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Electrodynamics on a Principal Bundle II

August 16, 2009

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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Electrodynamics on a Principal Bundle I

August 16, 2009

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 $d\Omega=[\omega,\Omega]$ .