Basic Notion: A different(ial) form of electromagnetism
and there was light.
This joke is famous in physics to the extent that it is printed on shirts and sold worldwide. The four equations on those shirts are known in the world of physics as the Maxwell equations, named after James Clerk Maxwell (1831-1879), the first physicist who wrote down those four equations. Nowadays those equations are taught to first year physics students, because of their relatively simple formulation and their importance in the world we live in. They describe the force of electromagnetism, and this force is responsible for the propagation of light, the charging of your phone and even the movement of your own muscles.
One of the advantages of the Maxwell equations is that they are written down in a coordinate-free way. This implies that you can always use those equations, regardless whether you are moving, accelerating or standing still. This is already a nice feature, but mathematical physicists have pushed the elegance of the formulation of the equations of electromagnetism even further. In order to do this, they used the concept of the exterior derivative d from differential geometry. The exterior derivative is a generalization of the normal derivative, the divergence, gradient and rotation. Instead of acting only on functions, the exterior derivative also acts on an object that not only gives a number (such as the derivative) at a point, but also takes directions into account: the differential form. Using those concepts, the Maxwell equations can be rewritten in the following way:
At first, the striking difference between the two formulations is that the latter is rewritten in a much shorter way, but with some more complex objects, such as the electromagnetic form F. Secondly, the first of those two new equations also has the benefit of telling us something about relation between the solutions and the topology of the corresponding space. Topology is a branch of mathematics that concerns the shape of objects. It tells what objects are the same, if we allow them to be squished, stretched, or moulded in a way that doesn't break or cut them. One famous example of this is a coffee cup, which is similar to a doughnut, because both objects have precisely one hole. In particular, the space in which you look of solutions contains topological information about the existence of potentials. A potential can be viewed as an object to which you can apply the exterior derivative to find the electromagnetic tensor F. In a concise formula, this is the same as
By the work of Georges de Rham (1903-1990), we know that every possible electromagnetic tensor F for a space that can be squished into one point has a potential A as above. For other space, such as three-dimensional Euclidean space with a puncture, this is not the case: there exist electromagnetic tensors in that space with no global potential.
This reformulation of the Maxwell equations is in some ways more elegant and shows that physics can inspire and support mathematics and vice versa, enriching both areas. Perhaps we should put those two equations on a shirt?
This blog post was written subsequent to a presentation in the Basic Notions seminar. For more information about this seminar, go to the seminar's webpage.