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 DonderWeyl (DW). The bivertical ( 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 socalled vertical multivectors generalizing Hamiltonian vector fields. The analogue of the Poisson bracket on forms is defined which leads to the structure of Zgraded Lie algebra on the socalled 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 nform H overlinevol ( overlinevol) is the spacetime 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:

Reports on Mathematical Physics
 Pub Date:
 February 1998
 DOI:
 10.1016/S00344877(98)801821
 arXiv:
 arXiv:hepth/9709229
 Bibcode:
 1998RpMP...41...49K
 Keywords:

 High Energy Physics  Theory;
 General Relativity and Quantum Cosmology;
 Mathematical Physics;
 Mathematics  Differential Geometry
 EPrint:
 45 pages, LaTeX2e, to appear in Reports on Mathematical Physics v. 41 No. 1 (1998)