2014•Asian Journal of Fuzzy and Applied MathematicsRequires access

Tensor Formulation of Dirac Equation through Divisors

Sylvestre Bulikunzira

Open publisher page 6 citations

Abstract

In previous works, spinor Dirac equation for half-spin particle has been written in tensor form, in the form of non-linear Maxwell's like equations through two complex isotropic vectors F=E+iH and F'=E'+iH' . It has been proved, that the fields E and H have the same properties as those of the electric and magnetic fields, which are components of a single electromagnetic field. In particular, the solution of these non-linear equations for free particle as well fulfils Maxwell's equations for vacuum (with zero at the right side). This gives us the right to interpret electron field as a system of two electromagnetic fields ( E,H ) and ( E',H' ),  propagating with the phase velocity equal to the velocity of light and the velocity of the particle is the group velocity. In the present work, we shall write these non-linear Maxwell's like equations, representing Dirac equation in covariant form through divisors. This form directly reveals their relativistic invariance and open the possibilities to future generalizations.

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What this paper is about

In previous works, spinor Dirac equation for half-spin particle has been written in tensor form, in the form of non-linear Maxwell's like equations through two complex isotropic vectors F=E+iH and F'=E'+iH' . It has been proved, that the fields E and H have the same properties as those of the electric and magnetic fields, which are components of a single electromagnetic field. In particular, the solution of these non-linear equations for free particle as well fulfils Maxwell's equations for vacuum (with zero at the right side). This gives us the right to interpret electron field as a system of two electromagnetic fields ( E,H ) and ( E',H' ),  propagating with the phase velocity equal to the velocity of light and the velocity of the particle is the group velocity. In the present work, we shall write these non-linear Maxwell's like equations, representing Dirac equation in covariant form through divisors. This form directly reveals their relativistic invariance and open the possibilities to future generalizations.

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Available abstract

In previous works, spinor Dirac equation for half-spin particle has been written in tensor form, in the form of non-linear Maxwell's like equations through two complex isotropic vectors F=E+iH and F'=E'+iH' . It has been proved, that the fields E and H have the same properties as those of the electric and magnetic fields, which are components of a single electromagnetic field. In particular, the solution of these non-linear equations for free particle as well fulfils Maxwell's equations for vacuum (with zero at the right side). This gives us the right to interpret electron field as a system of two electromagnetic fields ( E,H ) and ( E',H' ),  propagating with the phase velocity equal to the velocity of light and the velocity of the particle is the group velocity. In the present work, we shall write these non-linear Maxwell's like equations, representing Dirac equation in covariant form through divisors. This form directly reveals their relativistic invariance and open the possibilities to future generalizations.

Key concepts: Dirac equation, Maxwell's equations, Electromagnetic field, Electromagnetic tensor, Tensor (intrinsic definition), Physics, Mathematical physics, Covariant transformation

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