Free products in AQFT
Annales de l'Institut Fourier, Volume 69 (2019) no. 3, p. 1229-1258

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 Möbius 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.

Nous appliquons la construction du produit libre à diverses algèbres locales dans la théorie des champs quantiques algébriques.

Si l’on prend le produit libre d’une infinité d’inclusions modulaires unilatéral identiques avec endomorphisme canonique ergodique, on obtient une inclusion modulaire unilatéral avec endomorphisme canonique ergodique et commutant relatif trivial. D’autre part, si nous prenons des réseaux covariants de Möbius avec la propriété de trace, nous pouvons construire une inclusion d’algèbres libres de von Neumann avec un grand commutant relatif, en considérant soit une famille finie d’inclusions identiques, soit une famille infinie d’inclusions deux à deux inéquivalentes. Dans l’espace-temps bi-dimensionnel, nous construisons des triplets de Borchers avec un commutant relatif trivial en prenant des produits libres d’un nombre infini de triplets de Borchers identiques. Il est possible que les produits libres d’un nombre fini de triplets de Borchers soient associés au réseau de Haag–Kastler avec une matrice S non triviale et non asymptotiquement complète, mais la non-trivialité des algèbres à double cône reste une question ouverte.

Received : 2017-06-19
Revised : 2018-02-06
Accepted : 2018-03-13
Published online : 2019-06-03
DOI : https://doi.org/10.5802/aif.3269
Classification:  81T05,  81T40,  46L54
Keywords: Algebraic QFT, half-sided modular inclusions, conformal nets, free products
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     author = {Longo, Roberto and Tanimoto, Yoh and Ueda, Yoshimichi},
     title = {Free products in AQFT},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {3},
     year = {2019},
     pages = {1229-1258},
     doi = {10.5802/aif.3269},
     zbl = {07067430},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_3_1229_0}
}
Free products in AQFT. Annales de l'Institut Fourier, Volume 69 (2019) no. 3, pp. 1229-1258. doi : 10.5802/aif.3269. https://aif.centre-mersenne.org/item/AIF_2019__69_3_1229_0/

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