Poisson bundles over unordered configurations

TL;DR

This paper constructs a Poisson algebra bundle over the unordered configuration space of a manifold, using a novel 2-monoidal category structure to model multilocal observables in classical field theory.

Contribution

It introduces a new framework with a 2-monoidal category structure for vector bundles, enabling the construction of a Poisson 2-algebra bundle for classical field observables.

Findings

Develops a Poisson algebra bundle over unordered configurations

Defines a symmetric algebra structure using two tensor products

Provides a foundation for explicit observable descriptions in future work

Abstract

In this paper we construct a Poisson algebra bundle whose distributional sections are suitable to represent multilocal observables in classical field theory. To do this, we work with vector bundles over the unordered configuration space of a manifold $M$ and consider the structure of a $2$-monoidal category given by the usual (Hadamard) tensor product of bundles and a new (Cauchy) tensor product which provides a symmetrized version of the usual external tensor product of vector bundles on $M$. We use the symmetric algebras with respect to both products to obtain a Poisson 2-algebra bundle mimicking the construction of Peierls bracket from the causal propagator in field theory. The explicit description of observables from this Poisson algebra bundle will be carried out in a forthcoming paper.