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arXiv · 1602.00708

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

Marco Benini·Alexander Schenkel
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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.

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Marco Benini, Alexander Schenkel. 2016-02-01. Poisson algebras for non-linear field theories in the Cahiers topos. https://doi.org/10.1007/s00023-016-0533-2

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