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Intheoretical physics, theWeinberg–Witten (WW)theorem, proved bySteven Weinberg andEdward Witten, states that massless particles (either composite or elementary) with spinj > 1/2 cannot carry aLorentz-covariant current, while massless particles with spinj > 1 cannot carry a Lorentz-covariantstress-energy. The theorem is usually interpreted to mean that thegraviton (j = 2) cannot be a composite particle in a relativisticquantum field theory.

During the 1980s,preon theories,technicolor and the like were very popular and some people speculated that gravity might be anemergent phenomenon or thatgluons might becomposite. Weinberg and Witten, on the other hand, developed ano-go theorem that excludes, under very general assumptions, the hypothetical composite and emergent theories. Decades later new theories of emergent gravity are proposed and somehigh-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 ofLorentz covariance, however, usually leads to other problems.^{[citation needed]}

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

• A 3 + 1D QFT (quantum field theory) with aconserved4-vector current${\displaystyle J^{\mu }}$ (seefour-current) which isPoincaré covariant (andgauge invariant if there happens to be anygauge symmetry which hasn't beengauge-fixed) does not admitmassless particles withhelicity |h| > 1/2 that also have nonzero charges associated with the conserved current in question.
• A 3 + 1D QFT with a non-zero conservedstress–energy tensor${\displaystyle T^{\mu \nu }}$ which isPoincaré covariant (andgauge invariant if there happens to be anygauge symmetry which hasn't beengauge-fixed) does not admit massless particles with helicity |h| > 1.

The conserved chargeQ 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,${\displaystyle |p⟩}$ and${\displaystyle |p^{\prime }⟩}$, labeled by theirlightlike4-momenta. We shall consider the case in which${\displaystyle (p-p^{\prime })}$ isn't null, which means that the momentum transfer isspacelike. Letq be the eigenvalue of those states for the charge operatorQ, so that:

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

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

with${\displaystyle q\ne 0}$.

Let's transform to areference frame wherep moves along the positivez-axis andp′ moves along the negativez-axis. This is always possible for anyspacelike momentum transfer.

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

Ifh is nonzero, we need to specify the phases of states. In general, this can't be done in a Lorentz-invariant way (seeThomas precession), but theone particle Hilbert spaceis 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 thez-axis. So, unless |h| = 0 or 1/2, all of the components have to be zero.

Weinberg and Wittendid not assume the continuity

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

Rather, the authors argue that thephysical (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,${\displaystyle {\delta }^3({\overrightarrow{p}}^{\prime }-\overrightarrow{p})}$ in the first equation can be replaced by "smeared out"Dirac delta function, which corresponds to performing the${\displaystyle d^{3}x}$ 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${\displaystyle T^{\mu \nu }}$:

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

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

with${\displaystyle E\ne 0}$.

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

Note that this theorem also applies tofree field theories. If they contain massless particles with the "wrong" helicity/charge, they have to be gauge 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 flatMinkowski spacetime, then byNoether'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 theBelinfante–Rosenfeld stress–energy tensor which is gauge-invariant. As there is no fundamentaldiffeomorphism 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/emergentgraviton.

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

There are a number of ways to see why nonabelianYang–Mills theories in theCoulomb phase don't violate this theorem. Yang–Mills theories don't have any conserved 4-current associated with the Yang–Mills charges that are both Poincaré covariant and gauge invariant. Noether's theorem gives a current which is conserved and Poincaré covariant, but not gauge invariant. As |p> is really an element of theBRST cohomology, i.e. aquotient space, it is really an equivalence class of states. As such,${\displaystyle ⟨p^{\prime }|J|p⟩}$ is only well defined if J is BRST-closed. But ifJ isn't gauge-invariant, thenJ isn't BRST-closed in general. The current defined as${\displaystyle J^{\mu }(x)\equiv \frac{\delta }{\delta A_{\mu }(x)}{S}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}}$ is not conserved because it satisfies${\displaystyle D_{\mu }{J}^{\mu }=0}$ instead of${\displaystyle {\mathrm{\partial }}_{\mu }{J}^{\mu }=0}$ where D is thecovariant derivative. The current defined after a gauge-fixing like theCoulomb gauge is conserved but isn't Lorentz covariant.

Thegauge bosons associated withspontaneously broken symmetries are massive. For example, inQCD, we have electrically chargedrho mesons which can be described by an emergent hidden gauge symmetry which is spontaneously broken. Therefore, there is nothing in principle stopping us from having composite preon models ofW andZ bosons.

On a similar note, even though thephoton is charged under the SU(2) weak symmetry (because it is thegauge boson associated with a linear combination of weak isospin and hypercharge), it is also moving through a condensate of such charges, and so, isn't an exact eigenstate of the weak charges and this theorem doesn't apply either.

