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 Image may be NSFW.
Clik here to view. (the algebra of observables of some quantum system) and a Hamiltonian Image may be NSFW.
Clik here to view. $H \in A$ , we obtain a derivation Image may be NSFW.
Clik here to view. [-, 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 Image may be NSFW.
Clik here to view. is commutative, all commutators are trivial, and yet classical mechanics somehow takes a Hamiltonian Image may be NSFW.
Clik here to view. $H \in A$ and produces a notion of time evolution. How does that work? It turns out that for algebras of observables Image may be NSFW.
Clik here to view. of a classical system, we can think of Image may be NSFW.
Clik here to view. as the classical limit Image may be NSFW.
Clik here to view. $\hbar \to 0$ of a family Image may be NSFW.
Clik here to view. $A_{\hbar}$ of noncommutative algebras. While Image may be NSFW.
Clik here to view. is commutative, the noncommutativity of the family Image may be NSFW.
Clik here to view. $A_{\hbar}$ equips Image may be NSFW.
Clik here to view. 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.

Hochschild cohomology

In this post we consider associative, but not necessarily unital, algebras over a fixed field Image may be NSFW.
Clik here to view. of characteristic not equal to Image may be NSFW.
Clik here to view..

Let Image may be NSFW.
Clik here to view. be such an algebra. A formal deformation of Image may be NSFW.
Clik here to view. is an associative Image may be NSFW.
Clik here to view. $k[\epsilon]$ -bilinear product Image may be NSFW.
Clik here to view. $\star$ on Image may be NSFW.
Clik here to view. $A[[\epsilon]]$ (when Image may be NSFW.
Clik here to view. is noncommutative, Image may be NSFW.
Clik here to view. $\epsilon$ should be central) such that for Image may be NSFW.
Clik here to view. $a, b \in A$ we have

Image may be NSFW.
Clik here to view. $\displaystyle a \star b = ab + \epsilon (\text{other terms})$ .

In other words, quotienting by Image may be NSFW.
Clik here to view. $\epsilon$ we obtain the original product on Image may be NSFW.
Clik here to view.. For the purposes of understanding the classical limit, Image may be NSFW.
Clik here to view. should be thought of as the classical algebra of observables and Image may be NSFW.
Clik here to view. $A[[\epsilon]]$ , together with the deformed product Image may be NSFW.
Clik here to view. $\star$ , should be thought of as the quantum one. Roughly speaking, formal deformations of an algebra Image may be NSFW.
Clik here to view. describe the “formal neighborhood” of Image may be NSFW.
Clik here to view. in the moduli space of algebras.

Formal deformations turn out to be difficult to construct in general. As a first approximation, instead of quotienting by Image may be NSFW.
Clik here to view. $\epsilon$ we can quotient by Image may be NSFW.
Clik here to view. $\epsilon^2$ . A first-order deformation of Image may be NSFW.
Clik here to view. is an associative Image may be NSFW.
Clik here to view. $k[\epsilon]/\epsilon^2$ -bilinear product on Image may be NSFW.
Clik here to view. $A[\epsilon]/\epsilon^2$ satisfying the same condition as above. Explicitly, it is given by a product

Image may be NSFW.
Clik here to view. $\displaystyle a \star b = ab + \epsilon f(a, b)$

where Image may be NSFW.
Clik here to view. $f(a, b) : A^2 \to A$ is a Image may be NSFW.
Clik here to view.-bilinear map such that Image may be NSFW.
Clik here to view. $(a \star b) \star c = a \star (b \star c)$ , or equivalently

Image may be NSFW.
Clik here to view. $\displaystyle f(ab, c) + f(a, b) c = f(a, bc) + a f(b, c)$ .

The Image may be NSFW.
Clik here to view.-vector space of all such maps is denoted Image may be NSFW.
Clik here to view. Z^2(A, A) and called the space of Hochschild Image may be NSFW.
Clik here to view.-cocycles of Image may be NSFW.
Clik here to view. with coefficients in Image may be NSFW.
Clik here to view.. Roughly speaking, first-order deformations of Image may be NSFW.
Clik here to view. describe the “tangent space” to Image may be NSFW.
Clik here to view. in the moduli space of algebras.

