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Prog. Theor. Phys. Vol. 40 No. 5 (1968) pp. 1159-1180

Theory of Fields in Finsler Spaces. I

Yoshirô Takano

Department of Physics, Yokohama National University, Yokohama

(Received May 13, 1968)

Abstract:

The theory of fields in Finsler spaces, where the fields have internal freedom, is developed. In particular, the spinor field is discussed in detail and it is found necessary to introduce the gauge fields into the equation of the spinor field. The generalizations of Finsler spaces are also attempted.

URL : http://ptp.ipap.jp/link?PTP/40/1159/
DOI : 10.1143/PTP.40.1159

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References:

1. H. Yukawa, Phys. Rev. 77 (1950), 219[APS]; ibid. 80 (1950), 1047[APS].
2. H. Yukawa, Phys. Rev. 91 (1953), 415[APS]; ibid. 91 (1953), 416[APS]; Prog. Theor. Phys. 16 (1956), 688[PTP]; ibid. 31 (1964), 1167[PTP]; Rev. Mod. Phys. 29 (1957), 213[APS].
   Y. Katayama and H. Yukawa, Prog. Theor. Phys. 32 (1964), 366[PTP].
3. T. Takabayasi, Nuovo Cim. 33 (1964), 668; Prog. Theor. Phys. 34 (1965), 124[PTP].
   Y. Katayama, R. Yamada and H. Yukawa, Prog. Theor. Phys. 33 (1965), 541[PTP].
4. Finsler spaces may be regarded as the superposition of Riemann spaces corresponding to the various values of the directional variables.
   Refer to Y. Takano, Prog. Theor. Phys. 26 (1961), 304[PTP]; ibid. 27 (1962), 212[PTP]; ibid. 38 (1967), 1185[PTP]; ibid. 38 (1967), 1186[PTP].
5. J. I. Horváth, Nuovo Cim. Suppl. 9 (1958), 444; Acta Phys. et Chem. Szeged 7 (1961), 3; ibid. 9 (1963), 3.
6. Y. Takano, Prog. Theor. Phys. 32 (1964), 365[PTP].
7. References concerning the spinor field in Riemann spaces are shown in the following papers:
   W. L. Bade and H. Jehle, Rev. Mod. Phys. 25 (1953), 714[APS].
   A. Peres, Nuovo Cim. Suppl. 24 (1962), 389.
   See especially F. J. Belinfante, Physica 7 (1940), 305, [CrossRef]
   and H. Takeno, Tensor 7 (1957), 143; ibid. 7 (1957), 160; ibid. 8 (1958), 21.
8. J. Rayski, Nuovo Cim. 10 (1953), 1729.
9. C. N. Yang and R. L. Mills, Phys. Rev. 96 (1954), 191[APS].
   R. Utiyama, Phys. Rev. 101 (1956), 1597[APS].
   J. J. Sakurai, Ann. of Phys. 11 (1960), 1[CrossRef].
   G. R. Allcock, Nuovo Cim. 28 (1963), 858.
10. As to the classification of elementary particles by means of SU(3), see
    R. E. Behrends, J. Dreitlein, C. Fronsdal and W. Lee, Rev. Mod. Phys. 34 (1962), 1[APS].
11. As to Cartan spaces, refer to E. Cartan, Les espace métriques fondés sur la notion d'aire, Actualités 72 (Hermann, 1933).
12. S. Deser, Rev. Mod. Phys. 29 (1957), 417[APS].
13. As to Finsler geometry, refer to the following books:
    P. Finsler, Über Kurven und Flächen in allgemeiner Räumen, Dissertation, Göttingen (1918).
    E. Cartan, Les espace de Finsler, Actualités 79 (Hermann, 1934); Abh. aus dem Seminar für Vektor- und Tensoranalysis 4 (1937), 70.
    H. Rund, The Differential Geometry of Finsler Spaces (Springer, 1959).
    M. Matsumoto and T. Okada, “Connections in Finsler Spaces”, Seminar in Differential Geometry 4, Kyoto University (1965).
14. L. P. Eisenhart, Riemannian Geometry (Princeton, 1925).

Citing Article(s) :

1. Progress of Theoretical Physics Vol. 44 No. 1 (1970) pp. 272-283 :

   Quantum Field Theory of Complex Mass

   Hiroshi Yamamoto

2. Progress of Theoretical Physics Vol. 44 No. 4 (1970) pp. 1090-1091 :

   A Possible Model of a Divergenceless Nonlocal Quantum Field Theory

   D. F. Gallaher

3. Progress of Theoretical Physics Vol. 62 No. 1 (1979) pp. 295-296 :

   Indefinite Finsler Type Metrics and Their Peculiar Properties

   Hirohisa Ishikawa

4. Progress of Theoretical Physics Vol. 65 No. 6 (1981) pp. 2075-2078 :

   Some Structurological Features Induced by the Internal Variable in the Theory of Fields in Finsler Spaces

   Satoshi Ikeda

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