under construction
A Hamiltonian -vector field is the -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 -vector field to mapping spaces out of an -manifold yields an ordinary Hamiltonian vector field.
Let be a smooth manifold equipped with a degree differential form for , with the pair regarded as a multisymplectic manifold/n-plectic manifold.
Let moreover be a smooth function, to be regarded as an extended Hamiltonian function, hence a de Donder-Weyl Hamiltonian.
Then a Hamiltonian -vector field on is an -multivector field satisfying the analog of Hamilton's equations, namely the differential equation of differential 1-forms on
Let be a manifold to be regarded as a spacetime/worldvolume and let be a vector bundle, to be regarded as a field bundle. Assume for simplicity of notation that both and are in fact Cartesian spaces (which is always true locally). Write 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 and smooth functions on and for vector fields
the equation
is equivalent to the de Donder-Weyl equations
We have
and
The claimed equation comes from contracting in all but the 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 -vector fields are equivalently homorphisms from an -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 induces a Poisson bracket Lie n-algebra , given as follows:
in positive degree it has the degree -differential forms on
in degree 0 it has pairs of Hamiltonian n-forms with their Hamiltonian vector fields (such that )
and
whose n-ary Lie bracket for is non-trivial only on -tuples of degree-0 elements, where it is given by
and whose unary Lie bracket is given by the de Rham differential, .
This means that the -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 be a manifold to be regarded as a spacetime/worldvolume and let be a vector bundle, to be regarded as a field bundle. Assume for simplicity of notation that both and are in fact Cartesian spaces (which is always true locally). Write 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 for the corresponding Poisson bracket Lie n-algebra.
which induce the canonical map under the projection to vector fields on are equivalently tuples consisting of
on , such that the satisfy the de Donder-Weyl equation? for Hamiltonian
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 for the generators of the Grassmann algebra , then, by remark , is of the form
where the are vector fields on the extended phase space,
and is a smooth function. Again by remark , the Maurer-Cartan equation on this element is equivalent to
for all and all , 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 by a prequantum n-bundle :
regarded as an object in the slice (∞,1)-topos of that cohesive (∞,1)-topos of smooth ∞-groupoids, hence of the quantomorphism n-group)
In terms of this Lie integrated perspective, remark says that -Hamitlonian -vector fields are the infinitesimal n-morphisms in whose faces carry trivial labels.
Here for instance for a 2-morphism in looks, “viewed from the top”, like a diagram in 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 is a simplicial object which in simplicial degree has as elements the Maurer-Cartan elements of , where is the -dimensional simplex, regarded as a smooth manifold (with boundaries and corners).
Write then for the canonical basis of 1-forms on , 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 a pre-n-plectic manifold and the corresponding Poisson bracket Lie n-algebra then L-∞ algebra homomorphisms of the form
are in bijection to tuples consisting of vector fields on and differential forms and a smooth function on such that
holds.
The following is the higher/local analog of the symplectic Noether theorem.
For a pre-n-plectic manifold, (…)
let be a smooth function to be regarded as a de Donder-Weyl Hamiltonian and let be a Hamiltonian -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 to a homomorphism of the form
is equivalently the choice of a vector field on 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.