Poisson manifolds are a mathematical setup for classical mechanics with finitely many degrees of freedom.
A Poisson algebra is a commutative unital associative algebra , in this case over the field of real or complex numbers, equipped with a Lie bracket such that, for any , is a derivation of as an associative algebra.
A Poisson manifold is a real smooth manifold equipped with a Poisson structure. A Poisson structure is a Lie algebra bracket on the vector space of smooth functions on which together with the pointwise multiplication of functions makes it a Poisson algebra. As derivations of correspond to smooth tangent vector fields, for each there is a vector given by and called the Hamiltonian vector field corresponding to the function , 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 ; then .
This induces and is equivalently encoded by the structure of a Poisson Lie algebroid.
A morphism of Poisson manifolds is a morphism of smooth manifolds such that, for all , .
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; 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.
Given a symplectic manifold and given a Hamiltonian function , there is a Poisson bracket on the functions on the smooth path space (the “space of histories” or “space of trajectories”), for the closed interval, which is such that its symplectic leaves are each a copy of , but regarded as the space of initial conditions for evolution with respect to 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.
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|
Examples of sequences of local structures
|geometry||point||first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|smooth functions||derivative||Taylor series||germ||smooth function|
|curve (path)||tangent vector||jet||germ of curve||curve|
|smooth space||infinitesimal neighbourhood||formal neighbourhood||open neighbourhood|
|function algebra||square-0 ring extension||nilpotent ring extension/formal completion||ring extension|
|arithmetic geometry||finite field||p-adic integers||localization at (p)||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|