Electromagnetism as a gauge theory

Maxwell's equations specify the div \(\nabla\cdot\) and the curl \(\nabla\times\) of \(\boldsymbol{E}\) and \(\boldsymbol{B}\). The equations involving the cross product also have a term of a time derivative \(\partial_t\) of the other field.

Both div and curl can be written as exact differential forms and Stokes' theorem may be applied in the integral form of these equations to obtain a macroscopic interpretation on surfaces (in the case of div) and loops (in the case of curl). These experiments are Gauss's flux theorem, Faraday's law of induction, and Ampère's circuital law.

The equations contain a positive constant \(c\) which they predict to be the speed of light, i.e. the speed with which the wavefronts of electromagnetic radiation travel. This becomes an issue: these equations are written in a specific coordinate system \(\boldsymbol{x}\). With a different coordinate system \(\boldsymbol{x'}\) whose axes are parallel but whose center is at \(\boldsymbol{v}t\), there is a contradiction that \(c = c + v\). In other words, they seem to require a preferred origin with which to measure velocities. We have no preferred point of origin in nature, and this led others to look for a formulation of the equations in some affine space; Minkowski spacetime is the answer.

There is yet another quantity, aside from the wavefront velocity, that arises in the solutions of these equations: the phase of the waves, a unit complex number \(e^{i\theta}\). These numbers with multiplication form the group \(U(1)\). The number \(\theta\) varies with time, but yet again we have discovered a preferred origin since \(\theta\) is an element of the vector space \(\mathbb{R}\). In nature we can only meaningfully measure the difference \(\Delta\theta\) between two measurements of \(\theta\), and there is no distinguished \(0\) phase. Principal \(G\)-bundles are spaces with a group action where the identity element of the group is "forgotten"¹; this is the exact geometric concept required.

The electromagnetic field, or more correctly its potential, then becomes a principal connection of the principal \(U(1)\)-bundle. However, this formalism is only required if we decide to couple electromagnetism to a matter field: after all, to measure phases we must use matter interactions.

We will quickly show how the theory is formulated as a gauge theory, and in the next sections we will attempt to explain all of the mathematics that we introduced.

Lying in the background of the solution \(\boldsymbol{E}\) and \(\boldsymbol{B}\) are the electric potential \(\phi\) and the magnetic potential \(\boldsymbol{A}\). These potentials may be used to obtain \(\boldsymbol{E}\) and \(\boldsymbol{B}\) by differentiation, but they are not unique. They have experimental significance, as they can be measured, e.g. in the Aharonov–Bohm effect, and therefore should be considered central to the theory of electromagnetism. Given a principal connection \(A_\mu\) (also called a gauge) of the principal \(U(1)\)-bundle \(P = \coloneqq \mathcal{M}\times U(1)\), where \(\mathcal{M}\) is Minskowski spacetime, we may form the curvature \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\) (also called the field strength). The Yang-Mills \(L^2\) energy \(\int \|F\|_2~dV\) (this expression makes use of the metric) is minimized for curvatures with \(\partial^\mu F_{\mu\nu} = 0\). This equation together with the Bianchi identity for \(F_{\mu\nu}\) consist of Maxwell's equations. A solution is then a principal connection \(A_\mu\) satisfying these equations. Given any section \(s\) of \(P \to \mathcal{M}\) we may take the pullback \(s^*A\), and recall that for a trivialization \(\psi : \pi^{-1}(U) \to U \times G\) we have a local unit section \(s(x) = \psi^{-1}(x,1)\) of \(P \to \mathcal{M}\), which is the section we will use to pull back the connection on \(U\), which is what physicists call gauge fixing. The effect of the section \(s\) is to choose an origin for each copy of \(U(1)\) over \(\mathcal{M}\), and thus, together with coordinates \(x^\mu\), we can think of these choices as a coordinate chart for \(P\), on which we write down Maxwell's equations.

A Lie group \(G\) comes with the maps

for left and right multiplication, and conjugation, by \(g\in G\). Trivially, the left and right multiplication commute with one another, which is useful for proofs of other properties. Taking the tangent map, we obtain

and in particular

1. We can use left and right multiplication to lift any vector in \(T_1G\) into a vector field. In particular, for any \(\varepsilon \in T_1G\) define the vector field \(\xi_g \coloneqq TL_g\varepsilon\). It is a left-invariant vector field in the sense that \(TL_g\xi = \xi\) for any \(g\in G\). The collection makes up the Lie algebra of left-invariant vector fields \(\mathfrak{g}_l\), and similarly by using \(TR_g\varepsilon\) we have \(\mathfrak{g}_r\), the right-invariant vector fields. We may denote this lift by \(\varepsilon^{\#}\) for \(\varepsilon \in T_1G\) with the left or right action inferred by context.
   1. The tangent bundle \(TG\) is trivial: choose a basis \(\varepsilon_p\) of \(T_1G\) and lift it to global left-invariant sections \(\varepsilon^{\#}_p\) of \(TG\) that form a basis of its tangent spaces, thus \(TG \underset{G}{\cong} G\times\mathfrak{g}_l\).

