# Free products in AQFT

June 19, 2017

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.

Keywords:

*none*