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The relation between Maxwell, Dirac, and the Seiberg-Witten equations. (English) Zbl 1035.81016
Summary: We discuss unsuspected relations between Maxwell, Dirac, and the Seiberg-Witten equations. First, we present 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 task is presented and it permits to clarify some objections to the MDE which claim 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 (nonlinear) 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 result which can also be called MDE of the second kind. 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 physical interpretation for those equations is proposed.

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81R25 Spinor and twistor methods applied to problems in quantum theory
15A66 Clifford algebras, spinors
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