Electrodynamics on a Principal Bundle I

I want to switch gears and talk about some mathematical physics. Actually, I’m going to cross-post some exposition I wrote for a gauge theory seminar that we held at Stony Brook.

Maxwell’s equations in relativistically covariant form are

\partial_{\mu}F^{\mu\nu}=J^{\nu}
\partial_{[\lambda}F_{\mu\nu]}=0

Since F_{\mu\nu}=-F_{\nu\mu} we can define a 2-form F=F_{\mu\nu}dx^{\mu}dx^{\nu}. We can also define a 1-form J=J_\mu dx^\mu. Then we can re-express Maxwell’s equations using exterior differentiation and the Hodge star.

d*F=*J
dF=0

The continuity equation d*J=0 then follows from the inhomogeneous Maxwell equation. We expect from the homogeneous Maxwell equation that F=dA. In fact this is only true locally. This means that for every event m in our spacetime M there is an open set U with m\in U\subset M and a 1-form A_U on U with F|_U=dA_U. This follows from Poincare’s lemma.

We cannot say the A exists globally. For instance if F=sin\phi d\phi d\theta, the area form of the unit sphere in spherical coordinates, then dF=cos\phi d\phi d\phi d\theta=0 since d\phi d\phi=0 by antisymmetry of wedge product of 1-forms. Also, taking \Sigma to be the unit sphere, we know that \int_\Sigma F=4\pi. However, by Stokes’ Theorem, if F=dA then \int_\Sigma F=\int_\Sigma dA=\int_{\partial \Sigma} A=0\neq 4\pi. So, we cannot have F=dA globally.

Physically we interpret this as a magnetic monopole with magnetic charge 4\pi and worldline, the time axis, r=0. Mathematically, what is happening is that the complement of the time axis has nontrivial topology. Specifically its second de Rham cohomology is nontrivial. Intuitively, there is a kind of 2-dimensional, spherical “hole” in the complement of the time axis.

In addition to being nonglobal, the potential A is defined only up to addition of a closed 1-form since d(A+d\lambda)=dA=F. We would like to find a global mathematical object corresponding to the potential which doesn’t depend on our “choice of gauge”. This is our motivation for understanding connections on principal bundles.

We will assume G is a group of matrices. A principal G-bundle is a smooth surjection of manifolds \pi:P\to M with a free transitive right action R of G on P such that \pi^{-1}\pi(p)=pG and for any m\in M there is an open set U with m\in U\subset M and a diffeomorphism T_U=\pi\times t_U:\pi^{-1}(U)\to U\times G called a “local trivialization” such that t_U(pg)=t_U(p)g. Local trivializations correspond to the physical notion of “choice of gauge”.

Intuitively, P is a manifold composed of copies of the group G parametrized by the base space M. A good example is the boundary of the Mobius strip which can be thought of as a \mathbb{Z}/2-bundle over S^1.

A useful notion is that of a local section \sigma_U:U\to P with U an open set with U\subset M such that \pi(\sigma_U(m))=m. It can be shown that there is a canonical 1-1 correspondence between local sections \sigma_U and local trivializations T_U.

Define transition functions g_{UV}:U\cap V\to G by g_{UV}(m)=t_U(p)t_V(p)^{-1} where \pi(p)=m. This is well defined since t_U(pg)t_V(pg)^{-1}=t_U(p)gg^{-1}t_V(p)^{-1}=t_U(p)t_V(p)^{-1}. Transition functions correspond to the physical notion of “change of gauge”. We can relate any two local sections by \sigma_V=\sigma_U g_{UV}.

Let \mathfrak{g} be the Lie algebra for G. A connection \omega is a \mathfrak{g}-valued 1-form on P such that if If X\in\mathfrak{g} and \tilde{X} is the tangent field on P given by \tilde{X}_{p}=\frac{d}{dt}pe^{tX}|_{t=0}, then \omega(\tilde{X})=X. Also we require that R(g)^{*}(\omega)=g^{-1}\omega g.

We define local connections on M by \omega_U=\sigma_U^*\omega. Local connections are related by \omega_{V}=g_{UV}^{-1}\omega_{U}g_{UV}+g_{UV}^{-1}dg_{UV}.

We define curvature \Omega=d\omega+\frac{1}{2}[\omega,\omega] meaning \Omega(X,Y)=d\omega(X,Y)+\frac{1}{2}[\omega(X),\omega(Y)]. We can define local curvature by \Omega_U=\sigma_U^*\Omega. Local curvatures are then related by \Omega_V=g_{UV}^{-1}\Omega_U g_{UV}. The Bianchi identity says d\Omega=[\omega,\Omega].

We are now in a position to define electrodynamics on a principal bundle.

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