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Communications in Mathematical Physics

, Volume 268, Issue 3, pp 621–672 | Cite as

String-Localized Quantum Fields and Modular Localization

  • Jens Mund
  • Bert Schroer
  • Jakob YngvasonEmail author
Article

Abstract

We study free, covariant, quantum (Bose) fields that are associated with irreducible representations of the Poincaré group and localized in semi-infinite strings extending to spacelike infinity. Among these are fields that generate the irreducible representations of mass zero and infinite spin that are known to be incompatible with point-like localized fields. For the massive representations and the massless representations of finite helicity, all string-localized free fields can be written as an integral, along the string, of point-localized tensor or spinor fields. As a special case we discuss the string-localized vector fields associated with the point-like electromagnetic field and their relation to the axial gauge condition in the usual setting.

Keywords

Growth Order Modular Localization Wigner Representation Haag Duality Positive Energy Representation 
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.

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References

  1. 1.
    Abbott L.F. (1976) Massless particles with continuous spin indices. Phys. Rev. D 13, 2291–2294MathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Babujian H., Foerster A., Karowski M. (2006) Exact form factors in integrable quantum field theories: the scaling Z(N)-Ising model. Nucl. Phys. B 736, 169–198MathSciNetCrossRefADSzbMATHGoogle Scholar
  3. 3.
    Bisognano J.J., Wichmann E.H. (1975) On the duality condition for a Hermitian scalar field. J. Math. Phys. 16, 985–1007zbMATHMathSciNetCrossRefADSGoogle Scholar
  4. 4.
    Borchers H.-J., Buchholz D., Schroer B. (2001) Polarization-Free Generators and the S-Matrix. Commun. Math. Phys. 219, 125–140zbMATHMathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Bros J., Buchholz D. (1994) Towards a relativistic KMS-condition. Nucl. Phys. B 429, 291–318zbMATHMathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Bros J., Moschella U. (1996) Two-point functions and quantum fields in de Sitter universe. Rev. Math. Phys. 8, 327–391zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Brunetti R., Guido D., Longo R. (2002) Modular localization and Wigner particles. Rev. Math. Phs. 14, 759–786zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Buchholz D., D’Antoni C., Fredenhagen K. (1987) The universal structure of local algebras. Commun. Math. Phys. 111, 123–135zbMATHMathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Buchholz D., Fredenhagen K. (1982) Locality and the structure of particle states. Commun. Math. Phys 84, 1–54zbMATHMathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Buchholz D., Summers S.J. (2005) Quantum Statistics and Locality. Phys. Lett. A 337, 17–21MathSciNetCrossRefADSzbMATHGoogle Scholar
  11. 11.
    Buchholz D., Yngvason J. (1991) Generalized Nuclearity Conditions and the Split Property in Quantum Field theory. Lett. Math. Phys. 23, 159–167zbMATHMathSciNetCrossRefADSGoogle Scholar
  12. 12.
    Buchholz D., Yngvason J. (1994) Phys. Rev. Lett. 73, 613–616zbMATHMathSciNetCrossRefADSGoogle Scholar
  13. 13.
    Buchholz D., Wichmann E.H. (1986) Causal independence and the energy density of states in local quantum field theory. Commun. Math. Phys. 106, 321–344zbMATHMathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Chang S.-J. (1967) Lagrange Formulation for Systems with Higher Spin. Phys. Rev. 161, 1308–1315CrossRefADSGoogle Scholar
  15. 15.
    Dimock J. (2000) Locality in free string field theory. J. Math. Phys. 41, 40–61zbMATHMathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Doplicher S., Haag R., Roberts J.E. (1969) Fields, observables and gauge transformations II. Commun. Math. Phys. 15, 173–200zbMATHMathSciNetCrossRefADSGoogle Scholar
  17. 17.
    Doplicher S., Longo R. (1984) Standard and split inclusions of von Neumann algebras. Invent. Math. 75, 493–536zbMATHMathSciNetCrossRefADSGoogle Scholar
  18. 18.
    Duetsch, M., Schroer, B.: Massive vector mesons and gauge theory. J. Phys. A: Math. Gen. 30, 4317 (2000), and previous work by G. Scharf cited thereinGoogle Scholar
