Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 100, 39 pages      arXiv:1602.04696      https://doi.org/10.3842/SIGMA.2016.100

Bound State Operators and Wedge-Locality in Integrable Quantum Field Theories

Yoh Tanimoto ab
a) Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan
b) Institut für Theoretische Physik, Göttingen University, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

Received February 19, 2016, in final form October 10, 2016; Published online October 19, 2016

Abstract
We consider scalar two-dimensional quantum field theories with a factorizing $S$-matrix which has poles in the physical strip. In our previous work, we introduced the bound state operators and constructed candidate operators for observables in wedges. Under some additional assumptions on the $S$-matrix, we show that, in order to obtain their strong commutativity, it is enough to prove the essential self-adjointness of the sum of the left and right bound state operators. This essential self-adjointness is shown up to the two-particle component.

Key words: Haag-Kastler net; integrable models; wedge; von Neumann algebras; Hardy space; self-adjointness.

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References

  1. Alazzawi S., Deformations of quantum field theories and the construction of interacting models, Ph.D. Thesis, Universität Wien, 2014, arXiv:1503.00897.
  2. Bischoff M., Tanimoto Y., Construction of wedge-local nets of observables through Longo-Witten endomorphisms. II, Comm. Math. Phys. 317 (2013), 667-695, arXiv:1111.1671.
  3. Borchers H.-J., Buchholz D., Schroer B., Polarization-free generators and the $S$-matrix, Comm. Math. Phys. 219 (2001), 125-140, hep-th/0003243.
  4. Buchholz D., Lechner G., Modular nuclearity and localization, Ann. Henri Poincaré 5 (2004), 1065-1080, math-ph/0402072.
  5. Buchholz D., Lechner G., Summers S.J., Warped convolutions, Rieffel deformations and the construction of quantum field theories, Comm. Math. Phys. 304 (2011), 95-123, arXiv:1005.2656.
  6. Cadamuro D., Tanimoto Y., Wedge-local fields in integrable models with bound states, Comm. Math. Phys. 340 (2015), 661-697, arXiv:1502.01313.
  7. Cadamuro D., Tanimoto Y., Wedge-local fields in integrable models with bound states II: Diagonal $S$-matrix, Ann. Henri Poincaré, to appear, arXiv:1601.07092.
  8. Cadamuro D., Tanimoto Y., Modular nuclearity and the inverse scattering problem for integrable QFT with bound states, in preparation.
  9. Cadamuro D., Tanimoto Y., Wedge-local fields in the deformed sine-Gordon model, in preparation.
  10. Carpi S., Kawahigashi Y., Longo R., Weiner M., From vertex operator algebras to conformal nets and back, Mem. Amer. Math. Soc., to appear, arXiv:1503.01260.
  11. Driessler W., Fröhlich J., The reconstruction of local observable algebras from the Euclidean Green's functions of relativistic quantum field theory, Ann. Inst. H. Poincaré Sect. A 27 (1977), 221-236.
  12. Dybalski W., Tanimoto Y., Asymptotic completeness in a class of massless relativistic quantum field theories, Comm. Math. Phys. 305 (2011), 427-440, arXiv:1006.5430.
  13. Epstein H., Some analytic properties of scattering amplitudes in quantum field theory, in Particle Symmetries and Axiomatic Field Theory, Vol. 1, Axiomatic Field Theory, Gordon and Breach, New York - London - Paris, 1966, 1-128.
  14. Faddeev L.D., Volkov A.Yu., Discrete evolution for the zero modes of the quantum Liouville model, J. Phys. A: Math. Theor. 41 (2008), 194008, 12 pages, arXiv:0803.0230.
  15. Glimm J., Jaffe A., Quantum physics. A functional integral point of view, 2nd ed., Springer-Verlag, New York, 1987.
  16. Haag R., Local quantum physics. Fields, particles, algebras, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996.
  17. Konrady J., Almost positive perturbations of positive selfadjoint operators, Comm. Math. Phys. 22 (1971), 295-300.
  18. Lechner G., Polarization-free quantum fields and interaction, Lett. Math. Phys. 64 (2003), 137-154, hep-th/0303062.
  19. Lechner G., Construction of quantum field theories with factorizing $S$-matrices, Comm. Math. Phys. 277 (2008), 821-860, math-ph/0601022.
  20. Lechner G., Deformations of quantum field theories and integrable models, Comm. Math. Phys. 312 (2012), 265-302, arXiv:1104.1948.
  21. Lechner G., Algebraic constructive quantum field theory: integrable models and deformation techniques, in Advances in Algebraic Quantum Field Theory, Math. Phys. Stud., Springer, Cham, 2015, 397-448, arXiv:1503.03822.
  22. Lechner G., Schützenhofer C., Towards an operator-algebraic construction of integrable global gauge theories, Ann. Henri Poincaré 15 (2014), 645-678, arXiv:1208.2366.
  23. Longo R., Witten E., An algebraic construction of boundary quantum field theory, Comm. Math. Phys. 303 (2011), 213-232, arXiv:1004.0616.
  24. Mund J., An algebraic Jost-Schroer theorem for massive theories, Comm. Math. Phys. 315 (2012), 445-464, arXiv:1012.1454.
  25. Nelson E., Analytic vectors, Ann. of Math. 70 (1959), 572-615.
  26. Reed M., Simon B., Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press, New York - London, 1975.
  27. Rudin W., Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987.
  28. Rudin W., Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
  29. Schmüdgen K., On commuting unbounded selfadjoint operators. I, Acta Sci. Math. (Szeged) 47 (1984), 131-146.
  30. Schmüdgen K., Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics, Vol. 265, Springer, Dordrecht, 2012.
  31. Schroer B., Modular wedge localization and the $d=1+1$ formfactor program, Ann. Physics 275 (1999), 190-223, hep-th/9712124.
  32. Stein E.M., Weiss G., Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, Vol. 32, Princeton University Press, Princeton, N.J., 1971.
  33. Tanimoto Y., Construction of wedge-local nets of observables through Longo-Witten endomorphisms, Comm. Math. Phys. 314 (2012), 443-469, arXiv:1107.2629.
  34. Tanimoto Y., Construction of two-dimensional quantum field models through Longo-Witten endomorphisms, Forum Math. Sigma 2 (2014), e7, 31 pages, arXiv:1301.6090.
  35. Tanimoto Y., Self-adjointness of bound state operators in integrable quantum field theory, arXiv:1508.06402.

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