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Annales Henri Poincaré

, Volume 19, Issue 3, pp 937–958 | Cite as

The Bisognano–Wichmann Property on Nets of Standard Subspaces, Some Sufficient Conditions

  • Vincenzo MorinelliEmail author
Article

Abstract

We discuss the Bisognano–Wichmann property for localPoincaré covariant nets of standard subspaces. We provide a sufficient algebraic condition on the covariant representation ensuring the Bisognano–Wichmann and the duality properties without further assumptions on the net. We call it modularity condition. It holds for direct integrals of scalar massive and massless representations. We present a class of massive modular covariant nets not satisfying the Bisognano–Wichmann property. Furthermore, we give an outlook on the relation between the Bisognano–Wichmann property and the split property in the standard subspace setting.

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References

  1. 1.
    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)Google Scholar
  2. 2.
    Bisognano, J.J., Wichmann, E.H.: On the duality condition for quantum fields. J. Math. Phys. 17, 303–321 (1976)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bisognano, J.J., Wichmann, E.H.: On the duality condition for Hermitian scalar field. J. Math. Phys. 16, 985–1007 (1975)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brunetti, R., Guido, D., Longo, R.: Modular localization and Wigner particles. Rev. Math. Phys. 14(7 & 8), 759–786 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brunetti, R., Guido, D., Longo, R.: Group cohomology, modular theory and space-time symmetries. Rev. Math. Phys. 7, 57–71 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borchers, H.J.: On Poincaré transformations and the modular group of the algebra associated with a wedge. Lett. Math. Phys. 46(4), 295–301 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Borchers, H.J.: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Buchholz, D., Dreyer, O., Florig, M., Summers, S.J.: Geometric modular action and spacetime symmetry groups. Rev. Math. Phys. 12(4), 475–560 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Buchholz, D., D’antoni, C., Longo, R.: Nuclearity and thermal states in conformal field theory. Commun. Math. Phys. 270(1), 267–293 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Buchholz, D., Epstein, H.: Spin and statistics of quantum topological charges. Fysica 17, 329–343 (1985)Google Scholar
  11. 11.
    Buchholz, D., Wichmann, E.: Causal independence and the energy-level density of states in local quantum field theory. Commun. Math. Phys. 106(2), 321–344 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Conrady, F., Hnybida, J.: Unitary irreducible representations of SL(2,\({\mathbb{C}}\)) in discrete and continuous SU(1,1) bases. J. Math. Phys. 52, 012501 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Doplicher, S., Longo, R.: Standard and split inclusions of von Neumann algebras. Invent. Math. 75(3), 493–536 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics II. Commun. Math. Phys. 35, 49–85 (1974)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Eckmann, J.P., Osterwalder, K.: An application of Tomita’s theory of modular Hilbert algebras: duality for free Bose fields. J. Funct. Anal. 13, 1–12 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Figliolini, F., Guido, D.: On the type of second quantization factors. J. Oper. Theory 31, 229–252 (1994)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Foit, J.J.: Abstract twisted duality for quantum free Fermi fields. Publ. RIMS Kyoto Univ. 19, 729–74 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Guido, D., Longo, R.: An algebraic spin and statistics theorem. Commun. Math. Phys. 172, 517–533 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lechner, G., Longo, R.: Localization in nets of standard spaces. Commun. Math. Phys. 336, 27–61 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Leyland, P., Roberts, J.E., Testard, D.: Duality for quantum free fields. Marseille (1978) (unpublished manuscript) Google Scholar
  21. 21.
    Longo, R.: Lectures on conformal nets. Preliminary Lecture Notes that are Available at http://www.mat.uniroma2.it/~longo/Lecture-Notes.html
  22. 22.
    Longo, R.: Real Hilbert subspaces, modular theory, SL(2,\({\mathbb{R}}\)) and CFT. In: Von Neumann Algebras in Sibiu. Theta Ser. Adv. Math. vol. 10, pp. 33–91. Theta, Bucharest (2008)Google Scholar
  23. 23.
    Longo, R., Morinelli, V., Rehren, K.-H.: Where infinite spin particles are localizable. Commun. Math. Phys. 345, 587–614 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Longo, R., Witten, E.: An algebraic construction of boundary quantum field theory. Commun. Math. Phys. 303, 213–232 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mund, J.: The Bisognano–Wichmann theorem for massive theories. Ann. Henri Poincaré 2, 907–926 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Morinelli, V.: An algebraic condition for the Bisognano–Wichmann property. In: Proceedings of the 14th Marcel Grossmann Meeting—MG14, Rome (2015). arXiv:1604.04750 (to appear)
  27. 27.
    Neeb, K.-H., Olafsson, G.: Antiunitary representations and modular theory. In: Grabowska, K., Grabowski, J., Fialowski, A., Neeb, K.-H (eds.) 50th Sophus Lie Seminar. Banach Center Publication (2017)Google Scholar
  28. 28.
    Osterwalder, K.: Duality for free Bose fields. Commun. Math. Phys. 29, 1–14 (1973)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rieffel, M.A., Van Daele, A.: A bounded operator approach to Tomita–Takesaki theory. Pac. J. Math. 69, 187–221 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Rühl, W.: The Lorentz Group and Harmonic Analysis. W. A. Benjamin Inc, New York (1970)Google Scholar
  31. 31.
    Schroer, B., Wiesbrock, H.-W.: Modular theory and geometry. Rev. Math. Phys. 12(1), 139–158 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Varadarajan, V.S.: Geometry of Quantum Theory, 2nd edn. Springer, New York (1985)zbMATHGoogle Scholar
  33. 33.
    Yngvason, J.: A note on essential duality. Lett. Math. Phys. 31(2), 127–141 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma Tor VergataRomeItaly

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