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 the 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 overlinevol ( overlinevol) 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.
- Publication:
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Reports on Mathematical Physics
- Pub Date:
- February 1998
- DOI:
- 10.1016/S0034-4877(98)80182-1
- arXiv:
- arXiv:hep-th/9709229
- Bibcode:
- 1998RpMP...41...49K
- Keywords:
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- High Energy Physics - Theory;
- General Relativity and Quantum Cosmology;
- Mathematical Physics;
- Mathematics - Differential Geometry
- E-Print:
- 45 pages, LaTeX2e, to appear in Reports on Mathematical Physics v. 41 No. 1 (1998)