On the "equivalence" of the Maxwell and Dirac equations

TL;DR

The paper demonstrates that Maxwell's and Dirac's equations are fundamentally incompatible for equivalence, due to differences in their mathematical structures and physical implications, especially regarding spinor representations.

Contribution

It clarifies that Maxwell's equations cannot be reformulated as Dirac spinors, highlighting fundamental differences in their mathematical and physical structures.

Findings

Maxwell's spinor form depends on three complex components

Dirac's equation involves a four-component spinor with complex structure

No physically meaningful transformation exists between 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 \psi in the representation \vec{F} = \psi \vec{u} \bar{\psi} 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.