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.
[citation needed] Weinberg and Witten proved two separate results.
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.
Weinberg and Witten did not assume the continuity 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.
in the first equation can be replaced by "smeared out" Dirac delta function, which corresponds to performing the
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.
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.
So, the Weinberg–Witten theorem applies and we can't get a massless spin-2 (i.e. helicity ±2) composite/emergent graviton.
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.
The current defined after a gauge-fixing like the Coulomb gauge is conserved but isn't Lorentz covariant.
For example, in QCD, we have electrically charged rho mesons which can be described by an emergent hidden gauge symmetry which is spontaneously broken.
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.
In GR, we have diffeomorphisms and A|ψ> (over an element |ψ> of the BRST cohomology) only makes sense if A is BRST-closed.
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
As such, the dual theory doesn't suffer from any infraparticle problem or a continuous mass spectrum.
So, we can have spin-3/2 and spin-2 bound states with arbitrarily small masses but still not violate the theorem.
Two otherwise identical charged infraparticles moving with different velocities belong to different superselection sectors.
Then as Jμ(0) is a local neutral operator, it does not map between different superselection sectors.
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.
However, some of these theories can give rise to an approximate emergent Lorentz symmetry at low energies.
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.