On Field Theoretic Generalizations of a Poisson Algebra

Details On Field Theoretic Generalizations of a Poisson Algebra On Field Theoretic Generalizations of a Poisson Algebra

1997-10-08

arxiv.org/abs/hep-th/9710069v2

Rept.Math.Phys. 40 (1997) 225

Physics

High Energy Physics

High Energy Physics - Theory

10 pages, LaTeX2e. Missprint in Ref. 1 is corrected (hep-th/9709229 instead of ...029)

Scientific paper

10.1016/S0034-4877(97)85919-8

A few generalizations of a Poisson algebra to field theory canonically formulated in terms of the polymomentum variables are discussed. A graded Poisson bracket on differential forms and an $(n+1)$-ary bracket on functions are considered. The Poisson bracket on differential forms gives rise to various generalizations of a Gerstenhaber algebra: the noncommutative (in the sense of Loday) and the higher-order (in the sense of the higher order graded Leibniz rule). The $(n+1)$-ary bracket fulfills the properties of the Nambu bracket including the ``fundamental identity'', thus leading to the Nambu-Poisson algebra. We point out that in the field theory context the Nambu bracket with a properly defined covariant analogue of Hamilton's function determines a joint evolution of several dynamical variables.

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