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

Marco Benini, Alexander Schenkel
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