QHE: rotational invariance, no terms linear in E or B

[binbagsss]

'Let’s first see what all of this means in the context of d = 3 + 1 dimensions. If we have rotational invariance then we can’t write down any terms linear in E or B. The first terms that we can write down are instead ...'

Why is this? I don't understnad . My thoughts would be pictruing the set up needing to be rotationally invariant, and since E and B are perpendicular a linear term alone wouldn't do this, but this wouldn't explaiin why E.E and B.B are invariant, I don't really know what I'm talking about, as you can tell

many thanks in advancehttp://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf , page 145 David Tong notes QHE , chapter 5 (eq. 5.5)

 

[PeterDonis]

Why is this?

Because vectors are not invariant under rotations: the rotation makes them point in a different direction. But the magnitudes of vectors are invariant under rotations, since rotation doesn't change the length of a vector, only its direction.