# Is it possible to generalize the Maxwell equations to higher dimensions?

The usual Maxwell equations are for 3 spatial dimensions, right?

Is it possible to generalize them to 2 spatial dimensions or 4 spatial dimensions?

Maxwell's equation can be given in the form $$\text dF = 0$$ $$\text d\star F + J = 0$$ where $F$ is a 2-form and $J$ an $n-1$-form (a current density) which in principle can be generalised to any manifold (for physical reasons one might want to consider pseudo-Riemannian manifolds with signature $(+,-,\cdots,-)$). In the four dimensional theory one usually sets $G:=\star F$, which is also a 2-form and has a clear physical interpretation in terms of electric and magnetic field. In higher or lower dimensions it becomes an $(n-2)$-form and as far as I know there is no clear interpretation of the many "components" of $G$ in these other cases.

Regarding the solutions, if any, the behaviour they exhibit should depend strongly on the number of dimensions (as the classical example of the Laplace's equation in different dimensions shows).

• It would be good to mention whether the solutions to those two equations are as determined as in the 4D case (i.e. up to a solution of the $n$-dimensional wave equation) or whether they open up more freedom than one would naively expect. Jul 27, 2015 at 16:49
• It is an interesting question, but I don't know the answer on top of my head. As the dimension increases I would expect the gauge freedom to increase, but this'd need to be checked. Jul 27, 2015 at 16:50
• Any good reference? Jul 27, 2015 at 21:34
• WetSavannaAnimalakaRodVance has indicated Penrose's book "Road to reality". I have a copy of that book however not with me at the moment, but I'm sure it contains some really nice ideas worth having a look at. Jul 28, 2015 at 7:17
• @poisson I'm pretty sure Penrose doesn't go into gauge freedom with dimension in the reference I cited, I only suggested the reference as a wonderful motivation and geometric intuition for Phoenix's answer. Also, IIRC, Penrose wasn't particularly worried about dimension, he just put his arguments in a general, dimension-free way. Jul 28, 2015 at 23:46

You can generalize Maxwell's equations to an arbitrary number of dimensions by using either the tensor or differential form version, as the vector formalism does not help too much (For instance, in two dimensions, the magnetic field is a (pseudo) scalar field, not a vector field). The equations are then :

$\partial_\alpha F^{\alpha\beta} = \mu_0 J^\beta$

or

$d\star F = \mu_0 J$