Observables, Superselection Sectors and Gauge Groups

  • Claus Fredenhagen
Part of the NATO ASI Series book series (NSSB, volume 295)


Symmetries play an important role in quantum physics from its very beginning. There are two types of symmetry which must be distinguished, namely internal and external symmetries. External symmetries have an active interpretation; they transform states and observables such that certain structures are respected, e.g. transition probabilities, the Hamiltonian, the S-matrix etc.. Typical examples are the space time symmetries in a situation where space time is a priori given in the sense of classical physics. Internal symmetries, on the contrary, have only a passive interpretation; they do not transform the states of the system but merely change the description of the system. Examples are gauge transformations in gauge theories and diffeomorphisms in general relativity. Hence internal symmetries occur only in cases where the formulation of the theory contains some redundancy. It is tempting to remove this redundancy and to base the theoretical description exclusively on observables. Then the question arises whether the internal symmetry has an intrinsic meaning and may be recovered from the observables. Conversely, one may interpret the observed structure of the state space in terms of an internal symmetry and develop a redundant description where this symmetry acts explicitely. Such a redundant description might be useful for the investigation of perturbed theories where the symmetry is slightly broken, either spontaneously or dynamically.


Braid Group Double Cone Internal Symmetry Local Algebra Superselection Sector 
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.
    S. Doplicher, R. Haag, J.E. Roberts: Local Observables and Particle Statistics. Commun. Math. Phys.23,199(1971) and 35,49(1974)Google Scholar
  2. 2.
    H.-J. Borchers: Local Rings and the Connection of Spin with Statistics. Commun. Math. Phys. 1,281(1965)MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    D. Buchholz, K. Fredenhagen: Locality and the Structure of Particle States. Commun. Math. Phys.84,1(1982)MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    K. Fredenhagen, M. Marcu: Charged States in ℤ2 Gauge Theories. Commun. Math. Phys. 92 81 (1983)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    J. Fröhlich, P. A. Marchetti: Soliton Quantization in Lattice Field Theories. Commun. Math. Phys. 112 343 (1987)ADSMATHCrossRefGoogle Scholar
  6. 6.
    K. Szlachanyi: Non-Local Fields in the Z(2) Higgs Model: The Global Gauge Symmetry Breaking and the Confinement Problem. Commun. Math. Phys. 108 319 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    K. Fredenhagen, K.H. Rehren, B. Schroer: Superselection Sectors with Braid Group Statistics and Exchange Algebras. Commun. Math. Phys. 125,201(1989)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    R. Longo: Index for Subfactors and Statistics of Quantum Fields. Commun. Math. Phys.126,217(1989) and 130,285(1990)Google Scholar
  9. 9.
    J. Fröhlich, Gabbiani: Braid Statistics in Local Quantum Field Theory. Rev. Math. Phys. 2,251(1990)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    J. Fröhlich: New Superselection Sectors (“Soliton States”) in Two Dimensional Bose Quantum Field Theory Models. Commun. Math. Phys. 47,269(1976)ADSCrossRefGoogle Scholar
  11. 11.
    K. Fredenhagen: Generalizations of the Theory of Superselection Sectors. In “The Algebraic Theory of Superselection Sectors. Introduction and Recent Results”, D. Kastler (ed.), World Scientific 1990Google Scholar
  12. 12.
    S. Doplicher, J.E. Roberts: Endomorphisms of C *-algebras, Cross Products and Duality for Compact Groups. Ann. Math. 130 75 (1989)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    S. Doplicher, J.E. Roberts: Why there is a Field Algebra with a Compact Gauge Group Describing the Superselection Structure in Particle Physics. Commun. Math. Phys.131,51(1990)MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    H.J. Borchers, J. Yngvason: Positivity of Wightman functions and the Existence of Local Nets. Commun. Math. Phys. 127 607 (1990)MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    R. Haag, D. Kastler: An Algebraic Approach to Quantum Field Theory. J. Math. Phys.5,848(1964)MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    H.J. Borchers: On the Vacuum State in Quantum Field Theory. Commun. Math. Phys. 1 57 (1965)MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    H.J. Borchers: Energy and Momentum as Observables in Quantum Field Theory. Commun. Math. Phys. 2, 49 (1966)MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    H.-J. Borchers D. Buchholz,: The Energy Momentum Spectrum in Local Field Theories with Broken Lorentz Symmetry. Commun. Math. Phys. 97 169 (1985)MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    H.-J. Borchers: Locality and Covariance of the Spectrum. Bielefeld 1984Google Scholar
  20. 20.
    H.-J. Borchers: A Remark on a Theorem of B. Misra. Commun. Math. Phys. 4, 315 (1967)MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    M. Takesaki, W. Winnink: Local Normality in Quantum Statistical Mechanics. Commun. Math. Phys. 30 129 (1973)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    R. Haag, H. Narnhofer, U. Stein: On Quantum Field Theory in Gravitational Background. Commun. Math. Phys.94, 219 (1984)MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    D. Buchholz, C. D’Antoni, K. Fredenhagen:The Universal Structure of Local Algebras. Commun. Math. Phys. 111 123 (1987)ADSMATHCrossRefGoogle Scholar
  24. 24.
    V.F. Jones: Index of Subfactors. Invent.Math.72,1 (1983)MathSciNetADSMATHCrossRefGoogle Scholar
  25. 25.
    A. Ocneanu: Quantized Groups, String Algebras and Galois Theory for Algebras. Lond.Math.Soc., Lect. Notes Vol. 136 Evans, Takesaki (eds.),119–172(1989)Google Scholar
  26. 26.
    M. Pimsner, S. Popa: Entropy and Index for Subfactors. Ann. Sci. Ec. Norm. Sup. 19 57 (1986)MathSciNetMATHGoogle Scholar
  27. 27.
    H. Wenzl: Hecke Algebras of Type A n and Subfactors. Invent. Math. 92 349 (1988)MathSciNetADSMATHCrossRefGoogle Scholar
  28. 28.
    J.J. Bisognano, E.H. Wichmann: On the Duality Condition for a Hermitean Scalar Field. J. Math. Phys 17 303 (1975)MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    S. Doplicher, J.E. Roberts: Fields, Statistics and Non Abelian Gauge Groups. Commun. Math. Phys. 28 331 (1972)MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    J. Cuntz: Simple C *-Algebras Generated by Isometries. Commun. Math. Phys. 57 173 (1977)MathSciNetADSMATHCrossRefGoogle Scholar
  31. 31.
    R. Longo: Minimal Index and Braided Subfactors (preprint)Google Scholar
  32. 32.
    K.H. Rehren: Braid Group Statistics and their Superselection Rules. In “The Algebraic Theory of Superselection Sectors. Introduction and Recent Results”, D. Kastler (ed.), World Scientific 1990Google Scholar
  33. 33.
    L.K. Hadjiivanov, R.R. Paunov, I.T. Todorov:Quantum Group Extended Chiral p-Models. INRNE-TH-90–7 (preprint)Google Scholar
  34. 34.
    G. Mack, V. Schomerus: Quasi Quantum Group Symmetry and Local Braid Relations in the Conformal Ising Model. DESY 91–060 (preprint)Google Scholar
  35. 35.
    K.H. Rehren: Field Operators for Anyons and Plektons. DESY 91–043 (preprint)Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Claus Fredenhagen
    • 1
  1. 1.II. Institut für Theoretische PhysikUniversität HamburgGermany

Personalised recommendations