Free products in AQFT

TL;DR

This paper explores the application of free product constructions to local algebras in algebraic quantum field theory, revealing new structures and properties of inclusions and Borchers triples with implications for quantum field models.

Contribution

It introduces novel free product constructions in AQFT, demonstrating how they produce inclusions with large or trivial relative commutants and constructing Borchers triples with specific properties.

Findings

Infinite free products of identical inclusions yield trivial relative commutants.

Finite free products of Borchers triples can have nontrivial S-matrices.

Construction of Borchers triples with trivial relative commutant in 2D spacetime.

Abstract

We apply the free product construction to various local algebras in algebraic quantum field theory. If we take the free product of infinitely many identical half-sided modular inclusions with ergodic canonical endomorphism, we obtain a half-sided modular inclusion with ergodic canonical endomorphism and trivial relative commutant. On the other hand, if we take M\"obius covariant nets with trace class property, we are able to construct an inclusion of free product von Neumann algebras with large relative commutant, by considering either a finite family of identical inclusions or an infinite family of inequivalent inclusions. In two dimensional spacetime, we construct Borchers triples with trivial relative commutant by taking free products of infinitely many, identical Borchers triples. Free products of finitely many Borchers triples are possibly associated with Haag-Kastler net having S-matrix which is nontrivial and non asymptotically complete, yet the nontriviality of double cone algebras remains open.