I just finished my first upper-division E&M course and have a question. Why do the electric field and the magnetic field have different units if they are essentially rotations of each other? Moreover if the same particle, the photon is the force carrying particle in QED how can they have different units. Thank you!
That is only an issue of the system of units you are using. In natural units $c=1$ and $E$ and $B$ have the same units, as $E=cB$, so dimensionally $E=B$ "naturally".