Annales Henri Poincaré

, Volume 18, Issue 4, pp 1435–1464 | Cite as

Poisson Algebras for Non-Linear Field Theories in the Cahiers Topos

  • Marco BeniniEmail author
  • Alexander Schenkel


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.


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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany
  2. 2.Department of MathematicsHeriot-Watt UniversityEdinburghUK
  3. 3.Maxwell Institute for Mathematical SciencesEdinburghUK
  4. 4.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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