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fa.useful evaluation – What is the exact relationship between actual Poisson algebras and commutative $C^*$ algebras?

I’ve been educating myself quantum mechanics, and I spotted that I’m lacking one thing elementary. Namely, there are two photos that I do not know methods to reconcile:

- Quantum Mechanics generalizes Hamiltonian dynamics within the following sense. In classical mechanics the clique of compactly supported real-valued features on aspect area figure a Poisson algebra, the place the Lie bracket is induced by the isomorphism between the real-valued features on aspect area and the vector fields on aspect area induced by the symplectic figure. In any such status, any altenative of an observable (=an component of the Poisson algebra) can breathe the “Hamiltonian” and would induce Hamiltonian equations. In Quantum Mechanics one as an alternative begins with a probably non-commutative $C^*$ algebra, the place the Lie bracket is given by $[x,y]=xy-yx$, and any altenative of an observable (i.e., an component $x$ of $C^*$ satisfying $x=x^*$) can breathe known as the “Hamiltonian” and induce Hamiltonian equations.
- Quantum Mechanics can breathe seen as being within the context of non-commutative chance concept, during which perceive it generalizes the illustration of commutative $C^*$ algebras. In specific, one can cerebrate of the commutative $C^*$ algebra of complex-valued compactly uphold features on aspect area. In that status normalized states coincide to chance measures on aspect area.

The drawback is that I do not know methods to reconcile these two photos. Can one perceive the actual Poisson algebra as inducing a commutative $C^*$ algebra in some route? Would that route answer that the Lie bracket from the Poisson algebra change into $[x,y]=xy-yx$ within the induced $C^*$ algebra? If this kind of development would not labor, what’s the capable route of reconciling these two understanding? For instance, do normalized states of the actual Poisson algebra coincide to chance measures on the area of characters, just like the commutative $C^*$ algebra illustration?

I’m merely confused by this. I grasp that this pertains to the conception of quantization, however I do not grasp how that matches. My grasp is that quantization takes a actual Poisson algebra and finds methods to contort it to non-commutative $C^*$ algebras that narrate quantum mechanical analogues of that classical system. But I’m attempting to do one thing completely different: grasp the connection between the actual Poisson algebra status and commutative $C^*$ algebras that “describes the same thing” in some route.

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