Canonical structure of classical field theory in the polymomentum phase space

Abstract

Canonical structure of classical field theory in n dimensions is studied within the covariant polymomentum Hamiltonian formulation of De Donder-Weyl (DW). The bi-vertical (n + 1)-form, called polysymplectic, is put forward as a generalization of the symplectic form in mechanics. Although not given in intrinsic geometric terms differently than a certain coset it gives rise to an invariantly defined map between horizontal forms playing the role of dynamical variables and the so-called vertical multivectors generalizing Hamiltonian vector fields. The analogue of the Poisson bracket on forms is defined which leads to the structure of Z-graded Lie algebra on the so-called Hamiltonian forms for which the map above exists. A generalized Poisson structure appears in the form of what we call a "higher-order" and a right Gerstenhaber algebra. The equations of motion of forms are formulated in terms of the Poisson bracket with the DW Hamiltonian n-form H vol ( vol is the space-time volume form, H is the DW Hamiltonian function) which is found to be related to the operation of the total exterior differentiation of forms. A few applications and a relation to the standard Hamiltonian formalism in field theory are briefly discussed. *

Geometry of Hamiltonian n-vector fields in multisymplectic field theory

“…While Forger and Römer [4] work on the extended multisymplectic phase space P that generalises the doubly extended phase space of time dependent symplectic mechanics, Kanatchikov [7,8] uses a space that has one dimension less than P and can be interpreted as the parameter space of hypersurfaces of constant DW Hamiltonians. This space will be denotedP for the rest of this paper.…”

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“…(1.1), have been studied in our previous papers [27][28][29] to which we refer for more details. The analogue of the Poisson bracket in the DW formulation is deduced from the object, called the polysymplectic form, which in local coordinates can be written in the form 1 Ω = −dy a ∧ dp µ a ∧ ω µ and is viewed as a field theoretic generalization of the symplectic form within the DW formulation.…”

“…Note that, as a consequence, the 1 Strictly speaking this object is understood as the equivalence class of forms modulo the forms of the horizontal degree n, see [27] for more details. Henceforth we denote ω :…”

“…In fact, the singular theories from the point of view of the canonical formalism can be regular from the precanonical point of view (as e.g. the Nambu-Goto string [27]) or vice verse (as e.g. the Dirac spinor field [10]).…”

“…It is only recently that a proper Poisson bracket operation, which is defined on differential forms representing the dynamical variables and leads to a Poisson-Gerstenhaber algerba structure, has been found within the DW theory in [25][26][27][28][29] (see also [18,30,31] for recent generalizations). This progress has been accompanied and followed by further developments in "multisymplectic" generalizations of the symplectic geometry aimed at applications in field theory and the calculus of variations [38,39,[41][42][43][44] and in other geometric aspects of the Lagrangian and Hamiltonian formalism in field theory [32-37, 40, 45] which to a great extent are been so far basically ignored by the wider mathematical physics community.…”