On the "equivalence" of the Maxwell and Dirac equations

Abstract

It is shown that Maxwell's equation cannot be put into a spinor form that is equivalent to Dirac's equation. First of all, the spinor ψin the representation \vec{F} = ψ\vec{u} \barψ of the electromagnetic field bivector depends on only three independent complex components whereas the Dirac spinor depends on four. Second, Dirac's equation implies a complex structure specific to spin 1/2 particles that has no counterpart in Maxwell's equation. This complex structure makes fermions essentially different from bosons and therefore insures that there is no physically meaningful way to transform Maxwell's and Dirac's equations into each other.