Communications in Mathematical Physics

, Volume 336, Issue 1, pp 27–61 | Cite as

Localization in Nets of Standard Spaces

Article

Abstract

Starting from a real standard subspace of a Hilbert space and a representation of the translation group with natural properties, we construct and analyze for each endomorphism of this pair a local, translationally covariant net of standard subspaces, on the lightray and on two-dimensional Minkowski space. These nets share many features with low-dimensional quantum field theory, described by corresponding nets of von Neumann algebras.

Generalizing a result of Longo and Witten to two dimensions and massive multiplicity free representations, we characterize these endomorphisms in terms of specific analytic functions. Such a characterization then allows us to analyze the corresponding nets of standard spaces, and in particular to compute their minimal localization length. The analogies and differences to the von Neumann algebraic situation are discussed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ala13.
    Alazzawi, S.: Deformations of fermionic quantum field theories and integrable models. Lett. Math. Phys. 103, 37–58 (2013). http://arxiv.org/abs/1203.2058v1
  2. Ara63.
    Araki H.: A lattice of von Neumann algebras associated with the quantum theory of a free Bose field. J. Math. Phys. 4, 1343–1362 (1963)CrossRefADSMATHMathSciNetGoogle Scholar
  3. Ara99.
    Araki, H.: Mathematical theory of quantum fields. In: International Series of Monographs on Physics. Oxford University Press, Oxford (1999)Google Scholar
  4. AZ05.
    Araki H., Zsido L.: Extension of the structure theorem of Borchers and its application to half-sided modular inclusions. Rev. Math. Phys. 17, 491–543 (2005)CrossRefMATHMathSciNetGoogle Scholar
  5. BC12.
    Bostelmann H., Cadamuro D.: An operator expansion for integrable quantum field theories. J. Phys. A Math. Theor. 46, 095401 (2012)CrossRefADSMathSciNetGoogle Scholar
  6. BDL90.
    Buchholz D., D’Antoni C., Longo R.: Nuclear maps and modular structures. I. General properties. J. Funct. Anal. 88, 233–250 (1990)CrossRefMATHMathSciNetGoogle Scholar
  7. BFK06.
    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). http://arxiv.org/abs/hep-th/0609130
  8. BGL02.
    Brunetti, R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–786 (2002). http://arxiv.org/abs/math-ph/0203021
  9. Bis12.
    Bischoff, M.: Models in boundary quantum field theory associated with lattices and loop group models. Commun. Math. Phys. 315, 827–858 (2012). http://arxiv.org/abs/1108.4889
  10. BJM13.
    Barata, J.C.A., Jäkel, C.D., Mund, J.: The \({{\fancyscript P}(\varphi)_2}\) model on the de Sitter space (2013). Preprint. http://arxiv.org/abs/1311.2905v1
  11. BL04.
    Buchholz, D., Lechner, G.: Modular nuclearity and localization. Ann. Henri Poincaré 5, 1065–1080 (2004). http://arxiv.org/abs/math-ph/0402072
  12. BLM11.
    Bostelmann, H., Lechner, G., Morsella, G.: Scaling limits of integrable quantum field theories. Rev. Math. Phys. 23, 1115–1156 (2011). http://arxiv.org/abs/1105.2781
  13. BLS11.
    Buchholz, D., Lechner, G., Summers, S.J.: Warped convolutions, Rieffel deformations and the construction of quantum field theories. Commun. Math. Phys. 304, 95–123 (2011). http://arxiv.org/abs/1005.2656
  14. Boa54.
    Boas R.: Entire Functions. Academic Press, London (1954)MATHGoogle Scholar
  15. Bor92.
    Borchers, H.: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992). http://projecteuclid.org/euclid.cmp/1104248958
  16. BT13a.
    Bischoff, M., Tanimoto, Y.: Construction of wedge-local nets of observables through Longo–Witten endomorphisms. II. Commun. Math. Phys. 317, 667–695 (2013). http://arxiv.org/abs/1111.1671v1
  17. BT13b.
    Bischoff, M., Tanimoto, Y.: Integrable QFT and Longo–Witten endomorphisms. Ann. Henri Poincaré (2014). Preprint. doi: 10.1007/s00023-014-0337-1
  18. BW86.
    Buchholz, D., Wichmann, E.H.: Causal independence and the energy level density of states in local quantum field theory. Commun. Math. Phys. 106, 321 (1986). http://projecteuclid.org/euclid.cmp/1104115703
  19. BW92.
    Baumgärtel H., Wollenberg M.: Causal Nets of Operator Algebras. Akademie Verlag, Berlin (1992)MATHGoogle Scholar
  20. Der06.
    Derezinski, J.: Introduction to representations of canonical commutation and anticommutation relations. Lect. Notes Phys. 695, 63–143 (2006). http://arxiv.org/abs/math-ph/0511030v2
  21. DL84.
    Doplicher, S., Longo, R., Standard and split inclusions of von Neumann algebras. Invent. Math. 75, 493–536 (1984)Google Scholar
