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 $\mathbf{E}$ and $\mathbf{B}$ fields influence each other, by introducing a vector potential $A_{\mu }$ according to $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ Faraday's Law is satisfied kinematically, with the relation $(-g{)}^{-1/2}{\epsilon }^{\mu \nu \sigma \tau }{\nabla }_{\nu }{F}_{\sigma \tau }=0$ holding on every path in a variational procedure or path integral. In a space with torsion $Q_{\alpha \beta \gamma }$ the axial vector $S^{\mu }=(-g{)}^{1/2}{\epsilon }^{\mu \alpha \beta \gamma }{Q}_{\alpha \beta \gamma }$ serves as a chiral analog of $A_{\mu }$, and via variation with respect to $S_{\mu }$ one can derive Faraday's Law dynamically as a stationarity condition. With $S_{\mu }$ serving as an axial potential one is able to introduce magnetic monopoles without $S_{\mu }$ 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 $A_{\mu }$ is metricated and then convert it into a standard Riemannian geometry.

4 pages, revtex4. In this version the Weyl geometry connection is taken to be associated with an anti-Hermitian field $i{A}_{\mu }$ rather than with a Hermitian $A_{\mu }$. It is the anti-Hermitian connection that yields an electromagnetic $A_{\mu }$ that couples to a Dirac fermion in the standard minimally coupled $i{\partial }_{\mu }-A_{\mu }$ Hermitian way

References in corpus (1)

• Gravitational Analog of Faraday's Law via Torsion and a Metric with an Antisymmetric Part