The Locality Axiom in Quantum Field Theory and Tensor Products of $C^*$-algebras

TL;DR

This paper explores how the principle of locality in quantum field theory can be represented through tensor products of $C^*$-algebras, linking spacetime separation with algebraic commutativity.

Contribution

It demonstrates that the commutativity of observables in spacelike separated regions can be encoded via the tensorial structure of the associated $C^*$-algebras within locally covariant quantum field theory.

Findings

Commutativity of observables is represented by tensor products of $C^*$-algebras.

The split property is essential for the tensor product structure.

Minimal tensor product encodes spacelike separation independence.

Abstract

The prototype of mutually independent systems are systems which are localized in spacelike separated regions. In the framework of locally covariant quantum field theory we show that the commutativity of observables in spacelike separated regions can be encoded in the tensorial structure of the functor which associates unital $C^*$-algebras (the local observable algebras) to globally hyperbolic spacetimes. This holds under the assumption that the local algebras satisfy the split property and involves the minimal tensor product of $C^*$-algebras.