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CONSTRUCTION OF TWO-DIMENSIONAL QUANTUM FIELD MODELS THROUGH LONGO–WITTEN ENDOMORPHISMS

Part of: Scattering theory Quantum field theory; related classical field theories

Published online by Cambridge University Press:  10 April 2014

YOH TANIMOTO
YOH TANIMOTO*
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germanyyoh.tanimoto@theorie.physik.uni-goettingen.de

Abstract

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We present a procedure for constructing families of local, massive and interacting Haag–Kastler nets on the two-dimensional spacetime through an operator-algebraic method. A proof of existence of local observables is given without relying on modular nuclearity. By a similar technique, another family of wedge-local nets is constructed using certain endomorphisms of conformal nets recently studied by Longo and Witten.

MSC classification

Secondary: 81T05: Axiomatic quantum field theory; operator algebras 81T40: Two-dimensional field theories, conformal field theories, etc. 81U99: None of the above, but in this section
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e7
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author 2014

References

Babujian, H. and Karowski, M., ‘Towards the construction of Wightman functions of integrable quantum field theories’, in Proceedings of 6th International Workshop on Conformal Field Theory and Integrable Models, 19, 2004, 34–49.CrossRefGoogle Scholar
Bargmann, V., ‘On unitary ray representations of continuous groups’, Ann. of Math. (2) 59 (1954), 1–46.Google Scholar
Baumgärtel, H., Operator Algebraic Methods in Quantum Field Theory, (Akademie Verlag, Berlin, 1995).Google Scholar
Beisert, N. et al. , ‘Review of AdS/CFT integrability: an overview’, Lett. Math. Phys. 99 (2012), 3–32.Google Scholar
Bischoff, M. and Tanimoto, Y., ‘Integrable QFT and Longo–Witten endomorphisms’, Ann. Henri Poincare (2014) (to appear).CrossRefGoogle Scholar
Bischoff, M. and Tanimoto, Y., ‘Construction of wedge-local nets of observables through Longo–Witten endomorphisms. II’, Commun. Math. Phys. 317 (2013), 667–695.Google Scholar
Borchers, H.-J., ‘The CPT-theorem in two-dimensional theories of local observables’, Commun. Math. Phys. 143 (1992), 315–332.Google Scholar
Borchers, H. -J., Buchholz, D. and Schroer, B., ‘Polarization-free generators and the $S$ -matrix’, Commun. Math. Phys. 219 (2001), 125–140.CrossRefGoogle Scholar
Bratteli, O. and Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics. 2. Equilibrium States. Models in Quantum Statistical Mechanics, 2nd edn, Texts and Monographs in Physics (Springer-Verlag, Berlin, 1997).Google Scholar
Brunetti, R., Guido, D. and Longo, R., ‘Modular structure and duality in conformal quantum field theory’, Commun. Math. Phys. 156 (1993), 201–219.Google Scholar
Buchholz, D., D’Antoni, C. and Longo, R., ‘Nuclear maps and modular structures. I. General properties’, J. Funct. Anal. 88 (1990), 233–250.CrossRefGoogle Scholar
Buchholz, D. and Lechner, G., ‘Modular nuclearity and localization’, Ann. Henri Poincaré 5 (2004), 1065–1080.CrossRefGoogle Scholar
Buchholz, D., Lechner, G. and Summers, S. J., ‘Warped convolutions, Rieffel deformations and the construction of quantum field theories’, Commun. Math. Phys. 304 (2011), 95–123.Google Scholar
Castro-Alvaredo, O. A. and Fring, A., ‘Form factors from free fermionic Fock fields, the Federbush model’, Nucl. Phys. B 618 (2001), 437–464.CrossRefGoogle Scholar
