Canonical structure of classical field theory in the polymomentum phase space
Jul, 1997Citations per year
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 space-time dimensions the set of polymomenta is associated to the space-time derivatives of field variables. The polysymplectic -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 -form H\vol (\vol is the space-time volume form and 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, 9410238, 9511039).Note:
- 45 pages, LaTeX2e, to appear in Reports on Mathematical Physics v. 41 No. 1 (1998)
- field theory: classical
- Hamiltonian formalism
- differential forms: symplectic
- commutation relations
- phase space
- bibliography
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