I learned this while talking to a physics friend. It is unfortunate that the audience aren’t exposed to too much math in most physics classes, so that such elegant mathematical formalisms are rarely mentioned in class. Anyways, here is the interpretation of Maxwell’s equations in terms of \( \mathrm{U}(1) \)-bundles.
1. Maxwell’s equations#
When we first learn Maxwell’s equations, we learn a set of four equations, up to some constants: \[ \displaystyle \begin{cases} \nabla \cdot E = \rho, \\ \nabla \times E = -\frac{\partial B}{\partial t}, \\ \nabla \cdot B = 0, \\ \nabla \times B = \frac{\partial E}{\partial t} + J. \end{cases} \] Here, \( E \) is the electric field, \( B \) is the magnetic field, \( \rho \) is the charge density, and \( J \) is the current density.
Because \( \nabla \cdot B = 0 \), we can find a smooth function \( A : \mathbb{R}^4 \rightarrow \mathbb{R}^3 \) such that \( B = \nabla \times A \). (This is essentially the Poincaré lemma.) Then, we get \[ \displaystyle 0 = \nabla \times E + \frac{\partial B}{\partial t} = \nabla \times (E + \tfrac{\partial A}{\partial t}) \] and a second application of the Poincaré lemma shows that \( E + \frac{\partial A}{\partial t} = -\nabla \varphi \) for some \( \varphi : \mathbb{R}^4 \rightarrow \mathbb{R} \). This \( A \) is called the vector potential and \( \varphi \) is called the scalar potential. Of course, they are not uniquely determined by \( E \) and \( B \). In this sense, they are not really observable quantities of the physical system.
Using these potentials, we can rewrite Maxwell’s equations as \[ \displaystyle \begin{cases} B = \nabla \times A, \\ E = -\nabla \varphi - \frac{\partial A}{\partial t}, \\ \nabla \cdot (-\nabla \varphi - \frac{\partial A}{\partial t}) = \rho, \\ \nabla \times (\nabla \times A) - \frac{\partial}{\partial t} (- \nabla \varphi - \frac{\partial A}{\partial t}) = J. \end{cases} \] Let me not expand the whole thing.
There is a problem with non-uniqueness of these potentials \( A \) and \( \varphi \). Equations are hard to solve if there are too many solutions. So people sometimes impose additional constraints on \( A \) and \( \varphi \). This is called gauge fixing. For instance, the Lorentz gauge is the additional constraint \[ \displaystyle \nabla \cdot A = -\frac{\partial \varphi}{\partial t}. \]
2. Relativistic formulation#
The whole idea of special and general relativity is that time is not much different from space. There is no such thing as a fixed "time direction" in spacetime \( \mathbb{R}^4 \), but only a pseudo-Riemannian metric. If we look at \( \mathbb{R}^4 \) with metric \[ \displaystyle g _ {\mu\nu} = \begin{pmatrix} -1 & \displaystyle 0 & \displaystyle 0 & \displaystyle 0 \\ 0 & \displaystyle 1 & \displaystyle 0 & \displaystyle 0 \\ 0 & \displaystyle 0 & \displaystyle 1 & \displaystyle 0 \\ 0 & \displaystyle 0 & \displaystyle 0 & \displaystyle 1 \end{pmatrix}, \] isometries are the same as Lorentz transformations. Here, \( 0 \) is the time-like coordinate and \( 1, 2, 3 \) are the space-like coordinates.
So how is this related to Maxwell’s equations? People observed that if we define the electromagnetic tensor as \[ \displaystyle F _ {\mu\nu} = \begin{pmatrix} 0 & \displaystyle E _ x & \displaystyle E _ y & \displaystyle E _ z \\ -E _ x & \displaystyle 0 & \displaystyle -B _ z & \displaystyle B _ y \\ -E _ y & \displaystyle B _ z & \displaystyle 0 & \displaystyle -B _ x \\ -E _ z & \displaystyle -B _ y & \displaystyle B _ x & \displaystyle 0 \end{pmatrix}, \] then the equations can be written as \[ \displaystyle \begin{cases} \partial _ \mu F^{\mu 0} = \rho, \\ \partial _ m F _ {n0} + \partial _ n F _ {0m} + \partial _ 0 F _ {mn} = 0, \\ \partial _ 1 F _ {32} + \partial _ 2 F _ {13} + \partial _ 3 F _ {21} = 0, \\ \partial _ \mu F^{\mu n} = J^n. \end{cases} \] This means that if we define the four-current as \[ \displaystyle J^\mu = \begin{pmatrix} \rho \\ J _ x \\ J _ y \\ J _ z \end{pmatrix}, \] then the equations can be compressed to \[ \displaystyle \begin{cases} \partial _ \alpha F _ {\beta \gamma} + \partial _ \beta F _ {\gamma \alpha} + \partial _ \gamma F _ {\alpha \beta} = 0, \\ \partial _ \mu F^{\mu \nu} = J^\nu. \end{cases} \] The four-current really can be thought of as a current, if we consider a charge as moving in the time-like direction in \( 4 \) dimensions.
