Why doesn't the Weinberg-Witten theorem forbid colinear photons?
The Weinberg-Witten theorem tells us that any theory that has an effective graviton, i.e. a massless helicity-2 particle as a state in the free-particle Fock space, cannot have a gauge-invariant and <PERSON>-covariant stress-energy tensor that gives the graviton nonzero energy. This is intended as a no-go theorem ruling out composite gravitons, because if the theory can be expressed using only particles of spin $\le 1$ then it presumably will have such a tensor.
A composite graviton would be a bound state of lower-spin particles such as gauge bosons, with that sum of their spins in the direction of propagation equal to 2. My question is: why does the state need to be bound? Why are we only interested in states that can be called "particles"? QED, for instance, includes massless states of helicity 2: states with 2 photons that just happen to have the same direction and spin. They are not single particles, but they are part of the Hilbert space, and matrix elements exist for them. The argument of Weinberg-Witten would seem to apply. 298
Yet QED does have a covariant stress-energy tensor, and collinear two-photon states do have nonzero energy. Why isn't this forbidden?
I think I have a partial answer: the proof of the WW theorem derives a contradiction by writing down the tensor (at the origin) as an operator on the gauge-fixed Fock space, taking its matrix elements between single-graviton states of different momenta, and taking a limit as the momenta approach coincidence while we shift the <PERSON> frame to make the momenta equal and opposite. Thus it's not enough for the graviton to have nonzero expected energy; it must have nonzero matrix elements even between states of unequal momenta. Since we are talking about an operator at a point, this seems like a reasonable assumption. Yet it seems to me that this is where my "collinear photon" case falls out: We can (I think) write the electromagnetic stress energy tensor as a sum of term of the form $a^\dagger_k a_{k'}$, meaning that we only get a nonzero matrix element when at most one photon has different momenta between the two states. Since we want states with different directions for the momenta of the collinear particle pairs, we get zero and cannot derive a contradiction.
Is this correct? Or am I perhaps confusing myself by thinking about pure momentum states rather than normalizable wavepackets? This was done in the original proof, but perhaps it introduces problems with more than one particle?
Of course, the really important question is: what changes when the state is bound; i.e. an actual composite graviton? Is it possible that we have a new loophole for the WW theorem, where we can have composite gravitons as long as we somehow force the stress-energy tensor to de diagonal in the momentum? 298