From the Poincaré-Cartan Form to a Gerstenhaber Algebra of Poisson Brackets in Field Theory

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.