# Electromagnetism and differential forms

I am currently writing a Bachelor's thesis in theoretical physics, and since I like the interplay between mathematics and theoretical physics, I am writing about Maxwell's law in terms of differential forms. Maxwell's equations read $$dF=0$$ $$d\star F=\star J$$ for the $$F$$ the field tensor, $$d$$ exterior derivative and $$\star$$ the Hodge star operation. Does anyone know any interesting applications/advantages of writing Maxwell's equations like this, apart from their stunning beauty?

• Well, the obvious advantage is that this formulation doesn't depend on the choice of coordinates; by the way, I have written an introductory paper on exactly this topic that can be found here: github.com/gennaro-tedesco/MaPhy/tree/master/… – gented May 10 '19 at 8:38
• I totally agree :), which physical implications can this have? Can we do something with Dirac monopoles? – Lucas Smits May 10 '19 at 8:39
• It’s just a rewriting, it doesn’t contain any new physical content. Though, as you remark, in this form it is especially inviting to put monopoles there. – Michael Angelo May 10 '19 at 8:41
• Do you have any sources on monopoles? – Lucas Smits May 10 '19 at 8:53

By themselves, these equations are just a (indeed beautiful) reformulation of the standard Maxwell equations, so they don't contain new physics. However, they are useful in the sense that they stress the geometric and topological aspects, and they generalise nicely:

• $$A$$ and $$F$$ are properly connections resp. sections in certain bundles, and the geometric structure of electromagnetism becomes much more clear.
• For example, connections on bundle, are naturally defined only on local patches with certain transition functions. Thus, it should not come as a surprise that cyou can have $$F=\text{d}A=0$$, but $$\int A\neq 0$$, as in the Aharonov-Bohm effect. Similarly, charge quantisation due to magnetic monopoles comes out easily.
• The generalisation to non-Abelian gauge groups is (somewhat) straightforward.
• The connection $$A$$ defines the covariant derivative $$\text{d}+A$$. In differential geometry, the commutator of two derivatives is the curvature tensor -- in general relativity, this is the Riemann tensor, while in gauge theories, it's the field strength tensor.
• In many theories, e.g. string theory, you get higher-form gauge fields (i.e. two-forms, three-forms etc.), which work quite analogously.
• In higher-dimensional theories, gauge fields nicely connect with algebraic and differential topology, characteristic classes, index theorems and all that.

Some of these aspects are quite advanced. If you are interested, the book "Geometry, Topology and Physics" by Nakahra should provide a reasonably accessible introduction.

• Awesome book by Nakahara! – gented May 10 '19 at 13:06

Any theory when expressed in the language of differential forms is "coordinate-free". This means that the theory is true for any arbitrary choices of coordinate systems, i.e. invariant under general coordinate transformations. Such formulations are particularly useful when the theory is somehow related to geometry.

Now recall that electromagnetism is a gauge theory and a gauge theory is related to geometry through the fiber bundle formulation. This implies that such a formulation expresses electromagnetism in terms of geometry. This formulation, however, is important when electromagnetism is studied in the context of other theories like General Relativity, Quantum Field Theory, Mathematical Fluid Mechanics etc.

• Coordinate-free is not the same as valid in arbitrary coordinates. Of course a coordinate-free equation is valid in any coordinates, but the expression $\partial_\mu F^{\mu\nu}=J^\nu$ is too. – Danu May 10 '19 at 9:27