Weinberg-Witten Theorem

Prerequisites: Canonical Quantum Field Theory, Poincare Group. Full path (23 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

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). https://physics.stackexchange.com/a/15164/37286

Concrete

Statement. Weinberg and Witten (1980) proved two statements about massless one-particle states $|p,h\rangle$ with helicity $h$, in a theory with a Lorentz-invariant vacuum and Poincaré-covariant states:

1. If the theory has a conserved current $J^\mu$ that is a Lorentz four-vector and whose charge $Q = \int d^3x\, J^0$ is nonzero on the particle, then $|h| \leq 1/2$.
2. If the theory has a conserved energy-momentum tensor $T^{\mu\nu}$ that is a Lorentz tensor and whose integral $P^\mu = \int d^3x\, T^{0\mu}$ is the momentum operator, then $|h| \leq 1$.

Proof. We set $\hbar = c = 1$ and normalise states as $\langle p',h|p,h\rangle = \delta^3(\vec{p}^{\,\prime} - \vec p)$. If the particle has charge $q$, then $\langle p',h|Q|p,h\rangle = q\,\delta^3(\vec{p}^{\,\prime}-\vec p)$. Using translation invariance, $\langle p'|J^\mu(x)|p\rangle = e^{i(p-p')\cdot x}\langle p'|J^\mu(0)|p\rangle$, and Lorentz covariance, one finds $$\lim_{p' \to p} \langle p',h|J^\mu(0)|p,h\rangle = \frac{q\, p^\mu}{(2\pi)^3 E}, \qquad \lim_{p'\to p}\langle p',h|T^{\mu\nu}(0)|p,h\rangle = \frac{p^\mu p^\nu}{(2\pi)^3 E} .$$ Both limits are nonzero (the second always, the first if $q \neq 0$). So the matrix elements cannot vanish for all $p' \neq p$ near $p$.

Now take two different lightlike momenta $p$ and $p'$. Since $p + p'$ is timelike, there is a frame in which $\vec p$ points along $+z$ and $\vec{p}^{\,\prime}$ along $-z$. Consider a rotation $R(\theta)$ by angle $\theta$ about the $z$-axis. It leaves both momenta invariant and acts on the helicity states by phases: $$U(R)|p,h\rangle = e^{i\theta h}|p,h\rangle, \qquad U(R)|p',h\rangle = e^{-i\theta h}|p',h\rangle ,$$ where the opposite sign appears because $\vec{p}^{\,\prime}$ points along $-z$. Therefore $$\langle p',h|U(R)^{-1} J^\mu(0)\, U(R)|p,h\rangle = e^{2i\theta h}\langle p',h|J^\mu(0)|p,h\rangle .$$ On the other hand $U(R)^{-1}J^\mu U(R) = R^\mu{}_\nu J^\nu$. Decomposed into components with definite angular momentum about the $z$-axis ($J^0$, $J^3$ and $J^1 \pm i J^2$), a four-vector only picks up the phases $1$ and $e^{\pm i\theta}$. Comparing both sides for all $\theta$, the matrix element vanishes unless $2h \in \{0, \pm 1\}$. For the tensor $T^{\mu\nu}$, the possible phases are $e^{i m\theta}$ with $m \in \{0,\pm1,\pm2\}$, so the matrix element vanishes unless $|2h| \leq 2$. If $|h| > 1/2$ (respectively $|h|>1$), the matrix elements vanish for all $p' \neq p$, contradicting the nonzero limits above.

What it does and does not exclude.

• The photon ($|h| = 1$) carries no electric charge, so the first statement does not constrain it.
• Gluons carry colour charge and have $|h| = 1$. This is consistent because the conserved colour current of Yang-Mills theory is not gauge invariant, and in a physical Hilbert space it is not a Lorentz-covariant four-vector operator.
• The graviton has $|h| = 2$. In general relativity there is no Lorentz-covariant, gauge-invariant energy-momentum tensor for the gravitational field, only pseudotensors, so there is no contradiction.

• The theorem rules out massless spin-2 particles (and massless particles with $|h| > 1$ in general) as bound states in any theory with a Lorentz-covariant conserved $T^{\mu\nu}$, such as ordinary local quantum field theories in flat space. This constrains attempts to obtain gravity as an emergent phenomenon from such theories. Holographic dualities are consistent with it because the graviton lives in a spacetime of different dimension from the field theory.

• A good introduction to the theorem and its meaning can be found in "https://pure.mpg.de/rest/items/item_33005_2/component/file_33006/content" by Florian Loebbert

• see also this great post by Ron Maimon
• Another great discussion can be found in Schwartz' QFT book page 153ff.

Abstract

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).

https://arxiv.org/pdf/1612.00003.pdf

Why is it interesting?

The punchline of the Weinberg-Witten theorem is that massless particles with helicity $|h| > 1/2$ cannot carry a Lorentz-covariant conserved current, and massless particles with $|h| > 1$ cannot carry a Lorentz-covariant conserved energy-momentum tensor.

"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."https://arxiv.org/abs/1007.0435

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