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Prog. Theor. Phys. Vol. 47 No. 1 (1972) pp. 293-303

[ Full Text PDF : FREE ACCESS (690K) ]

A Note on the Interaction Representation for the Non-Linear Field of the Chiral Dynamics Type

Takao Okabayashi, Takashi Sasaki and Koji Yoshikawa

Department of Physics, University of Tokyo, Tokyo

(Received August 12, 1971)

Abstract:

Starting from the energy operator in the Heisenberg representation which is invariant under an internal group in the sense of the non-linear realization of a group on its subgroup, the Hamiltonian density in the interaction representation is derived by the most elementary method that is convenient to keep the order of operators. Then the integrability condition of the Tomonaga-Schwinger equation is confirmed, and the applicability of Matthew's theorem to our system is discussed. The normal dependent part of the Hamiltonian density is composed of two parts: One is necessary to make the normal independent part integrable, and the other dependes on the order of operators in the energy operator. Provided the order of operators in the energy operator is symmetrized, the second part does not come out and Matthew's theorem does hold. The connection of the order of operators and the invariance of the energy operator under the internal group is also discussed.


URL : http://ptp.ipap.jp/link?PTP/47/293/
DOI : 10.1143/PTP.47.293

[ Full Text PDF : FREE ACCESS (690K) ] Citation:

References:

  1. T. Okabayashi, Prog. Theor. Phys. 46 (1971), 634[PTP].
    This article will be cited as article I hereafter.
  2. R. Courant and K. O. Friedlichs, Supersonic Flow and Shock Waves (Interscience Pub., New York, 1948).
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  11. The T* convention was used in analyzing the S-matrix for non-linear fields in the following articles:
    K. Nishijima and T. Watanabe, Prog. Theor. Phys. 45 (1971), 949[PTP].
    T. Watanabe, Tokyo Univ. Prep. UT-109 July (1971).
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  15. S. N. Gupta, Proc. Phys. Soc. A 63 (1650), 681.
    K. Bleuler, Helv. Phys. Acta 23 (1950), 567.
    Such a modification of the Lorentz condition was proposed independently also by one of the authors (T. O.), Soryushiron Kenkyu (mimeographed circular in Japanese) 2 (1950), 49; ibid. 2 (1950), 53.

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 47 No. 5 (1972) pp. 1714-1721 :
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  3. Progress of Theoretical Physics Vol. 50 No. 2 (1973) pp. 661-679 :
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  5. Progress of Theoretical Physics Vol. 52 No. 5 (1974) pp. 1687-1701 :
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  6. Progress of Theoretical Physics Vol. 52 No. 6 (1974) pp. 1953-1970 :
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  7. Progress of Theoretical Physics Vol. 54 No. 3 (1975) pp. 878-892 :
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  8. Progress of Theoretical Physics Vol. 55 No. 4 (1976) pp. 1276-1287 :
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  9. Progress of Theoretical Physics Vol. 58 No. 3 (1977) pp. 943-958 :
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  10. Progress of Theoretical Physics Vol. 62 No. 1 (1979) pp. 201-213 :
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