CLASSICAL MECHANICS AS QUANTUM-MECHANICS WITH INFINITESIMAL-H

Abstract

We define the classical limit of quantum theory in the mathematical framework of nonstandard analysis, choosing h as an infinitesimal number. Up to corrections of infinitesimally small norm, bounded observables which change continuously on the standard (non-infinitesimal) phase space scale, are identified with functions on phase space. We discuss the convergence of commutators to Poisson brackets, and the quantum time evolution to the classical one. These results are also shown for the classical limit of spin systems, by choosing the spin as an infinite half-integer.