Of course, it has much to do:
Leonhard Euler's four-squares identity prefigures quaternion multiplication:
It is the very reason why quaternions have a multiplicative norm, i.e. the length
of a product of two quaternions equals the product of the lengths of the
quaternions. Multiplication with a unit quaternion performs an isoclinic
double-rotation in 4-dimensional space; multiplication with an arbitrary
quaternion a double-rotation plus a stretching by the length of the quaternion.
Taking the quaternion differential
of the 4-potential A yields directly
the electromagnetic field in both components E and B:
j
k
after rearranging the terms in lines 2, 3, and 4:
i
j
k
The first line identically vanishes under Lorenz gauge
;
The terms:
are the components of the electric field (under inversion of sign, due to
the Minkowski metric (+1,-1,-1,-1));
and the terms:
;
;
are
are the components of the magnetic field.
The result is then:
i.e. the quaternion differential of the 4-potential A yields both,
the source (E) and the curl (B) part of the electromagnetic
field.
Leonhard Euler's four-square identity offers the most direct access to,
and is probably the reason for Electrodynamics!
