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Journal of Mathematical Physics

On the classical limit of phase-space formulation of quantum mechanics: Entropy

Lipo Wang

Published Feb 1, 1986• 10 citations
Publisher

Abstract

The classical limits of phase-space formulation of quantum mechanics are studied. As a special example, some properties of both quantum mechanical and classical entropies are discussed in detail.
DOI
10.1063/1.527247

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We discuss the unresolved problem of proving rigorously that in the classical limit ℏ→0, the quantum-thermodynamic entropy functional tends to the classical entropy functional. We state rather restrictive conditions that define the general problem of finding a complete classical phase-space representation of quantum kinematics. Whether the problem admits of solutions remains an unresolved question. We discuss a physically interesting attempt to relate the structure in the classical limit ℏ→0 of the well-known Blokhintzev, Wigner, and Wehrl phase-space functions to the spectral expansion of the quantum state operator.
We study information measures in quantum mechanics, with the particular emphasis on providing a quantification of the notion of predictability and classicality. Our primary tool is the Shannon-Wehrl entropy I. We give a precise criterion for phase space classicality of states and argue that in view of this (a) I provides a good measure for the degree of deviation from classicality in closed systems and (b) $I\ensuremath{-}S (S$ the von Neumann entropy) plays the same role in open quantum system. We examine particular examples in nonrelativistic quantum mechanics. Finally we generalize the discussion into the field theory case, and (this being one of our main motivations) we comment on the field classicalization in early universe cosmology.
The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.
The concept of quantum phase space offers a view on quantum mechanics, which is different from the standard Hilbert space approach, but which more closely resembles the classical phase space. Due to the properties of quantum mechanics there are several equivalent quantum phase space descriptions, and one cannot always prefer one or another as they all have certain merits and drawbacks. For example, the Husimi-Kano Q function is a probability distribution and thus gives rise to entropic quantities, namely the Renyi-Wehrl entropies, of which several properties are known. The Wigner function, on the other hand, has an easier physical explanation, but may take negative values. In this article, we investigate entropic measures of localisation for a state in quantum phase space by using the Beckner-Brascamp-Lieb inequality to relate different phase-space functions.
The formalism of classical and quantum mechanics on phase space leads to symplectic and Heisenberg group representations, respectively. The Wigner functions give a representation of the quantum system using classical variables. The correspondence between the classical and the quantum criterion of separability for the system is obtained in terms of these functions. Entanglement is generic and separability is special. Some applications are discussed in commonly occuring examples and possibly in exotic systems.
Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket and a quasidensity operator that is not positive definite. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Quantum mechanics is then viewed as a limiting form of classical mechanics, as Planck's constant approaches zero, rather than the other way around. The forms of semiquantum approximations to classical mechanics, analogous to semiclassical approximations to quantum mechanics, are indicated.