On a similar note, it is possible to have a composite/emergent theory ofmassive gravity.

In GR, we have diffeomorphisms and A|ψ> (over an element |ψ> of the BRST cohomology) only makes sense if A is BRST-closed. There are no local BRST-closed operators and this includes any stress–energy tensor that we can think of.

As an alternate explanation, note that the stress tensor for pure GR vanishes (this statement is equivalent to the vacuum Einstein equation) and the stress tensor for GR coupled to matter is just the matter stress tensor. The latter is not conserved,${\displaystyle {\mathrm{\partial }}^{\mu }{T}_{\mu \nu }^{\text{matter}}\ne 0}$, but rather${\displaystyle {\mathrm{\nabla }}^{\mu }{T}_{\mu \nu }^{\text{matter}}=0}$ where${\displaystyle {\mathrm{\nabla }}^{\mu }}$ is the covariant derivative.

Ininduced gravity, the fundamental theory is also diffeomorphism invariant and the same comment applies.

If we take N=1chiralsuperQCD with N_c colors and N_fflavors with${\displaystyle N_f-2\ge N_c>\frac{2}{3}{N}_f}$, then by theSeiberg duality, this theory is dual to anonabelian${\displaystyle SU(N_f-N_c)}$ gauge theory which is trivial (i.e. free) in theinfrared limit. As such, the dual theory doesn't suffer from any infraparticle problem or a continuous mass spectrum. Despite this, the dual theory is still a nonabelian Yang–Mills theory. Because of this, the dual magnetic current still suffers from all the same problems even though it is an "emergent current". Free theories aren't exempt from the Weinberg–Witten theorem.

In aconformal field theory, the only truly massless particles are noninteractingsingletons (seesingleton field). The other "particles"/bound states have a continuousmass spectrum which can take on any arbitrarily small nonzero mass. So, we can have spin-3/2 and spin-2 bound states with arbitrarily small masses but still not violate the theorem. In other words, they areinfraparticles.

Two otherwise identical chargedinfraparticles moving with different velocities belong to differentsuperselection sectors. Let's say they have momentap′ andp respectively. Then asJ^μ(0) is a local neutraloperator, it does not map between different superselection sectors. So, is zero. The only way |p′'> and |p> can belong in the same sector is if they have the same velocity, which means that they are proportional to each other, i.e. a null or zero momentum transfer, which isn't covered in the proof. So, infraparticles violate the continuity assumption

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

This doesn't mean of course that the momentum of a charge particle can't change by some spacelike momentum. It only means that if the incoming state is a one infraparticle state, then the outgoing state contains an infraparticle together with a number of soft quanta. This is nothing other than the inevitablebremsstrahlung. But this also means that the outgoing state isn't a one particle state.

Obviously, a nonlocal charge does not have a local 4-current and a theory with a nonlocal 4-momentum does not have a local stress–energy tensor.

These theories are not Lorentz covariant. However, some of these theories can give rise to an approximate emergent Lorentz symmetry at low energies.

Superstring theory defined over a background metric (possibly with some fluxes) over a 10D space which is the product of a flat 4D Minkowski space and a compact 6D space has a massless graviton in its spectrum. This is an emergent particle coming from the vibrations of a superstring. Let's look at how we would go about defining the stress–energy tensor. The background is given by g (the metric) and a couple of other fields. Theeffective action is a functional of the background. TheVEV of the stress–energy tensor is then defined as thefunctional derivative

${\displaystyle T^{MN}(x)\equiv \frac{1}{\sqrt{-g}}\frac{\delta }{\delta g_{MN}(x)}\mathrm{\Gamma }[\text{background}].}$

The stress-energy operator is defined as avertex operator corresponding to this infinitesimal change in the background metric.

Not all backgrounds are permissible. Superstrings have to havesuperconformal symmetry, which is a super generalization ofWeyl symmetry, in order to be consistent but they are only superconformal when propagating over some special backgrounds (which satisfy theEinstein field equations plus some higher order corrections). Because of this, the effective action is only defined over these special backgrounds and the functional derivative is not well-defined. The vertex operator for the stress–energy tensor at a point also doesn't exist.

• Weinberg, Steven; Witten, Edward (1980). "Limits on massless particles".Physics Letters B.96 (1–2):59–62.Bibcode:1980PhLB...96...59W.doi:10.1016/0370-2693(80)90212-9.
• Jenkins, Alejandro (2006).Topics in particle physics and cosmology beyond the standard model (Thesis).arXiv:hep-th/0607239.Bibcode:2006PhDT........96J. (see Ch. 2 for a detailed review)

• Quantum field theory
• Quantum gravity
• Theorems in quantum mechanics
• No-go theorems
• Steven Weinberg

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