How are plane waves, $p$-forms, and Maxwell's equations related? [closed]

Question

I am very new to the concepts of $p$-forms and trying to get a better grasp of physicist use them to state Maxwell's equations.

Wikipedia has a picture of a plane wave http://en.m.wikipedia.org/wiki/Plane_wave

I presume this is used to represent for example electromagnetic radiation moving through space.

Here's my question: That picture of a plane wave looks like a 1-form. Does this similarity have anything to do with how physicists think about formulating Maxwell's equations in (what I think is called) covariant terms?

In other words, I am trying to see if the visual similarities can help me better understand these more advanced forms of the Maxwell equations beyond mere vector calculus.

Answer 1

As far as I can tell, there is no direct link between $p$-forms and plane waves. On the other hands the language of $p$-forms can be used to express Maxwell's equation in a geometrically concise form. It turns out that the electric field $\mathbf E$ and the magnetic induction $\mathbf B$ are "pieces" of a more general and geometrical object $F$, known as the electromagnetic tensor. Such object is an element of $\bigwedge^2 T^*M$, where $M$ denotes the 4-manifold of space-time. So it is a $2$-form field over $M$, sometimes called a skew-symmetric tensor. Let $\mathrm d$ be the exterior derivative, $\star$ the Hodge map and $J$ the 4-current 1-form, then Maxwell equations can be stated as

$$\mathrm{d}F = 0,\qquad \star\mathrm d \star F + J = 0$$

The first one is zero because there are no magnetic monopoles, so no magnetic monopole charge density and currents. In a theory with such entities, the above equations would become

$$\star\mathrm dF - J_m = 0,\qquad \star\mathrm d \star F + J = 0$$

(signs should be double-checked). In the vacuum we have instead

$$\mathrm dF = 0,\qquad \star\mathrm d \star F = 0$$

which are more symmetric, and encode the wave-like behaviour of electromagnetic radiation.