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Electrodynamics on a principal bundle III

By Eitan

Suppose we had a principal U(1)-bundle \pi:P\to M with a connection \omega with curvature \Omega.

The Lie algebra \mathfrak{u}(1) is just the set of imaginary numbers i\mathbb{R} with trivial Lie bracket {[},{]}=0. The local potential is a real-valued 1-form A_{U} defined by \omega_{U}=iA_{U}. The local field strength F_{U} is defined by \Omega_{U}=iF_{U}.

A change of gauge is given by g_{UV}=e^{i\lambda} with \lambda:U\cap V\to\mathbb{R}. We see that local connections are related by \omega_{V}=e^{-i\lambda}\omega_{U}e^{i\lambda}+e^{-i\lambda}de^{i\lambda}=\omega_{U}+id\lambda, so that local potentials are related by A_{V}=A_{U}+d\lambda. Local curvatures are related by \Omega_{V}=e^{-i\lambda}\Omega_{U}e^{i\lambda}=\Omega_{U}, so that local field strengths are related by F_{U}=F_{V}. This means that the field strength is globally defined on M.

By the Bianchi identity we have d\Omega=0 so dF=0, so the homogeneous Maxwell equation comes along for free. We can get the inhomogeneous Maxwell equation by requiring that d*F=*J.

Now, consider the action U(1) on \mathbb{C} given by multiplication e^{i\lambda}z. Associated to our principal U(1) bundle we get a vector bundle with fiber \mathbb{C} with an induced connection \nabla locally given by \nabla=d+\omega_U=d+iA_U. We will write sections of the associated bundle as \psi. We can define the d’Alembert operator \square=*\nabla*\nabla+\nabla*\nabla*. If we require the Klein-Gordon equation, \square\psi=m^2\psi, then we have a theory of a charged spin-0 particle coupled to electromagnetism.

In order to couple electromagnetism to more interesting particles like Dirac’s electron, we need to incorporate spin somehow.

This entry was posted on August 4, 2008 at 3:05 PM and is filed under Gauge theory, Geometry. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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