# 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 one-to-one) 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–05315-970 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–59078-970 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–59078-970 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–59078-970 Natal, RN. Thu .
"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–59078-970 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 one-to-one) 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 = {Thu Oct 15 00:00:00 EDT 2015},

month = {Thu Oct 15 00:00:00 EDT 2015}

}