Electrodynamics on a Principal Bundle II

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.

Consider the matrix group O(1,3), i.e. matrices B such that B^{T}\eta B=\eta where \eta=diag(1,-1,-1,-1), or equivalently \eta(Bv,Bw)=\eta(v,w) for any events v,w in Minkowski spacetime. This group has 4 connected components coming from det(B)=\pm1 and B_{00}>0 or B_{00}<0. The component containing the identity is called the proper, orthochronous Lorentz group L=L_{+}^{\uparrow}. Physically it contains all rotations, and boosts (Lorentz tranformations) and so dim(L)=6.

We can cover L by the simply connected group SL(2,\mathbb{C}), i.e. 2\times2 complex matrices A with det(A)=1. First we identify Minkowski spacetime \mathbb{R}^{4} with the space of 2\times2 Hermitian matrices, i.e. matrices H such that \overline{H}^{T}=H, in such a way that if H is the Hermitian matrix identified with the event x then det(H)=|x|^{2}. Then we can define a covering map \Lambda:SL(2,\mathbb{C})\to L by identifying \Lambda(A)x with AH\overline{A}^{T}. We have that \Lambda(A)\in L since
|\Lambda(A)x|^{2}=det(AH\overline{A}^{T})=det(A)det(H)det(A)=det(H)=|x|^{2}. It can be shown that \Lambda is a 2-1 homomorphism of Lie groups.

Now, there are two important irreducible representations for SL(2,\mathbb{C}) on \mathbb{C}^{2}, the “spin \frac{1}{2}” representations given by multiplication A\left(\begin{array}{c} z_{1}\\ z_{2}\end{array}\right) and multiplication by the adjoint \overline{A}^{T}\left(\begin{array}{c} z_{1}\\z_{2}\end{array}\right). The Dirac representation is the direct sum of these representations \left(\begin{array}{cc}A& 0\\ 0&\overline{A}^{T}\end{array}\right) \left(\begin{array}{c}z_{1}\\z_{2}\\z_{3}\\z_{4}\end{array}\right).

Let \pi:FM\to M be the orthonormal frame bundle for spacetime. Its fibers F_{m}M are ordered orthonormal bases of T_{m}M, or equivalently isometries p:\mathbb{R}^{4}\to T_{m}M. There is a right action of O(1,3) given by right composition pB which makes the frame bundle an O(1,3)-bundle. We say that M is space and time orientable iff FM has 4 components and a choice of component FM_{0} is a space and time orientation. Then the restriction \pi:FM_{0}\to M is an L-bundle.

The solder form is an \mathbb{R}^{4}-valued 1-form \phi on FM_{0} given by \phi_{p}(X)=p^{-1}(\pi_{*}(X)). The torsion of a connection \theta on FM_{0} is \Theta=d\phi+\theta\wedge\phi. It turns out that there is a unique connection whose torsion is \Theta=0. This is the Levi-Civita connection \theta.

A spin structure on M is a manifold SM and a smooth map \lambda:SM\to FM_{0} such that \pi\circ\lambda:SM\to M is an SL(2,\mathbb{C})-bundle with \lambda(pA)=\lambda(p)\Lambda(A). We can define a connection \tilde{\theta} on SM by \tilde{\theta}=\Lambda_{*}^{-1}\lambda^{*}\theta where \Lambda_{*} is the isomorphism of Lie algebras induced by \Lambda:SL(2,\mathbb{C})\to L.

Now consider sections \psi of the vector bundle associated to SM by the Dirac representation. Dirac’s idea was to introduce an operator \not\hspace{-4pt}D such that \not\hspace{-4pt}D^{2}=\square, i.e. the Dirac operator is the “square root” of the d’Alembert operator. A full understanding of the Dirac operator requires Clifford algebras, i.e the algebra generated over Minkowski space modulo v^{2}=\eta(v,v). It turns out that the smallest representation \gamma of this Clifford algebra is 4-dimensional which is why we need a 4-dimensional representation of SL(2,\mathbb{C}) as well. Then we can define the Dirac operator as \not\hspace{-4pt}D=\eta(\gamma,\nabla) where \nabla is the connection associated to \tilde{\theta} and we inner product them somehow.

In more detail for the d’Alembertian on Minkowski spacetime, \square=\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-\frac{\partial^{2}}{\partial z^{2}}, define

\not\hspace{-4pt}D=\left(\begin{array}{cccc} 1& 0& 0& 0\\ 0 & 1& 0& 0\\ 0 & 0& -1& 0\\ 0 & 0& 0& -1\end{array}\right)\frac{\partial}{\partial t}+\left(\begin{array}{cccc} 0& 0& 0& 1\\ 0 & 0& 1& 0\\ 0 & -1& 0& 0\\ -1& 0& 0& 0\end{array}\right)\frac{\partial}{\partial x}

+\left(\begin{array}{cccc} 0& 0& 0& -i\\ 0 & 0& i& 0\\ 0 & i& 0& 0\\ -i& 0& 0& 0\end{array}\right)\frac{\partial}{\partial y}+\left(\begin{array}{cccc}0& 0& 1& 0\\ 0 & 0& 0& -1\\ -1& 0& 0& 0\\ 0 & 1& 0& 0\end{array}\right)\frac{\partial}{\partial z}

We can work out that \not\hspace{-4pt}D^{2}=\square.

Then we demand that the Dirac equation holds, \not\hspace{-4pt}D\psi=m\psi. This gives us a theory of a spin-\frac{1}{2} particle, an electron or positron, but we have not yet coupled it to electromagnetism.

Right now, our notion of an electron is that it is a field which takes its values in a representation of the spin group Spin(1,3)=SL(2,\mathbb{C}). In order to couple to the electromagnetic field, we will rather think of the electron taking its values in a representation of the charged spin group Spin_C(1,3)=U(1)\times SL(2,\mathbb{C})/(\mathbb{Z}/2).

We can splice a G_{1}-bundle \pi_{1}:P_{1}\to M with a G_{2}-bundle \pi_{2}:P_{2}\to M. Define P=\{(p_{1},p_{2})\in P_{1}\times P_{2}:\pi_{1}(p_{1})=\pi_{2}(p_{2})\} and \pi:P\to M by \pi(p_{1},p_{2})=\pi_{1}(p_{1})=\pi_{2}(p_{2}). This is a G_{1}\times G_{2}-bundle with (p_{1},p_{2})(g_{1},g_{2})=(p_{1}g_{1},p_{2}g_{2}). Given connections \omega_{1},\omega_{2} on P_{1},P_{2}, we can define a connection \omega on P by \omega=\pi^{1*}\omega_{1}\oplus\pi^{2*}\omega_{2} with \pi^{i}:P\to P_{i} given by \pi^{i}(p_{1},p_{2})=p_{i}.

Splice together our U(1)-bundle P with SM and also splice \omega with \tilde{\theta}. Consider the representation of U(1)\times SL(2,\mathbb{C}) on \mathbb{C}^{4} given by combining the Dirac representation with multiplication by e^{i\lambda}. This structure is \mathbb{Z}/2-invariant so defines a Spin_C(1,3) -bundle. We get an associated vector bundle with an associated connection and Dirac operator \not\hspace{-4pt}D. A charged electron coupled to electromagnetism is then a section \psi for which the Dirac equation \not\hspace{-4pt}D\psi=m\psi holds.

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