2. \(G\) is realized as a subgroup of \(\Aut(\mathfrak{g}_r)\):

   \begin{equation} \label{eq:Ad} \begin{aligned} \Ad : G & \to \Aut(\mathfrak{g}_r) \\ \Ad g : \xi & \mapsto TL_g\xi. \end{aligned} \end{equation}

   This representation is called the adjoint representation. Similarly there is an representation on \(\mathfrak{g}_l\) by \(\xi \mapsto TR_{g^{-1}}~\xi\).

   1. Note that right-invariantly lifting \((TL_g \varepsilon^{\#})(1) = TC_g\varepsilon\) results back in \(TL_g \varepsilon^{\#}\) again. Thus \(\Ad g\) may also be viewed as a map \(T_1G \to T_1G\) given by \(TC_g \varepsilon\).

   2. Taking the tangent map of \eqref{eq:Ad} at the identity results in

      \begin{align} \ad : T_1G \to \End(\mathfrak{g}_r). \end{align}

      This is the adjoint representation of \(\mathfrak{g}_r\) in itself (by identifying \(T_1G\) with \(\mathfrak{g}_r\)).

3. The flow of a given invariant vector field \(\varepsilon^{\#}\) yields paths \(\Psi^g_t\in G\) about each specified \(g\in G\). The flow at \(g = 1\) is denoted by \(t \mapsto \exp(t\varepsilon^{\#})\). For right-invariant vector fields we have \(\Psi^g_t = \exp(t\epsilon^{\#})g\) and for left-invariant vector fields we have \(\Psi^g_t = g\exp(t\epsilon^{\#})\). The inverse of \(\exp(\varepsilon^{\#})\) around a neighborhood of \(0\) of \(T_1G\) is \(\log g\). Note that the logarithm cannot be defined on components other than the identity component of \(G\); and it may be multi-valued far away from the identity element of \(G\), consider e.g. \(z = e^{i\theta} \in U(1)\) with any \(\theta + 2\pi \mathbb{Z}\) mapping exponentially to the given \(z\).

   1. The flow induces an invariant vector field via

      \begin{align} \xi_g \coloneqq \left.\frac{d}{dt}\right|_{t=0} \Psi^g_t. \end{align}
   2. The flow induces a flow to any manifold \(Z\) acted upon by \(G\) (say on the right) via \(z(t) \coloneqq z_0\Psi^g_t\). Consequently the invariant vector fields of \(G\) lift to \(Z\). A chosen basis \(\varepsilon_p\) of \(T_1G\) corresponds to the fundamental vector fields of \(Z\).
   3. We may equivalently lift invariant vector fields without using the flow as follows: the right action \(R_z : G \to Z\) defined by \(R_z(g) \coloneqq zg\) yields, taking the tangent map, the map \(TR_z : T_1G \to T_zZ\). Thus \(\varepsilon \in T_1G\) maps to the vector field \(z \mapsto TR_z\varepsilon\).

A review of Lawson and Michelsohn [1].

Let \(V\) a vector space over the field \(k\) (for ease of analysis we assume \(k = \mathbb{R}\) or \(k = \mathbb{C}\) and the reader is advised to read [1] for general fields), and \(q : V \to k\) a quadratic form², viz. \(q(\lambda v) = \lambda^2 q(v)\) for \(v\in V\) and \(\lambda\in k\). It is associated with a symmetric form, called the polarization form, \(q(v,w) \coloneqq \frac 12 (q(v + w) - q(v) - q(w))\). On the whole this forms a category with \(f : (V,q) \to (V', q')\) when \(f : V \to V'\) and \(f^*q' = q\).

The Clifford algebra \(\Cl(V,q)\) is defined as

Aside from determining the squares of vectors, this construction determines the anticommutator of vectors:

The Clifford algebra is a functorial construction with \(f : (V, q) \to (V', q')\) inducing \(\Cl(f) : \Cl(V, q) \to \Cl(V', q')\). The orthogonal group \(O(V,q) \coloneqq \{f \in \GL(V) : f^*q = q\}\) hence has a representation on \(\Cl(V,q)\) via \(f \mapsto \Cl(f)\).

• ☐ Study Chapter 2 from Lawson
  • ☑ Study covering spaces from Hatcher, to understand Lemma 1.1 on \(\operatorname{Cov}_2(X) \cong H^1(X; \mathbb{Z}_2)\).
  • ☐ Read rest of chapter.
• ☐ Study Principal G-bundle appendix from Lawson, Appendix A
• ☐ Classifying spaces and Characteristic Classes from Lawson, Appendix B
• ☐ Orientation Classes and Thom Isomorphisms in K-Theory from Lawson, Appendix C
• ☐ \(\text{Spin}^c\)-manifolds from Lawson, Appendix D