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

Igor V. Kanatchikov
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Igor V. Kanatchikov: RWTH Aachen, Institut für Theoretische Physik

A chapter in Quantization, Coherent States, and Complex Structures, 1995, pp 173-183 from Springer

Abstract: 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 Poincaré-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.

Keywords: Poisson Bracket; Dynamical Variable; Schrodinger Equation; Poisson Algebra; Classical Field Theory (search for similar items in EconPapers)
Date: 1995
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4899-1060-8_19

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DOI: 10.1007/978-1-4899-1060-8_19

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