Skip to main content
SpringerLink
Log in

Spin and Statistics and First Principles

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

It was shown in the early seventies that, in Local Quantum Theory (that is the most general formulation of Quantum Field Theory, if we leave out only the unknown scenario of Quantum Gravity) the notion of Statistics can be grounded solely on the local observable quantities (without assuming neither the commutation relations nor even the existence of unobservable charged field operators); one finds that only the well known (para)statistics of Bose/Fermi type are allowed by the key principle of local commutativity of observables. In this frame it was possible to formulate and prove the Spin and Statistics Theorem purely on the basis of First Principles.

In a subsequent stage it has been possible to prove the existence of a unique, canonical algebra of local field operators obeying ordinary Bose/Fermi commutation relations at spacelike separations.

In this general guise the Spin–Statistics Theorem applies to Theories (on the four dimensional Minkowski space) where only massive particles with finite mass degeneracy can occur. Here we describe the underlying simple basic ideas, and briefly mention the subsequent generalisations; eventually we comment on the possible validity of the Spin–Statistics Theorem in presence of massless particles, or of violations of locality as expected in Quantum Gravity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Haag, R.: Local Quantum Physics: Fields, Particles, Algebras. Springer, Berlin (1994)

    Google Scholar 

  2. Araki, H.: Mathematical Theory of Quantum Fields. Oxford U.P., Oxford (1999)

    MATH  Google Scholar 

  3. Bisognano, J., Wichmann, E.: On the duality condition for a Hermitian scalar field. J. Math. Phys. 16, 985 (1975)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Buchholz, D., Doplicher, S., Longo, R., Roberts, J.E.: A new look at Goldstone’s theorem, Rev. Math. Phys., pp. 47–82, special Issue (1992)

  5. Buchholz, D., D’Antoni, C., Fredenhagen, K.: The universal structure of local algebras. Commun. Math. Phys. 111, 123 (1987)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Fredenhagen, K.: On the existence of antiparticles. Commun. Math. Phys. 79, 141 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  9. 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  MATH  MathSciNet  ADS  Google Scholar 

  10. Fredenhagen, K.: Superselection sectors with infinite statistical dimension. In: Taniguchi Symposium on Condensed Matter Theory, Lake Biwa, Kashikojima, Japan, Oct. 1993; DESY-94-071 (1994)

  11. Bertozzini, P., Conti, R., Longo, R.: Covariant sectors with infinite dimension and positivity of the energy. Commun. Math. Phys. 193, 471–492 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Carpi, S.: The Virasoro algebra and sectors with infinite statistical dimension. Ann. Henri Poincare 4, 601–611 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics II. Commun. Math. Phys. 35, 49–85 (1974)

    Article  MathSciNet  ADS  Google Scholar 

  14. Buchholz, D., Epstein, H.: Spin and statistics of quantum topological charges. Fizika 17, 329 (1985)

    Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Guido, D., Longo, R., Roberts, J.E., Verch, R.: Charged sectors, spin and statistics in quantum field theory on curved space–times. Rev. Math. Phys. 13, 125–198 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with Braid group statistics and exchange algebras. 1. General theory. Commun. Math. Phys. 125, 201 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Longo, R.: Index of subfactors and statistics of quantum fields. I. Commun. Math. Phys. 126, 217–247 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Guido, D., Longo, R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11–36 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Mund, J.: The spin-statistics theorem for anyons and plektons in d=2+1. Commun. Math. Phys. 286, 1159–1180 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Kuckert, B.: Spin, statistics, and reflections, I. Rotation invariance. Ann. Henri Poincare 6, 849–862 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Doplicher, S., Roberts, J.E.: Endomorphisms of C* algebras, cross products and duality for compact groups. Ann. Math. 130, 75–119 (1989)

    Article  MathSciNet  Google Scholar 

  23. Doplicher, S., Roberts, J.E.: A new duality theory for compact groups. Invent. Math. 98, 157–218 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Doplicher, S., Piacitelli, G.: Any compact group is a gauge group. Rev. Math. Phys. 14, 873–886 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Doplicher, S.: Local aspects of superselection rules. Commun. Math. Phys. 85, 73–86 (1982)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Doplicher, S., Longo, R.: Local aspects of superselection rules II. Commun. Math. Phys. 88, 399–409 (1983)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. Buchholz, D., Doplicher, S., Longo, R.: On Noether’s theorem in quantum field theory. Ann. Phys. 170, 1 (1986)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Buchholz, D., Doplicher, S., Longo, R., Roberts, J.E.: A new look at Goldstone’s theorem, Rev. Math. Phys. Special Issue, pp. 49–83 (1992)

