Poisson bundles over unordered configurations
Alessandra Frabetti, Olga Kravchenko, Leonid Ryvkin
TL;DR
The paper develops a covariant, geometrically natural framework for multilocal observables in relativistic classical field theory by placing vector bundles over the unordered configuration space $ ext{UConf}(M)$. It introduces a symmetric 2-monoidal structure with Hadamard $ ensor$ and Cauchy $oxtimes$ tensor products, builds Hadamard and Cauchy algebra bundles, and constructs the Cauchy–Hadamard 2-algebra bundle $oldsymbol{S}^{oxtimes} S^{ ensor}(V)$. A Poisson structure is defined via a kernel $k:(JE)^*oxtimes (JE)^* o ext{I}_{ ensor}$, extending to density-valued bundles and yielding a Poisson framework on sections; this connects to the Peierls bracket and causal propagators in field theory. The results lay the algebraic-geometric foundation for covariant multilocal observables, with distributional sections and explicit observable descriptions to be addressed in the companion paper. This framework promises a rigorous path from classical covariant observables to their quantum counterparts within a 2-categorical, configuration-space setting.
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.