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

Abstract

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 Poincaré-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.

arXiv commentLaTeX file, 11pp

References in corpus (1)

• On the Canonical Structure of the De Donder-Weyl Covariant Hamiltonian Formulation of Field Theory I. Graded Poisson brackets and equations of motion

Cited by in corpus (6)

• DeDonder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory
• Geometric (pre)quantization in the polysymplectic approach to field theory
• On the Geometry of Multi-Dirac Structures and Gerstenhaber Algebras
• A frame bundle generalization of multisymplectic field theories
• Schrödinger Functional of a Quantum Scalar Field in Static Space-Times from Precanonical Quantization
• p-Mechanics and De Donder-Weyl Theory