Weinberg–Witten theorem

Template:Short description Template:Multiple issues 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.

Background

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 some high-energy physicists are still using this theorem to try and refute 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.Template:Citation needed

Theorem

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

• A 3 + 1D QFT (quantum field theory) with a conserved 4-vector current $J^{\mu }$ (see four-current) which is Poincaré covariant (and gauge invariant if there happens to be any gauge symmetry which hasn't been gauge-fixed) does not admit massless particles with helicity |h| > 1/2 that also have nonzero charges associated with the conserved current in question.
• A 3 + 1D QFT with a non-zero conserved stress–energy tensor $T^{\mu \nu }$ which is Poincaré covariant (and gauge invariant if there happens to be any gauge symmetry which hasn't been gauge-fixed) does not admit massless particles with helicity |h| > 1.

A sketch of the proof

The conserved charge Q is given by $\int d^{3}x{J}^0$. We shall consider the matrix elements of the charge and of the current $J^{\mu }$ for one-particle asymptotic states, of equal helicity, $|p⟩$ and $|p^{\prime }⟩$, labeled by their lightlike 4-momenta. We shall consider the case in which $(p-p^{\prime })$ 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:

$\begin{array}{cc}q{\delta }^3({\overrightarrow{p}}^{\prime }-\overrightarrow{p})=⟨p^{\prime }|Q|p⟩& =\int d^{3}x⟨p^{\prime }|J^0(\overrightarrow{x},0)|p⟩\\ & =\int d^{3}x⟨p^{\prime }|e^{-i\overrightarrow{P}\cdot \overrightarrow{x}}{J}^0(0,0)e^{i\overrightarrow{P}\cdot \overrightarrow{x}}|p⟩\\ & =\int d^{3}x{e}^{i(\overrightarrow{p}-{\overrightarrow{p}}^{\prime })\cdot \overrightarrow{x}}⟨p^{\prime }|J^0(0,0)|p⟩=(2\pi {)}^3{\delta }^3({\overrightarrow{p}}^{\prime }-\overrightarrow{p})⟨p^{\prime }|J^0(0,0)|p⟩\end{array}$

where we have now made used of translational covariance, which is part of the Poincaré covariance. Thus:

$⟨p^{\prime }|J^0(0)|p⟩=\frac{q}{(2\pi {)}^3}$

with $q\ne 0$.

Let's transform to a reference frame where p moves along the positive z-axis and p′ moves along the negative z-axis. This is always possible for any spacelike momentum transfer.

In this reference frame, $⟨p^{\prime }|J^0(0)|p⟩$ and $⟨p^{\prime }|J^3(0)|p⟩$ change by the phase factor $e^{i(h-(-h))\theta }=e^{2ih\theta }$ under rotations by θ counterclockwise about the z-axis whereas $⟨p^{\prime }|J^1(0)+i{J}^2(0)|p⟩$ and $⟨p^{\prime }|J^1(0)-i{J}^2(0)|p⟩$ change by the phase factors $e^{i(2h+1)\theta }$ and $e^{i(2h-1)\theta }$ respectively.

If h is nonzero, we need to specify the phases of states. In general, this can't be done in a Lorentz-invariant way (see Thomas precession), but the one particle Hilbert space is Lorentz-covariant. So, if we make any arbitrary but fixed choice for the phases, then each of the matrix components in the previous paragraph has to be invariant under the rotations about the z-axis. So, unless |h| = 0 or 1/2, all of the components have to be zero.

Weinberg and Witten did not assume the continuity

$⟨p|J^0(0)|p⟩=\underset{p^{\prime }\to p}{lim}⟨p^{\prime }|J^0(0)|p⟩$.

Rather, the authors argue that the physical (i.e., the measurable) quantum numbers of a massless particle are always defined by the matrix elements in the limit of zero momentum, defined for a sequence of spacelike momentum transfers. Also, ${\delta }^3({\overrightarrow{p}}^{\prime }-\overrightarrow{p})$ in the first equation can be replaced by "smeared out" Dirac delta function, which corresponds to performing the $d^{3}x$ volume integral over a finite box.

The proof of the second part of theorem is completely analogous, replacing the matrix elements of the current with the matrix elements of the stress–energy tensor $T^{\mu \nu }$:

$p^{\mu }=\int d^{3}x{T}^{\mu 0}(\overrightarrow{x},0)$ and

$⟨p|T^{00}(0)|p⟩=\frac{E}{(2\pi {)}^3}$

with $E\ne 0$.

For spacelike momentum transfers, we can go to the reference frame where p′ + p is along the t-axis and p′ − p is along the z-axis. In this reference frame, the components of $⟨p^{\prime }|𝐓(0)|p⟩$ transforms as $e^{i(2h-2)\theta }$, $e^{i(2h-1)\theta }$,$e^{i(2h)\theta }$, $e^{i(2h+1)\theta }$ or $e^{i(2h+2)\theta }$ under a rotation by θ about the z-axis. Similarly, we can conclude that $|h|=0,\frac{1}{2},1$

Note that this theorem also applies to free field theories. If they contain massless particles with the "wrong" helicity/charge, they have to be gauge theories.

Ruling out emergent theories

What does this theorem have to do with emergence/composite theories?

If let's say gravity is an emergent theory of a fundamentally flat theory over a flat Minkowski spacetime, then by Noether's theorem, we have a conserved stress–energy tensor which is Poincaré covariant. If the theory has an internal gauge symmetry (of the Yang–Mills kind), we may pick the Belinfante–Rosenfeld stress–energy tensor which is gauge-invariant. As there is no fundamental diffeomorphism symmetry, we don't have to worry about that this tensor isn't BRST-closed under diffeomorphisms. So, the Weinberg–Witten theorem applies and we can't get a massless spin-2 (i.e. helicity ±2) composite/emergent graviton.

If we have a theory with a fundamental conserved 4-current associated with a global symmetry, then we can't have emergent/composite massless spin-1 particles which are charged under that global symmetry.

References

• Template:Cite journal
• Template:Cite thesis (see Ch. 2 for a detailed review)