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Archive for August, 2011

(Commutative) Poisson algebras are clearly very interesting, so it would be nice to have ways of constructing examples. We know that k[x, p] is a Poisson algebra with bracket uniquely defined by \{ x, p \} = 1; this describes a classical particle in one dimension, and is the classical limit of a quantum particle in one dimension (essentially the Weyl algebra).

More generally, if A, B are Poisson algebras, then the tensor product A \otimes_k B can be given a Poisson bracket given by extending

\displaystyle \{ a_1 \otimes b_1, a_2 \otimes b_2 \} = \{ a_1, a_2 \} \otimes b_1 b_2 + a_2 a_1 \otimes \{ b_1, b_2 \}

linearly. At least when A, B are unital, this Poisson algebra is the universal Poisson algebra with Poisson maps from A, B such that the images of elements of A Poisson-commute with the images of elements of B. In particular, it follows that k[x_1, p_1, ..., x_n, p_n] is a Poisson algebra with the bracket

\{ x_i, x_j \} = \{ p_i, p_j \} = 0, \{ x_i, p_j \} = \delta_{ij}.

This describes a classical particle in n dimensions, or n different classical particles in one dimension, and it is the classical limit of a quantum particle in n dimensions, or n different quantum particles in one dimension.

Today we’ll discuss the question of how one might go about constructing Poisson brackets more generally.

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In the previous post we described the Heisenberg picture of quantum mechanics, which can be phrased quite generally as follows: given a noncommutative algebra A (the algebra of observables of some quantum system) and a Hamiltonian H \in A, we obtain a derivation [-, H], 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 A is commutative, all commutators are trivial, and yet classical mechanics somehow takes a Hamiltonian H \in A and produces a notion of time evolution. How does that work? It turns out that for algebras of observables A of a classical system, we can think of A as the classical limit \hbar \to 0 of a family A_{\hbar} of noncommutative algebras. While A is commutative, the noncommutativity of the family A_{\hbar} equips A 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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