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An Algebraic Version of Haag’s Theorem

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Abstract

Under natural conditions (such as split property and geometric modular action of wedge algebras) it is shown that the unitary equivalence class of the net of local (von Neumann) algebras in the vacuum sector associated to double cones with bases on a fixed space-like hyperplane completely determines an algebraic QFT model. More precisely, if for two models there is a unitary connecting all of these algebras, then — without assuming that this unitary also connects their respective vacuum states or spacetime symmetry representations — it follows that the two models are equivalent. This result might be viewed as an algebraic version of the celebrated theorem of Rudolf Haag about problems regarding the so-called “interaction-picture” in QFT. Original motivation of the author for finding such an algebraic version came from conformal chiral QFT. Both the chiral case as well as a related conjecture about standard half-sided modular inclusions will be also discussed.

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References

  1. Araki, H.: Relative Hamiltonian for faithful normal states of a von Neumann algebra. Publ. Res. Inst. Math. Sci. 9, 165–209 (1973/74)

    Google Scholar 

  2. Araki H., Zsidó L.: Extension of the structure theorem of Borchers and its application to half-sided modular inclusions. Rev. Math. Phys. 17, 495–543 (2005)

    Article  Google Scholar 

  3. Bisognano J., Wichmann E.H.: On the duality for quantum fields. J. Math. Phys. 16, 985–1007 (1975)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Brunetti R., Guido D., Longo R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys. 156, 201–219 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Carpi S., Weiner M.: On the uniqueness of diffeomorphism symmetry in Conformal Field Theory. Commun. Math. Phys. 258, 203–221 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Eckmann J.P., Fröhlich J.: Unitary equivalence of local algebras in the quasifree representation. Ann. Inst. H. Poincaré Sect. A (N.S.) 20, 201–209 (1974)

    MATH  Google Scholar 

  7. Doplicher S., Longo R.: Standard and split inclusions of von Neumann algebras. Invent. Math. 75, 493–536 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Fredenhagen K., Jörß M.: Conformal Haag-Kastler nets, pointlike localized fields and the existence of operator product expansions. Commun. Math. Phys. 176, 541–554 (1996)

    Article  ADS  MATH  Google Scholar 

  9. Fröhlich J., Gabbiani F.: Operator algebras and conformal field theory. Commun. Math. Phys. 155, 569–640 (1993)

    Article  ADS  MATH  Google Scholar 

  10. Glimm J., Jaffe A.: The \({\lambda(\phi^4)_2}\) quantum field theory without cutoffs II: The field operators and the approximate vacuum. Ann. Math. 91, 362–401 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guido D., Longo R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11–35 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Guido D., Longo R., Wiesbrock H.-W.: Extensions of conformal nets and superselection structures. Commun. Math. Phys. 192, 217–244 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Haag, R.: Local Quantum Physics. 2nd ed. Berlin-Heidelberg-New York: Springer-Verlag, 1996

  14. Kawahigashi Y., Longo R., Müger M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001)

    Article  ADS  MATH  Google Scholar 

  15. Longo R.: Notes on algebraic invariants for noncommutative dynamical systems. Commun. Math. Phys. 69, 195–207 (1979)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Streater, R., Wightman, A.S.: PCT, Spin and Statistics, and all that. New York-Amsterdam: W.A. Benjamin, 1964

  17. Weiner, M.: Conformal covariance and related properties of chiral QFT. PhD thesis, Department of Mathematics, University of Rome “Tor Vergata”, 2005

  18. Wiesbrock H.W.: Half-Sided Modular Inclusions of von Neumann algebras. Commun. Math. Phys. 157, 83–92 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Wiesbrock H.W.: A note on strongly additive conformal field theory and half-sided modular conormal standard inclusions. Lett. Math. Phys. 31, 303–307 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Rome, Italy

    Mihály Weiner

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, H-1364, Budapest, Hungary

    Mihály Weiner

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  1. Mihály Weiner
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Correspondence to Mihály Weiner.

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Communicated by Y. Kawahigashi

On leave from the Alfréd Rényi Institute of Mathematics, Budapest Hungary.

Supported by the ERC Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory”.

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Weiner, M. An Algebraic Version of Haag’s Theorem. Commun. Math. Phys. 305, 469–485 (2011). https://doi.org/10.1007/s00220-011-1236-7

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  • DOI: https://doi.org/10.1007/s00220-011-1236-7

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