  19. 19.
    Epstein H., Glaser V. (1973) The role of locality in perturbation theory. Ann. Inst. H. Poincaré A 19, 211–295MathSciNetGoogle Scholar
  20. 20.
    Erler, D.G., Gross, D.J.: Locality , Causality and an Initial Value Formulation of Open Bosonic String Field Theory. http://arxiv.org/list/hep-th/0406199, 2004Google Scholar
  21. 21.
    Fassarella L., Schroer B. (2002) Wigner particle theory and local quantum physics. J. Phys. A 35, 9123–9164zbMATHMathSciNetCrossRefADSGoogle Scholar
  22. 22.
    Fermi E. (1932) Quantum Theory of Radiation. Rev. Mod. Phys. 4, 87–132zbMATHCrossRefADSGoogle Scholar
  23. 23.
    Fredenhagen K., Gaberdiel M., Rüger S.M. (1996) Scattering states of plektons (particles with braid group statistics) in 2+1 dimensional field theory. Commun. Math. Phys. 175, 319–355zbMATHCrossRefADSGoogle Scholar
  24. 24.
    Fredenhagen K., Rehren K.-H., Schroer B. (1992) Superselection sectors with braid group statistics and exchange algebras II: Geometric aspects and conformal covariance. Rev. Math. Phys. SI1, 113–157MathSciNetCrossRefGoogle Scholar
  25. 25.
    Fröhlich J. (1976) New super-selection sectors (“soliton-states”) in two dimensional Bose quantum field models. Commun. Math. Phys. 47, 269–310CrossRefADSGoogle Scholar
  26. 26.
    Fröhlich J., Gabbiani F. (1990) Braid Statistics in Local Quantum Field Theory. Rev. Math. Phys. 2, 251–353zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Haag R., (1996) Local quantum physics Second ed Texts and Monographs in Physics. Berlin-Heidelberg, SpringerGoogle Scholar
  28. 28.
    Hegerfeldt G.C. (1994) Causality Problems in Fermi’s Two Atom System. Phys. Rev. Lett. 72, 596–599zbMATHCrossRefADSGoogle Scholar
  29. 29.
    Hirata K. (1977) Quantization of Massless Fields with Continuous Spin. Prog. Theor. Phys. 58, 652–666zbMATHMathSciNetCrossRefADSGoogle Scholar
  30. 30.
    Iverson G.J., Mack G. (1971) Quantum fields and interactions of massless particles: The continuous spin case. Ann. Phys. 64, 211–253MathSciNetCrossRefADSGoogle Scholar
  31. 31.
    Joos H. (1962) Zur Darstellungstheorie der inhomogenen Lorentzgruppe als Grundlage quantenmechanischer Kinematik. Forts. der Phys. 10, 65–146zbMATHCrossRefGoogle Scholar
  32. 32.
    Jordan P. (1935) Zur Quantenelektrodynamik, I. Eichinvariante Operatoren. Zeits. für Phys. 95, 202zbMATHGoogle Scholar
  33. 33.
    ———,: Beiträge zur Neutrinotheorie des Lichts. Zeits. für Phys. 114, 229 (1937)Google Scholar
  34. 34.
    Klaiber B., (1968) The Thirring model. In: Barut A.O., Brittin W.E. (eds), Lectures in Theoretical Physics, Vol 10A. New York, Gordon and Breach, pp. 141–176Google Scholar
  35. 35.
    Lechner, G.: Towards the construction of quantum field theories from a factorizing S-matrix. http://arxiv. org//list/hep-th/0502184, 2005Google Scholar
  36. 36.
    Lechner, G.: An Existence Proof for Interacting Quantum Field Theories with a Factorizing S-Matrix. http://arxiv.org//list/math-ph/0602022, 2006Google Scholar
  37. 37.
    Leinaas J.M., Myrheim J. (1977) On the Theory of Identical Particles. Il Nuovo Cimento 37b: 1–23ADSGoogle Scholar
  38. 38.
    Leyland, P., Roberts, J., Testard, D.: Duality for Quantum Free Fields. Unpublished notes, CNRS , 1978Google Scholar
  39. 39.
    Licht A.L. (1966) Local States. J. Math. Phys. 7: 1656zbMATHMathSciNetCrossRefADSGoogle Scholar
  40. 40.
    Malament D., (1996) In defence of dogma: Why there cannot be a relativistic quantum mechanics of (localizable) particles. In: Clifton R.K. (ed), Perspectives of quantum reality. Dordrecht, KluwerGoogle Scholar
  41. 41.
    Mandelstam S. (1962) Quantum electrodynamics without potentials. Ann. Phys. 19, 1–24zbMATHMathSciNetCrossRefADSGoogle Scholar
  42. 42.
    Mourad, J.: Continuous spin and tensionless strings. http://arxiv.org//list/hep-th/0410009, 2004Google Scholar
  43. 43.
    Mund J. (1998) No-go theorem for ‘free’ relativistic anyons in d = 2 + 1. Lett. Math. Phys. 43, 319–328zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    ———,: The Bisognano-Wichmann theorem for massive theories. Ann. H. Poin. 2, 907–926 (2001)Google Scholar
  45. 45.
