On covariant Poisson brackets in classical field theory

Journal Article · Thu Oct 15 04:00:00 EDT 2015 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4932011· OSTI ID:22479546

Forger, Michael ^[1]; Salles, Mário O. ^[1]

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1. Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, BR–05315-970 São Paulo, SP (Brazil)

How to give a natural geometric definition of a covariant Poisson bracket in classical field theory has for a long time been an open problem—as testified by the extensive literature on “multisymplectic Poisson brackets,” together with the fact that all these proposals suffer from serious defects. On the other hand, the functional approach does provide a good candidate which has come to be known as the Peierls–De Witt bracket and whose construction in a geometrical setting is now well understood. Here, we show how the basic “multisymplectic Poisson bracket” already proposed in the 1970s can be derived from the Peierls–De Witt bracket, applied to a special class of functionals. This relation allows to trace back most (if not all) of the problems encountered in the past to ambiguities (the relation between differential forms on multiphase space and the functionals they define is not one-to-one) and also to the fact that this class of functionals does not form a Poisson subalgebra.

• OSTI ID: 22479546
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 10 Vol. 56; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
FIELD THEORIES
FUNCTIONALS
GEOMETRY