Skip to main content

Advertisement

SpringerLink
  1. Home
  2. Communications in Mathematical Physics
  3. Article
Where Infinite Spin Particles are Localizable
Download PDF
Your article has downloaded

Similar articles being viewed by others

Slider with three articles shown per slide. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide.

Light-front description of infinite spin fields in six-dimensional Minkowski space

22 August 2022

I. L. Buchbinder, S. A. Fedoruk & A. P. Isaev

Spin-1/2 “bosons” with mass dimension 3/2 and fermions with mass dimension 1 cannot represent physical particle states

28 October 2022

A. R. Aguirre, M. M. Chaichian, … B. L. Sánchez-Vega

A bound on massive higher spin particles

05 April 2019

Nima Afkhami-Jeddi, Sandipan Kundu & Amirhossein Tajdini

Massless Infinite Spin Representations

01 July 2020

I. L. Buchbinder, A. P. Isaev & S. Fedoruk

Massless Infinite Spin (Super)particles and Fields

01 May 2020

I. L. Buchbinder, A. P. Isaev & S. A. Fedoruk

On the spectrum of pure higher spin gravity

01 December 2020

Luis F. Alday, Jin-Beom Bae, … Carmen Jorge-Diaz

Lorentz-covariant spin operator for spin 1/2 massive fields as a physical observable

10 January 2023

Taeseung Choi & Yeong Deok Han

Generalized spinning particles on $${\mathcal {S}}^2$$ S 2 in accord with the Bianchi classification

02 March 2021

Anton Galajinsky

Massive higher spins: effective theory and consistency

17 October 2019

Brando Bellazzini, Francesco Riva, … Francesco Sgarlata

Download PDF
  • Open Access
  • Published: 29 October 2015

Where Infinite Spin Particles are Localizable

  • Roberto Longo1,
  • Vincenzo Morinelli1 &
  • Karl-Henning Rehren2 

Communications in Mathematical Physics volume 345, pages 587–614 (2016)Cite this article

  • 661 Accesses

  • 19 Citations

  • 7 Altmetric

  • Metrics details

Abstract

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 \({s\geq 2}\).

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Bisognano J.J., Wichmann E.H.: On the duality condition for quantum fields. J. Math. Phys. 17, 303–321 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  3. Borchers H.-J.: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Brunetti R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14(7 & 8), 759–786 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buchholz D., Fredenhagen K.: Locality and structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  7. Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics. I. Commun. Math. Phys. 23, 199–230 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Foit, J.J.: Abstract twisted duality for quantum free Fermi fields. Publ. RIMS Kyoto Univ. 19, 729–74 (1983)

  11. Guido D., Longo R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys. 148, 521–551 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Guido D., Longo R.: An algebraic spin and statistics theorem. Commun. Math. Phys. 172, 517–533 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Guido D., Longo R.: Natural energy bounds in quantum thermodynamics. Commun. Math. Phys. 218, 513–536 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Haag R.: Local Quantum Physics—Fields, Particles, Algebras, 2nd edn. Springer, New York (1996)

    MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Isola T.: Modular structure of the crossed product by a compact group dual. J. Oper. Theory 33, 3–31 (1995)

    ADS  MathSciNet  MATH  Google Scholar 

  17. Iverson G.J., Mack G.: Quantum fields and interaction of massless particles: the continuous spin case. Ann. Phys. 64, 211–253 (1971)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Köhler, Ch.: On the localization properties of quantum fields with zero mass and infinite spin. Ph.D. thesis, Vienna (2015)

  19. Lawson H.B. Jr, Michelsohn M.L.: Spin Geometry. Princeton University Press, Princeton (1989)

    MATH  Google Scholar 

  20. Lechner G., Longo R.: Localization in nets of standard spaces. Commun. Math. Phys. 336, 27–61 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Leyland, P., Roberts, J.E., Testard, D.: Duality for quantum free fields. Unpublished manuscript, Marseille (1978)

  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)

  23. Longo R.: An analogue of the Kac–Wakimoto formula and black hole conditional entropy. Commun. Math. Phys. 186, 451–479 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Longo, R.: Lectures on Conformal Nets. Preliminary lecture notes that are available at http://www.mat.uniroma2.it/~longo/Lecture_Notes.html

  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)

  26. Mund J.: The Bisognano–Wichmann theorem for massive theories. Ann. Henri Poincaré 2, 907–926 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Mund J., Schroer B., Yngvason J.: String-localized quantum fields from Wigner representations. Phys. Lett. B 596, 156–162 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Oksak A.I., Todorov I.T.: Invalidity of TCP theorem for infinite-component fields. Commun. Math. Phys. 11, 125–130 (1968)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Rieffel M.A., Van Daele A.: A bounded operator approach to Tomita–Takesaki theory. Pac. J. Math. 69, 187–221 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  30. Schroer, B.: Manuscripts on infinite spin and dark matter. arXiv:0802.2098v3, arXiv:0802.2098v4, arXiv:1306.3876v5

  31. Sewell G.L.: Quantum fields on manifolds: PCT and gravitationally induced thermal states. Ann. Phys. 141, 201–224 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  33. Takesaki, M.: Theory of Operator Algebras, I & II. Springer-Verlag, New York-Heidelberg (2002, 2003)

  34. Wigner E.P.: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149–204 (1939)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Yngvason, J.: Zur Existenz von Teilchen mit Masse 0 und unendlichem Spin in der Quantenfeldheorie. Diploma thesis, Göttingen (1969)

  36. Yngvason J.: Zero-mass infinite spin representations of the Poincaré group and quantum field theory. Commun. Math. Phys. 18, 195–203 (1970)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Zimmer R.J.: Ergodic Theory of Semisimple Lie Groups. Birkhäuser, Boston-Basel-Stuttgart (1984)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, 00133, Rome, Italy

    Roberto Longo & Vincenzo Morinelli

  2. Institut für Theoretische Physik, Universität Göttingen, 37077, Göttingen, Germany

    Karl-Henning Rehren

Authors
  1. Roberto Longo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vincenzo Morinelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Karl-Henning Rehren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karl-Henning Rehren.

Additional information

Communicated by Y. Kawahigashi

Supported in part by the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”, PRIN-MIUR and GNAMPA-INdAM.

Vincenzo Morinelli: Supported in part by PRIN-MIUR and GNAMPA-INdAM.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Longo, R., Morinelli, V. & Rehren, KH. Where Infinite Spin Particles are Localizable. Commun. Math. Phys. 345, 587–614 (2016). https://doi.org/10.1007/s00220-015-2475-9

Download citation

  • Received: 22 May 2015

  • Accepted: 09 July 2015

  • Published: 29 October 2015

  • Issue Date: July 2016

  • DOI: https://doi.org/10.1007/s00220-015-2475-9

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Unitary Representation
  • Spin Representation
  • Spin Particle
  • Double Cone
  • Vacuum Vector
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Over 10 million scientific documents at your fingertips

Switch Edition
  • Academic Edition
  • Corporate Edition
  • Home
  • Impressum
  • Legal information
  • Privacy statement
  • California Privacy Statement
  • How we use cookies
  • Manage cookies/Do not sell my data
  • Accessibility
  • FAQ
  • Contact us
  • Affiliate program

Not affiliated

Springer Nature

© 2023 Springer Nature Switzerland AG. Part of Springer Nature.