Skip to main content
SpringerLink
Log in

Free quantum fields in ten dimensions withSp(4,R) symmetry

  • Published:
Il Nuovo Cimento B (1971-1996)

Summary

We study a class of free local quantum field theories on a flat 10-dimensional space, assuming a symmetry with respect to the symplectic groupSp(4,R) acting on the coordinates through its adjoint representation. These theories describe «particles» that transform according to irreducible unitary representations of an inhomogeneousSp(4,R) group satisfying a spectral condition (positivity of the energy). We give the explicit form of these representations, we construct the corresponding quantum fields and we determine the field equations, the transformation laws, the (anti)commutation relations and the two-point Wightman functions. We discuss the local (anti)commutativity properties of the fields, that are in a strict sense weaker than the ones of the Lorentz-invariant theories, and we find a connection between «spin» and statistics.

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. Th. Kaluza:Sitzungsber. Preuss. Akad. Wiss. Berlin, Math. Phys. K,1, 966 (1921).

    Google Scholar 

  2. O. Klein:Z. Phys.,37, 895 (1926).

    Article  ADS  MATH  Google Scholar 

  3. C. A. Orzalesi:Fortschr. Phys.,29, 413 (1981).

    Article  MathSciNet  Google Scholar 

  4. W. Mecklenburg:Fortschr. Phys.,32, 207 (1984).

    Article  MathSciNet  Google Scholar 

  5. M. J. Duff, B. E. W. Nilsson andC. N. Pope:Phys. Rep. C,130, 12 (1986).

    Article  MathSciNet  Google Scholar 

  6. S. Kobayashi andK. Nomizu:Foundations of Differential Geometry (Wiley, New York, N.Y., 1969).

    MATH  Google Scholar 

  7. Y. Choquet-Bruhat:Géométrie différentielle et systèmes extérieurs (Dunod, Paris, 1968).

    MATH  Google Scholar 

  8. M. Toller:Nuovo Cimento B,44, 67 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  9. G. Cognola, R. Soldati, M. Toller, L. Vanzo andS. Zerbini:Nuovo Cimento B,54, 325 (1979).

    Article  MathSciNet  ADS  Google Scholar 

  10. Y. Ne’eman andT. Regge:Riv. Nuovo Cimento,1, n. 5 (1978).

    MathSciNet  Google Scholar 

  11. M. Toller:Nuovo Cimento B,64, 471 (1981).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. M. Toller:Int. J. Theor. Phys.,29, 963 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Weinberg:Phys. Lett. B,138, 47 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  14. I. E. Segal:Mathematical Cosmology and Extragalactic Astronomy (Academic Press, New York, N.Y., 1976).

    Google Scholar 

  15. E. B. Vinberg:Funct. Anal. Appl.,14, 1 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  16. S. M. Paneitz:J. Funct. Anal,43, 313 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Inönü andE. P. Wigner:Proc. Natl. Acad. Sci. U.S.A.,39, 510 (1953).

    Article  ADS  MATH  Google Scholar 

  18. F. Lurçat:Physics,1, 95 (1964).

    Google Scholar 

  19. M. Toller:Nuovo Cimento B,102, 261 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  20. M. Toller: University of Trento preprint UTF 271 (1992).

  21. Harish-Chandra:Trans. Am. Math. Soc.,119, 457 (1965).

    Article  MathSciNet  Google Scholar 

  22. A. Kirillov:Eléments de la théorie des représentations (Editions MIR, Moscou, 1974).

    Google Scholar 

  23. V. Bargmann:Ann. Math,59, 1 (1954).

    Article  MathSciNet  Google Scholar 

  24. F. Wilczek:Phys. Rev. Lett,49, 957 (1982).

    Article  MathSciNet  ADS  Google Scholar 

  25. J. Fröhlich andP. A. Marchetti:Lett. Math. Phys.,16, 347 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. E. P. Wigner:Ann. Math,40, 149 (1939).

    Article  MathSciNet  Google Scholar 

  27. G. Mackey:Bull. Am. Math. Soc.,69, 628 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Flato andC. Fronsdal:Lett. Math. Phys.,2, 421 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  29. J. Fang andC. Fronsdal:Lett. Math. Phys.,2, 391 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  30. C. Fronsdal:Phys. Rev. D,20, 848 (1979).

    Article  MathSciNet  ADS  Google Scholar 

  31. J. Williamson:Am, J. Math.,58, 141 (1936).

    Article  Google Scholar 

  32. V. Arnold:Méthodes mathématiques de la mécanique classique (Editions MIR, Moscou, 1976).

    Google Scholar 

  33. N. Burgoyne andR. Cushman:J. Algebra,44, 339 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  34. L. Gårding:Ann. Math.,48, 785 (1947).

    Article  MATH  Google Scholar 

  35. G. Jameson:Ordered Linear Spaces (Springer-Verlag, Berlin, 1970).

    MATH  Google Scholar 

  36. C. Chevalley andS. Eilenberg:Trans. Am. Math. Soc.,63, 85 (1948).

    Article  MathSciNet  Google Scholar 

  37. W. Greub, S. Halperin andR. Vanstone:Connections, Curvature and Cohomology, Vol. III (Academic Press, New York, N.Y., 1976).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica dell’Università, Trento

    M. Toller

  2. Gruppo collegato di Trento, INFN, Trento, Italia

    M. Toller

Authors
  1. M. Toller
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Toller, M. Free quantum fields in ten dimensions withSp(4,R) symmetry. Nuov Cim B 108, 245–274 (1993). https://doi.org/10.1007/BF02887487

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02887487

PACS

PACS

Search

Navigation