I admit I am a bit confused by your terminology, but here is how I learned it: Let $P$ be a $G$-principal bundle and $\Sigma$ a spacetime.

**gauge group**: The fibers of the $G$-principal bundle over the spacetime, i.e. the group $G$.
**(Local) group of gauge transformations**: The group of diffeomorphisms $t : P \rightarrow P$, which are fiber-preserving and $G$-equivariant, i.e. if $\pi : P \rightarrow \Sigma$ is the projection then $\pi \circ t = \pi$, and $t$ commute with the group action on $P$.

One can now, by transitivity of the group action on the fibers, define a function $g_t: P \rightarrow G$ by $t(p) = pg_t(p) \forall p \in P$, and such functions $g : P \rightarrow G$ conversely define a gauge transformation by $t_g(p) = pg(p)$ as long as they fulfill $g_t(ph) = h^{-1}g_t(p)h \forall h \in G$, so we have two alternative characterizations of local gauge transformations:

$\mathcal{G} = \{t |t \in \mathrm{Diff}(P) \wedge \pi \circ t = t \wedge t(ph) = t(p)h \forall h \in G\} = \{ g| g \in \mathrm{Maps}(P,G) \wedge g(ph) = h^{-1}g(p)h \forall h \in G \}$

The equivariant diffeomorphisms of $P$ are called local, since they apply a different group element to every spacetime point.

Now, the associated bundles are affected as follows: Let $\phi : \Sigma \rightarrow P \times_G V$ be a section of the associated bundle, i.e. a field. By a similar argument to the above, these are in bijection to $G$-equviariant functions $f_\phi : P \rightarrow V$ satisfying $f_\phi(pg) = \rho(g^{-1})f_\phi(p)$. This is esentially the reason why, in $\mathrm{U}(1)$ symmetry, a gauge transformation $\mathrm{e}^{\mathrm{i}\alpha(x)}$ acts on fields as $\phi(x) \mapsto \mathrm{e}^{-\mathrm{i}\alpha(x)} \phi(x)$.

So, you see, the local group of gauge transformations is much bigger that the global gauge group since it allows far more functions than just the constant ones. You can always clearly write down the global gauge group (it defines your theory!), but writing down the local one more explicit than I did above is hard. For $\mathrm{U}(1)$, however, it is just $\{x \mapsto \mathrm{e}^{\mathrm{i}\alpha(x)} x | \alpha : P \rightarrow \mathrm{U}(1) \text{is smooth (enough)}\}$, I think. Cases where the two groups coincide demand a spacetime that is a point, I would guess, but I am not wholly confident in that.

Also, all of this can be done classically, nothing about gauge theories is inherently quantum.

**EDIT**:

Alright, your edit was very helpful in discerning what is actually going on here.

Your *global* gauge group is what physicists call the group of gauge transformations. The *gauge group* of a a *gauge theory* is what you call a *local* gauge group (and what the nLab also calls the local gauge group). When physicists say **the** gauge group $\mathrm{SU}(N)$, they mean it is what you call the local gauge group.

The *global* gauge group of the nLab is just the group of transformations (not necessarily gauge transformations, terminology is terrible here, I know) that leaves all observables invariant, i.e. it is the group of symmetries of the theory (*not* the group of symmetries of the Lagrangian), the group of gauge transformations is naturally a subgroup of this. The difference is that this global gauge group can contain transformations that have not really something to do with the structure of the local gauge group, and can contain things which are not gauge transformations. This global gauge group can even exist if you have no explicit gauge theory, and is inherently a QFT concept.

In other news, you are right, your connection form $\omega$ is the gauge field $A$ of a physical gauge theory, and it transforms exactly like you wrote. Now, the problem with the gauge field is exactly that ugly transformation, so we construct the curvature transforming in the adjoint rep and call it the field strength $F$. The action of a pure (Yang-Mills) gauge theory is then (up to prefactors) given by

$$ \int_\Sigma \mathrm{Tr}_{ad}(F \wedge \star F)$$

since the action must be invariant under gauge transformations and the $\mathrm{Tr}_{ad}(F \wedge \star F)$ is pretty much the only object we can construct out of the gauge fields that is invariant *and* can be integrated over the spacetime.

When Maxwell formulated his equations he did so using quaternions, which BTW is a more elegant formalism, and Heaviside formulated them as we normally read them. Our standard vector forms of the Maxwell equations are more convenient for electrical engineering. These equations linearly add the electric and magnetic fields. This linear property is a signature of the abelian nature of the Lie group $U(1)$ for electrodynamics.

Let me argue this in somewhat more modern language with quantum mechanics. Suppose we have a quantum field (or wave) that transforms as $\psi(\vec r) \rightarrow e^{i\theta(\vec r)}\psi(\vec r)$. Now act on this with the differential operator $\hat p = -i\hbar\nabla$ and we find
$$
\hat p\psi(\vec r) = -i\hbar\nabla\psi(\vec r) = -i\hbar\left(\nabla\psi(\vec r) + i\psi(\vec r)\nabla\theta\right)
$$
We then see this does not transform in a homogeneous fashion, so we change the operator in to a covariant one by $\nabla \rightarrow \nabla - ie\vec A$ where $\vec A$ is the vector potential that subtracts out the $\nabla\theta$. We can now perform quantum mechanical calculations that include the electromagnetic field in a consistent manner.

