Communications in Mathematical Physics

, Volume 330, Issue 3, pp 935–972 | Cite as

New Light on Infrared Problems: Sectors, Statistics, Symmetries and Spectrum

  • Detlev BuchholzEmail author
  • John E. Roberts


A new approach to the analysis of the physical state space of a theory is presented within the general setting of local quantum physics. It also covers theories with long range forces, such as quantum electrodynamics. Making use of the notion of charge class, an extension of the concept of superselection sector, infrared problems are avoided by restricting the states to observables localized in a light cone. The charge structure of a theory can be explored in a systematic manner. The present analysis focuses on simple charges, thus including the electric charge. It is shown that any such charge has a conjugate charge. There is a meaningful concept of statistics: the corresponding charge classes are either of Bose or of Fermi type. The family of simple charge classes is in one-to-one correspondence with the irreducible unitary representations of a compact Abelian group. Moreover, there is a meaningful definition of covariant charge classes. Any such class determines a continuous unitary representation of the Poincaré group or its covering group satisfying the relativistic spectrum condition. The resulting particle aspects are also briefly discussed.


Massless Particle Double Cone Superselection Sector Hyperbolic Cone Infrared Problem 
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.
    Bargmann V.: On unitary ray representations of continuous groups. Ann. Math. 59, 1–46 (1954)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Borchers H.-J., Yngvason J.: From quantum fields to local von Neumann algebras. Rev. Math. Phys. SI 1, 15–47 (1992)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Buchholz D.: Collision theory for massless bosons. Commun. Math. Phys. 52, 147–173 (1977)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Buchholz D.: The physical state space of quantum electrodynamics. Commun. Math. Phys. 85, 49–71 (1982)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Buchholz D.: Gauss’ law and the infraparticle problem. Phys. Lett. B 174, 331–334 (1986)ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    Buchholz D., Doplicher S., Morchio G., Roberts J.E., Strocchi F.: A model for charges of electromagnetic type. In: Doplicher, S., Longo, R., Roberts, J.E., Zsido, L. (eds.) Operator Algebras and Quantum Field Theory, Rome 1996, pp. 647–660. International Press, Somerville (1997)Google Scholar
  7. 7.
    Buchholz D., Doplicher S., Morchio G., Roberts J.E., Strocchi F.: Quantum delocalization of the electric charge. Ann. Phys. 290, 53–66 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Buchholz D., Fredenhagen K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Camassa P.: Relative Haag duality for the free field in Fock representation. Ann. Henri Poincaré 8, 1433–1459 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Connes A., Stormer E.: Homogeneity of the state space of factors of type III1. J. Funct. Anal. 28, 187–196 (1987)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Doplicher S., Haag R., Roberts J.E.: Fields, observables and gauge transformations II. Commun. Math. Phys. 15, 173–200 (1969)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics I. Commun. Math. Phys. 23, 199–230 (1971)ADSCrossRefMathSciNetGoogle Scholar
  13. 13.
    Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics II. Commun. Math. Phys. 35, 49–85 (1974)ADSCrossRefMathSciNetGoogle Scholar
  14. 14.
    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)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    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)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Fröhlich J.: The charged sectors of quantum electrodynamics in a framework of local observables. Commun. Math. Phys. 66, 223–265 (1979)ADSCrossRefGoogle Scholar
  17. 17.
    Fröhlich J., Gabbiani F.: Braid statistics in local quantum field theory. Rev. Math. Phys. 2, 251–353 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Fröhlich J., Morchio G., Strocchi F.: Charged sectors and scattering states in electrodynamics. Ann. Phys. 119, 241–284 (1979)ADSCrossRefGoogle Scholar
  19. 19.
    Glimm J., Jaffe A.M.: Quantum Physics. A Functional Integral Point of View. Springer, New York (1987)Google Scholar
  20. 20.
    Haag R.: Local Quantum Physics. Fields, Particles, Algebras. Springer, Berlin, Heidelberg, New York (1992)CrossRefzbMATHGoogle Scholar
  21. 21.
    Haag R.: Local algebras. A look back at the early years and at some achievements and missed opportunities. Eur. Phys. J. H 35, 255–261 (2010)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Longo R.: Notes on algebraic invariants for non-commutative dynamical systems. Commun. Math. Phys. 69, 195–207 (1979)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Longo R.: Index of subfactors and statistics of quantum fields. II correspondences, Braid group statistics and Jones polynomial. Commun. Math. Phys. 130, 285–309 (1990)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Sadowski P., Woronowicz S.L.: Total sets in quantum field theory. Rep. Math. Phys. 2, 113–120 (1971)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Summers S.J., Wichmann E.H.: Concerning the condition of additivity in quantum field theory. Ann. Poincare Phys. Theor. 47, 113–124 (1987)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Takesaki M.: Theory of Operator Algebras I. Springer, Berlin, Heidelberg, New York (1979)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut für Theoretische Physik, Courant Centre “Higher Order Structures in Mathematics”Universität GöttingenGöttingenGermany
  2. 2.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly

Personalised recommendations