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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Communicated by Karl-Henning Rehren.
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Benini, M., Schenkel, A. Poisson Algebras for Non-Linear Field Theories in the Cahiers Topos. Ann. Henri Poincaré 18, 1435–1464 (2017). https://doi.org/10.1007/s00023-016-0533-2
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DOI: https://doi.org/10.1007/s00023-016-0533-2