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arXiv:quant-ph/9503023

Quantum dynamics with two Planck constants and the semiclassical limit

Lajos Diósi

Published Mar 28, 1995• 10 citations
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Abstract

The mathematical possibility of coupling two quantum dynamic systems having two different Planck constants, respectively, is investigated. It turns out that such canonical dynamics are always irreversible. Semiclassical dynamics is obtained by letting one of the two Planck constants go to zero. This semiclassical dynamics will preserve positivity, as expected, so an improvement of the earlier proposals by Aleksandrov and by Boucher and Traschen is achieved. Coupling of quantized matter to gravity is illustrated by a simplistic example.

DOI
10.48550/arxiv.quant-ph/9503023

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The mathematical possibility of coupling two quantum dynamic systems having two different Planck constants, respectively, is investigated. It turns out that such canonical dynamics are always irreversible. Semiclassical dynamics is obtained by letting one of the two Planck constants go to zero. This semiclassical dynamics will preserve positivity, as expected, so an improvement of the earlier proposals by Aleksandrov and by Boucher and Traschen is achieved. Coupling of quantized matter to gravity is illustrated by a simplistic example.
The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing h0h_0h0​. For the second of these two extreme values, introduced operatorial algebra becomes equivalent to the algebra of observables of quantum mechanical system defined in the standard manner by operators in the Hilbert space. For the vanishing Planck constant, the generalized algebra gives the operator formulation of classical mechanics since it is equivalent to the algebra of variables of classical mechanical system defined, as usually, by functions over the phase space. In this way, the semiclassical limit of kinematical part of quantum mechanics is established through the generalized operatorial framework.
We set up a semiclassical approximation which helps us clarify by means of several simple examples the rich variety of time scale in the quantum domain. The underlying structure of quantum and classical mechanics is so completly different that it is naive to expect to reach a classical regime by counting powers of the quantum scale variant Planck's over 2pi. We show although it is possible to define a time scale for nonclassical phenomena, but it is impossible to characterize quantum dynamics through a unique time scale, such as Ehrenfest's time. We use simple systems to critically discuss and illustrate these features of the quantum-classical limit.
We discuss the semiclassical limit of the entanglement for the class of closed pure systems. By means of analytical and numerical calculations, we obtain two main results: (i) the short-time entanglement does not depend on Planck's constant and (ii) the long-time entanglement increases as more semiclassical regimes are attained. On one hand, this result is in contrast with the idea that the entanglement should be destroyed when the macroscopic limit is reached. On the other hand, it emphasizes the role played by decoherence in the process of emergence of the classical world. We also found that, for Gaussian initial states, the entanglement dynamics may be described by an entirely classical entropy in the semiclassical limit.
In this paper we have investigated the classical limit in Bohmian quantum cosmology. It is observed that in the quantum regime where the quantum potential is greater than the classical one, one has an expansion in terms of negative powers of the Planck constant. But in the classical limit there are regions having positive powers of the Planck constant, and regions having negative powers and also regions having both. The conclusion is that the Bohmian classical limit cannot be obtained by letting the Planck constant goes to zero.
In this paper we have investigated the classical limit in Bohmian quantum cosmology. It is observed that in the quantum regime where the quantum potential is greater than the classical one, one has an expansion in terms of negative powers of the Planck constant. But in the classical limit there are regions having positive powers of the Planck constant, and regions having negative powers and also regions having both. The conclusion is that the Bohmian classical limit cannot be obtained by letting the Planck constant goes to zero.