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

Benini, Marco; Schenkel, Alexander

Abstract

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.

Citation

Benini, M., & Schenkel, A. (2017). Poisson algebras for non-linear field theories in the Cahiers topos. Annales Henri Poincaré, 18(4), 1435-1464. https://doi.org/10.1007/s00023-016-0533-2

• Journal Article Type: Article
• Acceptance Date: Oct 3, 2016
• Online Publication Date: Nov 17, 2016
• Publication Date: Apr 1, 2017
• Deposit Date: Mar 6, 2017
• Publicly Available Date: Mar 6, 2017
• Journal: Annales Henri Poincaré
• Print ISSN: 1424-0637
• Electronic ISSN: 1424-0661
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 18
• Issue: 4
• Pages: 1435-1464
• DOI: https://doi.org/10.1007/s00023-016-0533-2
• Keywords: non-linear classical field theory, synthetic differential geometry, Cahiers topos, Poisson algebras
• Public URL: https://nottingham-repository.worktribe.com/output/828058
• Publisher URL: http://link.springer.com/article/10.1007%2Fs00023-016-0533-2
• Related Public URLs: https://arxiv.org/abs/1602.00708
• Additional Information: The final publication is available at Springer via http://dx.doi.org/10.1007/s00023-016-0533-2.