no. 3
Free products in AQFT
[Produits libres dans l’AQFT]
Longo, Roberto1 ; Tanimoto, Yoh2 ; Ueda, Yoshimichi3
1 Dipartimento di Matematica Università di Roma “Tor Vergata” Via della Ricerca Scientifica 1, I-00133 Roma (Italy)
2 Dipartimento di Matematica Università di Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy)
3 Graduate School of Mathematics, Nagoya University Furocho, Chikusaku Nagoya, 464-8602 (Japan)
Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1229-1258.

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3269
Classification : 81T05, 81T40, 46L54
Keywords: Algebraic QFT, half-sided modular inclusions, conformal nets, free products
Mot clés : la théorie des champs quantiques algébriques, réseaux conformes inclusions modulaires unilatéral, produits libres

Longo, Roberto 1 ; Tanimoto, Yoh 2 ; Ueda, Yoshimichi 3

1 Dipartimento di Matematica Università di Roma “Tor Vergata” Via della Ricerca Scientifica 1, I-00133 Roma (Italy)
2 Dipartimento di Matematica Università di Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy)
3 Graduate School of Mathematics, Nagoya University Furocho, Chikusaku Nagoya, 464-8602 (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2019__69_3_1229_0,
     author = {Longo, Roberto and Tanimoto, Yoh and Ueda, Yoshimichi},
     title = {Free products in {AQFT}},
     journal = {Annales de l'Institut Fourier},
     pages = {1229--1258},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {3},
     year = {2019},
     doi = {10.5802/aif.3269},
     zbl = {07067430},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3269/}
}
TY  - JOUR
AU  - Longo, Roberto
AU  - Tanimoto, Yoh
AU  - Ueda, Yoshimichi
TI  - Free products in AQFT
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 1229
EP  - 1258
VL  - 69
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3269/
DO  - 10.5802/aif.3269
LA  - en
ID  - AIF_2019__69_3_1229_0
ER  - 
%0 Journal Article
%A Longo, Roberto
%A Tanimoto, Yoh
%A Ueda, Yoshimichi
%T Free products in AQFT
%J Annales de l'Institut Fourier
%D 2019
%P 1229-1258
%V 69
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3269/
%R 10.5802/aif.3269
%G en
%F AIF_2019__69_3_1229_0
Longo, Roberto; Tanimoto, Yoh; Ueda, Yoshimichi. Free products in AQFT. Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1229-1258. doi : 10.5802/aif.3269. https://aif.centre-mersenne.org/articles/10.5802/aif.3269/

[1] Alazzawi, Sabina; Lechner, Gandalf Inverse Scattering and Local Observable Algebras in Integrable Quantum Field Theories, Commun. Math. Phys., Volume 354 (2017) no. 3, pp. 913-956 (https://arxiv.org/abs/1608.02359) | DOI | MR | Zbl

[2] Araki, Huzihiro; Zsidó, László Extension of the structure theorem of Borchers and its application to half-sided modular inclusions, Rev. Math. Phys., Volume 17 (2005) no. 5, pp. 491-543 (https://arxiv.org/abs/math/0412061) | DOI | MR | Zbl

[3] Barnett, Lance Free product von Neumann algebras of type III , Proc. Am. Math. Soc., Volume 123 (1995) no. 2, pp. 543-553 (http://www.ams.org/journals/proc/1995-123-02/S0002-9939-1995-1224611-7/S0002-9939-1995-1224611-7.pdf) | DOI | MR | Zbl

[4] Bischoff, Marcel; Tanimoto, Yoh Integrable QFT and Longo-Witten endomorphisms, Ann. Henri Poincaré, Volume 16 (2015) no. 2, pp. 569-608 (https://arxiv.org/abs/1305.2171) | DOI | MR | Zbl

[5] Borchers, Hans-Jürgen The CPT-theorem in two-dimensional theories of local observables, Commun. Math. Phys., Volume 143 (1992) no. 2, pp. 315-332 (http://projecteuclid.org/euclid.cmp/1104248958) | DOI | MR | Zbl

[6] Borchers, Hans-Jürgen; Buchholz, Detlev; Schroer, Bert Polarization-free generators and the S-matrix, Commun. Math. Phys., Volume 219 (2001) no. 1, pp. 125-140 (https://arxiv.org/abs/hep-th/0003243) | DOI | MR | Zbl

