Communications in Mathematical Physics

, Volume 345, Issue 2, pp 587–614 | Cite as

Where Infinite Spin Particles are Localizable

  • Roberto Longo
  • Vincenzo Morinelli
  • Karl-Henning Rehren
Open Access


Particle states transforming in one of the infinite spin representations of the Poincaré group (as classified by E. Wigner) are consistent with fundamental physical principles, but local fields generating them from the vacuum state cannot exist. While it is known that infinite spin states localized in a spacelike cone are dense in the one-particle space, we show here that the subspace of states localized in any double cone is trivial. This implies that the free field theory associated with infinite spin has no observables localized in bounded regions. In an interacting theory, if the vacuum vector is cyclic for a double cone local algebra, then the theory does not contain infinite spin representations. We also prove that if a Doplicher–Haag–Roberts representation (localized in a double cone) of a local net is covariant under a unitary representation of the Poincaré group containing infinite spin, then it has infinite statistics. These results hold under the natural assumption of the Bisognano–Wichmann property, and we give a counter-example (with continuous particle degeneracy) without this property where the conclusions fail. Our results hold true in any spacetime dimension s + 1 where infinite spin representations exist, namely \({s\geq 2}\).


  1. 1.
    Araki H.: A lattice of von Neumann algebras associated with the quantum theory of a free Bose field. J. Math. Phys. 4, 1343–1362 (1963)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bisognano J.J., Wichmann E.H.: On the duality condition for quantum fields. J. Math. Phys. 17, 303–321 (1976)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Borchers H.-J.: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brunetti R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14(7 & 8), 759–786 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Buchholz D., Fredenhagen K.: Locality and structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Buchholz D., Porrmann M., Stein U.: Dirac versus Wigner: towards a universal particle concept in local quantum field theory. Phys. Lett. B 267, 377–381 (1991)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics. I. Commun. Math. Phys. 23, 199–230 (1971)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Eckmann J.P., Osterwalder K.: An application of Tomita’s theory of modular Hilbert algebras: duality for free Bose fields. J. Funct. Anal. 13, 1–12 (1973)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Foit, J.J.: Abstract twisted duality for quantum free Fermi fields. Publ. RIMS Kyoto Univ. 19, 729–74 (1983)Google Scholar
  11. 11.
    Guido D., Longo R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys. 148, 521–551 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Guido D., Longo R.: An algebraic spin and statistics theorem. Commun. Math. Phys. 172, 517–533 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Guido D., Longo R.: Natural energy bounds in quantum thermodynamics. Commun. Math. Phys. 218, 513–536 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Haag R.: Local Quantum Physics—Fields, Particles, Algebras, 2nd edn. Springer, New York (1996)MATHGoogle Scholar
  15. 15.
    Hislop P.D., Longo R.: Modular structure of the local algebras associated with the free massless scalar field theory. Commun. Math. Phys. 84, 71–85 (1982)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Isola T.: Modular structure of the crossed product by a compact group dual. J. Oper. Theory 33, 3–31 (1995)ADSMathSciNetMATHGoogle Scholar
  17. 17.
    Iverson G.J., Mack G.: Quantum fields and interaction of massless particles: the continuous spin case. Ann. Phys. 64, 211–253 (1971)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Köhler, Ch.: On the localization properties of quantum fields with zero mass and infinite spin. Ph.D. thesis, Vienna (2015)Google Scholar
  19. 19.
    Lawson H.B. Jr, Michelsohn M.L.: Spin Geometry. Princeton University Press, Princeton (1989)MATHGoogle Scholar
  20. 20.
    Lechner G., Longo R.: Localization in nets of standard spaces. Commun. Math. Phys. 336, 27–61 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Leyland, P., Roberts, J.E., Testard, D.: Duality for quantum free fields. Unpublished manuscript, Marseille (1978)Google Scholar
  22. 22.
    Longo, R.: On the spin-statistics relation for topological charges. In: Doplicher, S. et al. (eds.). Operator Algebras and Quantum Field Theory (Rome, 1996), pp. 661–669. Int. Press, Cambridge, MA (1997)Google Scholar
  23. 23.
    Longo R.: An analogue of the Kac–Wakimoto formula and black hole conditional entropy. Commun. Math. Phys. 186, 451–479 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Longo, R.: Lectures on Conformal Nets. Preliminary lecture notes that are available at
  25. 25.
    Longo, R.: Real Hilbert subspaces, modular theory, SL\({(2, \mathbb{R})}\) and CFT. In: Von Neumann algebras in Sibiu, pp. 33–91, Theta Ser. Adv. Math., 10, Theta, Bucharest (2008)Google Scholar
  26. 26.
    Mund J.: The Bisognano–Wichmann theorem for massive theories. Ann. Henri Poincaré 2, 907–926 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Mund J., Schroer B., Yngvason J.: String-localized quantum fields from Wigner representations. Phys. Lett. B 596, 156–162 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Oksak A.I., Todorov I.T.: Invalidity of TCP theorem for infinite-component fields. Commun. Math. Phys. 11, 125–130 (1968)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Rieffel M.A., Van Daele A.: A bounded operator approach to Tomita–Takesaki theory. Pac. J. Math. 69, 187–221 (1977)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Schroer, B.: Manuscripts on infinite spin and dark matter. arXiv:0802.2098v3, arXiv:0802.2098v4, arXiv:1306.3876v5
  31. 31.
    Sewell G.L.: Quantum fields on manifolds: PCT and gravitationally induced thermal states. Ann. Phys. 141, 201–224 (1982)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Streater R.F., Wightman A.S.: PCT, Spin and Statistics, and All that, 2nd edn. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City (1989)MATHGoogle Scholar
  33. 33.
    Takesaki, M.: Theory of Operator Algebras, I & II. Springer-Verlag, New York-Heidelberg (2002, 2003)Google Scholar
  34. 34.
    Wigner E.P.: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149–204 (1939)ADSMathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Yngvason, J.: Zur Existenz von Teilchen mit Masse 0 und unendlichem Spin in der Quantenfeldheorie. Diploma thesis, Göttingen (1969)Google Scholar
  36. 36.
    Yngvason J.: Zero-mass infinite spin representations of the Poincaré group and quantum field theory. Commun. Math. Phys. 18, 195–203 (1970)ADSMathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Zimmer R.J.: Ergodic Theory of Semisimple Lie Groups. Birkhäuser, Boston-Basel-Stuttgart (1984)CrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Roberto Longo
    • 1
  • Vincenzo Morinelli
    • 1
  • Karl-Henning Rehren
    • 2
  1. 1.Dipartimento di MatematicaUniversità di Roma Tor VergataRomeItaly
  2. 2.Institut für Theoretische PhysikUniversität GöttingenGöttingenGermany

Personalised recommendations