Independence Conditions for Nets of Local Algebras as Sheaf Conditions

TL;DR

This paper explores how topos-theoretic and sheaf-theoretic methods can be used to analyze independence conditions in algebraic quantum field theory, linking sheaf properties to physical independence of spacetime regions.

Contribution

It introduces a sheaf-theoretic reformulation of nets of operator algebras and relates sheaf conditions to independence criteria in quantum field theory.

Findings

Reformulation of nets as functors into ringed topoi

Connection between sheaf conditions and kinematical independence

Relation of C*-sheaf conditions to C*-independence

Abstract

We apply constructions from topos-theoretic approaches to quantum theory to algebraic quantum field theory. Thus a net of operator algebras is reformulated as a functor that maps regions of spacetime into a category of ringed topoi. We ask whether this functor is a sheaf, a question which is related to the net satisfying certain kinematical independence conditions. In addition, we consider a C*-algebraic version of Nuiten's recent sheaf condition, and demonstrate how it relates to C*-independence of the underlying net of operator algebras.