Canonical structure of classical field theory in the polymomentum phase space

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 Hvol (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.