highest weight representation and electromagnetic fields An electromagnetic field is given by 6 components (Ex,Ey,Ez,Bx,By,Bz).
Now this is a 6-dimensional irreducible representation of so(1,3) which
is a highest weight representation. So there should be a single function
S(t,x,y,z) (a "superpotential") corresponding to the highest weight and
such that all the other components are derived from it through lowering 
operators. A similar question can be asked for the electromagnetic 4-potential,
(\rho,Ax,Ay,Az); there should be a single function and lowering operators
to derive all components. Can someone provide a reference where this is is
discussed
 A: The 6-dimensional representation corresponding to the electromagnetic field is the realification of the symmetric square of the defining representation of $\mathrm{SL}(2,\mathbb{C})$, whose Lie algebra is isomorphic to that of $\mathrm{SO}(1,3)$.  The highest weight vector is the square of the highest weight vector of the defining representation.
Any book which treats electromagnetism in the "spinorial" language should discuss this: perhaps Penrose and Rindler "Spinors and spacetime"?
A: I think I see the discrepancy. The 6 components of the field then do NOT transform as
a 6 dimensional rep, even for a fixed spacetime point. The action is :
F(x) -> M(g) * F(g' x)    (F=field, M : 6x6 matrix)
which is an infinite dimensional rep. I don't see any finite reps here at all; so the
common statement "the electromagnetic field transforms as a 6 dimensional rep..." really
isn't true. That being said, is there any way to adapt the well structured theory of
highest weight representation to this infinite representation....
