The conformal group, its casimir operators, and a four-position operator

Конформная группа, операторы Каэимира и четырехмерный оператор положениявремени

Summary

We are investigating a class of representations of the conformal group which are physically interesting because there exists a position-time 4-vector operator within them and, related to this, because they are used in field transformation laws. We show that in these representations the three Casimir operators of the group have a very simple interpretation; we discuss when these representations are irreducible and we show that this 4-position operator can be expressed in terms of the generators of the group.

Riassunto

Si stanno facendo ricerche su una classe di rappresentazioni del gruppo conforme, che sono fisicamente interessanti perché all’interno di esse esiste un operatore quadrivettoriale spazio-temporale e, in relazione con ciò, perché sono usate nelle leggi di trasformazione dei campi. Si mette in luce come in queste rappresentazioni i tre operatori di Casimir del gruppo abbiano un’interpretazione molto semplice; si discute quando queste rappresentazioni sono irriducibili; e si fa vedere che questo operatore quadriposizionale può essere espresso per mezzo dei generatori del gruppo.

Реэюме

Мы исследуем класс представлений конформной группы, которые представляют фиэический интерес, потому что сушествует оператор, определяюший четырех-вектор положения-времени, в зтих представлениях, и которые свяэаны с зтой группой, так как они испольэуются в эаконах преобраэования полей. Мы покаэываем, что в зтих представлениях три оператора Каэимира зтой группы имеют очень простую интерпретацию. Мы обсуждаем вопрос, когда зти представления являются неприводимыми. Мы покаэываем, что зтот оператор, определяюший четырех-вектор положения-времени, может быть выражен в терминах генераторов группы.

This is a preview of subscription content, access via your institution.

References

  1. (1)

    See,e.g., De Sitter and Conformal Groups and Their Applications, edited byA. O. Barut andW. E. Brittin (Boulder, Col., 1971);A. Salam andJ. Strathdee:Phys. Rev.,184, 1760 (1969);D. J. Gross andJ. Wess:Phys. Rev. D,2, 753 (1970);S. Coleman andR. Jackiw:Ann. of Phys.,67, 552 (1971);R. Jackiw: MIT-Center for Theoretical Physics, Publication No. 236 (1971);K. Symanzik: DESY preprint No. 72/6 (1972);B. Schroer: talk given at theTopical Meeting on Outlook for Broken Conformal Symmetry in Elementary Particle Physics, Frascati, Rome, May 1972.

  2. (2)

    See,e.g.,S. D. Drell andT. M. Yan:Ann. of Phys.,66, 578 (1971) and references there.

    Article  ADS  Google Scholar 

  3. (3)

    See,e.g.,L. Gross:Journ. Math. Phys.,5, 687 (1964).

    MATH  Article  ADS  Google Scholar 

  4. (4)

    See,e.g.,A. O. Barut: inDe Sitter and Conformal Groups and Their Applications, edited byA. O. Barut andW. E. Brittin (Boulder, Col., 1971), p. 337, and references there.

  5. (5)

    G. Mack andA. Salam:Ann. of Phys.,53, 174 (1969).

    MathSciNet  Article  ADS  Google Scholar 

  6. (6)

    T. D. Newton andE. P. Wigner:Rev. Mod. Phys.,21, 400 (1949).

    MATH  Article  ADS  Google Scholar 

  7. (7)

    F. Gürsey: talk given at theCoral Gables Conference 1967, published as Report CNAEM 53, Çekmece Nuclear Research Center, Istanbul (1968).

    Google Scholar 

  8. (8)

    M. H. L. Pryce:Proc. Roy. Soc., A195, 62 (1948);A. J. Kalnay: inProblems in the Foundations of Physics, edited byM. Bunge (Berlin, 1971), p. 93, and references there.

    MathSciNet  Article  ADS  Google Scholar 

  9. (9)

    An operator is called « symmetric » if it is equal to its adjoint on their common domain of definition. A symmetric operator is called «self-adjoint » if its maximal domain of symmetry agrees with that of its adjoint. For more detail, see,e.g.,T. F. Jordan:Linear Operators for Quantum Mechanics (New York, N. Y., 1969), p. 30.

  10. (10)

    S. Adler andR. Dashen:Current Algebra (New York, N. Y., 1968), p. 7.

  11. (11)

    F. Rellich:Gött. Nachr., 107 (1946);J. Dixmier:Compositio Math.,13, 263 (1958);C. Folias et al.:Acta Sci. Math. Szeged,21, 78 (1960).

  12. (12)

    This ansatz is related to the most general transformation law under conformal transformations of fields obtained in ref. (5). The expressions obtained there appear to be slightly more general than the ansatz (4) because of the presence of an additional operatorx μ to be added to the expression (4d) forK μ . This additionalx μ , however, if unequal to zero, leads to a reducible representation of the conformal Lie algebra, at least for finite-dimensionalx μ . This is a consequence of Lemma 1 of Sect.4 in ref. (5). Therefore, we takex μ to be zero here.

    MathSciNet  Article  ADS  Google Scholar 

  13. (13)

    In the case of noncompact groups, like the conformal group, it is possible, however, that there are inequivalent irreducible representations for which the Casimir operators have the same values. The representations studied here actually provide an example of this. In this connection, seeG. Racah: inSpringer Tracts in Modern Physics, Vol.37, (Berlin, 1965), p. 52;G. Racah:Rend. Lincei,8, 108 (1950).

    Google Scholar 

  14. (14)

    Y. Murai:Progr. Theor. Phys.,9, 147 (1953).

    MATH  MathSciNet  Article  ADS  Google Scholar 

  15. (15)

    When writing up this paper, we noticed thatT. Yao:Journ. Math. Phys.,12, 315 (1971) has also calculated the Casimir operators ofSO 4,2 for certain representations of this group. His calculation was done in a basis whose elements correspond to a decomposition of the given representation into irreducible representations of an iso-Poincaré subgroup. This approach is different from ours, and, therefore, his results although similar in character cannot easily be compared to ours nor do they seem to lead to an immediate physical interpretation.

    MATH  Article  ADS  Google Scholar 

  16. (16)

    H. A. Kastrup:Phys. Rev.,143, 1021 (1966).

    MathSciNet  Article  ADS  Google Scholar 

  17. (17)

    See,e.g.,L. O’Rafeartaigh:Group representations in mathematics and physics, inLectures Notes in Physics, Vol.6 (Berlin, 1970).

  18. (18)

    N. Jacobson:Lie Algebras (New York, N. Y., 1962), Theorem 6, p. 166.

  19. (19)

    See ref. (7).

    Google Scholar 

  20. (20)

    For similar considerations, see the investigation of the question of self-adjointness of the operatori(d/dx) inN. L. Achieser andI. M. Glasmann:Theorie der linearen Operatoren in Hilbert-Raum, Sect.49 (Berlin, 1954).

Download references

Author information

Affiliations

Authors

Corresponding authors

Correspondence to S. Lagu or H. Laue.

Additional information

Research supported by the NRC.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lagu, S., Laue, H. The conformal group, its casimir operators, and a four-position operator. Nuov Cim A 20, 217–231 (1974). https://doi.org/10.1007/BF02727449

Download citation