Communications in Mathematical Physics

, Volume 294, Issue 2, pp 505–538 | Cite as

The CPT and Bisognano-Wichmann Theorems for Anyons and Plektons in d = 2 + 1

  • Jens MundEmail author


We prove the Bisognano-Wichmann and CPT theorems for massive theories obeying braid group statistics in three-dimensional Minkowski space. We start from first principles of local relativistic quantum theory, assuming Poincaré covariance and asymptotic completeness. The particle masses must be isolated points in the mass spectra of the corresponding charged sectors, and may only be finitely degenerate.


Twist Operator Asymptotic Completeness Superselection Sector Universal Covering Group Observable Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Araki, H.: Mathematical Theory of Quantum Fields. Int. Series of Monographs in Physics, no. 101, Oxford: Oxford University Press, 1999Google Scholar
  2. 2.
    Bisognano J.J., Wichmann E.H.: On the duality condition for a Hermitian scalar field. J. Math. Phys. 16, 985 (1975)zbMATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Bisognano J.J., Wichmann E.H.: On the duality condition for quantum fields. J. Math. Phys. 17, 303 (1976)CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Borchers H.J.: The CPT-theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Borchers H.J.: On Poincaré transformations and the modular group of the algebra associated with a wedge. Lett. Math. Phys. 46, 295–301 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Borchers H.J., Buchholz D., Schroer B.: Polarization-free generators and the S-matrix. Commun. Math. Phys. 219, 125–140 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Borchers, H.J., Yngvason, J.: On the PCT-theorem in the theory of local observables. In: Mathematical Physics in Mathematics and Physics (Siena) R. Longo, ed., Fields Institute Communications, Vol. 30, Providence, RI: Amer. Math. Soc., 2001, pp. 39–64Google Scholar
  8. 8.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 1. Second ed., TMP, New York: Springer, 1987Google Scholar
  9. 9.
    Brunetti R., Guido D., Longo R.: Modular structure and duality in conformal field theory. Commun. Math. Phys. 156, 201–219 (1993)zbMATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Brunetti R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–786 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Buchholz D., Epstein H.: Spin and statistics of quantum topological charges. Fysica 17, 329–343 (1985)Google Scholar
  12. 12.
    Buchholz D., Fredenhagen K.: Locality and the structure of particle states. Commun. Math. Phys 84, 1–54 (1982)zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Doplicher S., Haag R., Roberts J.E.: Fields, observables and gauge transformations I. Commun. Math. Phys. 13, 1–23 (1969)zbMATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics I. Commun. Math. Phys. 23, 199 (1971)CrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics II. Commun. Math. Phys. 35, 49–85 (1974)CrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Doplicher S., Roberts J.E.: Fields, statistics and non-Abelian gauge groups. Commun. Math. Phys. 28, 331–348 (1972)CrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Doplicher S., Roberts J.E.: Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131, 51–107 (1990)zbMATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Fredenhagen K.: On the existence of antiparticles. Commun. Math. Phys. 79, 141–151 (1981)CrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Fredenhagen, K.: Structure of Superselection Sectors in Low Dimensional Quantum Field Theory. Proceedings (Lake Tahoe City), L.L. Chau, W. Nahm, eds., New York: Plenum, 1991Google Scholar
  20. 20.
    Fredenhagen, K.: Generalizations of the theory of superselection sectors. In: The Algebraic Theory of SuperSelection Sectors. Introduction and recent results. D. Kastler, ed., Singapore: World Scientific, 1990Google Scholar
  21. 21.
    Fredenhagen K., Gaberdiel M., Rüger S.M.: Scattering states of plektons (particles with braid group statistics) in 2+1 dimensional field theory. Commun. Math. Phys. 175, 319–355 (1996)zbMATHCrossRefADSGoogle Scholar
  22. 22.
    Fredenhagen K., Rehren K.-H., Schroer B.: Superselection sectors with braid group statistics and exchange algebras I: General theory. Commun. Math. Phys. 125, 201–226 (1989)zbMATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    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. SI1, 113–157 (1992)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Fröhlich J., Marchetti P.A.: Quantum field theories of vortices and anyons. Commun. Math. Phys. 121, 177–223 (1989)zbMATHCrossRefADSGoogle Scholar
  25. 25.
    Fröhlich J., Marchetti P.A.: Spin-statistics theorem and scattering in planar quantum field theories with braid statistics. Nucl. Phys. B 356, 533–573 (1991)CrossRefADSGoogle Scholar
  26. 26.
    Frölich, J., Kerler, T.: Quantum Groups, Quantum Categories, and Quantum Field Theory. Lecture Notes in Mathematics, Vol. 1542, Berlin: Springer, 1993Google Scholar
  27. 27.
    Fuchs J., Ganchev A., Vecsernyés P.: Rational Hopf algebras: Polynomial equations, gauge fixing, and low dimensional examples. Int. J. Mod. Phys. A 10, 3431–3476 (1995)zbMATHCrossRefADSGoogle Scholar
  28. 28.
