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

Abstract

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 $Ψ$(a representative on a given spinorial frame of a Dirac-Hestenes spinor field (DHSF)) the equation $F=Ψγ_{21} \simΨ$, 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 $Ψ$ 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.