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Generalized Particle Statistics in Two-Dimensions: Examples from the Theory of Free Massive Dirac Field

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In the framework of algebraic quantum field theory we analyze the anomalous statistics exhibited by a class of automorphisms of the observable algebra of the two-dimensional free massive Dirac field, constructed by fermionic gauge group methods. The violation of Haag duality, the topological peculiarity of a two-dimensional space-time and the fact that unitary implementers do not lie in the global field algebra account for strange behaviour of statistics, which is no longer an intrinsic property of sectors. Since automorphisms are not inner, we exploit asymptotic abelianness of intertwiners in order to construct a braiding for a suitable C *-tensor subcategory of End(\(\fancyscript{A}\)). We define two inequivalent classes of path connected bi-asymptopias, selecting only those sets of nets which yield a true generalized statistics operator.

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References

  1. Adler C. (1996) Braid group statistics in two-dimensional quantum field theory. Rev. Math. Phys. 7:907–924

    MATH  Google Scholar 

  2. Buchholz, D., Doplicher, S., Morchio, G., Roberts, J.E., Strocchi, F.: Asymptotic abelianness and braided tensor C *-categories. http://arxiv.org/list/math-ph/0209038,2002

  3. Buchholz D., Lechner G. (2004) Modular nuclearity and localization. Annales Henri Poincare 5:1065–1080

    Article  MATH  MathSciNet  Google Scholar 

  4. Baumgärtel H., Jurke M., Lledó F. (2002) Twisted duality of the CAR-algebra. J. Math. Phys. 43:4158–4179

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Carey A.L., Ruijsenaars S.N.M. (1987) On fermionic gauge groups, current algebras and Kac-Moody algebras. Acta Appl. Math. 10, 1–86

    Article  MATH  MathSciNet  Google Scholar 

  6. Carey A.L., Hurst C.A., O’Brien D.M. (1982) Automorphisms of the canonical anticommutation relation and index theory. J. Funct. Anal. 48, 360–393

    Article  MATH  MathSciNet  Google Scholar 

  7. Doplicher S., Haag R., Roberts J.E. (1969) Fields, observables and gauge transformations I. Commun. Math. Phys. 1, 1–23

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Doplicher S., Haag R., Roberts J.E. (1969) Fields, observables and gauge transformations II. Commun. Math. Phys. 15, 173–200

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Doplicher S., Haag R., Roberts J.E. (1971) Local observables and particle statistics I. Commun. Math. Phys. 23, 199–230

    Article  MathSciNet  ADS  Google Scholar 

  10. Doplicher S., Haag R., Roberts J.E. (1974) Local observables and particle statistics II. Commun. Math. Phys. 35, 49–85

    Article  MathSciNet  ADS  Google Scholar 

  11. Doplicher, S., Roberts, J.E.: C *-algebras and duality for compact groups: why there is a compact group of internal symmetries in particle physics. Proceedings of the International Conference on Mathematical Physics, Marseille (1986), Singapore: World Scientific, 1987

  12. Doplicher S., Roberts J.E. (1990) Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131, 51–107

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras II: geometric aspects and conformal covariance. Rev. Math. Phys., Special Issue, 113–157 (1992)

  14. Köberle R., Marino E.C. (1983) Duality, mass spectrum and vacuum expectation values. Phys. Lett. B126, 475–480

    ADS  Google Scholar 

  15. Mueger M. (1998) Superselection structure of massive quantum field theories in 1+1 dimensions. Rev. Math. Phys. 10:1147–1170

    Article  MATH  MathSciNet  Google Scholar 

  16. Mueger M. (1998) Quantum double actions on operator algebras and orbifold quantum field theories. Commun. Math. Phys. 181, 137–181

    Article  MATH  ADS  Google Scholar 

  17. Mund J. (1998) No-go theorem for ‘free’ relativistic anyons in d=2+1. Lett. Math. Phys. 43, 319–328

    Article  MATH  MathSciNet  Google Scholar 

  18. Pressley A., Segal G. (1986) Loop groups. Oxford, Clarendon Press

    Google Scholar 

  19. Roberts, J.E.: Lectures on algebraic quantum field theory. In: The algebraic theory of superselection sectors: Introduction and recent results. Singapore: World Scientific, 1990

  20. Ruijsenaars S.N.M. (1982) The Wightman axioms for the fermionic Federbush model. Commun. Math. Phys. 87, 181–228

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Ruijsenaars S.N.M. (1989) Index formulas for generalized Wiener-Hopf operators and boson-fermion correspondence in 2N dimensions. Commun. Math. Phys. 124, 553–593

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Schroer B. (1992) Scattering properties of anyons and plektons. Nucl. Phys. B369:478–498

    Article  MathSciNet  ADS  Google Scholar 

  23. Schroer B. (2006) Two-dimensional models as testing ground for principles and logarithmic structures. Ann. Phys. 321, 435–479

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Schroer B., Swieca J.A. (1977) Spin and statistics of quantum kinks. Nucl. Phys. B121, 505–513

    Article  ADS  Google Scholar 

  25. Wilczek F. (1983) Quantum mechanics of fractional spin particles. Phys. Rev. Lett. 49:957–1149

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “La Sapienza”, P.le Aldo Moro 2, 00185, Roma, Italy

    Dario Salvitti

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  1. Dario Salvitti
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Correspondence to Dario Salvitti.

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Communicated by Y. Kawahigashi

Dedicated to the memory of Sabrina Picucci

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Salvitti, D. Generalized Particle Statistics in Two-Dimensions: Examples from the Theory of Free Massive Dirac Field. Commun. Math. Phys. 269, 473–492 (2007). https://doi.org/10.1007/s00220-006-0140-z

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