Canonical Structure of Classical Field Theory in the Polymomentum Phase Space

Abstract

Canonical structure of the space-time symmetric analogue of the Hamiltonian formalism in field theory based on the De Donder-Weyl (DW) theory is studied. In $n$ space-time dimensions the set of $n$ polymomenta is associated to the space-time derivatives of field variables. The polysymplectic $(n+1)$-form generalizes the simplectic form and gives rise to a map between horizontal forms playing the role of dynamical variables and vertical multivectors generalizing Hamiltonian vector fields. Graded Poisson bracket is defined on forms and leads to the structure of a Z-graded Lie algebra on the subspace of the so-called Hamiltonian forms for which the map above exists. A generalized Poisson structure arises in the form of what we call a ``higher-order'' and a right Gerstenhaber algebra. Field euations and the equations of motion of forms are formulated in terms of the graded Poisson bracket with the DW Hamiltonian $n$-form $H\vol$ ($\vol$ is the space-time volume form and $H$ is the DW Hamiltonian function). A few applications to scalar fields, electrodynamics and the Nambu-Goto string, and a relation to the standard Hamiltonian formalism in field theory are briefly discussed. This is a detailed and improved account of our earlier concise communications (hep-th/9312162, hep-th/9410238, and hep-th/9511039).