Equations of Motion in the Inhomogeneous SL(6, C) Space and a Model which Generates the Lorentz World

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The Dirac equation, which is invariant with respect to the translation in a 72-dimensional space L_<72> and to the SL(6, C) group and the parity reflection, is introduced. It is shown that the equation is a set of homogeneous and simultaneous equations of the hyperbolic type. Therefore there does not appear such a ghost as has an imaginary frequency. Some difficulties, which will be brought about with the generalization of the space dimmension, are illustrated. To overcome the difficulties we introduce a gauge theoretical formulation of the 72 translational invariances. It is shown that there is a couple of gauge fields without which the particle can never propagate, that is, the propagation of the particle is induced by the interaction with the gauge fields introduced. It is also shown that there exists a dissymmetric solution to the equation of motion for the gauge fields. This solution makes the particle to propagate only on a restricted manifold in L_<72>, which is regarded as the Lorentz space. By this argument the physical foundation of the non-covariant equation of motion, introduced by Salam et al. and Sakita and Wali, is obtained. Finally we show that one of the gauge fields is the gravitational field.
理論物理学刊行会の論文
1965-10-25