nLab Poisson manifold

Idea

Poisson manifolds are a mathematical setup for classical mechanics with finitely many degrees of freedom.

Definition

A Poisson algebra is a commutative unital associative algebra $A$, in this case over the field of real or complex numbers, equipped with a Lie bracket $\{,\} \colon A\otimes A\to A$ such that, for any $f\in A$, $\{ f,-\} \colon A\to A$ is a derivation of $A$ as an associative algebra.

A Poisson manifold is a real smooth manifold $M$ equipped with a Poisson structure. A Poisson structure is a Lie bracket $\{,\} \colon C^\infty(M)\times C^\infty(M)\to C^\infty(M)$ on the vector space of smooth functions on $M$ which together with the pointwise multiplication of functions makes it a Poisson algebra. As derivations of $C^\infty(M)$ correspond to smooth tangent vector fields, for each $f\in C^\infty(M)$ there is a vector $X_f$ given by $X_f(g)=\{f,g\}$ and called the Hamiltonian vector field corresponding to the function $f$, which is viewed as a classical hamiltonian function.

Alternatively a Poisson structure on a manifold is given by a choice of smooth antisymmetric bivector called a Poisson bivector $P\in\Lambda^2 T M$; then $\{f,g\}:=\langle d f\otimes d g, P\rangle$.

This induces and is equivalently encoded by the structure of a Poisson Lie algebroid.

A morphism $h \colon M\to N$ of Poisson manifolds is a smooth function between the underlying smooth manifolds which preserves the Poisson brackets, in that for all $f, g \in C^\infty(N)$, $\{f\circ h, g\circ h\}_M = \{f,g\}_N$.

Examples

Various

Example

Every manifold admits the trivial Poisson structure for which the Poisson bracket simply vanishes on all elements.

Example

Every symplectic manifold carries a natural Poisson structure. Given a smooth function $f: X \to \mathbb{R}$, the symplectic form $\omega$ induces Hamiltonian vector field $H_f: X \to T X$. We can then define the Poisson bracket as

$$\{f, g\} = H_f g = \omega(H_f, H_g).$$

See below for more. However, such Poisson manifolds are very special. It is a basic theorem that Poisson structures on a manifold are equivalent to the smooth foliations of the underlying manifold such that each leaf is a symplectic manifold.

Example

The dual to a finite-dimensional Lie algebra has a natural structure of a Poisson manifold, the Lie-Poisson structure. Its leaves are called coadjoint orbits.

Example

Given a symplectic manifold $(X,\omega)$ and given a Hamiltonian function $H \colon X \longrightarrow \mathbb{R}$, there is a Poisson bracket on the functions on the smooth path space $[I,X]$ (the “space of histories” or “space of trajectories”), for $I = [0,1]$ the closed interval, which is such that its symplectic leaves are each a copy of $X$, but regarded as the space of initial conditions for evolution with respect to $H$ with a source term added. For more on this see at off-shell Poisson bracket.

Example

Every local action functional which admits a Green's function for its equations of motion defines the Peierls bracket on covariant phase space (where in fact it is symplectic) and also “off-shell” on all of configuration space, where it is a genuine Poisson bracket, the canonocal Poisson bracket of the corresponding prequantum field theory.

Pre-symplectic manifolds and infinitesimal quantomorphisms

We discuss the traditional definition of the Poisson bracket of a (pre-)symplectic manifold $(X,\omega)$, and then show how it may equivalently be understood as the algebra of infinitesimal symmetries of any of the prequantizations of $(X,\omega)$. For more on this see at Poisson bracket Lie n-algebra and at geometry of physics – prequantum geometry.

Definition

Let $X$ be a smooth manifold. A closed differential 2-form $\omega \in \Omega_{cl}^2(X)$ is a symplectic form if it is non-degenerate in that the kernel of the operation of contracting with vector fields

$$\iota_{(-)}\omega \colon Vect(X) \longrightarrow \Omega^1(X)$$

is trivial: $ker(\iota_{(-)}\omega) = 0$.

If $\omega$ is just closed with possibly non-trivial kernel, we call it a presymplectic form. (We do not require here the dimension of the kernel restricted to each tangent space to be constant.)

