From the Poincaré-Cartan form to a Gerstenhaber algebra of Poisson brackets in field theory

Details From the Poincaré-Cartan form to a Gerstenhaber algebra of Poisson brackets in field theory From the Poincaré-Cartan form to a Gerstenhaber algebra of Poisson brackets in field theory

1995-11-06

arxiv.org/abs/hep-th/9511039v1

Physics

High Energy Physics

High Energy Physics - Theory

LaTeX file, 11pp

Scientific paper

We review the recent generalization of the basic structures of classical analytical mechanics to field theory within the framework of the De Donder-Weyl (DW) covariant canonical theory. We start from the Poincar\'e-Cartan form and construct the analogue of the symplectic form -- the polysymplectic form of degree n+1, where n is the dimension of the space-time. The dynamical variables are represented by differential forms and the polysymplectic form leads to the definition of the Poisson brackets on forms. The Poisson brackets equip the exterior algebra of dynamical variables with a structure of a "higher-order" Gerstenhaber algebra. We also briefly outline a possible approach to field quantization which proceeds from the DW Hamiltonian formalism and the Poisson brackets of forms.

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