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Prog. Theor. Phys. Vol. 43 No. 5 (1970) pp. 1343-1363

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Formal Theory of Non-Linear Realization of a Group on Its Sub-group*

Takao Okabayashi and Tadashi Watanabe

Department of Physics, University of Tokyo, Tokyo

(Received October 17, 1969)

Abstract:

Groups with a sub-group are classified from the point of view of the theory of non-linear realization of a group G on its sub-group H. Starting from Lie's differential equations, we develop a general prescription for the realization by taking two typical cases; the one is the case of a self-conjugate preferred (or boost) field and the other a conjugate pair of preferred firlds case. It is shown that the realization is possible only if the homogeneous space G/H is symmetric. The formulation and almost all te results are completely general and the properties of individual groups are confined to some constants which appear in several identities on the generators and structure constants of the sub-group. The theory is applied to the groups SU(2) ×SU(2) and SU(3).


URL : http://ptp.ipap.jp/link?PTP/43/1343/
DOI : 10.1143/PTP.43.1343


* For an editorial reason, this paper was separated into two parts. The first part is published in Vol. 43 No. 4. (See Ref. 1.)

[ Full Text PDF : FREE ACCESS (1045K) ] Citation:

References:

  1. The first part of this paper; T. Okabayashi and T. Watanabe, Prog. Theor. Phys. 43 (1970), 1085[PTP].
    J. Schwinger, Phys. Lett. B 24 (1967), 473[CrossRef]; Phys. Rev. 167 (1968), 1432[APS].
  2. S. Weinberg, Phys. Rev. 166 (1968), 1568[APS].
  3. For instance, K. Hiida, Y. Ohnuki and Y. Yamaguchi, Prog. Theor. Phys. Suppl. Extra Number (1968), 337[PTP].
  4. S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969), 2239[APS].
    C. G. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969), 2247[APS].
  5. A. Salam and J. Strathdee, Phys. Rev. 184 (1969), 1750, [APS] ibid. 184 (1969), 1760[APS].
  6. Kh. Khristov and D. Stoyanov, Preprint of Inst. Theor. Phys., Kiev, ITP/68/56.
  7. See, for instance, G. Racah, “Group Theory and Spectroscopy”, Lecture at the Institute for Advanced Study 1951.
  8. E. Cartan, Thèse Paris 1894: Bull. Soc. Math. de France 41 (1913), 53.
  9. E. Cartan, Bull. Soc. Math. de France 54 (1926), 214; ibid. 55 (1927), 114.
    As to non-compact groups see M. Berger, Ann. Sci. Ecole Norm. Sup. (3) 74 (1957), 85.
  10. M. Goldhaber, Phys. Rev. 92 (1953), 1279[APS]; ibid. 101 (1956), 433[APS].
  11. S. Sakata, Prog. Theor. Phys. 16 (1956), 686[PTP].
  12. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961), 345[APS].
    T. Okabayashi, K. Tanabe and M. Yamada, Prog. Theor. Phys. 32 (1964), 619[PTP].

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 43 No. 4 (1970) pp. 1085-1104 :
    Formal Theory of Non-Linear Realization of a Group on Its Sub-group*
    Takao Okabayashi and Tadashi Watanabe
  2. Progress of Theoretical Physics Vol. 45 No. 1 (1971) pp. 243-250 :
    On the Gauge Fields in the Non-Linear Realization of a Group on Its Sub-Group
    Takao Okabayashi and Tadashi Watanabe
  3. Progress of Theoretical Physics Vol. 45 No. 1 (1971) pp. 251-262 :
    Currents in the Non-Linear Realization of a Group on Its Sub-Group
    Takao Okabayashi
  4. Progress of Theoretical Physics Vol. 46 No. 2 (1971) pp. 634-658 :
    A Quantum Theory of the Non-Linear Realization of a Group on Its Sub-Group
    Takao Okabayashi
  5. Progress of Theoretical Physics Vol. 47 No. 1 (1972) pp. 293-303 :
    A Note on the Interaction Representation for the Non-Linear Field of the Chiral Dynamics Type
    Takao Okabayashi, Takashi Sasaki and Koji Yoshikawa
  6. Progress of Theoretical Physics Vol. 47 No. 4 (1972) pp. 1343-1366 :
    A Model for the Broken SU(3) Symmetry Based on the Non-Linear Realization of the Group
    Takao Okabayashi
  7. Progress of Theoretical Physics Vol. 47 No. 5 (1972) pp. 1714-1721 :
    Breakdown of Symmetry in the Quantum Theory of the Non-Linear Realization of a Group on Its Sub-Group
    Takao Okabayashi
  8. Progress of Theoretical Physics Vol. 47 No. 6 (1972) pp. 2077-2084 :
    Covariant T* Products and Schwinger Terms
    Tadashi Watanabe
  9. Progress of Theoretical Physics Vol. 48 No. 4 (1972) pp. 1375-1394 :
    An Ambiguity in Particle Interpretation of Gauge Field of the Yang-Mills Type
    Takao Okabayashi
  10. Progress of Theoretical Physics Vol. 50 No. 2 (1973) pp. 661-679 :
    A Quantum Theory of a Conjugate Pair of Preferred Fields Interacting with a Spinor Field. I
    Takao Okabayashi
  11. Progress of Theoretical Physics Vol. 51 No. 2 (1974) pp. 592-599 :
    A Note on Covariant Derivatives in the Quantum Field Theory
    Takao Okabayashi
  12. Progress of Theoretical Physics Vol. 52 No. 6 (1974) pp. 1953-1970 :
    A Quantum-Theoretical Lagrangian Formalism for Quasi-Linear Field Theories. II
    Takao Okabayashi and Hiroyuki Kikugawa
  13. Progress of Theoretical Physics Vol. 62 No. 1 (1979) pp. 201-213 :
    T*-Product Convention for Quasi-Linear Systems and for Transverse Non-Abelian Gauge Fields
    Takao Okabayashi

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