Poisson algebras and the classical limit

In the previous post we described the Heisenberg picture of quantum mechanics, which can be phrased quite generally as follows: given a noncommutative algebra (the algebra of observables of some quantum system) and a Hamiltonian , we obtain a derivation , which is (up to some scalar multiple) the infinitesimal generator of time evolution. This is a natural and general way to start with an algebra and an energy function and get a notion of time evolution which automatically satisfies conservation of energy.

However, if is commutative, all commutators are trivial, and yet classical mechanics somehow takes a Hamiltonian and produces a notion of time evolution. How does that work? It turns out that for algebras of observables of a classical system, we can think of as the classical limit of a family of noncommutative algebras. While is commutative, the noncommutativity of the family equips with the extra structure of a Poisson bracket, and it is this Poisson bracket which allows us to describe time evolution.

Today we’ll describe one way to formalize the notion of taking the classical limit using the deformation theory of algebras. We’ll see how Poisson brackets pop out along the way, as well as the relevance of the lower Hochschild cohomology groups.

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