anderscolingustafson wrote:quickfur,
Is an argument produced by the cross product the same as a magnetic charge?
I'm not sure I understood your question.
Maxwell's equations describe (classical) electromagnetism, and two of them connect (the strength and direction of) the electric field with (the strength and direction of) the magnetic field. These two equations basically state that the strength of the electric field is proportional to the change in the magnetic field, and vice versa, and that the direction of one is perpendicular to the other. Or more precisely, a magnetic field in a certain direction induces a
rotation of the electric field in the perpendicular plane, and vice versa.
There is a fundamental dependence on space being 3D implicit in these equations, in that 3D is the only dimension where a vector uniquely defines a rotation (i.e., the vector points in the direction of the rotational axis). In 2D, rotations can only happen around a point, so when you have a vector, there is no rotation possible around it. In 4D, on the other hand, fixing a single vector does not uniquely define a 2D rotational plane; it only determines a 3D hyperplane, and there are 3 principal rotations (along with all their oblique combinations thereof) possible. So, a single vector is insufficient to determine the plane of rotation. So basically, in 2D Maxwell's equations will most of the time have no solution (or contradictory solutions), whereas in 4D, it does not have unique solutions (it is under-determined), so it cannot fully describe what happens to one field when the other changes.
Basically, electromagnetism as we know it is strongly bound to space being 3D; to make it work in 4D would require fundamental changes that will probably make it unrecognizable as electromagnetism to us.