Weinberg–Witten theorem

In theoretical physics, the Weinberg–Witten (WW) theorem, proved by Steven Weinberg and Edward Witten, states that massless particles (either composite or elementary) with spin j > 1/2 cannot carry a Lorentz-covariant current, while massless particles with spin j > 1 cannot carry a Lorentz-covariant stress-energy. The theorem is usually interpreted to mean that the graviton (j = 2) cannot be a composite particle in a relativistic quantum field theory.

During the 1980s, preon theories, technicolor and the like were very popular and some people speculated that gravity might be an emergent phenomenon or that gluons might be composite. Weinberg and Witten, on the other hand, developed a no-go theorem that excludes, under very general assumptions, the hypothetical composite and emergent theories. Decades later new theories of emergent gravity are proposed and mainstream high-energy physicists are still using this theorem to "debunk" such theories. Because most of these emergent theories aren't Lorentz covariant, the WW theorem doesn't apply. The violation of Lorentz covariance, however, usually leads to other problems.

Weinberg and Witten proved two separate results. According to them, the first is due to Sidney Coleman, who did not publish it:

The conserved charge Q is given by ${\displaystyle \int d^{3}x\,J^{0}}$. We shall consider the matrix elements of the charge and of the current ${\displaystyle J^{\mu }}$ for one-particle asymptotic states, of equal helicity, ${\displaystyle |p\rangle }$ and ${\displaystyle |p'\rangle }$, labeled by their lightlike 4-momenta. We shall consider the case in which ${\displaystyle (p-p')}$ isn't null, which means that the momentum transfer is spacelike. Let q be the eigenvalue of those states for the charge operator Q, so that:

...
Wikipedia