HOME    |    BROWSE    |    SEARCH    |    SUBSCRIPTION   |    SUBMISSION    |    REGISTRATION

Quick Search:
Author: Title/Abstract: Vol./No: Page:
PTP Online Top > Volume 70 (1983) > Number 6
| Next Article >

Prog. Theor. Phys. Vol. 70 No. 6 (1983) pp. 1648-1665

[ Full Text PDF : FREE ACCESS (743K) ]

Quaternionic Formulation of Dirac Theory in Special and General Relativity

Katsusada Morita

Department of Physics, Nagoya University, Nagoya 464

(Received August 22, 1983)

Abstract:

It is shown that Dirac theory can be formulated also in the quaternion form at the Lagrangian level. Crucial to our formulation is that a spinor-quaternion, which may be regarded as 2 ×2 matrix-valued complex fields, is decomposed into left ideals in a Lorentz invariant manner. The new formalism is applied to gauge theories. Assuming that not only the space-time symmetry but also the internal symmetry are described by Hamilton's quaternions, we are led to the previous algebraic formulation of the standard SU (3) ×SU(2)L ×SU(2)R ×U(1) theory with the required family structure of quarks and leptons including spin. The gauge principle with respect to the space-time symmetry, on the other hand, naturally leads to the quaternionic formulation of the Poinca'e gauge theory for spin-1/2 particles. The most general gravitational Lagrangian of Hayashi is presented in the quaternion form without referring to the principle of irreducibility.


URL : http://ptp.ipap.jp/link?PTP/70/1648/
DOI : 10.1143/PTP.70.1648

[ Full Text PDF : FREE ACCESS (743K) ] Citation:

See Also:

References:

  1. F. Klein, Physik. Zeitschr. 12 (1911), 17.
  2. O. F. Fischer, Philos. Mag. 27 (1939), 375; ibid. 30 (1940), 135.
  3. C. Lanczos, Z. Phys. 57 (1929), 447; ibid. 57 (1929), 474.
    See, also, A. W. Conway, Proc. R. Soc. London A 162 (1937), 135.
  4. P. Rastall, Rev. Mod. Phys. 36 (1964), 820[APS].
    A related work was also done by P. G. Bergmann, Phys. Rev. 107 (1957), 624[APS].
  5. In another context of special relativity and electromangetism, see E. Mignani, Proceedings of the First Session of the Interdisciplinary Seminars on Tachyons and Related Topics, ed. E. Recami (North-Holland Pub. Comp., 1978), p. 67 and references therein.
    See, also K. Imaeda, Nuovo Cim. B 32 (1976), 138.
  6. K. Morita, Prog. Theor. Phys. 67 (1982), 1860[PTP].
  7. K. Morita, Prog. Theor. Phys. 68 (1982), 2159[PTP].
  8. S. Weinberg, Phys. Rev. Lett. 19 (1967), 1264[APS].
    A. Salam, Weak and Electromagnetic Interactions, Elementary Particle Theory, ed. N. Svartholm (John Wiley and Sons, Inc., New York, London, Sydney, 1968), p. 367.
    S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D 2 (1970), 1285[APS].
    M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973), 652[PTP].
  9. S. Weinberg, Phys. Rev. Lett. 29 (1972), 1698[APS].
    J. C. Pati and A. Salam, Phys. Rev. D 10 (1974), 275[APS].
    R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11 (1975), 566[APS]; ibid. 11 (1975), 2588[APS].
    G. Senjanović and R. N. Mohapatra, Phys. Rev. D 12 (1975), 1502[APS].
    E. Ma, Nucl. Phys. B 121 (1976), 421[CrossRef].
    A. De Rújula, H. Georgi and S. L. Glashow, Ann. of Phys. 109 (1977), 242[CrossRef].
    R. N. Mohapatra and D. P. Sidhu, Phys. Rev. D 16 (1977), 2843[APS].
    M. S. Mani, J. C. Pati and A. Salam, Phys. Rev. D 17 (1978), 2510[APS].
    I. Liede, J. Maalampi and M. Roos, Nucl. Phys. B 146 (1978), 157[CrossRef].
    G. Senjanović, Nucl. Phys. B 153 (1979), 334.
  10. As for QCD, see, for instance, W. Marciano and H. Pagels, Phys. Rep. C 36 (1978), 138.
  11. R. Utiyama, Phys. Rev. 101 (1956), 1597[APS].
  12. T. W. Kibble, J. Math. Phys. 2 (1961), 212[CrossRef].
  13. K. Hayashi and T. Nakano, Prog. Theor. Phys. 38 (1967), 491[PTP].
  14. K. Hayashi, Prog. Theor. Phys. 39 (1968), 494[PTP].
  15. K. Hayashi and A. Bregman, Ann. of Phys. 75 (1973), 562[CrossRef].
  16. F. W. Hehl, P. Von der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys. 48 (1976), 393[APS].
  17. K. Hayashi and T. Shirafuji, Prog. Theor. Phys. 64 (1980), 866[PTP]; ibid. 64 (1980), 883[PTP].
  18. C. N. Yang and R. L. Mills, Phys. Rev. 96 (1954), 191[APS].
  19. K. Hayashi and T. Shirafuji, Phys. Rev. D 19 (1979), 3524[APS].
  20. D. Møller, K. Dan. Vidensk. Mat. -Fys. Skr. 1 (1961), 1.
  21. K. Hayashi and T. Shirafuji, Prog. Theor. Phys. 57 (1977), 302[PTP].
  22. R. Utiyama and B. S. Dewitt, J. Math. Phys. 3 (1962), 608[CrossRef].
    K. S. Stelle, Phys. Rev. D 16 (1977), 953[APS].

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 72 No. 5 (1984) pp. 1056-1059 :
    Quaternions and Simple D=4 Supergravity
    Katsusada Morita
  2. Progress of Theoretical Physics Vol. 75 No. 1 (1986) pp. 220-223 :
    A Role of Quaternions in the Dirac Theory
    Katsusada Morita
  3. Progress of Theoretical Physics Vol. 85 No. 1 (1991) pp. 157-168 :
    Quaternion Gauge Theory of Dyonic Fields
    P. S. Bisht, O. P. S. Negi and B. S. Rajput
  4. Progress of Theoretical Physics Vol. 90 No. 1 (1993) pp. 219-236 :
    Quaternions and Non-Commutative Geometry
    Katsusada Morita
  5. Progress of Theoretical Physics Vol. 117 No. 3 (2007) pp. 501-532 :
    Quaternions, Lorentz Group and the Dirac Theory
    Katsusada Morita
  6. Progress of Theoretical Physics Vol. 128 No. 6 (2012) pp. 1107-1139 :
    Quaternionic Variational Formalism for General Relativity in Riemann and Riemann-Cartan Space-Times
    Katsusada Morita

[PTP HOME] [BROWSE] [SEARCH] [SUBSCRIPTION] [SUBMISSION] [REGISTRATION]

Copyright © Progress of Theoretical Physics 2013 All rights reserved.