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 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.
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.
We discuss the traditional definition of the Poisson bracket of a (pre-)symplectic manifold , and then show how it may equivalently be understood as the algebra of infinitesimal symmetries of any of the prequantizations of . For more on this see at Poisson bracket Lie n-algebra and at geometry of physics -- prequantum geometry.
is trivial: .
If 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.)
If is such that there exists at least one Hamiltonian for it then it is called a Hamiltonian vector field. Write
When 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 be a presymplectic manifold. Write
Define a bilinear map
for the resulting Lie algebra. In the case that is symplectic, then and hence in this case
Moreover, since is connected, these Hamiltonians are unique up to a choice of constant function. Write for the unit constant function, then the nontrivial Poisson brackets between the basis vector fields are
This is called the Heisenberg algebra.
More generally, the Hamiltonian vector fields corresponding to quadratic Hamiltonians, i.e. degree-2 polynomials in the and , generate the affine symplectic group of . The freedom to add constant terms to Hamiltonians gives the extended affine symplectic group.
for the canonical choice of differential 1-form satisfying
Since is contractible as a topological space, every circle bundle over it is necessarily trivial, and hence any choice of 1-form such as may canonically be thought of as being a connection on the trivial -principal bundle. As such this is a prequantization of .
in the presence of a choice for the right condition to ask for is that there is such that
For more on this see also at prequantized Lagrangian correspondence.
Notice then the following basic but important fact.
For a presymplectic manifold and a 1-form such that then for the condition
Using Cartan's magic formula and by the prequantization condition the we have
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
with Lie bracket
Then by (1) the linear map
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:
an open cover ;
on all ;
on all ;
on all .
The quantomorphism Lie algebra of this is
The condition on the infinitesimal quantomorphisms, togther with the Cech-Deligne cocycle condition says that on
and hence that there is a globally defined function such that . This shows that the map is an isomrophism of vector spaces.
|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||germ of a space||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|