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Prog. Theor. Phys. Vol. 46 No. 2 (1971) pp. 634-658

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A Quantum Theory of the Non-Linear Realization of a Group on Its Sub-Group

Takao Okabayashi

Depaartment of Physics, University of Tokyo, Tokyo

(Received February 8, 1971)

Abstract:

A quantum theory of the non-linear realization is proposed for the groups with a sub-group which can be realized non-linearly by a single self-conjugate preferred field. In the non-linear realization of a group on its sub-group, the displacement operators in Minkowski space can be written down at once as a sum of two terms. Not only the coefficients of the two terms but also the equal-time commutation relations between field operators and the equation of motion can be determined by the property of the displacement operators. Furthermore, all of them can be proved to be covariant under the internal group. Thus we can construct a quantum theory of the non-linear realization in a closed form without referring to the Lagrangian formalism or the variation method, where it is difficult to maintain the order of the operators correctly throughout all steps of manipulations. But it is argued that the Fock space cannot be spanned without violating the symmetry. Our method can be applied to any non-linear system once the energy-momentum tensor is specified. The difference between non-linear systems and linear ones and that between the Lagrangian formalism and ours are also discussed.


URL : http://ptp.ipap.jp/link?PTP/46/634/
DOI : 10.1143/PTP.46.634

[ Full Text PDF : FREE ACCESS (1477K) ] Citation:

References:

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Citing Article(s) :

  1. 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
  2. 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
  3. 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
  4. 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
  5. Progress of Theoretical Physics Vol. 50 No. 1 (1973) pp. 277-289 :
    Quantization for Non-Linear Lagrangian
    Teruya Ohtani
  6. 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
  7. Progress of Theoretical Physics Vol. 51 No. 2 (1974) pp. 592-599 :
    A Note on Covariant Derivatives in the Quantum Field Theory
    Takao Okabayashi
  8. Progress of Theoretical Physics Vol. 52 No. 5 (1974) pp. 1687-1701 :
    A Quantum-Theoretical Lagrangian Formalism for Quasi-Linear Field Theories. I
    Takao Okabayashi and Hiroyuki Kikugawa
  9. 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
  10. Progress of Theoretical Physics Vol. 54 No. 3 (1975) pp. 878-892 :
    On Subsidiary Condition in Quantum Theory of Non-Abelian Gauge Fields
    Takao Okabayashi and Yasuo Yoshihuku
  11. Progress of Theoretical Physics Vol. 58 No. 3 (1977) pp. 943-958 :
    Local Gauge Invariance of Non-Abelian Gauge Field Theory
    Takao Okabayashi and Norio Nakagawa

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