We now consider the covariant differential one-form ${\bf D} = {\bf d} - ie{\bf A}$ such that ${\bf d} = dx\cdot\nabla$ and ${\bf A} = {\vec A}\cdot dx$. We can now look as the action of ${\bf D}\wedge {\bf D}$ on a unit or constant test function
$$
{\bf D}\wedge {\bf D}\odot\mathbb I = {\bf d}\wedge{\bf d} \odot\mathbb I - e^2{\bf A}\wedge{\bf A} \odot\mathbb I - ie{\bf d}\wedge{\bf A} \odot\mathbb I - ie{\bf A}\wedge{\bf d} \odot\mathbb I ,
$$
where the wedge product of a p-form with itself is zero and elementary manipulations gives
$$
{\bf D}\wedge {\bf D}\odot\mathbb I = ie\left({\bf d}\wedge{\bf A}\right) \odot\mathbb I .
$$
Breaking out the differential form on the vector potential gives the magnetic field by $B = -\nabla\times A$
that the vector potential one-form wedged with itself is zero is a signature of the abelian or $U(1)$ symmetry of the electromagnetic field. For other gauge fields there is a color index, where there are different sorts of charges, and so ${\bf A}\wedge{\bf A}$ is nonzero and this is a signature of the nonabelian nature of other gauge fields, in particular the weak and strong nuclear forces.

This is done in a three dimensional or nonrelativistic manner, and of course this must be generalized to relativistic QM or the Dirac equation for the quantum dynamics of a fermion. So this is a bit of an elementary introduction. The main upshot though is the reason electromagnetism is abelian or $U(1)$ is there is only one electric charge $e$, with positive and negative values, while other gauge fields have an array of color-charges.

## Best Answer

Yes, you can formulate Maxwell's equations as a classical $U(1)$ gauge theory without using the word "quantum" at all. In non-relativistic notation, we introduce a scalar potential $\phi$ and a vector potential $\vec{A}$ which are related to the electric $\vec{E}$ and magnetic $\vec{B}$ fields via \begin{eqnarray} -\nabla \phi - \frac{\partial \vec{A}}{\partial t} &=& \vec{E} \\ \nabla \times \vec{A} &=& \vec{B} \end{eqnarray} From these definitions, it is easy to check that $\vec{E}$ and $\vec{B}$ are unchanged if we transform $\phi$ and $\vec{A}$ by \begin{eqnarray} \phi &\rightarrow& \phi - \frac{\partial \lambda}{\partial t}\\ \vec{A} &\rightarrow& \vec{A} + \nabla \lambda \end{eqnarray} for an arbitrary (smooth) function $\lambda$. This is gauge invariance. The above set of statements can be reformulated in much more sophisticated ways, including saying that $(\phi,\vec{A})$ is a connection on a $U(1)$ fiber bundle.

The potential formalism was known and used well before quantum electrodynamics was discovered. The main reason for introducing the potentials classically, is calculational convenience. Instead of needing to solve for the 6 components of $\vec{E}$ and $\vec{B}$, we only need to solve for 4 components (1 in $\phi$ and 3 in $\vec{A}$).

In fact it is even better than this because we can use gauge invariance to massage the form of the equations into standard forms. For instance, we can always choose the function $\lambda$ such that the sourced Maxwell's equations take the form of four

decoupledstandard wave equations (theLorenz gauge) \begin{eqnarray} -\frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} + \nabla^2 \phi &=& 4\pi \rho \\ -\frac{1}{c^2} \frac{\partial^2 \vec{A}}{\partial t^2} + \nabla^2 \vec{A} &=& 4\pi \vec{J} \end{eqnarray} where $\rho$ is the charge density and $\vec{J}$ is the current density. These can be solved formally using Green's function methods, and the expressions for the electric and magnetic fields can be obtained by differentiating the potentials. This method of solution is easier than trying to directly solve for $\vec{E}$ and $\vec{B}$, because the sourced Maxwell's equations do not take the form of decoupled copies of the wave equation in standard form in those variables (the components are mixed together and there are a mix of different kinds of second spatial derivatives).Also note that we need to solve fewer independent equations using the potential formalism. Since $\nabla \cdot \nabla \times \vec{A}=0$ identically, Maxwell's equation $\nabla \cdot B=0$ is automatically satisfied if we solve for the vector potential. Similarly, the equation $\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$ is automatically satisfied by the potential, which you can verify using the identity $\nabla \times \nabla \phi=0$ and the definition $\vec{B}=\nabla \times \vec{A}$.

Finally, as others have noted, using the potentials lets you express Maxwell's theory in terms of a local Lagrangian. This is useful at a formal level; for example it lets you derive a stress-energy tensor in a systematic way, and to explain conservation of charge as a consequence of Noether's theorem applied to the

global$U(1)$ symmetry ($\lambda={\rm const}$, which leaves the potentials invariant but under which charged particles and fields will transform).Incidentally, to make a comment that isn't directly in response to your question, but which I think is relevant: it's worth understanding why the formulation of a $U(1)$ gauge theory is useful in quantum theory. It actually is not necessary to formulate Maxwell's equations as a gauge theory quantum mechanically, but it is

convenient, because thelocalLagrangian formulation makes the locality of physical results manifest. However, this comes at the cost of needing to establish that the unphysical parts of the gauge field do not contribute to the final answer, established using an elaborate set of relations known as Ward-Takahashi identities, and makes unitarity difficult to establish. In other formulations (helicity-spinor variables), locality is not obvious but other properties of the theory such as unitarity are.