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

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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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Authors and Affiliations

  1. Institut für Mathematik, Universität Potsdam, Karl-Liebknecht-Str. 24-25, 14476, Potsdam, Germany

    Marco Benini

  2. Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh, EH14 4AS, UK

    Alexander Schenkel

  3. Maxwell Institute for Mathematical Sciences, Edinburgh, UK

    Alexander Schenkel

  4. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    Alexander Schenkel

Authors
  1. Marco Benini
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  2. Alexander Schenkel
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Correspondence to Marco Benini.

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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

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