Poisson manifold



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


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

A Poisson manifold is a real smooth manifold MM equipped with a Poisson structure. A Poisson structure is a Lie algebra bracket {,}:C (M)×C (M)C (M)\{,\}:C^\infty(M)\times C^\infty(M)\to C^\infty(M) on the vector space of smooth functions on MM which together with the pointwise multiplication of functions makes it a Poisson algebra. As derivations of C (M)C^\infty(M) correspond to smooth tangent vector fields, for each fC (M)f\in C^\infty(M) there is a vector X fX_f given by X f(g)={f,g}X_f(g)=\{f,g\} and called the Hamiltonian vector field corresponding to the function ff, 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 PΛ 2TMP\in\Lambda^2 T M; then {f,g}:=dfdg,P\{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:MNh:M\to N of Poisson manifolds is a morphism of smooth manifolds such that, for all f,gC (N)f,g\in C^\infty(N), {fh,gh} M={f,g} N\{f\circ h, g\circ h\}_M = \{f,g\}_N.




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


Every symplectic manifold carries a natural Poisson structure 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.


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.


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


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,ω)(X,\omega), and then show how it may equivalently be understood as the algebra of infinitesimal symmetries of any of the prequantizations of (X,ω)(X,\omega). For more on this see at Poisson bracket Lie n-algebra and at geometry of physics -- prequantum geometry.


Let XX be a smooth manifold. A closed differential 2-form ωΩ cl 2(X)\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

ι ()ω:Vect(X)Ω 1(X) \iota_{(-)}\omega \colon Vect(X) \longrightarrow \Omega^1(X)

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

If ω\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.)


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

ι vω+dH=0. \iota_{v} \omega + d H = 0 \,.

If vVect(X)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,ω)Vect(X) HamVect(X,\omega) \hookrightarrow Vect(X)

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


When ω\omega is symplectic then, evidently, there is a unique Hamiltonian vector field, def. 2, 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 fuctions but pairs of a Hamiltonian vector field and a compatible Hamiltonian.


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

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

for the linear subspace of the direct sum of Hamiltonian vector fields, def. 2, and smooth functions on those pairs (v,H)(v,H) for which HH is a Hamiltonian for vv

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

Define a bilinear map

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


[(v 1,H 1),(v 2,H 2)]([v 1,v 2],ι v 2ι v 1ω), [(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][v_1,v_2] is the standard Lie bracket on vector fields. Write

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

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

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

Let X= 2nX = \mathbb{R}^{2n} and let ω= i=1 ndp idq i\omega = \sum_{i = 1}^n d p_i \wedge d q^i for {q i} i=1 n\{q^i\}_{i = 1}^n the canonical coordinates on one copy of n\mathbb{R}^n and {p i} i=1 n\{p_i\}_{i = 1}^n that on the other (“canonical momenta”). Hence let (X,ω)(X,\omega) be a symplectic vector space of dimension 2n2n, regarded as a symplectic manifold.

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

ι q iω=dp i \iota_{\partial_{q^i}} \omega = - d p_i
ι p iω=+dq i. \iota_{\partial_{p_i}} \omega = + d q^i \,.

Moreover, since XX is connected, these Hamiltonians are unique up to a choice of constant function. Write iC (X)\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][( p i,q i),( q j,p j)]=δ j i(0,i)=δ j ii. [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}\{q^i\} and {p i}\{p_i\}, generate the affine symplectic group of (X,ω)(X,\omega). The freedom to add constant terms to Hamiltonians gives the extended affine symplectic group.

Example 6 serves to motivate a more conceptual origin of the definition of the Poisson bracket in def. 3.



θ i=1 np idq iΩ 1( 2n) \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

dθ=ω. d \theta = \omega \,.

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

Since 2n\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 θ\theta may canonically be thought of as being a connection on the trivial U(1)U(1)-principal bundle. As such this θ\theta is a prequantization of ( 2n, i=1 ndp idq i)(\mathbb{R}^{2n}, \sum_{i=1}^n d p_i \wedge d q^i).

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

θθ+dα \theta \mapsto \theta + d\alpha

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

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

vω=0 \mathcal{L}_v \omega = 0

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

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

For more on this see also at prequantized Lagrangian correspondence.

Notice then the following basic but important fact.