[7] Bostelmann, Henning; Lechner, Gandalf; Morsella, Gerardo Scaling limits of integrable quantum field theories, Rev. Math. Phys., Volume 23 (2011) no. 10, pp. 1115-1156 (https://arxiv.org/abs/1105.2781) | DOI | MR | Zbl

[8] Buchholz, Detlev Product states for local algebras, Commun. Math. Phys., Volume 36 (1974), pp. 287-304 (http://projecteuclid.org/euclid.cmp/1103859773) | DOI | MR | Zbl

[9] Buchholz, Detlev; D’Antoni, Claudio; Fredenhagen, Klaus The universal structure of local algebras, Commun. Math. Phys., Volume 111 (1987) no. 1, pp. 123-135 (http://projecteuclid.org/euclid.cmp/1104159470) | DOI | MR | Zbl

[10] Buchholz, Detlev; D’Antoni, Claudio; Longo, Roberto Nuclear maps and modular structures. I. General properties, J. Funct. Anal., Volume 88 (1990) no. 2, pp. 233-250 | DOI | MR | Zbl

[11] Buchholz, Detlev; D’Antoni, Claudio; Longo, Roberto Nuclearity and Thermal States in Conformal Field Theory, Commun. Math. Phys., Volume 270 (2007) no. 1, pp. 267-293 (https://arxiv.org/abs/math-ph/0603083) | DOI | MR | Zbl

[12] Buchholz, Detlev; Lechner, Gandalf Modular nuclearity and localization, Ann. Henri Poincaré, Volume 5 (2004) no. 6, pp. 1065-1080 (https://arxiv.org/abs/math-ph/0402072) | DOI | MR | Zbl

[13] Camassa, Paolo; Longo, Roberto; Tanimoto, Yoh; Weiner, Mihály Thermal States in Conformal QFT. II, Commun. Math. Phys., Volume 315 (2012) no. 3, pp. 771-802 (https://arxiv.org/abs/1109.2064) | DOI | MR | Zbl

[14] Dabrowski, Yoann; Dykema, Kenneth J.; Mukherjee, Kunal The simplex of tracial quantum symmetric states, Stud. Math., Volume 225 (2014) no. 3, pp. 203-218 (https://arxiv.org/abs/1401.4692) | DOI | MR | Zbl

[15] D’Antoni, Claudio; Longo, Roberto; Rădulescu, Florin Conformal nets, maximal temperature and models from free probability, J. Oper. Theory, Volume 45 (2001) no. 1, pp. 195-208 (https://arxiv.org/abs/math/9810003) | MR | Zbl

[16] Dykema, Ken; Mukherjee, Kunal KMS quantum symmetric states, J. Math. Phys., Volume 58 (2017) no. 1, 012103, 12 pages (Art. ID 012103, 12 pages, https://arxiv.org/abs/1609.01225) | MR | Zbl

[17] Dykema, Kenneth J. Factoriality and Connes’ invariant T(ℳ) for free products of von Neumann algebras, J. Reine Angew. Math., Volume 450 (1994), pp. 159-180 (https://arxiv.org/abs/funct-an/9302005) | DOI | MR | Zbl

[18] Florig, Martin On Borchers’ theorem, Lett. Math. Phys., Volume 46 (1998) no. 4, pp. 289-293 (https://dx.doi.org/10.1023/A:1007546507392) | DOI | MR | Zbl

[19] Glimm, James; Jaffe, Arthur Quantum physics. A functional integral point of view, Springer, 1987, xxii+535 pages (https://books.google.com/books?id=VSjjBwAAQBAJ) | DOI | MR | Zbl

[20] Guido, Daniele; Longo, Roberto; Wiesbrock, Hans-Werner Extensions of conformal nets and superselection structures, Commun. Math. Phys., Volume 192 (1998) no. 1, pp. 217-244 (https://arxiv.org/abs/hep-th/9703129) | DOI | MR | Zbl

[21] Houdayer, Cyril; Ueda, Yoshimichi Asymptotic structure of free product von Neumann algebras, Math. Proc. Camb. Philos. Soc., Volume 161 (2016) no. 3, pp. 489-516 | DOI | MR | Zbl

[22] Kähler, Ralf; Wiesbrock, Hans-Werner Modular theory and the reconstruction of four-dimensional quantum field theories, J. Math. Phys., Volume 42 (2001) no. 1, pp. 74-86 | DOI | MR | Zbl

