Operator Formulation of Classical Mechanics

By making use of the Weyl–Wigner–Groenewold–Moyalassociation rules, acommutative product and a new quantum bracket are constructed in the spaceof operators F(H). In this way, an isomorphism between the Lie algebra ofclassical observables (with Poisson bracket) and the Lie algebra of quantumobservables with this new bracket is established. By these observations, aformulation of classical mechanics in F(H) is obtained and is shown to be theħ → 0 limit of the Heisenberg-picture formulation of quantum mechanics.