nLab
Poisson manifold

Contents

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 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 df\otimes dg, 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.

Examples

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; 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,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.

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.

Properties

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:

algebrageometry
Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
AQFTFQFT
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 neighbourhoodopen 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 September 15, 2013 18:13:13 by Urs Schreiber (89.204.137.43)