Electromagnetic Waves in N-d Spacetime

[DonSoreno]

I've found another interesting text on the Spacetime (Clifford) Algebra.
https://arxiv.org/abs/1411.5002

I want to adapt their formalism to the 4d case and beyond.


For an EM wave in a vacuum, the authors use this equation:
∂ F = ∂ · F(x) + ∂ ∧ F(x) = 0

Where F is the Electromagnetic Bivector:*
F = E₁e01 E₂e02 E₃e03 + B₁₂e12 + B₁₃e13 + B₂₃e23

And ∂ is the first derivative (with respect to space and time): ( d/dt; d/dx1; d/dx2; d/dx3).


Now this can be generalized to higher dimensions:
For 4d:
F = E₁e01 E₂e02 E₃e03 + E₄e04 + B₁₂e12 + B₁₃e13 + B₁₄e14 + B₂₃e23 + ...
(4 electric components, 6 magnetic components)
















*In this formalism both electric and magnetic field are bivectors.
Some other Clifford Algebra-based formalisms use scalars for time, and vectors for space, and thus have electric vector E = E1 e1 + E2 e2 + E3
and magnetic bivectors B1 e23 + B2 e13 + B3 e12
(Those approaches do not nicely generalize to the relativist framework, where we could otherwise just model Lorentz Boosts as hyperbolic rotations.)

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[DonSoreno]

In this approach I used the EM potential directly, so the relation between E and B is clear.

I'm not sure if the EM potentials themselves are correct, since this isn't a very common approach.
However, I think the Electric & Magnetic components should work out to some extend.