The classical limit of quantum mechanics as a Lie algebra contraction

Abstract

The classical limit of a quantum system with one degree of freedom is examined in terms of the contraction of the underlying non-compact kinematical algebra w1 (the Weyl-Heisenberg algebra) to the three-dimensional Abelian algebra t3. An appropriate definition of the contraction of a Lie algebra and of a sequence of its representations is given. For some quantum systems with simple explicitly integrable dynamics, it is shown how the classical Poisson bracket and classical trajectories are obtained in the limit. Each trajectory is associated with a one-dimensional representation of t3 within a direct integral of such representations in a Hilbert space. Each of the corresponding generalized one dimensional subspaces is stable under the action of the limiting dynamics, and a superselection rule arises naturally between any two such subspaces.