# maxwell's equations and hodge theory

How is Hodge theory of harmonic forms related to maxwell's equations.Atiyah says that Hodge was directly motivated by considerations of maxwell's equations while commenting on donaldson.

-

## 1 Answer

In the absence of electric current, Maxwell's equations say precisely that the electromagnetic potential is a harmonic 1-form; see Wikipedia. This is in a space-time manifold, so it isn't the usual Hodge theory. Being harmonic in space-time is a wave equation, not an electrostatics equation. But if your space-time is a product of a Riemannian space manifold and a time direction, then time invariance of the electromagnetic potential reduces the equations of motion to harmonicity in the sense of the Hodge Laplacian.

-
Actually it's the fieldstrength which is harmonic. Maxwell's equations in vacuo are dF =0 \qquad\text{and}\qquad d\star F = 0$, where$F\$ is the fieldstrength 2-form. –  José Figueroa-O'Farrill Mar 11 '13 at 8:52
Oops. As José Figueroa-O'Farrill points out, it is the field strength, not the potential, which is harmonic. –  Ben McKay Mar 11 '13 at 10:12
...potential is also harmonic in Lorentz gauge –  Y Macdisi Mar 11 '13 at 21:21