Symmetries

The symmetries of this field are generated by the following:
(x, y, z, t) → (x−a, y−b, z−ct, ct) for arbitrary reals a, b, c.
(x, y, z, t) → (−x, −y, −z, t); B → −B; E → −E

The presumed symmetries of space must also be complied with. The natures of Gibb’s notation ensures that equations written in that notation is properly symmetric. If we shift our solution in the z direction by π/2 and rotate it by π/2 about the Z axis we get:
E = (0, cos(z − ct), 0)
B = (−cos(z − ct), 0, 0).

The equations are linear and the sum of two solutions also solves the equations. Considering that e^ib = cos b + i sin b the sum is:
E = (e, f, 0)
B = (−f, e, 0)
where e + if = e^{i(z − ct)}.
Here e and f are real numbers but i(z − ct) is complex.

This solution reveals a helical symmetry:
(x, y, z, t) → (x (cos p) + y (sin p), − x (sin p) + y(cos p), z + p, t)
This is circularly polarized light.