  In this paper we discuss some unusual and unsuspected relations between
Maxwell, Dirac and the Seiberg-Witten equations. First we investigatethe
Maxwell-Dirac equivalence (MDE) of the first kind. Crucial to that proposed
equivalence is the possibility of solving for $\Psi$(a representative on a
given spinorial frame of a Dirac-Hestenes spinor field (DHSF)) the equation
$F=\Psi \gamma_{21} \sim{\Psi}$, where F is a given electromagnetic field. Such
task is presented in this paper and it permits to clarify some possible
objections to the MDE which claims that no MDE may exist, because F has six
(real) degrees of freedom and $\Psi$ has eight (real) degrees of freedom. Also,
we review the generalized Maxwell equation describing charges and monopoles.
The enterprise is worth even if there is no evidence until now for magnetic
monopoles, because there are at least two faithful field equations that have
the form that equations. One is the generalized Hertz potential field equation
associated with Maxwell theory and the other is a (non linear) equation
satisfied by the 2-form field, which is part of a representative of a DHSF that
solves the Dirac-Hestenes equation for a free electron. This is a new and
surprising result, which can also be called MDE of the second kind. It strongly
suggests that the electron is a composed system with more elementary
  "charges" of the electric and magnetic types. Finally, we use the MDE of the
first kind together with a reasonable hypothesis to give a derivation of the
famous Seiberg-Witten equations on Minkowski spacetime.
