Weinberg-Witten theorem

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Steven Weinberg and Edward Witten consider the so-called emergent theories to be misguided. During the 80's, preon theories, technicolor and the like were very popular and some people were speculating that gravity might be an emergent phenomena or that gluons might be composite. So, they came up with 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 the mainstream high-energy physicists are still using this theorem to "debunk" such theories. Because most of these emergent theories aren't Lorentz covariant, the WW theorem doesn't apply. The violation of Lorentz covariance however leads to other problems.

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[edit] Theorem

[edit] A sketch of the proof

Let's assume that the assumptions of the first theorem are satisfied. By the charge Q, we mean \int d^3x\, J^0 of course. Let's look at the S-matrix element (assuming the S-matrix exists, which is a nontrivial assumption in theories without a mass gap; however, we're only interested in one particle to one particle states, which doesn't presuppose the existence of an S-matrix) \langle p'|J^\mu(0)|p\rangle where |p> and |p′> are the one-particle states of the massless charged particle with helicity h and 4-momentum p and p′ respectively. Let's assume that the sign of the helicity of both particles are the same (after all, we can't rule out self-dual particles with only one sign for the helicity! CPT theorem???). Of course p and p′ lie on the boundary of the forward light cone. Let's look at the case where p − p' isn't a null vector (i.e. lightlike momentum transfer), which means that the momentum transfer is spacelike.

Since we know that

q\delta^3(\vec{p'}-\vec{p})=\langle p'|Q|p\rangle = \int d^3x\, \langle p'|J^0(\vec{x},0)|p\rangle =\int d^3x\, \langle p'|e^{-i\vec{P}\cdot\vec{x}}J^0(0,0)e^{i\vec{P}\cdot\vec{x}}|p\rangle =\int d^3x\, e^{i(\vec{p}-\vec{p'})\cdot \vec{x}} \langle p'|J^0(0,0)|p\rangle = (2\pi)^3\delta^3(\vec{p'}-\vec{p})\langle p'|J^0(0,0)|p\rangle

where we have now made used of the translational covariance part of the Poincaré covariance. And so,

\langle p'|J^0(0)|p\rangle =\frac{q}{(2\pi)^3}

with q\neq 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′|J0(0)|p> and <p'|J3(0)|p> changes by the phase factor ei(h − ( − h))θ = e2ihθ under rotations by θ counterclockwise about the z-axis whereas <p'|J1(0)+iJ2(0)|p> and <p'|J1(0)-iJ2(0)|p> change by the phase factors ei(2h + 1)θ and ei(2h − 1)θ respectively.

If h is nonzero, we need to specify the phases of |p>. This can't be done in a Lorentz-invariant way (see Thomas precession), but the one particle Hilbert space is Lorentz-covariant and so it still makes sense to speak of the Lorentz invariance of the S-matrix. 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.

If we assume that

\langle p|J^0(0)|p\rangle =\lim_{p'\rightarrow p}\langle p'|J^0(0)|p\rangle,

(this is a dangerous assumption! see the section below), then from the fact that \langle p|J^0(0)|p\rangle = \frac{q}{(2\pi)^3}, \langle p'|J^\mu(0)|p\rangle can't be zero for all spacelike momentum transfers (This is because the case of zero momentum transfer is the limit of a sequence of spacelike momentum transfers). But this is only possible if |h|=0,\frac{1}{2}.

Similarly, for theorem 2,

p^\mu=\int d^3x\, T^{\mu 0}(\vec{x},0) and
\langle p|T^{0 0}(0)|p\rangle =\frac{E}{(2\pi)^3}

with E\neq 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 \langle p'|\mathbf{T}(0)|p\rangle transforms as ei(2h − 2)θ, ei(2h − 1)θ,ei(2h)θ, ei(2h + 1)θ or ei(2h + 2)θ 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.

[edit] Ruling out emergent theories

What has this theorem got 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 let's say 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.

[edit] Theories where the theorem is inapplicable

[edit] Nonabelian gauge theories

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, \langle p'|J|p\rangle is only well defined if J is BRST-closed. But if J isn't gauge-invariant, then J isn't BRST-closed in general. The current defined as J^\mu(x)\equiv\frac{\delta}{\delta A_\mu(x)}S_\mathrm{matter} is not conserved because it satisfies DμJμ = 0 instead of \partial_\mu J^\mu=0 where D is the covariant derivative. The current defined after a gauge-fixing like the Coulomb gauge is conserved but isn't Lorentz covariant.

[edit] Spontaneously broken gauge theories

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.

[edit] Massive gravity

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

[edit] General relativity

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.

[edit] Induced gravity

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

[edit] Seiberg duality

If we take N=1 chiral super QCD with Nc colors and Nf flavors with N_f-2 \ge N_c > \frac{2}{3}N_f, then by the Seiberg duality, this theory is dual to a nonabelian SU(Nf − Nc) gauge theory which is trivial (i.e. free) in the infrared 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"(It can't be defined in terms of the fundamental fields. Only operators which are magnetically neutral in the dual description can be defined in terms of the "fundamental" field operators and only nonlocally with respect to them. But this is completely besides the point.). Free theories aren't exempt from the Weinberg-Witten theorem (they still have to be gauge theories) (This is also besides the point but this model is only free in the IR).

[edit] Conformal field theory

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.

[edit] 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' | Jμ(0) | p > 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

\langle p|J^0(0)|p\rangle =\lim_{p'\rightarrow p}\langle p'|J^0(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.

[edit] Theories with nonlocal charges

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.

[edit] Acoustic metric theories and analog model of gravity

These theories are not Lorentz covariant. However, some of these theories can give rise to an approximate emergent Lorentz symmetry at low energies so that we can both have the cake and eat it too.

[edit] Superstring theory

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

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

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

Unfortunately (or fortunately), 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.

[edit] References

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