  22. Dur70.
    Duren, P.: Theory of H p spaces. In: Dover Books on Mathematics. Dover Publications Inc, New York (1970)Google Scholar
  23. Gar07.
    Garnett J.: Bounded Analytic Functions. Springer, New York (2007)Google Scholar
  24. Haa96.
    Haag, R.: Local Quantum Physics—Fields, Particles, Algebras, 2nd edn. Springer, New York (1996)Google Scholar
  25. Ing34.
    Ingham, A.E.: A note on Fourier transforms. J. Lond. Math. Soc. 1, 29–32 (1934)Google Scholar
  26. KL04.
    Kawahigashi, Y., Longo, R.: Classification of local conformal nets: case c < 1. Ann. Math. 160, 493–522 (2004). http://arxiv.org/abs/math-ph/0201015
  27. KL06.
    Kawahigashi, Y., Longo, R.: Local conformal nets arising from framed vertex operator algebras. Adv. Math. 206, 729–751 (2006). http://arxiv.org/abs/math/0407263v2
  28. Kuc00.
    Kuckert B.: Localization regions of local observables. Commun. Math. Phys. 215, 197–216 (2000)CrossRefADSMATHMathSciNetGoogle Scholar
  29. Lec03.
    Lechner, G.: Polarization-free quantum fields and interaction. Lett. Math. Phys. 64, 137–154 (2003). http://arxiv.org/abs/hep-th/0303062
  30. Lec08.
    Lechner, G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008). http://arxiv.org/abs/math-ph/0601022
  31. Lec12.
    Lechner, G.: Deformations of quantum field theories and integrable models. Commun. Math. Phys. 312, 265–302 (2012). http://arxiv.org/abs/1104.1948
  32. LS14.
    Lechner, G., Schützenhofer, C.; Towards an operator-algebraic construction of integrable global gauge theories. Ann. Henri Poincaré 15, 645–678 (2014). http://arxiv.org/abs/1208.2366v1
  33. LST13.
    Lechner, G., Schlemmer, J., Tanimoto, Y.: On the equivalence of two deformation schemes in quantum field theory. Lett. Math. Phys. 103, 421–437 (2013)Google Scholar
  34. LRT78.
    Leylands, P., Roberts, J.E., Testard, D.: Duality for quantum free fields (1978). PreprintGoogle Scholar
  35. Lon08.
    Longo, R.: Lectures on conformal nets—part 1. In: Von Neumann Algebras in Sibiu, pp. 33–91. Theta (2008). http://www.mat.uniroma2.it/~longo/Lecture_Notes_files/LN-Part1.pdf
  36. LR04.
    Longo, R., Rehren, K.: Local fields in boundary conformal QFT. Rev. Math. Phys. 16, 909 (2004). http://arxiv.org/abs/math-ph/0405067
  37. LR12.
    Longo, R., Rehren K.: Boundary quantum field theory on the interior of the Lorentz hyperboloid. Commun. Math. Phys. 311, 769–785 (2012). http://arxiv.org/abs/1103.1141
  38. LW11.
    Longo, R., Witten, E.: An algebraic construction of boundary quantum field theory. Commun. Math. Phys. 303, 213–232 (2011). http://arxiv.org/abs/1004.0616
  39. MSY06.
    Mund, J., Schroer, B., Yngvason, J.: String-localized quantum fields and modular localization. Commun. Math. Phys. 268, 621–672 (2006). http://arxiv.org/abs/math-ph/0511042
  40. Pla13.
    Plaschke, M.: Wedge local deformations of charged fields leading to anyonic commutation relations. Lett. Math. Phys. 103, 507–532 (2013). http://arxiv.org/abs/1208.6141v1
  41. RR94.
    Rosenblum M., Rovnyak J.: Topics in Hardy Classes and Univalent Functions. Birkhäuser, Boston (1994)CrossRefMATHGoogle Scholar
  42. RS72.
    Reed M., Simon B.: Methods of Modern Mathematical Physics I—Functional Analysis. Academic Press, London (1972)Google Scholar
  43. RS75.
    Reed M., Simon B.: Methods of Modern Mathematical Physics II—Fourier Analysis. Academic Press, London (1975)MATHGoogle Scholar
  44. Sch97.
    Schroer, B.: Modular localization and the bootstrap-formfactor program. Nucl. Phys. B 499, 547–568 (1997). http://arxiv.org/abs/hep-th/9702145v1
  45. Sch99.
    Schroer, B.: Modular wedge localization and the d = 1 + 1 formfactor program. Ann. Phys. 275, 190–223 (1999). http://arxiv.org/abs/hep-th/9712124
  46. Smi92.
    Smirnov F.A.: Form Factors in Completely Integrable Models of Quantum Field Theory. World Scientific, Singapore (1992)CrossRefMATHGoogle Scholar
  47. SW71.
    Stein E.M., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)MATHGoogle Scholar
  48. SW00.
    Schroer, B., Wiesbrock, H.: Modular constructions of quantum field theories with interactions. Rev. Math. Phys. 12, 301–326 (2000). http://arxiv.org/abs/hep-th/9812251
  49. Tan12.
    Tanimoto, Y.: Construction of wedge-local nets of observables through Longo–Witten endomorphisms. Commun. Math. Phys. 314, 443–469 (2012). http://arxiv.org/abs/1107.2629
  50. Wie92.
    Wiesbrock, H.: A comment on a recent work of Borchers. Lett. Math. Phys. 25, 157–160 (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany
  2. 2.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly

Personalised recommendations