Types of quantum field thories
Multisymplectic geometry is a generalization of symplectic geometry in the context of variational calculus and mechanical systems in which the symplectic form is generalized from a closed 2-form to a closed -form, for – the n-plectic form.
It is closely related to the de Donder-Weyl formalism of variational calculus. In the context of quantization it is meant to provide a refinement of geometric quantization which is well-adapted to -dimensional quantum field theory. However, details of the multisymplectic quantization procedure remain under investigation.
We comment a bit on how to, presumably, think of multisymplectic geometry from the nPOV, in the context of higher geometric quantization. Readers may want to skip ahead to traditional technical discussion at Extended phase spacehase space).
in the multisymplectic extended phase space of an -dimensional field theory a state is not just a point, but an -dimensional subspace.
See also n-plectic geometry.
In as far as symplectic geometry encodes Hamiltonian mechanics, multisymplectic geometry may be regarded as resolving the symplectic geometry of the Hamiltonian mechanics of classical field theory: the kinematics of an -dimensional field theory may be encoded in an degree symplectic form.
In this application to physics, multisymplectic geometry is also known as the covariant symplectic approach to field theory (e.g. section 2 here).
The idea is that under a suitable fiber integration multisymplectic geometry becomes ordinary symplectic form on the ordinary phase space of the theory, similar to, and in fact as a special case of, how for instance a line bundle on a loop space with a 2-form Chern class may arise by transgression from a bundle gerbe down on the original space, with a 3-form class.
By effectively undoing this implicit transgression in the ordinary Hamiltonian mechanics of classical field theory, multisymplectic geomtry provides a general framework for a geometric, covariant formulation of classical field theory, where covariant formulation means that spacelike and timelike directions on a given space-time be treated on equal footing.
We discuss here the refinement in multisymplectic geometry of the covariant phase spaces of classical field theory/prequantum field theory from (pre-)symplectic manifolds of initial value data in a Cauchy surface to multisymplectic manifolds of local initial value data.
Recall that an ordinary phase space of a physical system is a symplectic manifold whose points correspond to the states of the system. The extended phase space of an -dimensional quantum field theory is a multisymplectic space whose points correspond to pairs consisting of
a point in the field theory’s parameter space – an “event”;
a state of the theory “at that event”.
So extended phase spaces localizes the information about states : a point in here encodes not just the entire state of the system, but remembers explcitly what that state is like over any point in parameter space.
Beware of the standard source of confusion here when correlating this formalism with actual physics: the physical spacetime that we inhabit may be given either by or by :
in the description of the quantum mechanics of objects propagating in our physical spacetime, subject to forces exerted by fixed background gauge fields (such as electrons propagating in our particle accelerator, subject to the electromagnetic field in the accelerator tube), physical spacetime is identifid with target space , while is the worldvolume of the object that propagates through . The field configurations on are really the maps that determine how the object sits in spacetime.
in quantum mechanics of fields on spacetime, such as the quantized electromagnetic field in a laser, it is which represents physical spacetime, and is some abstract space, for instance a smooth version of the classifying space , so that a field configuration encodes a line bundle with connection that encodes a configuration of the electromagnetic field.
Beware that in low dimensions one often distinguishes between the space of configurations and that of trajectories or histories . This comes from the case where for a particle propagating on the maps are the possible configurations of the particle at a given parameter times, while maps are the trajectories. But for the higher dimensional and extended field theories under discussion here, this distinction becomes a bit obsolete and trajectories become just a special case of configurations.
In the non-covariant approach one would try to consider the a cotangent bundle of the configuration space as phase space . Contrary to that, in the covariant approach one considers the much smaller space instead. This is then called the covariant configuration space or covariant configuration bundle.
Write for the first order jet bundle of the configuration space bundle . Its fiber over are equivalence classses of germs of sections at , where two germs are identified if their first derivatives coincide.
Given a vector bundle over a smooth manifold of dimension , the affine dual first jet bundle (or often just dual first jet bundle for short) is the vector bundle whose fiber at is the set of affine maps
Given , with , the dual jet bundle is isomorphic to a particular vector sub-bundle of the -form bundle . To see this, first consider the following
Given a point , a tangent vector is said to be vertical if . Define
to be the subbundle of the -form bundle whose fiber at consists of all such that
for all vertical vectors . Sections of are called -horizontal -forms.