Of course, this is not really the space we’re interested in. If we actually want to understand first-order deformations of Image may be NSFW.
Clik here to view., then it would be a good idea to identify Image may be NSFW.
Clik here to view.-cocycles that give isomorphic deformations. Hence suppose Image may be NSFW.
Clik here to view. f_1, f_2 are Image may be NSFW.
Clik here to view.-cocycles describing deformations Image may be NSFW.
Clik here to view. $\star_1, \star_2$ which are isomorphic in the sense that there is a Image may be NSFW.
Clik here to view. $k[\epsilon]$ -linear map Image may be NSFW.
Clik here to view. $\phi : A[\epsilon]/\epsilon^2 \to A[\epsilon]/\epsilon^2$ sending one to the other which reduces to the identity Image may be NSFW.
Clik here to view. $\bmod \epsilon$ . Writing Image may be NSFW.
Clik here to view. $\phi = I + \epsilon T$ , we want

Image may be NSFW.
Clik here to view. $\displaystyle (a + \epsilon Ta) \star_1 (b + \epsilon Tb) = a \star_2 b + \epsilon T(a \star_2 b)$

which after simplification gives

Image may be NSFW.
Clik here to view. $\displaystyle f_2(a, b) - f_1(a, b) = (Ta) b + a (Tb) - T(ab)$ .

In other words, Image may be NSFW.
Clik here to view. f_1, f_2 give isomorphic deformations if and only if Image may be NSFW.
Clik here to view. f_2 - f_1 lies in the subspace of Image may be NSFW.
Clik here to view.-cocycles of the form Image may be NSFW.
Clik here to view. (Ta) b + a (Tb) - T(ab) for some Image may be NSFW.
Clik here to view.-linear map Image may be NSFW.
Clik here to view. $T : A \to A$ . These Image may be NSFW.
Clik here to view.-cocycles are precisely the Image may be NSFW.
Clik here to view.-coboundaries Image may be NSFW.
Clik here to view. B^2(A, A) ; as Image may be NSFW.
Clik here to view.-cocycles, they describe the deformations isomorphic to the trivial deformation given by Image may be NSFW.
Clik here to view. $A[\epsilon]/\epsilon^2$ with its usual product. The quotient space Image may be NSFW.
Clik here to view. Z^2(A, A) / B^2(A, A) , the second Hochschild cohomology Image may be NSFW.
Clik here to view. H^2(A, A) , is the desired space which parameterizes first-order deformations of Image may be NSFW.
Clik here to view..

It’s worth mentioning at this point that Hochschild cohomology can be computed from a cochain complex beginning

Image may be NSFW.
Clik here to view. $0 \to A \xrightarrow{d_1} \text{Hom}_k(A, A) \xrightarrow{d_2} \text{Hom}_k(A \otimes A, A) \xrightarrow{d_3} ...$

where Image may be NSFW.
Clik here to view. d_1 sends an element Image may be NSFW.
Clik here to view. $a \in A$ to a map Image may be NSFW.
Clik here to view. $b \mapsto ab - ba$ and Image may be NSFW.
Clik here to view. d_2 sends a map Image may be NSFW.
Clik here to view. $f : A \to A$ to a map Image may be NSFW.
Clik here to view. $a \otimes b \mapsto f(a) b + a f(b) - f(ab)$ (at least up to a sign I may have wrong). The kernel of Image may be NSFW.
Clik here to view. d_3 is the space of Hochschild Image may be NSFW.
Clik here to view.-cocycles, and its quotient by the image of Image may be NSFW.
Clik here to view. d_2 gives Image may be NSFW.
Clik here to view. H^2(A, A) as above.

But there are two smaller cohomology groups we can describe reasonably concretely as well. The kernel of Image may be NSFW.
Clik here to view. d_1 defines zeroth Hochschild cohomology

Image may be NSFW.
Clik here to view. $\displaystyle H^0(A, A) \cong Z(A)$

which is precisely the center of Image may be NSFW.
Clik here to view.. The kernel of Image may be NSFW.
Clik here to view. d_2 defines the space of Hochschild Image may be NSFW.
Clik here to view.-cocycles Image may be NSFW.
Clik here to view. $Z^1(A, A) \cong \text{Der}(A, A)$ , which is precisely the space of derivations Image may be NSFW.
Clik here to view. $A \to A$ . The image of Image may be NSFW.
Clik here to view. d_1 defines the space of Hochschild Image may be NSFW.
Clik here to view.-coboundaries Image may be NSFW.
Clik here to view. B^1(A, A) , which are given by inner derivations Image may be NSFW.
Clik here to view. $b \mapsto [a, b]$ (named by analogy with inner automorphisms). The quotient Image may be NSFW.
Clik here to view. Z^1(A, A)/B^1(A, A) defines the first Hochschild cohomology