[23] Lechner, Gandalf Polarization-free quantum fields and interaction, Lett. Math. Phys., Volume 64 (2003) no. 2, pp. 137-154 (https://arxiv.org/abs/hep-th/0303062) | DOI | MR | Zbl

[24] Lechner, Gandalf Construction of quantum field theories with factorizing S-matrices, Commun. Math. Phys., Volume 277 (2008) no. 3, pp. 821-860 (https://arxiv.org/abs/math-ph/0601022) | DOI | MR | Zbl

[25] Lechner, Gandalf Algebraic constructive quantum field theory: Integrable models and deformation techniques, Advances in Algebraic Quantum Field Theory, Springer, 2015, pp. 397-448 (https://arxiv.org/abs/1503.03822) | DOI | MR | Zbl

[26] Lechner, Gandalf; Longo, Roberto Localization in nets of standard spaces, Commun. Math. Phys., Volume 336 (2015) no. 1, pp. 27-61 | DOI | MR | Zbl

[27] Longo, Roberto Solution of the factorial Stone-Weierstrass conjecture. An application of the theory of standard split W * -inclusions, Invent. Math., Volume 76 (1984) no. 1, pp. 145-155 (https://eudml.org/doc/143121) | DOI | MR | Zbl

[28] Longo, Roberto Simple injective subfactors, Adv. Math., Volume 63 (1987) no. 2, pp. 152-171 | DOI | MR | Zbl

[29] Longo, Roberto Notes for a quantum index theorem, Commun. Math. Phys., Volume 222 (2001) no. 1, pp. 45-96 (https://arxiv.org/abs/math/0003082) | DOI | MR | Zbl

[30] Takesaki, Masamichi Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, 125, Springer, 2003, xxii+518 pages (https://books.google.com/books?id=-4GyR1VlQz4C) | MR | Zbl

[31] Tanimoto, Yoh Construction of Wedge-Local Nets of Observables Through Longo-Witten Endomorphisms, Commun. Math. Phys., Volume 314 (2012) no. 2, pp. 443-469 (https://arxiv.org/abs/1107.2629) | DOI | MR | Zbl

[32] Tanimoto, Yoh Noninteraction of Waves in Two-dimensional Conformal Field Theory, Commun. Math. Phys., Volume 314 (2012) no. 2, pp. 419-441 (https://arxiv.org/abs/1107.2662) | DOI | MR | Zbl

[33] Tanimoto, Yoh Construction of two-dimensional quantum field models through Longo-Witten endomorphisms, Forum Math. Sigma, Volume 2 (2014), e7, 31 pages (Art. ID e7, 31 pages, https://arxiv.org/abs/1301.6090) | DOI | MR | Zbl

[34] Ueda, Yoshimichi Factoriality, type classification and fullness for free product von Neumann algebras, Adv. Math., Volume 228 (2011) no. 5, pp. 2647-2671 | DOI | MR | Zbl

[35] Ueda, Yoshimichi On type III 1 factors arising as free products, Math. Res. Lett., Volume 18 (2011) no. 5, pp. 909-920 | DOI | MR | Zbl

[36] Voiculescu, Dan Symmetries of some reduced free product C * -algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) (Lecture Notes in Mathematics), Volume 1132, Springer, 1985, pp. 556-588 | DOI | MR | Zbl

[37] Voiculescu, Dan; Dykema, Kenneth J.; Nica, Alexandru Free random variables, CRM Monograph Series, 1, American Mathematical Society, 1992, vi+70 pages (https://books.google.com/books?id=r7BJAwAAQBAJ) | MR | Zbl

[38] Wiesbrock, Hans-Werner Half-sided modular inclusions of von-Neumann-algebras, Commun. Math. Phys., Volume 157 (1993) no. 1, pp. 83-92 (https://projecteuclid.org/euclid.cmp/1104253848) | MR | Zbl

[39] Wiesbrock, Hans-Werner Modular intersections of von Neumann algebras in quantum field theory, Commun. Math. Phys., Volume 193 (1998) no. 2, pp. 269-285 (https://www-sfb288.math.tu-berlin.de/pub/Preprints/preprint193.ps.gz) | DOI | MR | Zbl

Cité par Sources :