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In some textbooks, the Maxwell equations are stated in a very simple mathematical form (up to multiplicative constants coming from the system of units being used):

$$ \begin{array} \mbox{d}F =0, \\ \delta F = j, \end{array}$$

where $d$ and $\delta$ are the exterior differential and co-differential on Minkowski spacetime, respectively. This form carries over to curved spacetime.

However, we know (see for example here) that F is not simply a two-form field in the cotangent bundle of the Minkowski spacetime, but is a section of a U(1) associated bundle also over Minkowski spacetime.

So why does $F$ actually carry Lorentz indices, which are typical for de Rham bundle/complex, and why is it therefore acted on by the $d$ and $\delta$?

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2 Answers 2

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The curvature form on a $G$-principal bundle $P\to M$ is a $\mathfrak{g}$-valued 2-form that is equivariant and horizontal, and hence descends to a well-defined global $\mathfrak{g}$-valued 2-form on $M$. Since it is a 2-form, it has 2 ordinary Lorentz indices.

To see this, pick any trivializations $\phi_i : U_i\to M$ and identity sections $s_i : U_i \to U_i\times G\subset P, x\mapsto (x,1)$ over them and try to glue the pullbacks $s_i^\ast F$ together to a form on $M$. You will see that equivariance and horizontality make it so that the gluing is consistent. For this reason equivariant horizontal forms are also called basic or tensorial forms.

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  • $\begingroup$ I am just noting here that the curvature form descends to local $\mathfrak g$ valued forms on each trivialization that are related by the images of the transition functions in the adjoint representation or it descends globally into an $\mathrm{Ad}(P)$ valued 2-form, but the Lie valued form is only global if the bundle is globally trivial. $\endgroup$ May 25, 2020 at 21:47
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If it were a group $G$ other that ${\rm U}(1)$, $F$ would carry other indices and have the form $$ F= \frac 12 \hat \lambda_a F^a_{\mu \nu} \,dx^\mu \wedge dx^\nu $$ where $\hat \lambda_a$ are generators of the Lie algebra of the group $G$. As a result $F$ is Lie-algebra-valued two form, and is the curvature form of the of the principal $G$-bundle $P$. If one chooses a represesentation space $V$ for $G$, the abstract generators $\hat \lambda_a$ become matrices acting on $V$ and $F$ now becomes the curvature on the associated bundle $V\times_G P$.

Because ${\rm U}(1)$ is a Abelian, and hence all its irrreps are one dimensional, the single matrix $\lambda$ is just a numbers and we usually don't bother to include an explicit $\hat \lambda$ in the formula for $F$. Instead people write $qF$ where $q$ is the charge of the particle. The integer valued charge $q$ labels the ${\rm U}(1)$ representation $e^{i\theta}\mapsto e^{iq\theta}$.

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