    ———,: Modular localization of massive particles with “any” spin in d=2+1. J. Math. Phys. 44, 2037–2057 (2003)Google Scholar
  46. 46.
    ———,: String-Localized Covariant Quantum Fields. To appear in Rigorous Quantum Field Theory, Birkhauser Publishing (2006), Basel; available at http://arxiv.org//list/hep-th/0502014, 2005Google Scholar
  47. 47.
    Mund J., Schroer B., Yngvason J. (2004) String–localized quantum fields from Wigner representations. Phys. Lett. B 596, 156–162MathSciNetADSGoogle Scholar
  48. 48.
    Newton T.D., Wigner E.P. (1949) Localized States for Elementary Systems. Rev. Mod. Phys. 21, 400–406zbMATHCrossRefADSGoogle Scholar
  49. 49.
    Perez J.F., Wilde I.F. (1977) Localization and causality in relativistic quantum mechanics. Phys. Rev. 16, 315–317CrossRefADSGoogle Scholar
  50. 50.
    Polchinski J., (1998) String Theory Vol I and II. Cambridge, Cambridge Univ PressGoogle Scholar
  51. 51.
    Reed M., Simon B., (1975) Methods of modern mathematical physics II. New York, Academic PresszbMATHGoogle Scholar
  52. 52.
    Reeh H., Schlieder S. (1961) Bemerkungen zur unitäräquivalenz von Lorentzinvarianten Feldern. Nuovo Cimento 22, 1051–1068MathSciNetCrossRefGoogle Scholar
  53. 53.
    Rieffel M.A., Van Daele A. (1977) A bounded operator approach to Tomita-Takesaki theory. Pac. J. Math. 1, 187–221MathSciNetGoogle Scholar
  54. 54.
    Savvidy G. (2005) Tensionless strings, correspondence with SO(D,D) sigma model. Phys. Lett. B 615, 285–290MathSciNetADSGoogle Scholar
  55. 55.
    Schroer B. (1999) Modular Wedge Localization and the d=1+1 Formfactor Program. Ann. Phys. 295, 190–223MathSciNetADSGoogle Scholar
  56. 56.
    Schroer B. (2005) Constructive proposals based on the crossing property and the lightfront holography. Ann. Phys. 319, 48zbMATHMathSciNetCrossRefADSGoogle Scholar
  57. 57.
    Steinmann O. (1982) A Jost-Schroer theorem for string fields. Commun. Math. Phys. 87, 259–264zbMATHMathSciNetCrossRefADSGoogle Scholar
  58. 58.
    Streater R.F., Wightman A.S., (1964) PCT, spin and statistics, and all that. New York, W.A. Benjamin InczbMATHGoogle Scholar
  59. 59.
    Streater R.F., Wilde I.F. (1970) Fermion states of a Bose field. Nucl. Phys. B24, 561CrossRefADSGoogle Scholar
  60. 60.
    Strocchi F. (1969). Phys. Rev 166, 1302–1307MathSciNetCrossRefADSGoogle Scholar
  61. 61.
    Summers S. (1990) On the independence of local algebras in quantum field theory. Rev. Math. Phys. 2, 201–247zbMATHMathSciNetCrossRefGoogle Scholar
  62. 62.
    Weinberg, S.: What is quantum field theory, and what did we think it is? http://arxiv.org/list/ hepth/9702027, 1997Google Scholar
  63. 63.
    Weinberg S., (1995) The Quantum Theory of Fields I. Cambridge, Cambridge University PressGoogle Scholar
  64. 64.
    Weinberg S. (1964) Feynman Rules For Any Spin. Phys. Rev. 133: B1318–30MathSciNetCrossRefADSGoogle Scholar
  65. 65.
    Werner R. (1987) Local preparability of States and the Split Property in Quantum Field Theory. Lett. Math. Phys. 13, 325–329zbMATHMathSciNetCrossRefADSGoogle Scholar
  66. 66.
    Wigner E.P. (1948) Relativistische Wellengleichungen. Z. Physik 124, 665–684zbMATHMathSciNetCrossRefADSGoogle Scholar
  67. 67.
    Wilczek F. (1982) Quantum Mechanics of Fractional-Spin Particles. Phys. Rev. Lett. 49, 957–1149MathSciNetCrossRefADSGoogle Scholar
  68. 68.
    Wilson K.G. (1974) Confinement of Quarks. Phys. Rev. D 10, 2445–2459CrossRefADSGoogle Scholar
  69. 69.
    Yngvason J. (1970) Zero-mass infinite spin representations of the Poincaré group and quantum field theory. Commun. Math. Phys. 18, 195–203zbMATHMathSciNetCrossRefADSGoogle Scholar
  70. 70.
    Yngvason J. (2005) The Role of Type III Factors in Quantum Field Theory. Rep. Math. Phys. 55, 135–147MathSciNetCrossRefzbMATHADSGoogle Scholar
  71. 71.
    Mund, J., Schroer, B., Yngvason, Y.: http://arxiv.org/abs/math-ph/0511042, 2005Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Departamento de Física, ICEUniversidade Federal de Juiz de ForaJuiz de ForaBrazil
  2. 2.CBPFRio de JaneiroBrazil
  3. 3.Institut für Theoretische PhysikFU-BerlinBerlinGermany
  4. 4.Erwin Schrödinger Institute for Mathematical PhysicsViennaAustria
  5. 5.Institut für Theoretische PhysikUniversität WienViennaAustria

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