Torsion, magnetic monopoles and Faraday's law via a variational principle

Abstract

Even though Faraday's Law is a dynamical law that describes how changing E and B fields influence each other, by introducing a vector potential Aμ according to Fμν = ∂νAν - ∂νAμ Faraday's Law is satisfied kinematically, with the relation (-g)-1/2μνστ ∇νFστ = 0 holding on every path in a variational procedure or path integral. In a space with torsion Qαβγ the axial vector Sμ = (-g)1/2μαβγQαβγ serves as a chiral analog of Aμ, and via variation with respect to Sμ one can derive Faraday's Law dynamically as a stationarity condition. With Sμ serving as an axial potential one is able to introduce magnetic monopoles without Sμ needing to be singular or have a non-trivial topology. Our analysis permits torsion and magnetic monopoles to be intrinsically Grassmann, which could explain why they have never been detected. Our procedure permits us to both construct a Weyl geometry in which Aμ is metricated and then convert it into a standard Riemannian geometry.