# Poisson algebras for non-linear field theories in the Cahiers topos

February 01, 2016

We develop an approach to construct Poisson algebras for non-linear scalar
field theories that is based on the Cahiers topos model for synthetic
differential geometry. In this framework the solution space of the field
equation carries a natural smooth structure and, following Zuckerman's ideas,
we can endow it with a presymplectic current. We formulate the Hamiltonian
vector field equation in this setting and show that it selects a family of
observables which forms a Poisson algebra. Our approach provides a clean
splitting between geometric and algebraic aspects of the construction of a
Poisson algebra, which are sufficient to guarantee existence, and analytical
aspects that are crucial to analyze its properties.

open access link
doi:10.1007/s00023-016-0533-2

@article{Benini:2016nqj,
author = "Benini, Marco and Schenkel, Alexander",
title = "{Poisson algebras for non-linear field theories in the
Cahiers topos}",
journal = "Annales Henri Poincare",
volume = "18",
year = "2017",
number = "4",
pages = "1435-1464",
doi = "10.1007/s00023-016-0533-2",
eprint = "1602.00708",
archivePrefix = "arXiv",
primaryClass = "math-ph",
reportNumber = "EMPG-16-03",
SLACcitation = "%%CITATION = ARXIV:1602.00708;%%"
}

Keywords:

non-linear classical field theory, synthetic differential geometry, Cahiers topos, Poisson algebras