  29. Buchholz, D., Verch, R.: Scaling algebras and renormalization group in algebraic quantum field theory. Rev. Math. Phys. 7, 1195–1240 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  30. D’Antoni, C., Morsella, G., Verch, R.: Scaling algebras for charged fields and short distance analysis for localizable and topological charges. Ann. Henri Poincare 5, 809–870 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  31. Bostelmann, H., D’Antoni, C., Morsella, G.: Scaling algebras and pointlike fields: A Nonperturbative approach to renormalization. Commun. Math. Phys. 285, 763–798 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  32. Conti, R., Morsella, G.: Scaling limit for subsystems and Doplicher-Roberts reconstruction. Ann. Henri Poincare 10, 485–511 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  33. Deligne, P.: Categories tannakiennes. Grothendieck Festschrift II, 111–195 (1991)

    Google Scholar 

  34. Halvorson, H., Mueger, M.: Algebraic quantum field theory. In: Handbook of the Philosophy of Physics. Elsevier/North Holland, Amsterdam (2006). arXiv:math-ph/0602036

    Google Scholar 

  35. Buchholz, D., Doplicher, S., Morchio, G., Roberts, J.E., Strocchi, F.: A model for charges of electromagnetic type. In: Operator Algebras and Quantum Field Theory, International Press, pp. 647–660 (1997)

  36. Buchholz, D., Doplicher, S., Morchio, G., Roberts, J.E., Strocchi, F.: Quantum delocalization of the electric charge. Ann. Phys. 290, 53–66 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. Buchholz, D., Doplicher, S., Morchio, G., Roberts, J.E., Strocchi, F.: Asymptotic Abelianness and braided tensor c*-categories. Prog. Math. 251, 49–64 (2007)

    Article  MathSciNet  Google Scholar 

  38. Buchholz, D.: Collision theory for massless bosons. Comm. Math. Phys. 52, 147–173 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  39. Doplicher, S., Fredenhagen, K., Roberts, J.E.: The quantum structure of spacetime at the Planck scale and quantum fields. Commun. Math. Phys. 172, 187–220 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  40. Doplicher, S., Fredenhagen, K., Roberts, J.E.: Spacetime quantization induced by classical gravity. Phys. Lett. B 331, 39–44 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  41. Piacitelli, G.: Nonlocal theories: New rules for old diagrams. J. High Energy Phys. 0408, 031 (2004) arXiv:hep-th/0403055

    Article  MathSciNet  ADS  Google Scholar 

  42. Bahns, D., Doplicher, S., Fredenhagen, K., Piacitelli, G.: On the unitarity problem in space-time noncommutative theories. Phys. Lett. B 533, 178–181 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  43. Bahns, D., Doplicher, S., Fredenhagen, K., Piacitelli, G.: Ultraviolet finite quantum field theory on quantum space-time. Commun. Math. Phys. 237, 221–241 (2003)

    MATH  MathSciNet  ADS  Google Scholar 

  44. Piacitelli, G.: Twisted covariance as a non invariant restriction of the fully covariant DFR model. arXiv:0902.0575 [hep-th]

  45. Doplicher, S.: Space-time and fields: A quantum texture. In: Karpacz 2001, New developments in fundamental interaction theories; pp. 204–213. arXiv:hep-th/0105251v2

  46. Doplicher, S.: Quantum field theory on quantum spacetime. J. Phys. Conf. Ser. 53, 793–798 (2006) arXiv:hep-th/0608124

    Article  Google Scholar 

  47. Doplicher, S., Fredenhagen, K.: (in preparation)

  48. Doplicher, S.: The measurement process in local quantum theory and the EPR paradox. arXiv:0908.0480

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, University of Rome “La Sapienza”, 00185, Roma, Italy

    Sergio Doplicher

Authors
  1. Sergio Doplicher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergio Doplicher.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Doplicher, S. Spin and Statistics and First Principles. Found Phys 40, 719–732 (2010). https://doi.org/10.1007/s10701-009-9337-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-009-9337-2

Keywords

Search

Navigation