The standard linear one-dimensional wave equation is (d^2/dt^2 – d^2/dx^2) w = 0. If the propagation speed is c, then one puts a factor of (1/c^2) in front of the time-derivative term. The solution to the wave equation is a superposition of a forward propagating and a backward propagating function of the form F(x+ct) + G(x-ct).
The linear wave equation on quaternions is the same with d^2/dx^2 replaced with the Laplacian. If Dirac operator is the square root of the Laplacian, then the wave equation on quaternions can be factored as:
(1/c d/dt + Dirac) ( 1/c d/dt – Dirac ) Psi = 0
The factorization introduced 2×2 quaternion matrices in the Dirac operator because the quaternions have real four dimensions, and the Clifford algebra of R^4 is the algebra of 2×2 matrices with quaternion entries. Thus Psi must be pairs of quaternions.
Now the Dirac operator swaps the positive and negative spinors, the two components of the pairs of spinors. Thus if Psi = (A,B), the factorization contains the equations
1/c d/dt A + Dirac(B) = 0
1/c d/dt B – Dirac(A) = 0.
These are the Maxwell’s equations with curl replaced by the Dirac operator. This observation is not very rigorous but is helpful for understanding the meaning of Maxwell’s equations nonetheless because they suggest the rule of thumb at least that says: the linear wave equation on quaterions has as a square root the Maxwell’s equations on spinors.
The wave equation on quaternions is equivariant by multiplying the coordinate on the right by an SU(2) element, and therefore so is its square root system, which implies that that system (which are Maxwell’s equations with Dirac replacing curl) are PROJECTIVE lifts and their solutions are right equivariant. Therefore these equations define functions on the quaternionic projective line — functions E, B: HP^1 -> H, the electromagnetic fields.