In words, an -horizontal -form is one which has at most one “leg” not along . This is made very explicit in the proof of the following proposition.
It suffices to work locally with respect to a good open cover, so we reduce the statement to the special case of the sigma model i.e. the trivial bundle over . By the assumoption that admits an orientation we may pick a volume form .
and second, wedge products of cotangent vectors of type and a single one of type :
If , this basis gives an isomorphism
The volume form on also determines isomorphisms
We thus have obtained an isomorphism
On the other hand, the trivialization gives an isomorphism of affine spaces
which has the side-effect of exhibiting on the structure of a vector space. Since we’ve identified
with , an affine map from to is just an element of plus a constant. So, we obtain
This gives a specific vector bundle isomorphism , as desired.
In these coordinates the canonical -form on is:
and the multisymplectic form is
This definition was proposed in (Forger-Romero 04, section 2.5).
The second term in prop. 2 is what is traditionally called the Legendre transform in multisymplectic geometry/de Donder-Weyl formalism. Def. 7 may be regarded as explaining the conceptual role of this expression, in particular in view of the following proposition.
Given a local Lagrangian , the pullback of the canonical pre-n-plectic form , def. 4, along the Legendre transform , def. 7, to the first jet bundle is the sum of the Euler-Lagrange equation and the canonical symplectic form from covariant phase space formalism:
It follows that
This statement is essentially the content of (Forger-Romero 04, equation (54) and theorem 1). In the above form in terms of variational bicomplex notions this statement has been amplified by Igor Khavkine.
We write out the multisymplectic geometry corresponding to a free field theory.
The canonical momenta are
where in the last line we adopted the notation of remark 4.
This shows that the canonical multisymplectic form is the “covariant symplectic potential current density” which is induced by the free field Lagrangian.
See also (Forger Romero 04, section 3.2).
parameter space: , the worldline;
target space: , some manifold – spacetime;
configuration bundle: ;
jet bundle: .
for a local patch with coordinate functions , there are canonically induced coordinates on written .
Here a collection of vaues is a position of the particle and is a velocity of the particle. Notice that in this covariant approach these are not positions and velocities “at a given time”. Rather, a point in specified a parameter time and a corresponding position and velocity.
Let be a local patch of with canonical coordinates .
The canonical 2-form on the extended phase space in this case is traditionally locally written as
A field configuration of the electromagnetic field is a line bundle with connection on . If we assume the corresponding bundle to be trivial, then this is just a 1-form on . So in this simplified case we can take
parameter space: some 2-dimensional manifold – the worldsheet – for instance models a closed string propagating without interaction.
target space. spacetime;
covariant configuration bundle .
For simplicity we will only consider the case where is the cylinder and is -dimensional Minkowski spacetime, . A solution of the classical bosonic string is then a map which is a critical point of the area subject to certain boundary conditions.
Equivalently, by exploiting symmetries in the variational problem, one can describe solutions by equipping with its standard Minkowski metric and then solving the dimensional field theory specified by the Lagrangian density
Here are standard coordinates on and is the Minkowski metric on , while are the coordinates of the map in and is the Minkowski metric on . The corresponding Euler-Lagrange equation is just a version of the wave equation:
The space can be thought of as a trivial bundle over , and the graph of a function is a smooth section of . We write the coordinates of a point as . Let be the first jet bundle of . We may regard as a vector bundle whose fiber over is .
The Lagrangian density for the string can be defined as a smooth function on :
which depends in this example only on the fiber coordinates .
From the Lagrangian , the de Donder-Weyl Hamiltonian can be constructed via a Legendre transform. It is given as follows:
where are defined implicitly by , and are coordinates on the fiber . Note that differs from the standard (non-covariant) Hamiltonian density for a field theory:
Let be a section of and let be a smooth section of restricted to with fiber coordinates . It is then straightforward to show that is a solution of the Euler-Lagrange equations if and only if and satisfy the following system of equations:
This system of equations is a generalization of Hamilton’s equations for the point particle.
