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
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1$\begingroup$ There is something not quite right in this question. The value of the electromagnetic field at a point defines a vector in a six-dimensional real irrep of the Lorentz group, but the electromagnetic field itself defines a section through a homogeneous bundle over Minkowski spacetime associated to that irrep, and the space of sections is an infinite-dimensional representation which is not highest weight and hence I don't see how to arrive at this "superpotential". $\endgroup$– José Figueroa-O'FarrillCommented Jul 26, 2011 at 0:03
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2 Answers
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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"?
answered Jul 25, 2011 at 23:47
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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....