Maxwell's equations of Electromagnetism in 2+1 spacetime dimensions

Question

What would be different in the theory of electromagnetism if instead of considering the equations of Maxwell in 3+1 spacetime dimensions, one would consider 2+1 spacetime dimensions?

Answer 1

The electromagnetic field in our physical universe is best described by an antisymmetric tensor in spacetime. It has six independent components, three of which are space-space (magnetic field) and three time-space (electric field).

In a two-dimensional space, spacetime would be three dimensions total. An antisymmetric tensor will have three independent components. Two would be time-space, making an electric field. It's just a 2D vector in 2D space, no surprise. The remaining component is space-space, and is pseudo-scalar. This is the magnetic field.

If Edwin Abbott's Flatland beings were to explore electromagnetism, they'd find electric fields to be vectors and magnetic fields to be scalar, without meaningful direction. Except magnetic fields would change sign in mirror-imaged arrangements of charges and fields - something like how we have to replace neutrinos with antineutrinos when considering spin seen in a mirror.

Maxwell's equations are normally taken to be four in number, but in relativity using the antisymmetric tensor, can be best understood as two. Even those can be combined into one, using Clifford Algebra, but that's going somewhat off topic, except to note that the one feature of all of Maxwell's equations is that they are about derivatives of the field. In 3D we have dot products and cross products. With a (pseudo) scalar magnetic field in 2D, there's no curl or divergence. We only have a gradient.

We can imagine things intuitively if we expand the two dimensional physics (using x,y) into three by smearing everything out along a third dimension (z). Point charges in 2D become line charges, all parallel to z. The electric field as ordinarily imagined in 3D physics, will have no z components. The magnetic field in 3D will be a (pseudo) vector with only a z component - always parallel to the line charges. B will vary from place to place in the x,y plane but for any (x,y) will be constant along z. We can see that such a field will have a curl in 3D, which will always be a vector with only x,y components. We can also contemplate curl and divergence of the electric field. What happens in a 2D universe is identical to taking a slice at z=0 (or any constant) of this 3D model.

So, to examine the four individual Maxwell equations:

• The divergence of the electric field remains the same, just one dimension less.

• The time derivative of the electric field, relating to the curl of the magnetic field, and with a current density source term, remains except we now use the gradient of the magnetic "scalar" seen by Flatlanders.

• The time derivative of the magnetic field, relating to the curl of the electric field, also remains, but now the "curl" of the electric field is a pseudo-scalar.

• The equation that says magnetic field lines do not end - the divergence is always zero - vanishes! Thinking about the 3D line-charge model, the divergence of the magnetic field with only z components, constant along z, gives zero, and nothing meaningful to the 2D Flatlander physicists can be said.