Abstract
Recently, large families of two-dimensional quantum field theories with factorizing S-matrices have been constructed by the operator-algebraic methods, by first showing the existence of observables localized in wedge-shaped regions. However, these constructions have been limited to the class of S-matrices whose components are analytic in rapidity in the physical strip. In this work, we construct candidates for observables in wedges for scalar factorizing S-matrices with poles in the physical strip and show that they weakly commute on a certain domain. We discuss some technical issues concerning further developments, especially the self-adjointness of the candidate operators here and strong commutativity between them.
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Alazzawi S.: Deformations of fermionic quantum field theories and integrable models. Lett. Math. Phys. 103(1), 37–58 (2013)
Alazzawi, S.: Deformations of quantum field theories and the construction of interacting models. (2014). Ph.D. thesis, Universität Wien. http://arxiv.org/abs/1503.00897
Babujian, H., 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, vol. 19, pp. 34–49 (2004)
Babujian H.M., Foerster A., Karowski M.: The form factor program: a review and new results—the nested SU(N) off-shell Bethe ansatz. SIGMA 2, 082 (2006)
Babujian H., Foerster A., Karowski M.: Exact form factors in integrable quantum field theories: the scaling Z(N)-Ising model. Nucl. Phys. B 736(3), 169–198 (2006)
Bischoff M.: Models in boundary quantum field theory associated with lattices and loop group models. Commun. Math. Phys. 315(3), 827–858 (2012)
Bischoff M., Tanimoto Y.: Construction of wedge-local nets of observables through Longo–Witten endomorphisms. II. Commun. Math. Phys. 317(3), 667–695 (2013)
Bischoff M., Tanimoto Y.: Integrable QFT and Longo–Witten endomorphisms. Ann. Henri Poincaré 16(2), 569–608 (2015)
Borchers H.-J.: The CPT-theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143(2), 315–332 (1992)
Borchers H.J., Buchholz D., Schroer B.: Polarization-free generators and the S-matrix. Commun. Math. Phys. 219(1), 125–140 (2001)
Bostelmann H., Cadamuro D.: An operator expansion for integrable quantum field theories. J. Phys. A 46(9), 095401,25 (2013)
Bostelmann H., Cadamuro D.: Characterization of local observables in integrable quantum field theories. Commun. Math. Phys. 337(3), 1199–1240 (2015)
Buchholz D., D’Antoni C., Longo R.: Nuclear maps and modular structures. I. General properties. J. Funct. Anal. 88(2), 233–250 (1990)
Buchholz D., Lechner G.: Modular nuclearity and localization. Ann. Henri Poincaré 5(6), 1065–1080 (2004)
Buchholz D., Lechner G., Summers S.J.: Warped convolutions, Rieffel deformations and the construction of quantum field theories. Commun. Math. Phys. 304(1), 95–123 (2011)
Cadamuro, D.: A characterization theorem for local operators in factorizing scattering models. Ph.D. thesis, Universität Göttingen (2012). http://arxiv.org/abs/1211.3583
Cadamuro, D., Tanimoto, Y.: (2015) (in preparation)
Cardy J.L., Mussardo G.: S-matrix of the Yang-Lee edge singularity in two dimensions. Phys. Lett. B 225(3), 275–278 (1989)
Dorey, P.: Exact S-matrices. In: Conformal field theories and integrable models (Budapest, 1996). Lecture Notes in Physics, vol. 498, pp. 85–125. Springer, Berlin (1997)
Driessler, W., Fröhlich, J.: The reconstruction of local observable algebras from the euclidean green’s functions of relativistic quantum field theory. In: Annales de L’Institut Henri Poincare Section Physique Theorique, vol. 27, pp. 221–236 (1977)
Dybalski W., Tanimoto Y.: Asymptotic completeness in a class of massless relativistic quantum field theories. Commun. Math. Phys. 305(2), 427–440 (2011)
