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.
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.
Weinberg and Witten proved two separate results. According to them, the first is due to Sidney Coleman, who did not publish it:
J\mu
T\mu
The conserved charge Q is given by
\intd3xJ0
J\mu
|p\rangle
|p'\rangle
(p-p')
\begin{align} q\delta3(\vec{p}'-\vec{p})=\langlep'|Q|p\rangle&=\intd3x\langlep'|J0(\vec{x},0)|p\rangle\\ &=\intd3x\langlep'|e-i\vec{P ⋅ \vec{x}}J0(0,0)ei\vec{P ⋅ \vec{x}}|p\rangle\\ &=\intd3xei(\vec{p-\vec{p}') ⋅ \vec{x}}\langlep'|J0(0,0)|p\rangle=(2\pi)3\delta3(\vec{p}'-\vec{p})\langlep'|J0(0,0)|p\rangle \end{align}
where we have now made used of translational covariance, which is part of the Poincaré covariance. Thus:
\langlep'|J0(0)|p\rangle=
q | |
(2\pi)3 |
with
q ≠ 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,
\langlep'|J0(0)|p\rangle
\langlep'|J3(0)|p\rangle
ei(h-(-h))\theta=e2ih\theta
\langlep'|J1(0)+iJ2(0)|p\rangle
\langlep'|J1(0)-iJ2(0)|p\rangle
ei(2h+1)\theta
ei(2h-1)\theta
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
\langlep|J0(0)|p\rangle=\limp' → \langlep'|J0(0)|p\rangle
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,
\delta3(\vec{p}'-\vec{p})
d3x
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
p\mu=\intd3xT\mu(\vec{x},0)
\langlep|T0(0)|p\rangle=
E | |
(2\pi)3 |
with
E ≠ 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
\langlep'|T(0)|p\rangle
ei(2h-2)\theta
ei(2h-1)\theta
ei(2h)\theta
ei(2h+1)\theta
ei(2h+2)\theta
|h|=0, | 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.
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.
There are a number of ways to see why nonabelian Yang–Mills theories in the Coulomb 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 the BRST cohomology, i.e. a quotient space, it is really an equivalence class of states. As such,
\langlep'|J|p\rangle
| ||||
J |
Smatter
D\muJ\mu=0
\partial\muJ\mu=0
The gauge bosons associated with spontaneously broken symmetries are massive. For example, in QCD, we have electrically charged rho 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 of W and Z bosons.
On a similar note, even though the photon is charged under the SU(2) weak symmetry (because it is the gauge 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 of massive 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,
\partial\mu
matter | |
T | |
\mu\nu |
≠ 0
\nabla\mu
matter | |
T | |
\mu\nu |
=0
\nabla\mu
In induced gravity, the fundamental theory is also diffeomorphism invariant and the same comment applies.
If we take N=1 chiral super QCD with Nc colors and Nf flavors with
Nf-2\geNc>
2 | |
3 |
Nf
SU(Nf-Nc)
In a conformal field theory, the only truly massless particles are noninteracting singletons (see singleton field). The other "particles"/bound states have a continuous mass 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 are infraparticles.
Two otherwise identical charged infraparticles moving with different velocities belong to different superselection sectors. Let's say they have momenta p′ and p respectively. Then as Jμ(0) is a local neutral operator, it does not map between different superselection sectors. So,
<p'|J\mu(0)|p>
\langlep|J0(0)|p\rangle=\limp' → \langlep'|J0(0)|p\rangle
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 inevitable bremsstrahlung. 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. The effective action is a functional of the background. The VEV of the stress–energy tensor is then defined as the functional derivative
TMN(x)\equiv
1 | |
\sqrt{-g |
The stress-energy operator is defined as a vertex operator corresponding to this infinitesimal change in the background metric.
Not all backgrounds are permissible. Superstrings have to have superconformal symmetry, which is a super generalization of Weyl symmetry, in order to be consistent but they are only superconformal when propagating over some special backgrounds (which satisfy the Einstein 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.