Weinberg–Witten theorem

Remarks

In both parts one also assumes that the matrix elements $\langle p', h \rvert J^\mu(0) \lvert p, h \rangle$ (or those of $T^{\mu\nu}$) tend to their forward values as $p' \to p$. The proof compares two facts about these matrix elements for lightlike $p$ and $p'$. The forward limit is fixed by the charge (or the momentum) and is nonzero. Work in the frame where $\mathbf{p}' = -\mathbf{p}$. A rotation by $\theta$ about the axis of $\mathbf{p}$ fixes both momenta and multiplies the matrix element by a phase $e^{\pm 2ih\theta}$. A vector or a tensor can only produce phases with $\abs{2h} \le 1$ or $\abs{2h} \le 2$, so the matrix element vanishes for larger $\abs{h}$, which contradicts the limit.

Examples: the photon has $\abs{h} = 1$ and carries no charge, so part 1 does not forbid it. The graviton has $\abs{h} = 2$, so by part 2 a theory with a graviton has no Lorentz-covariant conserved stress-energy tensor. In general relativity the gravitational energy is indeed not given by a covariant tensor, which is consistent with the theorem.

Going deeper

For a student

The theorem is a tool for ruling out ideas, such as an emergent graviton in a theory with an ordinary Lorentz-covariant stress tensor. Its loopholes are known. Gauge theories have currents that are not gauge invariant. In theories where spacetime itself emerges, as in AdS/CFT, the graviton does not live in the same spacetime as the stress tensor. It also shows that gluons, which have helicity $\pm 1$ and color charge, cannot have a gauge-invariant Lorentz-covariant conserved color current.