under construction
A Hamiltonian $n$-vector field is the $n$-dimensional analog of a Hamiltonian vector field as one passes from symplectic geometry to multisymplectic geometry/n-plectic geometry. Roughly, the transgression of a Hamiltonian $n$-vector field to mapping spaces out of an $(n-1)$-manifold yields an ordinary Hamiltonian vector field.
Let $X$ be a smooth manifold equipped with a degree $(n+1)$ differential form $\omega \in \Omega^{n+1}(X)$ for $n \in \mathbb{N}$, with the pair $(X,\omega)$ regarded as a multisymplectic manifold/n-plectic manifold.
Let moreover $H \;\colon\; X \longrightarrow \mathbb{R}$ be a smooth function, to be regarded as an extended Hamiltonian function, hence a de Donder-Weyl Hamiltonian.
Then a Hamiltonian $n$-vector field on $X$ is an $n$-multivector field $\mathbf{v} \in \Gamma(\wedge^{n} T X)$ satisfying the analog of Hamilton's equations, namely the differential equation of differential 1-forms on $X$
Let $\Sigma$ be a manifold to be regarded as a spacetime/worldvolume and let $E \to \Sigma$ be a vector bundle, to be regarded as a field bundle. Assume for simplicity of notation that both $\Sigma$ and $E$ are in fact Cartesian spaces (which is always true locally). Write $j^1 E$ for the corresponding jet bundle]], equipped with the canonical “kinetic” n-plectic form which in local coordinates reads
as discussed at multisymplectic geometry.
Then for $\phi^a$ and $\phi^a_{,i}$ smooth functions on $\Sigma$ and for vector fields
the equation
is equivalent to the de Donder-Weyl equations
We have
and
The claimed equation comes from contracting in $\mathbf{d}\phi^a_{,i} \wedge \mathbf{d}\phi^a \wedge (\iota_{\partial_i}vol_\Sigma)$ all but the $i$th vector with the contracted volume form. The remaining contractions are then
and these cancel against the horizontal derivative contributions form above.
We discuss here how Hamiltonian $n$-vector fields are equivalently homorphisms from an $n$-dimensional surface into the Poisson bracket Lie n-algebra associated with the given n-plectic geometry.
By the discussion at n-plectic geometry, a (pre-)n-plectic manifold $(X,\omega)$ induces a Poisson bracket Lie n-algebra $\mathfrak{pois}(X,\omega)$, given as follows:
in positive degree $k$ it has the degree $(n-k)$-differential forms on $X$
in degree 0 it has pairs $(v,A)$ of Hamiltonian n-forms $A$ with their Hamiltonian vector fields $v$ (such that $\mathbf{d}A = \iota_v \omega$)
and
whose n-ary Lie bracket for $n \geq 2$ is non-trivial only on $n$-tuples of degree-0 elements, where it is given by
and whose unary Lie bracket is given by the de Rham differential, $\{-\} = \mathbf{d}$.
This means that the $n$-extended Hamilton equation of def. reads in terms of the Poisson bracket Lie n-algebra equivalently thus (hgp 13, remark 2.5.10):
Let now $\Sigma$ be a manifold to be regarded as a spacetime/worldvolume and let $E \to \Sigma$ be a vector bundle, to be regarded as a field bundle. Assume for simplicity of notation that both $\Sigma$ and $E$ are in fact Cartesian spaces (which is always true locally). Write $(j^1 E)^\ast$ for the coreresponding dual jet bundle, equipped with the canonical “kinetic” n-plectic form which in local coordinates reads
as discussed at multisymplectic geometry. Write $\mathfrak{pois}(X,\omega)$ for the corresponding Poisson bracket Lie n-algebra.
which induce the canonical map under the projection to vector fields on $\Sigma$ are equivalently tuples consisting of
a smooth function $H$;
$n$ vector fields $v_i$
on $(j^1 E)^\ast$, such that the $(v_i)$ satisfy the de Donder-Weyl equation? for Hamiltonian $H$
By the discussion at Maurer-Cartan element, a homomorphism as indicated is equivalently an element
of total degree 1, which satifies the Maurer-Cartan equation
If we suggestively write $\mathbf{d}\sigma^i$ for the generators of the Grassmann algebra $\wedge^\bullet(\mathbb{R}^n)$, then, by remark , $\mathcal{J}$ is of the form
where the $v_i$ are vector fields on the extended phase space,
and $H$ is a smooth function. Again by remark , the Maurer-Cartan equation on this element is equivalent to
for all $1 \leq k \leq n-1$ and all $\{i_j\}_{j = 1}^k$, and
The condition on the projection means that
This way the last equation is the de Donder-Weyl equation?. But then
We can further interpret this in local prequantum field theory as follows:
By (hgp 13) the Poisson bracket Lie n-algebra is the Lie differentiation of the smooth automorphism n-group of any prequantization of $(X,\omega)$ by a prequantum n-bundle $\nabla$:
regarded as an object in the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}$ of that cohesive (∞,1)-topos $\mathbf{H}$ of smooth ∞-groupoids, hence of the quantomorphism n-group)
In terms of this Lie integrated perspective, remark says that $H$-Hamitlonian $n$-vector fields are the infinitesimal n-morphisms in $\left(\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}\right)^{\Box_n}$ whose faces carry trivial labels.
Here for instance for $n = 2$ a 2-morphism in $\left(\mathbf{H}_{/\mathbf{B}^2 U(1)_{conn}}\right)^{\Box_2}$ looks, “viewed from the top”, like a diagram in $\mathbf{H}$ of the form
with a big 3-homotopy filling this pyramid.
The equation of remark is the Maurer-Cartan equation exhibiting an infinitesimal such situation.
More in detai: the Lie integration of $\mathfrak{pois}(X,\omega)$ is a simplicial object which in simplicial degree $n$ has as elements the Maurer-Cartan elements of $\Omega^\bullet(\Sigma_n)\otimes \mathfrak{pois}(X,\omgea)$, where $\Sigma_n = \Delta^n$ is the $n$-dimensional simplex, regarded as a smooth manifold (with boundaries and corners).
Write then $\mathbf{d}\sigma^i$ for the canonical basis of 1-forms on $\Sigma_n$, then consider an element in there is of the form
This satisfying the Maurer-Cartan equation
then is equivalent to
which is again the de Donder-Weyl-Hamilton equation of motion.
For $(X,\omega)$ a pre-n-plectic manifold and $\mathfrak{poiss}(X,\omega)$ the corresponding Poisson bracket Lie n-algebra then L-∞ algebra homomorphisms of the form
are in bijection to tuples consisting of $k$ vector fields $(v_1, \cdots, v_k)$ on $X$ and differential forms $\{J_{i_1 \cdots i_l}\}$ and a smooth function $H$ on $X$ such that
holds.
The following is the higher/local analog of the symplectic Noether theorem.
For $(X,\omega)$ a pre-n-plectic manifold, (…)
let $H \in C^\infty(X)$ be a smooth function to be regarded as a de Donder-Weyl Hamiltonian and let $(v_1, \cdots, v_n)$ be a Hamiltonian $n$-vector field, hence a solution to the Hamilton-de Donder-Weyl equation of motion. By prop. this corresponds to an L-∞ algebra map of the form
(higher symplectic Noether theorem)
The extension of $((v_1, \cdots, v_n), H)$ to a homomorphism of the form
is equivalently the choice of a vector field $v_0$ on $X$ such that
The specific condition of def. appears as equation (4) in
Last revised on October 2, 2013 at 02:19:42. See the history of this page for a list of all contributions to it.