Abstract
We consider the generalization of the basic structures of classical analytical mechanics to field theory within the framework of the De Donder-Weyl (DW) co-variant canonical theory. We start from the Poincare-Cartan form and construct the analogue of the symplectic form — the polysymplectic form of degree (n + 1), n is the dimension of the space-time. The dynamical variables are represented by differential forms and the polysymplectic form leads to a natural definition of the Poisson brackets on forms. The Poisson brackets equip the exterior algebra of dynamical variables with the structure of a “higher-order” Gerstenhaber algebra. We also briefly discuss a possible approach to field quantization which proceeds from the DW Hamiltonian formalism and the Poisson brackets of forms.
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