Werner, R.F., Wolff, M.P.H.
Classical Mechanics as Quantum Mechanics with Infinitesimal $\hbar$
(28K, LaTeX)
ABSTRACT. We develop an approach to the classical limit of quantum theory
using the mathematical framework of nonstandard analysis. In this
framework infinitesimal quantities have a rigorous meaning, and the
quantum mechanical parameter $\hbar$ can be chosen to be such an
infinitesimal. We consider those bounded observables which are
transformed continuously on the standard (non-infinitesimal) scale
by the phase space translations. We show that, up to corrections of
infinitesimally small norm, such continuous elements form a
commutative algebra which is isomorphic to the algebra of classical
observables represented by functions on phase space. Commutators of
differentiable quantum observables, divided by $\hbar$, are
infinitesimally close to the Poisson bracket of the corresponding
functions. Moreover, the quantum time evolution is infinitesimally
close to the classical time evolution. Analogous results are shown
for the classical limit of a spin system, in which the half-integer
spin parameter, i.e.\ the angular momentum divided by $\hbar$, is
taken as an infinite number.