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S1t. The classical limit via coherent states
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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