As explained above, the covariant phase space for the bosonic string is the dual jet bundle , and this space is equipped with a canonical 2-form whose exterior derivative is a multisymplectic 3-form. Using the isomorphism
a point in gets coordinates . In terms of these coordinates,
The multisymplectic structure on is thus
So, the variable may be considered as canonically conjugate to the area form .
As before, let be a section of and let be a smooth section of restricted to . Consider the submanifold with coordinates:
Note that is constructed from , and from the constraint . This constraint is analogous to the one that is used in finding constant energy solutions in the extended phase space approach to classical mechanics. At each point in , a tangent bivector can be defined as
Explicit computation reveals that the submanifold is generated by solutions to Hamilton’s equations if and only if
There is also the notion of
Which is different, but related…
There is in this sense 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 -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 -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 Hamiltonian 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 -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, -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 way.
Not much is known about the interpretation of Poisson forms of form degree between zero and n-1. However, as -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 -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 integral.
This discussion so far concerns 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 .
A comprehensive source on covariant field theory with plenty of further references is
Much of the material in the section on covariant field theoryhase space) is based on this.
Other discussions include
The relation to covariant phase space methods is discussed in
Other texts include
Much of the above survey of recent developments and of the following list of references is reproduced from the web-page
References mentioned above are
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 M. J. Gotay, J. Isenberg, J. E. Marsden: Momentum maps and classical relativistic fields. I: Covariant field theory arXiv:physics/9801019v2.
 M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations. I: Covariant Hamiltonian formalism In M. Francaviglia (ed.), Mechanics, analysis and geometry: 200 years after Lagrange Amsterdam etc.: North-Holland (1991), 203–235.
 M. de Leon, D. Martin de Diego, A. Santamaria-Merino, Symmetries in Classical Field Theory arXiv:math-ph/0404013.
 M. J. Gotay, J. E. Marsden: Stress-energy-momentum tensors and the Belinfante–Rosenfeld formula Contemp. Math. vol. 132, AMS, Providence, 1992, 367–392.
 M. Forger, H. Römer, Currents and the energy-momentum tensor in classical field theory: a fresh look at an old problem Ann. Phys. (N.Y.) 309 (2004) 306–389. arXiv:hep-th/0307199.
 A. Lew, J. E. Marsden, M. Ortiz, M. West, An overview of variational integrators In L. P. Franca (ed.), Finite Element Methods: 70’s and Beyond. Barcelona (2003).
 C. Paufler, H. Römer, The Geometry of Hamiltonian -vector fields in Multisymplectic Field Theory J. Geom. Phys. 44, No.1(2002), 52–69. arXiv:math-ph/0102008.
 Y. Kosmann-Schwarzbach: Derived brackets. Lett. Math. Phys. 69 (2004) 61–87 arXiv:math.DG/0312524.
 M. Forger, C. Paufler, H. Römer: The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory Rev. Math. Phys. 15 (2003) 705 arXiv:math-ph/0202043.
 M. Forger, C. Paufler, H. Römer, Hamiltonian Multivector Fields and Poisson Forms in Multisymplectic Field Theory arXiv:math-ph/0407057.
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 M. O. Salles, Campos Hamiltonianos e Colchete de Poisson na Teoria Geométrica dos Campos , PhD thesis, IME-USP, June 2004.
 M. J. Gotay, J. M. Nester, Generalized constraint algorithm and special presymplectic manifolds In G. E. Kaiser, J. E. Marsden, Geometric methods in mathematical physics, Proc. NSF-CBMS Conf., Lowell/Mass. 1979, Berlin: Springer-Verlag, Lect. Notes Math. 775 (1980) 78–80.
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Higher order moment maps are considered in
I.V. Kanatchikov, DeDonder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory , (arXiv:hep-th/9810165)
I.V. Kanatchikov, Geometric (pre)quantization in the polysymplectic approach to field theory , (arXiv:hep-th/0112263).
Kanatchikov’s “algebra of observables” is what he calls a “higher-order Gerstenhaber algebra”. (The “bracket” in this structure fails to satisfy Leibniz’s rule as a derivation of the product.) The relationship between it and the Lie superalgebra of observables constructed by Forger, Paufler, and Roemer is discussed in this paper:
and (Forger-Romero 04) above.
Kanatchikov’s formalism was used by S.P. Hrabak to propose a multisymplectic refinement of BV-BRST formalism. See there for more details.