On covariant Poisson brackets in classical field theory
Abstract
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 onetoone) and also to the fact that this class of functionals does not form a Poisson subalgebra.
 Authors:
 Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, BR–05315970 São Paulo, SP (Brazil)
 (Brazil)
 Publication Date:
 OSTI Identifier:
 22479546
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 10; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FIELD THEORIES; FUNCTIONALS; GEOMETRY
Citation Formats
Forger, Michael, Salles, Mário O., and Centro de Ciências Exatas e da Terra, Universidade Federal do Rio Grande do Norte, Campus Universitário – Lagoa Nova, BR–59078970 Natal, RN. On covariant Poisson brackets in classical field theory. United States: N. p., 2015.
Web. doi:10.1063/1.4932011.
Forger, Michael, Salles, Mário O., & Centro de Ciências Exatas e da Terra, Universidade Federal do Rio Grande do Norte, Campus Universitário – Lagoa Nova, BR–59078970 Natal, RN. On covariant Poisson brackets in classical field theory. United States. doi:10.1063/1.4932011.
Forger, Michael, Salles, Mário O., and Centro de Ciências Exatas e da Terra, Universidade Federal do Rio Grande do Norte, Campus Universitário – Lagoa Nova, BR–59078970 Natal, RN. 2015.
"On covariant Poisson brackets in classical field theory". United States.
doi:10.1063/1.4932011.
@article{osti_22479546,
title = {On covariant Poisson brackets in classical field theory},
author = {Forger, Michael and Salles, Mário O. and Centro de Ciências Exatas e da Terra, Universidade Federal do Rio Grande do Norte, Campus Universitário – Lagoa Nova, BR–59078970 Natal, RN},
abstractNote = {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 onetoone) and also to the fact that this class of functionals does not form a Poisson subalgebra.},
doi = {10.1063/1.4932011},
journal = {Journal of Mathematical Physics},
number = 10,
volume = 56,
place = {United States},
year = 2015,
month =
}

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