We can also drag the potentials into the picture. Define the four-potential as \[ \displaystyle A^\mu = \begin{pmatrix} \varphi \\ A _ x \\ A _ y \\ A _ z \end{pmatrix}, \] and playing around with the equations shows that \[ \displaystyle F _ {\mu\nu} = \partial _ \mu A _ \nu - \partial _ \nu A _ \mu. \] At the end, the two equations \[ \displaystyle \begin{cases} F _ {\mu\nu} = \partial _ \mu A _ \nu - \partial _ \nu A _ \mu, \\ \partial _ \mu F^{\mu \nu} = J^\nu \end{cases} \] encapsulates all of Maxwell’s equations.
3. Principal \( \mathrm{U}(1) \)-bundles#
Let’s consider the manifold \( M = \mathbb{R}^4 \), and a principal \( \mathrm{U}(1) \)-bundle \( E \) on \( M \). Because \( \mathrm{U}(1) = S^1 \simeq K(\mathbb{Z}, 1) \) implies \( B \mathrm{U}(1) \simeq K(\mathbb{Z}, 2) \) and \( H^2(M; \mathbb{Z}) = 0 \), the bundle \( E \) is necessarily a trivial bundle. So just let \( E = M \times \mathrm{U}(1) \).
We will now consider an Ehresmann connection on this bundle. This is a \( 1 \)-form on \( M \) with values in the Lie algebra \( \mathfrak{u}(1) \). More precisely, it is a section of the bundle \( T^\ast M \) tensor producted with some associated vector bundle. But because the Lie group \( \mathrm{U}(1) \) is abelian, the associated vector bundle is trivial, and \( \mathfrak{u}(1) \cong \mathbb{R} \). That is, a connection on \( E \) is just a \( 1 \)-form on \( M \).
Let \( A _ \mu = A _ \mu dx^\mu \) be an arbitrary connection on \( E \). Then we can define a curvature associated to this connection. This is a \( 2 \)-form, taking in two vectors, and spits out the holonomy around an infinitesimal parallelogram made by the two vectors. The holonomy again lies in the Lie algebra, which is \( \mathbb{R} \) in our case. If you work out this in formulas, you will get \[ \displaystyle F _ {\mu\nu} = \partial _ \mu A _ \nu - \partial _ \nu A _ \mu \] for the expression for the curvature tensor.
Well, now this looks familiar, doesn’t it? I’m not sure if it is merely a convenient coincidence or a historical choice of notation, but \( F \) as in the curvature tensor is the same \( F \) as in the electromagnetic tensor, and \( A \) as in the connection is the same \( A \) as in the vector potential.
Vector potential \( A _ \mu \) — Connection \( A _ \mu \)
Electromagnetic tensor \( F _ {\mu\nu} \) — Curvature tensor \( F _ {\mu\nu} \)
With this perspective, we may consider the electromagnetic field \( F _ {\mu\nu} \) as the curvature of some connection on a principal \( \mathrm{U}(1) \)-bundle. This problem with gauge can be interpreted as there being no canonical way of trivializing the bundle, and hence gauge fixing is just a choice of trivialization.
Then we might ask what \( \partial _ \mu F^{\mu\nu} = J^\nu \) corresponds to. Well, I don’t know what a good analogy would be. One problem we have is that in general, \( \partial _ \mu F^{\mu\nu} \) is not a well-defined tensor, as it depends on the choice of coordinates. But maybe we can change it to something like \( \nabla _ \mu F^{\mu\nu} = J^\nu \) and there is a nice interpretation.