Definition

Given a presymplectic manifold $(X, \omega)$, then a Hamiltonian for a vector field $v \in Vect(X)$ is a smooth function $H \in C^\infty(X)$ such that

$$\iota_{v} \omega + d H = 0 \,.$$

If $v \in Vect(X)$ is such that there exists at least one Hamiltonian for it then it is called a Hamiltonian vector field. Write

$$HamVect(X,\omega) \hookrightarrow Vect(X)$$

for the $\mathbb{R}$-linear subspace of Hamiltonian vector fields among all vector fields

Remark

When $\omega$ is symplectic then, evidently, there is a unique Hamiltonian vector field, def. , associated with every Hamiltonian, i.e. every smooth function is then the Hamiltonian of precisely one Hamiltonian vector field (but two different Hamiltonians may still have the same Hamiltonian vector field uniquely associated with them). As far as prequantum geometry is concerned, this is all that the non-degeneracy condition that makes a closed 2-form be symplectic is for. But we will see that the definitions of Poisson brackets and of quantomorphism groups directly generalize also to the presymplectic situation, simply by considering not just Hamiltonian functions but pairs of a Hamiltonian vector field and a compatible Hamiltonian.

Definition

Let $(X,\omega)$ be a presymplectic manifold. Write

$$Ham(X,\omega) \hookrightarrow HamVect(X,\omega) \oplus C^\infty(X)$$

for the linear subspace of the direct sum of Hamiltonian vector fields, def. , and smooth functions on those pairs $(v,H)$ for which $H$ is a Hamiltonian for $v$

$$Ham(X,\omega) \coloneqq \left\{ (v,H) | \iota_v \omega + d H = 0 \right\} \,.$$

Define a bilinear map

$$[-,-] \;\colon\; Ham(X,\omega) \otimes Ham(X,\omega) \longrightarrow Ham(X,\omega)$$

by

$$[(v_1,H_1), (v_2,H_2)] \coloneqq ([v_1,v_2], \iota_{v_2}\iota_{v_1} \omega) \,,$$

called the Poisson bracket, where $[v_1,v_2]$ is the standard Lie bracket on vector fields. Write

$$\mathfrak{Pois}(X,\omega) \coloneqq (Ham(X,\omega),[-,-])$$

for the resulting Lie algebra. In the case that $\omega$ is symplectic, then $Ham(X,\omega) \simeq C^\infty(X)$ and hence in this case

$$\mathfrak{Pois}(X,\omega) \simeq (C^\infty(X),[-,-]) \,.$$

Example

Let $X = \mathbb{R}^{2n}$ and let $\omega = \sum_{i = 1}^n d p_i \wedge d q^i$ for $\{q^i\}_{i = 1}^n$ the canonical coordinates on one copy of $\mathbb{R}^n$ and $\{p_i\}_{i = 1}^n$ that on the other (“canonical momenta”). Hence let $(X,\omega)$ be a symplectic vector space of dimension $2n$, regarded as a symplectic manifold.

Then $Vect(X)$ is spanned over $C^\infty(X)$ by the canonical bases vector fields $\{\partial_{q^i}, \partial_{p^i}\}$. These basis vector fields are manifestly Hamiltonian vector fields via

$$\iota_{\partial_{q^i}} \omega = - d p_i$$

$$\iota_{\partial_{p_i}} \omega = + d q^i \,.$$

Moreover, since $X$ is connected, these Hamiltonians are unique up to a choice of constant function. Write $\mathbf{i} \in C^\infty(X)$ for the unit constant function, then the nontrivial Poisson brackets between the basis vector fields are

$$[q^i, p_j] \coloneqq [(-\partial_{p_i}, q^i), (\partial_{q^j}, p_j)] = - \delta_j^i (0, \mathbf{i}) = - \delta_j^i \mathbf{i} \,.$$

This is called the Heisenberg algebra.

More generally, the Hamiltonian vector fields corresponding to quadratic Hamiltonians, i.e. degree-2 polynomials in the $\{q^i\}$ and $\{p_i\}$, generate the affine symplectic group of $(X,\omega)$. The freedom to add constant terms to Hamiltonians gives the extended affine symplectic group.

Example serves to motivate a more conceptual origin of the definition of the Poisson bracket in def. .

Example

Write

$$\theta \coloneqq \sum_{i = 1}^n p_i d q_i \in \Omega^1(\mathbb{R}^{2n})$$

for the canonical choice of differential 1-form satisfying

$$d \theta = \omega \,.$$

If we regard $\mathbb{R}^{2n} \simeq T^\ast \mathbb{R}^n$ as the cotangent bundle of the Cartesian space $\mathbb{R}^n$, then this is what is known as the Liouville-Poincaré 1-form.

