In this chapter we discuss theories which are rigorous in their formulation but which in order to be useful need to be modified by introducing approximations of some kind. The approximations we are interested in are those which involve introduction of classical mechanical concepts, that is, the classical picture and/or classical mechanical equations of motion in part of the system. At this point, we wish to distinguish between “the classical picture,” which is obtained by taking the classical limit ħ → 0 and the appearance of “classical equations of motion.” The latter may be extracted from the quantum mechanical formulation without taking the classical limit—but, as we shall see later by introducing a certain parametrization of quantum mechanics. Thus there are two ways of introducing classical mechanical concepts in quantum mechanics. In the first method, the classical limit is defined by taking the limit ħ → 0 either in all degrees of freedom (complete classical limit) or in some degrees of freedom (semi-classical theories). We note in passing that the word semi-classical has been used to cover a wide variety of approaches which have also been referred to as classical S-matrix theories, quantum-classical theories, classical path theory, hemi-quantal theory, Wentzel Kramer-Brillouin (WKB) theories, and so on. It is not the purpose of this book to define precisely what is behind these various acronyms. We shall rather focus on methods which we think have been successful as far as practical applications are concerned and discuss the approximations and philosophy behind these. In the other approach, the ħ-limit is not taken—at least not explicitly— but here one introduces “classical” quantities, such as, trajectories and momenta as parameters, and derives equations of motion for these parameters. The latter method is therefore one particular way of parameterizing quantum mechanics. We discuss both of these approaches in this chapter. The Feynman path-integral formulation is one way of formulating quantum mechanics such that the classical limit is immediately visible [3]. Formally, the approach involves the introduction of a quantity S, which has a definition resembling that of an action integral [101].
Classical Chaos Described by a Density Matrix
