Multisymplectic geometry

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 [1]. For a recent exposition on the differential geometry of this construction, see [2]. Multiphase space, together with a closed, nondegenerate differential (n+1)-form, the multisymplectic form, is an example of a multisymplectic manifold [3]. 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 [4] 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 [7].

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 [8] 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 [9] 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 [10], 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 [11]. In particular, (n-1)-forms in this context can be shown to arise exactly from the covariantly conserved currents of Noether symmetries [11], 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 (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 [8], 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") [12] 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 [13]. With the choice of a hypersurface, a constraint analysis [14] à la Dirac [15,16] can be performed [17]. Again, the necessary breaking of general covariance raises the need for an alternative formulation of all this [18]; first attempts have been made to carry out a Marsden-Weinstein reduction [19] for multisymplectic manifolds with symmetries [20]. However, not very much is known about how to quantize a multisymplectic geometry, see [21] for an approach using a path integral.

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 [22]. A definition of the covariant Legendre transform and the multiphase space has been given for this case [3].

References
[1] J. Kijowski, W. Szczyrba: A Canonical Structure For Classical Field Theories. Commun. Math. Phys. 46 (1976) 183.
[2] M. J. Gotay, J. Isenberg, J. E. Marsden: Momentum maps and classical relativistic fields. I: Covariant field theory. [arXiv:physics/9801019v2].
[3] 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.
[4] M. de Leon, D. Martin de Diego, A. Santamaria-Merino: Symmetries in Classical Field Theory. [arXiv:math-ph/0404013].
[5] M. J. Gotay, J. E. Marsden: Stress-energy-momentum tensors and the Belinfante-Rosenfeld formula. Contemp. Math. vol. 132, AMS, Providence, 1992, 367-392.
[6] 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].
[7] 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).
[8] C. Paufler, H. Römer: The Geometry of Hamiltonian n-vector fields in Multisymplectic Field Theory. J. Geom. Phys. 44, No.1(2002), 52-69. [arXiv: math-ph/0102008].
[9] Y. Kosmann-Schwarzbach: Derived brackets. Lett. Math. Phys. 69 (2004) 61-87 [arXiv:math.DG/0312524].
[10] 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].
[10] M. Forger, C. Paufler, H. Römer: Hamiltonian Multivector Fields and Poisson Forms in Multisymplectic Field Theory. [arXiv:math-ph/0407057].
[11] M. J. Gotay: A multisymplectic framework for classical field theory and the calculus of variations. II: Space + time decomposition. Differ. Geom. Appl. 1(4) (1991), 375-390.
[12] M. O. Salles: Campos Hamiltonianos e Colchete de Poisson na Teoria Geométrica dos Campos, PhD thesis, IME-USP, June 2004.
M. Forger, S. V. Romero: Covariant Poisson Brackets in Geometric Field Theory, Commun. Math. Phys. 256 (2005), 375-410. [arXiv:math-ph/0408008].
[13] 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.
[14] P. A. M. Dirac: Lectures on Quantum Mechanic. Belfer Graduate School of Science, Yeshiva University, N.Y., 1964.
[15] M. Henneaux, C. Teitelboim: Quantization of Gauge systems. Princeton University Press, 1992.
[16] M. J. Gotay, J. Isenberg, J. E. Marsden, R. Montgomery: Momentum Maps and Classical Relativistic Fields II: Canonical Analysis of Field Theories. (2004) [arXiv:math-ph/0411032].
[17] N. P. Landsman: Against the Wheeler-DeWitt equation. Class. Quan. Grav. 12 (1995) L119-L124. [arXiv:gr-qc/9510033].
[18] J. E. Marsden, A. Weinstein: Reduction of symplectic manifolds with symmetry. Rept. Math. Phys. 5 (1974) 121-130.
[19] F. Munteanu, A. M. Rey, M. Salgado: The Günther's formalism in classical field theory: momentum map and reduction. J. Math. Phys. 45, No. 5 (2004) 1730-1750.
[20] D. Bashkirov, G. Sardanashvily: Covariant Hamiltonian Field Theory. Path Integral Quantization. [arXiv:hep-th/0402057].
[21] D. J. Saunders: The Geometry of Jet Bundles. Lond. Math. Soc. Lect. Notes Ser. 142, Cambr. Univ. Pr., Cambridge, 1989.

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