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

February 01, 2016

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.

Keywords:

non-linear classical field theory, synthetic differential geometry, Cahiers topos, Poisson algebras