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
$\begingroup$
$\endgroup$
3
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.
-
$\begingroup$ 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. $\endgroup$ Mar 11, 2013 at 8:52
-
$\begingroup$ Oops. As José Figueroa-O'Farrill points out, it is the field strength, not the potential, which is harmonic. $\endgroup$ Mar 11, 2013 at 10:12
-
4$\begingroup$ ...potential is also harmonic in Lorentz gauge $\endgroup$ Mar 11, 2013 at 21:21