Image may be NSFW.
Clik here to view. $\displaystyle H^1(A, A) \cong \text{Der}(A, A)/\text{Inn}(A, A)$

which is precisely the space of outer derivations of Image may be NSFW.
Clik here to view..

Introducing the lower Hochschild cohomology groups gives Hochschild cohomology extra structure. For reasons I don’t particularly understand, if Image may be NSFW.
Clik here to view. D_1, D_2 are two derivations, then

Image may be NSFW.
Clik here to view. f(a, b) = D_1(a) D_2(b)

is a Hochschild Image may be NSFW.
Clik here to view.-cocycle. Moreover, if either of the derivations Image may be NSFW.
Clik here to view. D_i is inner, then Image may be NSFW.
Clik here to view. turns out to be a Image may be NSFW.
Clik here to view.-coboundary. So the above map descends to a “cup product”

Image may be NSFW.
Clik here to view. $H^1(A, A) \otimes H^1(A, A) \to H^2(A, A)$ .

Hodge decomposition

Now suppose Image may be NSFW.
Clik here to view. is commutative. Taking the opposite algebra of a first-order deformation gives another first-order deformation; concretely, if Image may be NSFW.
Clik here to view. f(a, b) is a Image may be NSFW.
Clik here to view.-cocycle, then Image may be NSFW.
Clik here to view. f(b, a) is also a Image may be NSFW.
Clik here to view.-cocycle. Then (and this is where the hypothesis that Image may be NSFW.
Clik here to view. $\text{char}(k) \neq 2$ is important) every Image may be NSFW.
Clik here to view.-cocycle admits a canonical decomposition

Image may be NSFW.
Clik here to view. $\displaystyle f(a, b) = \frac{f(a, b) + f(b, a)}{2} + \frac{f(a, b) - f(b, a)}{2}$

into symmetric and alternating Image may be NSFW.
Clik here to view.-cocycles. Moreover, since Image may be NSFW.
Clik here to view. is commutative, trivial deformations of Image may be NSFW.
Clik here to view. are commutative, so the Image may be NSFW.
Clik here to view.-coboundaries all lie in the symmetric subspace. Hence we can identify a direct summand Image may be NSFW.
Clik here to view. H^2_s(A, A) of the second Hochschild cohomology consisting of the quotient of the symmetric Image may be NSFW.
Clik here to view.-cocycles by the Image may be NSFW.
Clik here to view.-coboundaries. This is known as second Harrison cohomology, and describes commutative deformations of Image may be NSFW.
Clik here to view..

On the other hand, since all Image may be NSFW.
Clik here to view.-coboundaries are symmetric, it follows that the alternating Image may be NSFW.
Clik here to view.-cocycles map injectively into Hochschild cohomology. Moreover, since

Image may be NSFW.
Clik here to view. $\displaystyle a \star b - b \star a = \epsilon (f(a, b) - f(b, a))$

is a commutator, it follows that alternating Image may be NSFW.
Clik here to view.-cocycles are biderivations (derivations in each variable separately). On the other hand, it is straightforward to verify that any alternating biderivation is a Image may be NSFW.
Clik here to view.-cocycle. It follows that we have a “Hodge decomposition”

Image may be NSFW.
Clik here to view. $\displaystyle H^2(A, A) \cong \text{ABDer}(A, A) \oplus H^2_s(A, A)$ .

Notably, Image may be NSFW.
Clik here to view. $\text{ABDer}(A, A)$ is itself a space of functions rather than a quotient of a space of functions by another space of functions, so it is a little more concrete to work with than Hochschild or Harrison cohomology. In addition, biderivations are uniquely determined by what they do to generators of Image may be NSFW.
Clik here to view., so it is easier to write down biderivations than general Image may be NSFW.
Clik here to view.-cocycles.

