The algebra of observables in classical mechanics

Pöttinger, Jürgen

Yehuda Roth Roth

Emile Grgin, Aage Petersen

In classical and in quantum mechanics physical variables play a dual role as observables and as generators of infinitesimal transformations in the invariance groups. We show that if the Lie algebra of generators is central simple, the observable-generator duality restricts the structure of the algebra of observables to two cases: a commutative, associative algebra as in classical mechanics, or a central simple special Jordan algebra as in quantum mechanics.

Holger Neumann

Franklin E. Schroeck

The classical limit ℏ→0 of stochastic quantum mechanics is investigated. An algebra of stochastic classical observables is canonically associated to the algebra of stochastic quantum observables with a product inherited from sequential instruments and also with a Lie commutator. It is shown that the sharp-point limit (limit of no stochasticity), which implies ℏ→0, yields the usual algebra of classical observables.