Since $\mathbb{R}^{2n}$ is contractible as a topological space, every circle bundle over it is necessarily trivial, and hence any choice of 1-form such as $\theta$ may canonically be thought of as being a connection on the trivial $U(1)$-principal bundle. As such this $\theta$ is a prequantization of $(\mathbb{R}^{2n}, \sum_{i=1}^n d p_i \wedge d q^i)$.

Being thus a circle bundle with connection, $\theta$ has more symmetry than its curvature $\omega$ has: for $\alpha \in C^\infty(\mathbb{R}^{2n}, U(1))$ any smooth function, then

$$\theta \mapsto \theta + d\alpha$$

is the gauge transformation of $\theta$, leading to a different but equivalent prequantization of $\omega$.

Hence while a vector field $v$ is said to preserve $\omega$ (is a symplectic vector field) if the Lie derivative of $\omega$ along $v$ vanishes

$$\mathcal{L}_v \omega = 0$$

in the presence of a choice for $\theta$ the right condition to ask for is that there is $\alpha$ such that

$$\mathcal{L}_v \theta = d \alpha \,.$$

For more on this see also at prequantized Lagrangian correspondence.

Notice then the following basic but important fact.

Proposition

For $(X,\omega)$ a presymplectic manifold and $\theta \in \Omega^1(X)$ a 1-form such that $d \theta = \omega$ then for $(v,\alpha) \in Vect(X)\oplus C^\infty(X)$ the condition

$$\mathcal{L}_v \theta = d \alpha$$

is equivalent to the condition that makes $H \coloneqq \iota_v \theta - \alpha$ a Hamiltonian for $v$ according to def. :

$$\iota_v \omega + d (\iota_v \theta - \alpha ) = 0 \,.$$

Moreover, the Poisson bracket, def. , between two such Hamiltonian pairs $(v_i, \alpha_i -\iota_v \theta)$ is equivalently given by the skew-symmetric Lie derivative of the corresponding vector fields on the $\alpha_i$:

(1)$$\iota_{[v_1,v_2]} \theta - \iota_{v_2}\iota_{v_1}\omega = \mathcal{L}_{v_1} \alpha_2 - \mathcal{L}_{v_2} \alpha_1$$

Proof

Using Cartan's magic formula and by the prequantization condition $d \theta = \omega$ the we have

$$\begin{aligned} \mathcal{L}_v \theta &= \iota_v d\theta + d \iota_v \theta \\ & = \iota_v\omega + d \iota_v \theta \end{aligned} \,.$$

This gives the first statement. For the second we first use the formula for the de Rham differential and then again the definition of the $\alpha_i$

$$\begin{aligned} \iota_{v_2}\iota_{v_1} \omega & = \iota_{v_2}\iota_{v_1} d\theta \\ & = \iota_{v_1} d \iota_{v_2} \theta - \iota_{v_2} d \iota_{v_1} \theta - \iota_{[v_1,v_2]} \theta \\ & = \iota_{v_1} d \alpha_2 - \iota_{v_1} \iota_{v_2}\omega - \iota_{v_2} d \alpha_1 + \iota_{v_2} \iota_{v_1}\omega - \iota_{[v_1,v_2]} \theta \\ & = 2 \iota_{v_2} \iota_{v_1}\omega + \mathcal{L}_{v_1} \alpha_2 -\mathcal{L}_{v_2} \alpha_1 - \iota_{[v_1,v_2]} \theta \end{aligned} \,.$$

Corollary

For $(X,\omega)$ a presymplectic manifold with $\theta \in \Omega^1(X)$ such that $d \theta = \omega$, consider the Lie algebra

$$\mathfrak{QuantMorph}(X,\theta) = \left\{ (v,\alpha) | \mathcal{L}_v \theta = d \alpha \right\} \subset Vect(X) \oplus C^\infty(X)$$

with Lie bracket

$$[(v_1,\alpha_1), (v_2,\alpha_2)] = ([v_1,v_2], \mathcal{L}_{v_1}\alpha_2 - \mathcal{L}_{v_2}\alpha_1) \,.$$

Then by (1) the linear map

$$(v,H) \mapsto (v, \iota_v \theta - H)$$

is an isomorphism of Lie algebras

$$\mathfrak{Pois}(X,\omega) \stackrel{\simeq}{\longrightarrow} \mathfrak{QuantMorph}(X,\theta)$$

from the Poisson bracket Lie algebra, def. .

