A Five Dimensional Unification of the Vierbein and Electromagnetic Fields
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概要
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A Kaluza-Klein type unification of the vierbein and electromagnetic fields is developed on the basis of the principal fiber bundle over the four dimensional space-time with the group U(1). A parameter r which is directly related to the "length" of the fifth dimensional side of the bundle space in introduced. A spinor field φ on the bundle space is introduced, and a pullback ψ of φ by a cross section to the base space is identified with the Dirac field. A spinor field ψ(≠φ) on the bundle space is defined from φ, and its five dimensional Lagrangian density which leads to the Dirac equation for ψ is given. The vierbein fields and electromagnetic vector potential are unified into the "funfbein" fields on the bundle space. The most general Lagrangian density of the funfbein fields, which is a function of these fields and of their first derivatives and is quadratic in the derivatives, is given. It has four parameters in addition to r and is reduced to a linear sum of the Lagrangian densities of the vierbein fields and of the electromagnetic fields. Restrictions on the parameters imposed by experimental data and by a physical requirement are given. We discuss also a Lagrangian density which is reducible to the Einstein-Maxwell Lagrangian density. We can safely say that the bundle space is a five dimensional Lorentzian manifold and that the length of its fifth dimensional side is of the order of the Planck length or less. Any Dirac field defined as a pullback of the spinor field on the bundle space is expected to have an electric dipole moment with the magnitude φc√<x/8>, if the gravitational and electromagnetic fields are described by the Einstein-Maxwell theory.
- 理論物理学刊行会の論文
- 1982-06-25
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