We consider a canonical ensemble of q oscillators. Using classical realization for q oscillator algebra,17 we calculate, for a small real q, the partition function and thermodynamic potentials F, E and S. We show that F reaches the minimum and E and S the maximum (as functions of the deformation parameter q) when q = 1 (for the classical oscillator). We argue about possible far-reaching consequences of this fact. As an application we obtain a first quantum correction to Planck's black body radiation law. We introduce the slightly deformed oscillator (SDO) model, which provides us with a significant amount of information about the system. When q = 1, our results are shown to coincide with classical results.
Nonlinearity of electromagnetic field vibrations described by q-oscillators is shown to produce essential dependence of second correlation functions on intensity and deformation of Planck distribution. Experimental tests of such nonlinearity are suggested. 1 on leave from Lebedev Physical Institute, Moscow 1 In the framework of investigations on quantum groups [1] the special case of the Heisenberg-Weyl algebra was first considered by Biedenharn [2] and Macfarlane [3]. They introduced ”q-oscillator operators ” starting from a deformed commutation relation containing a dimensionless parameter q. The physical sense of the quantun q-oscillator up to now was not clarified. In [4] a generalization to several modes has been analyzed. An attempt to connect quantum q-oscillator with relativistic oscillator model has been discussed in [5]. The q-oscillators and quantum group SUq(2) have also been considered
The interpretation of the q-deformed 1-D quantum harmonic oscillator is investigated for two definitions of q-deformation.This investigation is achieved by using Zaslavskii's method to obtain the Heisenberg equations of motion (quantum Liouville equations) in the undeformed phase space.These quantum Liouville equations exhibit a non-commutative geometry produce from the existence of the dilatation operator which is inherent in the q-deformation process.The classical limits of these equations are obtained by applying a special classical limiting condition to produce the classical Liouville equations of the q-deformed oscillator.These classical Liouville equations are solved by using the method of characteristics in order to obtain the classical probability distribution functions for this system.The 2-D and 3-D behaviors of these functions were then investigated using a computer visualization method.The results of the mathematical derivations together with the computer visualization method show that the classical limit of the quantum Liouville equations for the q-deformed 1-D quantum harmonic oscillator are statistical in nature where the nonlinearity parameter for the q-deformed oscillator is connected with h .This result conforms to that obtained by Ghosh et al. for the undeformed 1-D quantum harmonic oscillator.The obtained classical probability distribution functions exhibit whorl shapes that evolve with time in phase space that are similar to the shapes obtained for the 1-D classical q-deformed oscillator.These whorl shapes in phase space are similar to those introduced by Milburn for the 1-D classical anharmonic oscillator.This similarity results from the fact that the anharmonicity itself represents a kind of deformation with a frequency that is a function of amplitude.
Abstract Phenomenological models aiming to join gravity and quantum mechanics often predict effects that are potentially measurable in refined low-energy experiments. For instance, modified commutation relations between position and momentum, that account for a minimal scale length, yield a dynamics that can be codified in additional Hamiltonian terms. When applied to the paradigmatic case of a mechanical oscillator, such terms, at the lowest order in the deformation parameter, introduce a weak intrinsic nonlinearity and, consequently, deviations from the classical trajectory. This point of view has stimulated several experimental proposals and realizations, leading to meaningful upper limits to the deformation parameter. All such experiments are based on classical mechanical oscillators, i.e., excited from a thermal state. We remark indeed that decoherence, that plays a major role in distinguishing the classical from the quantum behavior of (macroscopic) systems, is not usually included in phenomenological quantum gravity models. However, it would not be surprising if peculiar features that are predicted by considering the joined roles of gravity and quantum physics should manifest themselves just on purely quantum objects. On the basis of this consideration, we propose experiments aiming to observe possible quantum gravity effects on macroscopic mechanical oscillators that are preliminary prepared in a high purity state, and we report on the status of their realization. Graphical abstract
The definition of the specific heat for a damped quantum system is a subtle issue. A possible approach is based on an effective partition function defined as the ratio of the partition functions of system plus bath and of the bath alone. For the free damped particle it has been shown, however, that the ensuing specific heat may become negative for appropriately chosen environments. We argue that this quantity should be interpreted as the change of the specific heat of the heat bath when the system degrees of freedom are coupled to it. While this difference may become negative, the involved specific heats themselves are always positive and therefore thermodynamic stability is always guaranteed. For a damped harmonic oscillator, instead of negative values, under appropriate conditions one can observe a dip in the difference of specific heats as a function of temperature. Minimal models containing a single oscillator heat bath are employed to elucidate the temperature dependence of the difference of specific heats. Finally, we comment on the consequences for the interpretation of the density of states based on the effective partition function.