Example. If Image may be NSFW.
Clik here to view. D_1, D_2 are two derivations Image may be NSFW.
Clik here to view. $A \to A$ , then Image may be NSFW.
Clik here to view. $\frac{D_1(a) D_2(b) - D_1(b) D_2(a)}{2}$ is an alternating biderivation. Note that this is just the image of the cup product in Image may be NSFW.
Clik here to view. $\text{ABDer}(A, A)$ .

As it turns out, a large class of commutative algebras, the “smooth algebras,” have the property that Image may be NSFW.
Clik here to view. H^2_s(A, A) = 0 (so have no nontrivial commutative first-order deformations). For such algebras, Image may be NSFW.
Clik here to view. H^2(A, A) can be understood very concretely as the space of alternating biderivations. I believe such algebras include the algebras of functions on a smooth affine variety, but I don’t understand these issues well yet.

Poisson algebras

A second-order deformation of Image may be NSFW.
Clik here to view. is an associative Image may be NSFW.
Clik here to view. $k[\epsilon]/\epsilon^3$ -bilinear product on Image may be NSFW.
Clik here to view. $A[\epsilon]/\epsilon^3$ such that quotienting by Image may be NSFW.
Clik here to view. $\epsilon$ we obtain the original product on Image may be NSFW.
Clik here to view.. Writing such a deformation as

Image may be NSFW.
Clik here to view. $\displaystyle a \star b = ab + \epsilon f_1(a, b) + \epsilon^2 f_2(a, b)$

its first-order part Image may be NSFW.
Clik here to view. f_1(a, b) is necessarily a Hochschild Image may be NSFW.
Clik here to view.-cocycle, but one which satisfies an additional condition: the second-order part of the identity is Image may be NSFW.
Clik here to view. $(a \star b) \star c - a \star (b \star c) = 0$ gives

Image may be NSFW.
Clik here to view. $\displaystyle f_1(f_1(a, b), c) - f_1(a, f_1(b, c)) = f_2(a, bc) + a f_2(b, c) - f_2(ab, c) - f_2(a, b) c$ .

Functions Image may be NSFW.
Clik here to view. $A^{\otimes 3} \to A$ of the form on the RHS are Hochschild Image may be NSFW.
Clik here to view.-coboundaries, and in fact this condition can be interpreted to mean that the associator of Image may be NSFW.
Clik here to view. f_1 must be zero in Image may be NSFW.
Clik here to view. H^3(A, A) (which it would take us too far afield to define here). In particular, if Image may be NSFW.
Clik here to view. H^3(A, A) = 0 , then every first-order deformation of Image may be NSFW.
Clik here to view. extends to a second-order deformation.

The above condition on Image may be NSFW.
Clik here to view. f_1(a, b) implies another condition which we are more interested in for the time being. Namely, the commutator

Image may be NSFW.
Clik here to view. $\displaystyle [a, b] = \epsilon (f_1(a, b) - f_1(b, a)) + \epsilon^2 (f_2(a, b) - f_2(b, a))$

has first-order part Image may be NSFW.
Clik here to view. $\epsilon (f_1(a, b) - f_1(b, a)) = \epsilon \{ a, b \}$ , and after a quick calculation, it turns out that the fact that Image may be NSFW.
Clik here to view. [a, b] satisfies the Jacobi identity implies that Image may be NSFW.
Clik here to view. $\{ a, b \}$ does as well.

This suggests the following definition. A Poisson bracket on Image may be NSFW.
Clik here to view. is a Lie bracket Image may be NSFW.
Clik here to view. $\{ a, b \}$ which is also a biderivation. In formulas, Image may be NSFW.
Clik here to view. $\{ a, b \} : A \otimes A \to A$ is a Image may be NSFW.
Clik here to view.-bilinear map satisfying

Image may be NSFW.
Clik here to view. $\{ a, a \} = 0$
Image may be NSFW.
Clik here to view. $\{ a, bc \} = \{ a, b \} c + b \{ a, c \}$
Image may be NSFW.
Clik here to view. $\{ a, \{ b, c \} \} = \{ \{ a, b \}, c \} + \{ b, \{ a, c \} \}$ .

