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

TL;DR

This paper introduces a novel method using the Cahiers topos to construct Poisson algebras for non-linear scalar field theories, providing a clear separation of geometric, algebraic, and analytical components.

Contribution

It develops a synthetic differential geometry framework for Poisson algebra construction in non-linear field theories, highlighting the geometric and algebraic aspects distinctly.

Findings

Constructs Poisson algebras for non-linear scalar fields within the Cahiers topos.

Provides a natural smooth structure on the solution space of field equations.

Identifies a family of observables forming a Poisson algebra.

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.