94-388 Werner, R.F., Wolff, M.P.H.

Classical Mechanics as Quantum Mechanics with Infinitesimal $\hbar$ (28K, LaTeX) Dec 5, 94

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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.

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