    Guido D., Longo R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys. 148, 521–551 (1992)zbMATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Guido D., Longo R.: An algebraic spin and statistics theorem. Commun. Math. Phys. 172, 517 (1995)zbMATHCrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Guido D., Longo R.: Natural energy bounds in quantum thermodynamics. Commun. Math. Phys. 218, 513–536 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  31. 31.
    Haag, R.: Local Quantum Physics, Second ed., Texts and Monographs in Physics, Berlin-Heidelberg: Springer, 1996Google Scholar
  32. 32.
    Hepp, K.: On the connection between Wightman and LSZ quantum field theory, In: Axiomatic Field Theory, M. Chretien and S. Deser, eds., Brandeis University Summer Institute in Theoretical Physics 1965, Vol. 1, London-NewYork: Gordon and Breach, 1966, pp. 135–246Google Scholar
  33. 33.
    Jost, R.: The general theory of quantized fields. Providence, RI: Amer. Math. Soc., 1965Google Scholar
  34. 34.
    Kuckert B.: A new approach to spin & statistics. Lett. Math. Phys. 35, 319–331 (1995)zbMATHCrossRefMathSciNetADSGoogle Scholar
  35. 35.
    Kuckert B.: Two uniqueness results on the Unruh effect and on PCT-symmetry. Commun. Math. Phys. 221, 77–100 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  36. 36.
    Longo R.: An analogue of the Kac-Wakimoto formula and black hole conditional entropy. Commun. Math. Phys. 186, 451–479 (1997)zbMATHCrossRefMathSciNetADSGoogle Scholar
  37. 37.
    Mack G., Schomerus V.: Conformal field algebras with quantum symmetry from the theory of superselection sectors. Commun. Math. Phys. 134, 139–196 (1990)zbMATHCrossRefMathSciNetADSGoogle Scholar
  38. 38.
    Mund, J.: Quantum Field Theory of Particles with Braid Group Statistics in 2+1 Dimensions. Ph.D. thesis, Freie Universität Berlin, 1998Google Scholar
  39. 39.
    Mund J.: The Bisognano-Wichmann theorem for massive theories. Ann. H. Poinc. 2, 907–926 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Mund J.: Modular localization of massive particles with “any” spin in d=2+1. J. Math. Phys. 44, 2037–2057 (2003)zbMATHCrossRefMathSciNetADSGoogle Scholar
  41. 41.
    Mund J.: Borchers’ commutation relations for sectors with braid group statistics in low dimensions. Ann. H. Poinc. 10, 19–34 (2009)CrossRefMathSciNetGoogle Scholar
  42. 42.
    Mund J.: The spin statistics theorem for anyons and plektons in d=2+1. Commun. Math. Phys. 286, 1159–1180 (2009)zbMATHCrossRefMathSciNetADSGoogle Scholar
  43. 43.
    Mund J., Schroer B., Yngvason J.: String–localized quantum fields from Wigner representations. Phys. Lett. B 596, 156–162 (2004)MathSciNetADSGoogle Scholar
  44. 44.
    O’Neill B.: Semi–Riemannian Geometry. Academic Press, New York (1983)zbMATHGoogle Scholar
  45. 45.
    Pauli, W.: Exclusion principle, Lorentz group and reflection of space-time and charge, In: Niels Bohr and the Development of Physics, W. Pauli, ed., Oxford: Pergamon Press, 1955, p. 30Google Scholar
  46. 46.
    Rehren, K.-H.: Braid group statistics and their superselection rules. In: The Algebraic Theory of Superselection Sectors, D. Kastler, ed., Singapore: World Scientific, 1990Google Scholar
  47. 47.
    Rehren K.-H.: Spacetime fields and exchange fields. Commun. Math. Phys. 132, 461–483 (1990)zbMATHCrossRefMathSciNetADSGoogle Scholar
  48. 48.
    Rehren K.-H.: Field operators for anyons and plektons. Commun. Math. Phys. 145, 123 (1992)zbMATHCrossRefMathSciNetADSGoogle Scholar
  49. 49.
    Rehren, K.-H.: Weak C* Hopf symmetry. In: Group Theoretical Methods in Physics, A. Bohm, H.-D. Doebner, and P. Kielanowski, eds.), Lecture Notes in Physics, Vol. 504, Berlin: Heron Press, 1997, pp. 62–69Google Scholar
  50. 50.
    Roberts J.E.: Local cohomology and superselection structure. Commun. Math. Phys. 51, 107–119 (1976)zbMATHCrossRefADSGoogle Scholar
  51. 51.
    Roberts, J.E.: Net cohomology and its applications to field theory. In: Quantum Fields – Algebras, Processes, L. Streit, ed., Wien-New York: Springer, 1980, pp. 239–268Google Scholar
  52. 52.
    Roberts, J.E.: Lectures on algebraic quantum field theory. In: The Algebraic Theory of Superselection Sectors. Introduction and Recent Results, D. Kastler, ed., Singapore-River Edge, NJ-London-Hong Kong: World Scientific, 1990, pp. 1–112Google Scholar
  53. 53.
    Schroer, B.: Modular theory and symmetry in QFT. In: Mathematical Physics towards the 21st Century, R.N. Sen and A. Gersten, eds., Beer-Sheva: Ben-Gurion of the Negev Press, Israel, 1994Google Scholar
  54. 54.
    Schroer B., Wiesbrock H.-W.: Modular theory and geometry. Rev. Math. Phys. 12, 139–158 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Steinmann O.: A Jost-Schroer theorem for string fields. Commun. Math. Phys. 87, 259–264 (1982)zbMATHCrossRefMathSciNetADSGoogle Scholar
  56. 56.
    Strǎtilǎ S.: Modular Theory in Operator Algebras. Abacus Press, Tunbridge Wells (1981)Google Scholar
  57. 57.
    Unruh W.G.: Notes on black hole evaporation. Rev. Math. Phys. 14, 870–892 (1976)Google Scholar
  58. 58.
    Varadarajan, V.S.: Geometry of Quantum Theory, Vol. II, New York: Van Nostrand Reinhold Co., 1970zbMATHGoogle Scholar
  59. 59.
    Wilczek F.: Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957–1149 (1982)CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidade Federal de Juiz de ForaJuiz de ForaBrazil

Personalised recommendations