The Relation between Maxwell, Dirac and the Seiberg-Witten Equations

In this paper we discuss some unusual and unsuspected relations between Maxwell, Dirac and the Seiberg-Witten equations. First we investigate what is now known as the 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) the equation $F=\psi\gamma_{21}\tilde{\psi}$, where $F$ is a given electromagnetic field. Such non trivial 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 of the generalized Maxwell equations. One is the generalized Hertz potential field equation (which we discuss in detail) associated with Maxwell theory and the other is a (non linear) equation (of the generalized Maxwell type) satisfied by the 2-form field part of a Dirac-Hestenes spinor field 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. This finding may eventually account for the recent claims that the electron has been splited into two electrinos. 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. A suggestive physical interpretation for those equations is also given.