Classical Mechanics as Quantum Mechanics with Innnitesimal { H

We develop an approach to the classical limit of quantum theory using the mathematical framework of nonstandard analysis. In this framework innnitesimal quantities have a rigorous meaning, and the quantum mechanical parameter h can be chosen to be such an innnitesimal. We consider those bounded observables which are transformed continuously on the standard (non-innnitesimal) scale by the phase space translations. We show that, up to corrections of innnitesimally 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 diier-entiable quantum observables, divided by h, are innnitesimally close to the Poisson bracket of the corresponding functions. Moreover, the quantum time evolution is innnitesimally 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 h, is taken as an innnite number. 81Q20 Semiclassical techniques including WKB and Maslov methods 46S20 Nonstandard functional analysis 81S30 Phase space methods including Wigner distributions