A Poisson algebra is an algebra equipped with a Poisson bracket. Any noncommutative algebra has a canonical Poisson bracket given by the commutator Image may be NSFW.
Clik here to view. ab - ba , but significantly, even if Image may be NSFW.
Clik here to view. is commutative, it can still admit a nontrivial Poisson bracket if it admits any second-order deformations which are noncommutative to first order.

Poisson algebras are the natural setting for the most general form of the Heisenberg picture. Given a Poisson algebra Image may be NSFW.
Clik here to view. and a Hamiltonian Image may be NSFW.
Clik here to view., the derivation Image may be NSFW.
Clik here to view. $\{ -, H \}$ is now the infinitesimal generator of time evolution. The procedure above, where we took the first-order part of a noncommutative deformation, also elegantly explains how Poisson brackets on a quantum system descend to Poisson brackets on a classical system.

Hamiltonian mechanics

Let’s see how this works more explicitly. We return to the setting of a quantum particle in one dimension in a potential. Recall that in this case the Hamiltonian is given by

Image may be NSFW.
Clik here to view. $\displaystyle H = \frac{p^2}{2m} + V(x)$

where Image may be NSFW.
Clik here to view. x, V(x) are multiplication operators and Image may be NSFW.
Clik here to view. $p = - i \hbar \frac{d}{dx}$ . Suppose for simplicity that Image may be NSFW.
Clik here to view. V(x) is a polynomial in Image may be NSFW.
Clik here to view.. Then all of the operators we care about are more or less contained in the algebra of operators generated by Image may be NSFW.
Clik here to view. x, p subject to the relation

Image may be NSFW.
Clik here to view. $\displaystyle [x, p] = i \hbar$ .

Taking the classical limit as above with Image may be NSFW.
Clik here to view. $\epsilon = i \hbar$ we obtain the commutative algebra Image may be NSFW.
Clik here to view. $\mathbb{C}[x, p]$ with Poisson bracket uniquely defined by Image may be NSFW.
Clik here to view. $\{ x, p \} = 1$ and extended via the Leibniz rule. Recall that Image may be NSFW.
Clik here to view. $\{ a, b \} = D_1(a) D_2(b) - D_1(b) D_2(a)$ is a Poisson bracket for any two derivations Image may be NSFW.
Clik here to view. D_1, D_2 ; this particular Poisson bracket can be written

Image may be NSFW.
Clik here to view. $\displaystyle \{ a, b \} = \frac{\partial a}{\partial x} \frac{\partial b}{\partial p} - \frac{\partial b}{\partial x} \frac{\partial a}{\partial p}$

for all Image may be NSFW.
Clik here to view. $a, b \in \mathbb{C}[x, p]$ . We find that

Image may be NSFW.
Clik here to view. $\displaystyle \{ x, H \} = \frac{\partial H}{\partial p} = \frac{p}{m}$

Image may be NSFW.
Clik here to view. $\displaystyle \{ p, H \} = - \frac{\partial H}{\partial x} = -V'(x)$ .

Since we divided by Image may be NSFW.
Clik here to view. $i \hbar$ , the infinitesimal generator of time evolution is now given by Image may be NSFW.
Clik here to view. $\{ -, H \}$ , and so we have recovered Hamilton’s equations (as well as Newton’s second law, again) from the Heisenberg picture.

Note that the Poisson bracket defined above naturally extends to the algebra Image may be NSFW.
Clik here to view. $C^{\infty}(\mathbb{R}^2)$ of smooth functions on the classical phase space Image may be NSFW.
Clik here to view. $\{ (x, p) \in \mathbb{R}^2 \}$ of a classical particle in one dimension. More generally, given a smooth manifold Image may be NSFW.
Clik here to view., there is a natural Poisson algebra structure on the algebra Image may be NSFW.
Clik here to view. $C^{\infty}(T^{\ast}(M))$ of smooth functions on the cotangent bundle of Image may be NSFW.
Clik here to view. induced by a canonical symplectic form. In other words, the cotangent bundle is a Poisson manifold.

A natural question to ask about Poisson manifolds, and more generally about commutative Poisson algebras, is whether they admit a unique formal deformation whose first-order part gives the corresponding Poisson bracket; this is then the unique deformation quantization of the classical system described by the Poisson algebra. That this is true for Poisson manifolds is a deep result of Kontsevich.

Poisson algebras and the classical limit

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