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Weinberg-Witten Theorem


Higher spin particles have to be coupled to conserved currents, and there are no conserved currents of high spin in quantum field theories. The only conserved currents are vector currents associated with internal symmetries, the stress-energy tensor current, the angular momentum tensor current, and the spin-3/2 supercurrent, for a supersymmetric theory.

This restriction on the currents constrains the spins to 0,1/2 (which do not need to be coupled to currents), spin 1 (which must be coupled to the vector currents), spin 3/2 (which must be coupled to a supercurrent) and spin 2 (which must be coupled to the stress-energy tensor).



Frequently occurring scientific expressions will be abbreviated: quantum field theory (QFT). local quantum physics (LQP), point-like (pl), string-like (sl), string-local quantum field theory (SLFT), power-counting bound (pcb), spontaneous symmetry breaking (SSB), string theory (ST), the Becchi-RouetStora-Tyutin gauge formalism (BRST).


This does not only lead to the sl replacement of the missing pl massless potential but it also defuses a No-Go theorem by Weinberg and Witten claiming that massless energy-momentum tensors do not exist for s ≥ 2 [37]. The correct statement is that pl conserved massless E-M tensors do not exist; they have to be replaced by sl E-M tensors which are different as densities but lead to the same global charges (generators of the Poincare group).

Why is it interesting?

The punchline of the Weinberg-Witten theorem is that there are no interacting theories of massless particles of spin greater than 2.

"The Weinberg–Witten theorem states that a massless particle of spin strictly greater than one cannot possess an energy-momentum tensor $T_{\mu \nu}$ which is both Lorentz covariant and gauge invariant. Of course, this no-go theorem does not preclude gravitational interactions. In the spin-two case, it implies that there cannot exist any gauge-invariant energy-momentum tensor for the graviton."

theorems/weinberg-witten_theorem.txt · Last modified: 2022/10/13 21:15 by