nLab
Hamiltonian form

Definition

For $\omega \in {\Omega }^{n+1}(X)$ an n-plectic geometry, and for $v\in \Gamma (X)$ a vector field, a Hamiltonian form for $v$ is, if it exists, a differential form $h\in {\Omega }^n(X)$ such that

$${\iota }_v\omega =dh.$$\iota_{v} \omega = d h \,.

For $n=1$ this reduces to the notion of a Hamiltonian function on a symplectic manifold.

If a Hamiltonian form for $v$ exists then $v$ is called a Hamiltonian vector field.

The Hamiltonian forms are the local classical observables/prequantum observables in higher prequantum field theory, often called local currents. They form the Poisson-bracket Lie n-algebra of local observables.

Related concepts

higher and integrated Kostant-Souriau extensions

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $𝔾$-principal ∞-connection)

$$(\Omega 𝔾)\mathrm{FlatConn}(X)\to \mathrm{QuantMorph}(X,\nabla )\to \mathrm{HamSympl}(X,\nabla )$$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)

$n$	geometry	structure	unextended structure	extension by	quantum extension
$\mathrm{\infty }$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega 𝔾)$-flat ∞-connections on $X$	quantomorphism ∞-group
1	symplectic geometry	Lie algebra	Hamiltonian vector fields	real numbers	Hamiltonians under Poisson bracket
1		Lie group	Hamiltonian symplectomorphism group	circle group	quantomorphism group
2	2-plectic geometry	Lie 2-algebra	Hamiltonian vector fields	line Lie 2-algebra	Poisson Lie 2-algebra
2		Lie 2-group	Hamiltonian 2-plectomorphisms	circle 2-group	quantomorphism 2-group
$n$	n-plectic geometry	Lie n-algebra	Hamiltonian vector fields	line Lie n-algebra	Poisson Lie n-algebra
$n$		smooth n-group	Hamiltonian n-plectomorphisms	circle n-group	quantomorphism n-group

(extension are listed for sufficiently connected $X$)