Skip to main content
SpringerLink
Log in

Conformal invariance and conservation laws for relativistic wave equations for zero rest mass

  • Published:
Il Nuovo Cimento (1955-1965)

Summary

It is shown that the relativistic wave equations obtained byGårding for particles of zero rest mass and arbitrary spin are invariant under the conformal transformations of space-time. Conservation laws corresponding to this invariance are obtained. They are formally identical with those found by Bessel-Hagen for the electromagnetic field, with the exception of those found for the scalar field. For this case, the conservation laws contain extra terms which arise from the fact that the trace of the energy-momentum tensor is not zero.

Riassunto

Si dimostra che le equazioni d’onda relativistiche ottenute daGårding per particelle di massa a riposo nulla e spin arbitrario sono invarianti nelle trasformazioni conformi dello spazio-tempo. Si ottengono leggi di conservazione corrispondenti a detta invarianza. Sono formalmente identiche a quelle trovate da Bessel-Hagen per il campo elettromagnetico, ad ecoezione di quelle trovate per il campo scalare. Per questo caso le leggi di conservazione contengono termini aggiuntivi che originano dal fatto che la traccia del tensore energia-impulso non è zero.

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. E. Cunningham:Proc. Lond. Math. Soc.,8, 77 (1910).

    Article  MathSciNet  Google Scholar 

  2. H. Bateman:Proc. Lond. Math Soc.,8, 223 (1910).

    Article  MathSciNet  MATH  Google Scholar 

  3. E. Bessel-Hagen:Math. Ann.,84, 258 (1921).

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Noether:Nachr. Akad. Wiss. Göttingen, Math.-Physik. Kl. 235 (1918). See also E. L. Hill:Rev. Mod. Phys.,23, 253 (1951).

  5. P. A. M. Dirac:Ann. of Math.,37, 429 (1936).

    Article  MathSciNet  Google Scholar 

  6. J. Schouten andJ. Haantjes:Proc. Kon. Akad. v. Wet.,39, 1059 (1936).

    Google Scholar 

  7. W. Pauli:Helm. Phys. Acta,13, 204 (1940).

    MathSciNet  Google Scholar 

  8. H. J. Bhabha:Proc. Camb. Phil. Soc.,32, 622 (1936).

    Article  ADS  Google Scholar 

  9. L. Gårding:Proc. Camb. Phil. Soc.,41, 49 (1945).

    Article  ADS  Google Scholar 

  10. P. A. M. Dirac:Proc. Roy. Soc. London (A),155, 447 (1936).

    Article  ADS  Google Scholar 

  11. M. Fierz:Helv. Phys. Acta,12, 3 (1939); 13, 45 (1940).

    Article  Google Scholar 

  12. B. L. van der Waerden:Nachr. Akad. Wiss. Göttingen, Math.-physik. Kl., 100 (1929).

  13. O. Laporte andG. E. Uhlenbeck:Phys. Rev.,37, 1380 (1931). See alsoW. L. Bade andH. Jehle:Rev. Mod. Phys.,25, 714 (1953).

    Article  ADS  Google Scholar 

  14. H. J. Bhabha:Rev. Mod. Phys.,21, 451 (1949), i.e., the time-reversing or antichronous transformations are included in the term. The proper (improper) transformations are those for which det (Ljk)= + 1 (-1). (Spinor representations of the antichronous transformations have been given byS. Watanabe:Phys. Bev.,84, 1008 (1951) and Ken-iti Goto:Prog. Theor. Phys.,6, 990 (1951).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. S. Lie andF. Engel:Theorie der Transformationsgruppen (Leipzig, 1893), vol. III, p. 347 ff.

  16. For details of this and later calculations, seeJ. A. McLennan Jr.:On the Invariance Properties and Conservation Laws for Relativistic Wave Equations, Particularyl jor Zero Best Mass (Lehigh University, Thesis, 1952).

  17. G. Wentzel:Quantum Theory of Fields (New York, 1949).

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Lehigh University, Bethlehem, Pennsylvania

    J. A. McLennan

Authors
  1. J. A. McLennan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

A short account of some of the results of this paper was given in: J. A. McLennan Jr. and P. Havas:Phys. Rev.,87, 898 (1952).

This research was done while the author was holder of an A.E.C. Pre-Doctora Fellowship, 1950–1952.

Rights and permissions

Reprints and permissions

About this article

Cite this article

McLennan, J.A. Conformal invariance and conservation laws for relativistic wave equations for zero rest mass. Nuovo Cim 3, 1360–1379 (1956). https://doi.org/10.1007/BF02785012

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02785012

Search

Navigation