# Maxwell's equations in differential form in 2-space+1-time dimensions

How does one write maxwell's equation in 2+1 dimensions? It becomes particularly interesting as the components of 2 forms and 1 form are 3. Are there any sources for this?

## 2 Answers

The relativistic formulation of electromagnetism repackages Maxwell's equations as

$$F^{a} = \int d^{3}x\,j^{b}F^{a}{}_{b}$$, where $$F_{ab}$$ is the Maxwell Tensor given by $$F_{ab} = \nabla_{[a}A_{b]}$$, and $$A$$ is the electromagnetic potential. In Four dimensions, it can be shown that the first equation reduces to the normal Lorentz force law, and that the Maxwell Tensor is $$F_{0i} = E_{i}$$ and $$F_{ij} = B^{k}\epsilon_{ijk}$$, where $$\epsilon_{ijk}$$ is the standard Levi-Civita symbol.

Then, generalizing to 2+1 dimensions is easy, because all of these equations just cross over. All that changes is that your vector $$A$$ only has three components, which will mean that you have a 2-dimensional electric field, and a one-dimensional "magnetic" field.

The relativistic version of Maxwell's equations, $$\partial_\mu \, F^{\mu\nu} = J^\nu$$ (plus the dual equation, see wikipedia if that doesn't ring a bell), works fine with any number of spatial dimensions. Of course, $$F^{\mu\nu}$$ is the field strength tensor and $$J^\mu$$ is the relativistic current.

Here's where it gets fun. In 2+1 dimensions $$F^{\mu\nu} = \left( \begin{array}{ccc} 0 & -E_1 & -E_2 \\ E_1 & 0 & B \\ E_2 & -B & 0 \end{array} \right) .$$ So the electric field is a 2d vector $$\vec{E}=(E_1, E_2)$$, but there's only one component of the magnetic field: $$B$$ is a scalar (actually a pseudoscalar)!

That's because fundamentally, $$E$$ is a spatial vector and $$B$$ is a spatial "bivector". In 3d a bivector happens to be the same as a "psuedovector", so we tend to incorrectly think of $$B$$ as a vector quantity. But in other numbers of dimensions that's just not the case. To get a much deeper feel for this, I'd recommend checking out the geometric algebra formulation of electrodynamics (e.g. Ch. 7 of Doran and Lasenby).