Fring A., Mussardo G., Simonetti P.: Form factors of the elementary field in the Bullough–Dodd model. Phys. Lett. B 307(1–2), 83–90 (1993)
Fröhlich J., Seiler E.: The massive Thirring–Schwinger model (QED2): convergence of perturbation theory and particle structure. Helv. Phys. Acta 49(6), 889–924 (1976)
Grosse, H., Lechner, G.: Wedge-local quantum fields and noncommutative Minkowski space. J. High Energy Phys. 11:012,26 (2007)
Haag, R.: Local quantum physics. In: Texts and Monographs in Physics, Fields, Particles, Algebras, 2nd edn. Springer, Berlin, (1996)
Korff, C.: Lie algebraic structures in integrable models, affine Toda field theory. Ph.D. Thesis, Freie Universität Berlin (2000). http://arxiv.org/abs/hep-th/0008200
Lechner G.: Polarization-free quantum fields and interaction. Lett. Math. Phys. 64(2), 137–154 (2003)
Lechner, G.: On the construction of quantum field theories with factorizing S-matrices. Ph.D. Thesis, Universität Göttingen (2006). http://arxiv.org/abs/math-ph/0611050
Lechner G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277(3), 821–860 (2008)
Lechner G.: Deformations of quantum field theories and integrable models. Commun. Math. Phys. 312(1), 265–302 (2012)
Lechner G., Longo R.: Localization in nets of standard spaces. Commun. Math. Phys. 336(1), 27–61 (2015)
Lechner G., Schlemmer J., Tanimoto Y.: On the equivalence of two deformation schemes in quantum field theory. Lett. Math. Phys. 103(4), 421–437 (2013)
Lechner G., Schützenhofer C.: Towards an operator-algebraic construction of integrable global gauge theories. Ann. Henri Poincaré 15(4), 645–678 (2014)
Longo, R.: Real Hilbert subspaces, modular theory, \({{\rm SL}(2, \mathbf{R})}\) and CFT. In: Von Neumann algebras in Sibiu: Conference Proceedings, pp. 33–91. Theta, Bucharest (2008)
Longo R., Witten E.: An algebraic construction of boundary quantum field theory. Commun. Math. Phys. 303(1), 213–232 (2011)
Quella, T.: Formfactors and locality in integrable models of quantum field theory in 1+1 dimensions. Diploma thesis, Freie Universität Berlin (1999) (in German). http://www.thp.uni-koeln.de/~tquella/1999QuellaDiploma
Reed, M., Simon, B.: Methods of modern mathematical physics. II. In: Fourier Analysis, Self-Adjointness. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1975)
Rudin W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)
Schroer B.: Modular localization and the bootstrap-formfactor program. Nucl. Phys. B 499(3), 547–568 (1997)
Schroer B.: Modular wedge localization and the d = 1 + 1 formfactor program. Ann. Phys. 275(2), 190–223 (1999)
Smirnov, F.A.: Form factors in completely integrable models of quantum field theory. In: Advanced Series in Mathematical Physics, vol. 14. World Scientific Publishing Co. Inc., River Edge (1992)
Takesaki, M.: Theory of operator algebras. II. In: Encyclopaedia of Mathematical Sciences, Operator Algebras and Non-commutative Geometry, 6, vol. 125. Springer, Berlin (2003)
Tanimoto Y.: Construction of wedge-local nets of observables through Longo–Witten endomorphisms. Commun. Math. Phys. 314(2), 443–469 (2012)
Tanimoto Y.: Construction of two-dimensional quantum field models through Longo–Witten endomorphisms. Forum Math. Sigma 2, e7 (2014)
Tanimoto, Y.: Self-adjointness of bound state operators in integrable quantum field theory. (2015). arXiv:1508.06402
Zamolodchikov A.B., Zamolodchikov A.B.: Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Ann. Phys. 120(2), 253–291 (1979)
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Communicated by Y. Kawahigashi
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Cadamuro, D., Tanimoto, Y. Wedge-Local Fields in Integrable Models with Bound States. Commun. Math. Phys. 340, 661–697 (2015). https://doi.org/10.1007/s00220-015-2448-z
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DOI: https://doi.org/10.1007/s00220-015-2448-z