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

vθ=dα \mathcal{L}_v \theta = d \alpha

is equivalent to the condition that makes Hι vθαH \coloneqq \iota_v \theta - \alpha a Hamiltonian for vv according to def. 2:

ι vω+d(ι vθα)=0. \iota_v \omega + d (\iota_v \theta - \alpha ) = 0 \,.

Moreover, the Poisson bracket, def. 3, between two such Hamiltonian pairs (v i,α iι vθ)(v_i, \alpha_i -\iota_v \theta) is equivalently given by the skew-symmetric Lie derivative of the corresponding vector fields on the α i\alpha_i:

(1)ι [v 1,v 2]θι v 2ι v 1ω= v 1α 2 v 2α 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

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

vθ =ι vdθ+dι vθ =ι vω+dι vθ. \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 α i\alpha_i

ι v 2ι v 1ω =ι v 2ι v 1dθ =ι v 1dι v 2θι v 2dι v 1θι [v 1,v 2]θ =ι v 1dα 2ι v 1ι v 2ωι v 2dα 1+ι v 2ι v 1ωι [v 1,v 2]θ =2ι v 2ι v 1ω+ v 1α 2 v 2α 1ι [v 1,v 2]θ. \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} \,.

For (X,ω)(X,\omega) a presymplectic manifold with θΩ 1(X)\theta \in \Omega^1(X) such that dθ=ωd \theta = \omega, consider the Lie algebra

𝔔𝔲𝔞𝔫𝔱𝔐𝔬𝔯𝔭𝔥(X,θ)={(v,α)| vθ=dα}Vect(X)C (X) \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,α 1),(v 2,α 2)]=([v 1,v 2], v 1α 2 v 2α 1). [(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)(v,ι vθH) (v,H) \mapsto (v, \iota_v \theta - H)

is an isomorphism of Lie algebras

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

from the Poisson bracket Lie algebra, def. 3.

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:


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

  1. an open cover {U iX} i\{U_i \to X\}_i;

  2. 1-forms {θ iΩ 1(U i)}\{\theta_i \in \Omega^1(U_i)\};

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

such that

  1. dθ i=ω| U id \theta_i = \omega|_{U_i} on all U iU_i;

  2. θ j=θ i+dlogg ij\theta_j = \theta_i + d log g_{ij} on all U ijU_{i j};

  3. g ijg jk=g ikg_{i j} g_{j k} = g_{i k} on all U ijkU_{i j k}.

The quantomorphism Lie algebra of this is

𝔔𝔲𝔞𝔫𝔱𝔐𝔬𝔯𝔭𝔥(X,{U i},{g ij,θ i})={(v,{α i})| vlogg ij=α jα i, vθ i=dα i}Vect(X)(iC (U i)) \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,{(α 1) i}),(v 2,{(α 2) i})]([v 1,v 2],{ v 1(α 2) i v 2(α 1) i}). [(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\}) \,.

For (X,ω)(X,\omega) a presymplectic manifold and (X,{U i},{g ij,θ i})(X,\{U_i\},\{g_{i j}, \theta_i\}) a prequantization, def. 4, the linear map

(v,H)(v,{ι vθ iH| U i}) (v,H) \mapsto (v, \{\iota_v \theta_i - H|_{U_i}\})

constitutes an isomorphism of Lie algebras

𝔓𝔬𝔦𝔰(X,ω)𝔔𝔲𝔞𝔫𝔱𝔐𝔬𝔯𝔭𝔥(X,{U i},{g ij,θ i}). \mathfrak{Pois}(X,\omega) \stackrel{\simeq}{\longrightarrow} \mathfrak{QuantMorph}(X,\{U_i\},\{g_{i j}, \theta_i\}) \,.

The condition vlogg ij=α jα i\mathcal{L}_v log g_{i j} = \alpha_j - \alpha_i on the infinitesimal quantomorphisms, togther with the Cech-Deligne cocycle condition dlogg ij=θ jθ id log g_{i j} = \theta_j - \theta_i says that on U ijU_{i j}

ι vθ jα j=ι vθ iα i \iota_v \theta_j - \alpha_j = \iota_v \theta_i - \alpha_i

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

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


Deformation quantization

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

duality between algebra and geometry in physics:

Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Revised on May 28, 2015 16:13:11 by Urs Schreiber (