Derivation of Maxwell-Like Equations from the Quaternionic Dirac's Equation

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Arbab, Arbab I.
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Expanding the ordinary Dirac’s equation, 1 c @ @t + ~α · ~∇ψ + imc ¯h ψ = 0, in quaternionic form has yielded Maxwell-like field equations. As in the Maxwell’s formulation, the particle fields are represented by a scalar, ψ0 and a vector ~ψ . The analogy with Maxwell’s equations requires that ψ0 = −cβ ~α · ~ψ , ~ED = c2~α × ~ψ , and ~BD = ~α ψ0 + cβ ~ψ . An alternative solution suggests that monopole-like behaviour accompanies Dirac’s field. In this formulation a field-like representation of Dirac’s particle is derived. It is shown that when the vector field of the particle, ~ψ , is normal to the vector ~α, Dirac’s field represents a medium with maximal conductivity. The energy flux (Poynting vector) of the Dirac’s fields is found to flow in opposite direction to the particle’s motion. An equivalent symmetrised Maxwell’s equations are introduced. A longitudinal (scalar) wave traveling at speed of light is found to accompany magnetic charges flow. This wave is not affected by presence of electric charges and currents.
Derivation, Maxwell, equations, quaternionic Dirac's equation