Doplicher, S. and Longo, R., ‘Standard and split inclusions of von Neumann algebras’, Invent. Math. 75 (1984), 493–536.CrossRefGoogle Scholar
Doplicher, S. and Roberts, J. E., ‘A new duality theory for compact groups’, Invent. Math. 98 (1989), 157–218.CrossRefGoogle Scholar
Doplicher, S. and Roberts, J. E., ‘Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics’, Commun. Math. Phys. 131 (1990), 51–107.Google Scholar
Dybalski, W. and Tanimoto, Y., ‘Asymptotic completeness in a class of massless relativistic quantum field theories’, Commun. Math. Phys. 305 (2011), 427–440.Google Scholar
Gabbiani, F. and Fröhlich, J., ‘Operator algebras and conformal field theory’, Commun. Math. Phys. 155 (1993), 569–640.Google Scholar
Haag, R., Local Quantum Physics: Fields, Particles, Algebras, 2nd edn, Texts and Monographs in Physics (Springer-Verlag, Berlin, 1996).Google Scholar
Lechner, G., ‘Polarization-free quantum fields and interaction’, Lett. Math. Phys. 64 (2003), 137–154.Google Scholar
Lechner, G., ‘Construction of quantum field theories with factorizing $S$ -matrices’, Commun. Math. Phys. 277 (2008), 821–860.Google Scholar
Lechner, G., ‘Deformations of quantum field theories and integrable models’, Commun. Math. Phys. 312 (2011), 265–302.Google Scholar
Lechner, G., Schlemmer, J. and Tanimoto, Y., ‘On the equivalence of two deformation schemes in quantum field theory’, Lett. Math. Phys. 103 (2012), 421–437.CrossRefGoogle Scholar
Lechner, G. and Schützenhofer, C., ‘Towards an operator-algebraic construction of integrable global gauge theories’, Ann. Henri Poincaré 15 (4) (2014), 645–678.CrossRefGoogle Scholar
Longo, R., ‘Real Hilbert subspaces, modular theory, $\mathrm{SL}(2,\mathbf{R})$ and CFT’, in Von Neumann Algebas in Sibiu: Conference Proceedings, Theta, Bucharest, 2008, 33–91.Google Scholar
Longo, R. and Witten, E., ‘An algebraic construction of boundary quantum field theory’, Commun. Math. Phys. 303 (2011), 213–232.CrossRefGoogle Scholar
Müger, M., ‘Superselection structure of massive quantum field theories in $1+1$ -dimensions’, Rev. Math. Phys. 10 (1998), 1147–1170.Google Scholar
Ruijsenaars, S. N. M., ‘Scattering theory for the Federbush, massless Thirring and continuum Ising models’, J. Funct. Anal. 48 (1982), 135–171.CrossRefGoogle Scholar
Ruijsenaars, S. N. M., ‘The Wightman axioms for the fermionic Federbush model’, Commun. Math. Phys. 87 (1982/83), 181–228.Google Scholar
Schroer, B., Localization and nonperturbative local quantum physics 2012, arXiv:hep-th/9805093.Google Scholar
Smirnov, F. A., Form Factors in Completely Integrable Models of Quantum Field Theory. Advanced Series in Mathematical Physics, vol. 14 (World Scientific Publishing Co. Inc., River Edge, NJ, 1992).Google Scholar
Takesaki, M., Theory of Operator Algebras. II, Encyclopaedia of Mathematical Sciences 125 (Springer-Verlag, Berlin, 2003), Operator Algebras and Non-Commutative Geometry, 6.Google Scholar
Tanimoto, Y., ‘Construction of wedge-local nets of observables through Longo–Witten endomorphisms’, Commun. Math. Phys. 314 (2012), 443–469.Google Scholar
Tanimoto, Y., ‘Noninteraction of waves in two-dimensional conformal field theory’, Commun. Math. Phys. 314 (2012), 419–441.Google Scholar
Weiner, M., ‘An algebraic version of Haag’s theorem’, Commun. Math. Phys. 305 (2011), 469–485.CrossRefGoogle Scholar