Abstract
This paper is a first attempt to explore the relationship between classical and quantum mechanics from a group-theoretical point of view. We deal here with the algebraic aspects of the sets of classical and quantum observables in the framework of the algebraic structures associated with finite-dimensional Lie algebras. In particular, we investigate the canonical structure of the quotient fields predicted by the Gel’fand–Kirillov and Vergne conjectures in order to study the types of observables that emerge from a given Lie algebra.
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