On Field Theoretic Generalizations of a Poisson Algebra

I. V. Kanatchikov

doi:10.1016/s0034-4877(97)85919-8

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arXiv · hep-th/9710069

On Field Theoretic Generalizations of a Poisson Algebra

I. V. Kanatchikov

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Abstract

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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I. V. Kanatchikov. 1997-10-10. On Field Theoretic Generalizations of a Poisson Algebra. https://doi.org/10.1016/s0034-4877(97)85919-8

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