This shows that for exact pre-symplectic forms the Poisson bracket Lie algebra is secretly the Lie algebra of infinitesimal symmetries of any of its prequantizations. In fact this holds true also when the pre-symplectic form is not exact:

Definition

For $(X,\omega)$ a presymplectic manifold, a Cech-Deligne cocycle $(X,\{U_i\},\{g_{i j}, \theta_i\})$ for a prequantization of $(X,\omega)$ is

1. an open cover $\{U_i \to X\}_i$;

2. 1-forms $\{\theta_i \in \Omega^1(U_i)\}$;

3. smooth function $\{g_{i j} \in C^\infty(U_{i j}, U(1))\}$

such that

1. $d \theta_i = \omega|_{U_i}$ on all $U_i$;

2. $\theta_j = \theta_i + d log g_{ij}$ on all $U_{i j}$;

3. $g_{i j} g_{j k} = g_{i k}$ on all $U_{i j k}$.

The quantomorphism Lie algebra of this is

$$\mathfrak{QuantMorph}(X,\{U_i\},\{g_{i j}, \theta_i\}) = \left\{ (v, \{\alpha_i\}) | \mathcal{L}_v log g_{i j} = \alpha_j - \alpha_i \,, \mathcal{L}_v \theta_i = d \alpha_i \right\} \subset Vect(X) \oplus \left(\underset{i}{\bigoplus} C^\infty(U_i)\right)$$

with bracket

$$[(v_1, \{(\alpha_1)_i\}), (v_2, \{(\alpha_2)_i\})] \coloneqq ([v_1,v_2], \{\mathcal{L}_{v_1}(\alpha_2)_i - \mathcal{L}_{v_2} (\alpha_1)_i\}) \,.$$

Proposition

For $(X,\omega)$ a presymplectic manifold and $(X,\{U_i\},\{g_{i j}, \theta_i\})$ a prequantization, def. , the linear map

$$(v,H) \mapsto (v, \{\iota_v \theta_i - H|_{U_i}\})$$

constitutes an isomorphism of Lie algebras

$$\mathfrak{Pois}(X,\omega) \stackrel{\simeq}{\longrightarrow} \mathfrak{QuantMorph}(X,\{U_i\},\{g_{i j}, \theta_i\}) \,.$$

Proof

The condition $\mathcal{L}_v log g_{i j} = \alpha_j - \alpha_i$ on the infinitesimal quantomorphisms, togther with the Cech-Deligne cocycle condition $d log g_{i j} = \theta_j - \theta_i$ says that on $U_{i j}$

$$\iota_v \theta_j - \alpha_j = \iota_v \theta_i - \alpha_i$$

and hence that there is a globally defined function $H \in C^\infty(X)$ such that $\iota_v \theta_i - \alpha_i = H|_{U_i}$. This shows that the map is an isomrophism of vector spaces.

Now over each $U_i$ the the situation for the brackets is just that of corollary implied by (1), hence the morphism is a Lie homomorphism.

Properties

Deformation quantization

Kontsevich formality implies that every Poisson manifold has a family of deformation quantizations, parameterized by the Grothendieck-Teichmüller group.

Related concepts

• Poisson tensor

• Poisson geometry

• Poisson cohomology

• Poisson reduction

• isotropic submanifold, coisotropic submanifold

• symplectic leaf

• symplectic realization

• symplectic dual pair

• Poisson Lie algebroid

• deformation quantization

  • Moyal quantization

• higher Poisson structure

  • Nambu bracket

  • Poisson n-algebra

  • Poisson bracket Lie n-algebra

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization

References

The notion is due to:

• André Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, Journal of Differential Geometry 12 2 (1977) 253–300 [doi:10.4310/jdg/1214433987]

Further early discussion:

• Alan Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 3 (1983) 523-557 [doi:10.4310/jdg/1214437787]

Monograph on Poisson geometry:

• Marius Crainic, Rui Loja Fernandes, Lectures on Poisson Geometry, Graduate Studies in Mathematics 217, Amer. Math. Soc. (2021) [ISBN:978-1-4704-6430-1]

See also:

• Wikipedia, Poisson manifold

Discussion in Lagrangian field theory includes:

• Marco Benini, Alexander Schenkel, Poisson algebras for non-linear field theories in the Cahiers topos, Annales Henri Poincar'e, 18(4), 1435-1464 (2017) (arXiv:1602.00708)