Multisymplectic geometry constitutes the general framework for a
geometric, covariant formulation of classical field theory. Here,
covariant formulation means that spacelike and timelike directions
on a given space-time be treated on equal footing. With this
principle, one can construct a covariant form of the Legendre
transformation which associates to every field variable as many
conjugated momenta, the multimomenta , as there are space-time
dimensions. These, together with the field variables, those of
n-dimensional space-time, and an extra variable, the energy
variable, span the multiphase space . For a recent exposition on the
differential geometry of this construction, see .
Multiphase space, together with a closed, nondegenerate differential
(n+1)-form, the multisymplectic form, is an example of a
multisymplectic manifold . Among the achievements
of the multisymplectic approach is a geometric formulation of the
relation of infinitesimal symmetries and covariantly conserved
quantities (Noether's theorem), see
 for a recent review, and
[5,6] for a clarification of
the improvement techniques ("Belinfante-Rosenfeld formula") of the
energy-momentum tensor in classical field theory. Multisymplectic
geometry also provides convenient sets of variational integrators
for the numerical study of partial differential equations
Since in multisymplectic geometry, the symplectic 2-form of
classical mechanics is replaced by a closed differential form of
higher tensor degree, multivector fields and differential forms have
their natural appearance. (See  for an interpretation of multivector fields
as describing solutions to field equations infinitesimally.)
Multivector fields form a graded Lie
algebra with the Schouten bracket (see
 for a review and unified viewpoint).
Using the multisymplectic (n+1)-form, one can construct a new
bracket for the differential forms, the Poisson forms
, generalizing a well-known formula
for the Poisson brackets related to a symplectic 2-form. A
remarkable fact is that in order to establish a Jacobi identity, the
multisymplectic form has to have a potential, a condition that is
not seen in symplectic geometry. Further, the admissible
differential forms, the Poisson forms, are subject to the
rather strong restrictions on their dependence on the multimomentum
variables . In particular,
(n-1)-forms in this context can be shown to arise exactly from the
covariantly conserved currents of Noether symmetries
, which allows their pairing with
spacelike hypersurfaces to yield conserved charges in a geometric
Not much is known about the interpretation of Poisson forms of form degree between zero and n-1. However, as (n-1)-forms describe vector fields
and hence collections of lines [2, 10], and as (certain) functions describe n-vector fields and hence collections of bundle sections , it seems natural to speculate that the intermediate forms may be useful for the branes of String theory.
The Hamiltonian, infinite dimensional formulation of classical field
theory requires the choice of a spacelike hypersurface ("Cauchy
surface")  which manifestly breaks the general
covariance of the theory at hand. For (n-1)-forms, the above
mentioned new bracket reduces to the Peierls-deWitt bracket after
integration over the spacelike hypersurface . With
the choice of a hypersurface, a constraint analysis
 à la Dirac [15,16] can be performed . Again, the necessary breaking
of general covariance raises the need for an alternative formulation of all this
; first attempts have been made to carry out a
Marsden-Weinstein reduction  for
multisymplectic manifolds with symmetries
. However, not very much is known
about how to quantize a multisymplectic geometry, see
 for an approach using a path
So far, everything was valid for field theories of first order, i.e.
where the Lagrangian depends on the fields and their first
derivatives. Higher order theories can be reduced to first order
ones for the price of introducing auxiliary fields. A direct
treatment would involve higher order jet bundles
. A definition of the covariant Legendre
transform and the multiphase space has been given for this case
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Maintained by C. Mertzlufft-Paufler. Last modified 29 March 2007.