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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

flag	asked Jul 25 2011 at 23:04 r n tsai 21●1

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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". – José Figueroa-O'Farrill Jul 26 2011 at 0:03

José Figueroa-O'Farrill · Answer 1 · 2011-07-25 23:47:47Z UTC

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"?

r n tsai · Answer 2 · 2011-07-26 20:12:08Z UTC

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....

highest weight representation and electromagnetic fields

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