-------------------------------------------- S1t. The classical limit via coherent states -------------------------------------------- One method for producing classical mechanics from a quantum theory is by looking at coherent states of the quantum theory. The standard (Glauber) coherent states have a localized probability distribution in classical phase space? whose center follows the classical equations of motion when the Hamiltonian is quadratic in positions and momenta. (For nonquadratic Hamiltonians, this only holds approximately over short times. For example, for the 2-body problem with a 1/r^2 interaction, Glauber coherent states are not preserved by the dynamics. In this particular case, there are, however, alternative SO(2,4)-based coherent states that are preserved by the dynamics, smeared over Kepler-like orbits. The reason is that the Kepler 2-body problem -- and its quantum version, the hydrogen atom -- are superintegrable systems with the large dynamical symmetry group SO(2,4).) In general, roughly, coherent states form a nice orbit of unit vectors of a Hilbert space H under a dynamical symmetry group G with a triangular decomposition, such that the linear combinations of coherent states are dense in H, and the inner product phi^*psi of coherent states phi and psi can be calculated explicitly in terms of the highest weight representation theory of G. The diagonal of the N-th tensor power of H (coding systems with N-fold quantum numbers) has coherent states phi_N (labelled by the same classical phase space as the original coherent states, and orresponding to the N-fold highest weight) with inner product phi_N^*psi_N=(phi^*psi) N and for N --> inf, one gets a good classical limit. For the Heisenberg group, phi^*psi is a 1/hbar-th power, and the N-th power corresponds to replacing hbar by hbar/N. Thus one gets the standard classical limit. Basic literature on relations between coherent states and the classical limit, based on irreducible unitary representations of Lie groups includes the book A. M. Perelomov, Generalized Coherent States and Their Applications, Springer-Verlag, Berlin, 1986. and the paper L. Yaffe, Large N limits as classical mechanics, Rev. Mod. Phys. 54, 407--435 (1982) Both references assume that the Lie group is finite-dimensional and semisimple. This excludes the Heisenberg group, in terms of which the standard (Glauber) coherent states are usually defined. However, the Heisenberg group has a triangular decomposition, and this suffices to apply Perelomov's theory in spirit. The online book Arnold Neumaier, Dennis Westra, Classical and Quantum Mechanics via Lie algebras, http://lanl.arxiv.org/abs/0810.1019 contains a general discussion of the relations between classical mechanics and quantum mechanics, and discusses in Chapter 16 the concept of a triangular decomposition of Lie algebras and a summary of the associated representation theory (though in its present version not the general relation to coherent states). For other relevant approaches to a rigorous classical limit, see the online sources http://www.projecteuclid.org/Dienst/Repository/1.0/Disseminate/euclid.cmp/1103859040/body/pdf http://www.univie.ac.at/nuhag-php/bibtex/open_files/si80_SIMON!!!.pdf http://arxiv.org/abs/quant-ph/9504016 http://arxiv.org/pdf/math-ph/9807027