## 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.