Poisson manifolds are a mathematical setup for classical mechanics with finitely many degrees of freedom.
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 $\{,\}:A\otimes A\to A$ such that, for any $f\in A$, $\{ f,-\}:A\to A$ 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 algebra bracket $\{,\}:C^\infty(M)\times C^\infty(M)\to C^\infty(M)$ 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 $h:M\to N$ of Poisson manifolds is a morphism of smooth manifolds such that, for all $f,g\in C^\infty(N)$, $\{f\circ h, g\circ h\}_M = \{f,g\}_N$.
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 $(X,\omega)$ 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.
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 $(X,\omega)$, 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.
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
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.)
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
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
for the $\mathbb{R}$-linear subspace of Hamiltonian vector fields among all vector fields
When $\omega$ 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 functions but pairs of a Hamiltonian vector field and a compatible Hamiltonian.
Let $(X,\omega)$ be a presymplectic manifold. Write
for the linear subspace of the direct sum of Hamiltonian vector fields, def. 2, and smooth functions on those pairs $(v,H)$ for which $H$ is a Hamiltonian for $v$
Define a bilinear map
by
called the Poisson bracket, where $[v_1,v_2]$ is the standard Lie bracket on vector fields. Write
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
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
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
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 6 serves to motivate a more conceptual origin of the definition of the Poisson bracket in def. 3.
Write
for the canonical choice of differential 1-form satisfying
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
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
in the presence of a choice for $\theta$ the right condition to ask for is that there is $\alpha$ such that
For more on this see also at prequantized Lagrangian correspondence.
Notice then the following basic but important fact.
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
is equivalent to the condition that makes $H \coloneqq \iota_v \theta - \alpha$ a Hamiltonian for $v$ according to def. 2:
Moreover, the Poisson bracket, def. 3, 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 $\alpha_i$:
Using Cartan's magic formula and by the prequantization condition $d \theta = \omega$ 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 $\alpha_i$
For $(X,\omega)$ a presymplectic manifold with $\theta \in \Omega^1(X)$ such that $d \theta = \omega$, consider the Lie algebra
with Lie bracket
Then by (1) the linear map
is an isomorphism of Lie algebras
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 $(X,\omega)$ a presymplectic manifold, a Cech-Deligne cocycle $(X,\{U_i\},\{g_{i j}, \theta_i\})$ for a prequantization of $(X,\omega)$ is
an open cover $\{U_i \to X\}_i$;
1-forms $\{\theta_i \in \Omega^1(U_i)\}$;
smooth function $\{g_{i j} \in C^\infty(U_{i j}, U(1))\}$
such that
$d \theta_i = \omega|_{U_i}$ on all $U_i$;
$\theta_j = \theta_i + d log g_{ij}$ on all $U_{i j}$;
$g_{i j} g_{j k} = g_{i k}$ on all $U_{i j k}$.
The quantomorphism Lie algebra of this is
with bracket
For $(X,\omega)$ a presymplectic manifold and $(X,\{U_i\},\{g_{i j}, \theta_i\})$ a prequantization, def. 4, the linear map
constitutes an isomorphism of Lie algebras
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 $U_{i j}$
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 1 implied by (1), hence the morphism is a Lie homomorphism.
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:
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
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$\leftarrow$ 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 |