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

Autoren: Marco Benini, Alexander Schenkel (2017)

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.

Zeitschrift:
Ann. Henri Poincare
Verlag:
Springer
Seiten:
1435-1464
Band:
18, no. 4

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