Lubos On Fundamental Particle Spins

In one of his usual cocky posts, Lubos argues (more aggressively than most physicists in the field would be willing to) that there must be a spin-2 graviton, and that there cannot be fundamental particles with intrinsic spin greater than 2.  His statements on these subjects are in italics below (with his most over the top statement in bold and italics).

About Gravitons

The Higgs boson became the first discovered spinless elementary particle, one with j=0 . Leptons


and quarks have j=1/2 . The photon, gluon, W-boson, Z-boson – gauge bosons – carry j=1 .

And a j=2 graviton has to exist because we know that there exist gravitational waves (see e.g. 1993

Physics Nobel Prize) and because all energy at the frequency ω is inevitably packaged into quanta of

energy E=ℏω , because of the most universal laws of quantum mechanics. Why?

If all expectation values etc.

⟨ψ|L|ψ⟩

are demanded to be periodic with period 2π/ω , it follows that |ψ⟩ must be periodic with this period,

up to an overall phase. But if |ψ⟩ is a linear superposition of various energy eigenstate terms whose

time dependence is exp(Et/iℏ) , it follows that between t=0 and t=2π/ω , the relative phases



must return to the original value which means that Ei−Ej=Nℏω for any pair of allowed

eigenvalues Ei,Ej . If the two states included in the superposition differ by an addition of a particle

or particles, the particle(s) must have E=Nℏω for N∈Z .

Again, if you don't understand the argument above sufficiently clearly so that you have eradicated all doubts about the existence of gravitons, I kindly ask you to stop reading because you're not qualified to study or discuss the allowed spins of elementary particles.

Comment:  The main caveat to this observation is that nothing in general relativity requires gravity to be a force transmitted by a quantum mechanical particle and general relativity indeed, assumes a mechanism rooted in the geometry of space-time instead. 

Wave-like behavior in a model with bosonic force carriers does imply a graviton of some sort, very likely a spin-2 graviton.  If string theory is right, there must be a massless spin-2 graviton.  But, waves can arise without particle mediated forces as well. 

The assumption that gravity has a boson exchange mechanism comparable to that of the electromagnetic, strong nuclear and weak nuclear forces is unproven and faces the serious obstacle that naive efforts to fit gravity into a quantum mechanical form with a graviton carrier have produced non-renormalizable theories that can't be rigorously proven to be finite at all and can't be used to make calculations, at the very least. 

It also isn't obvious that loop quantum gravity theories that quantitize space-time, rather than simply dropping a quantum field theory into a background space-time, necessarily implies a spin-2 graviton. Some such theories do, but not necessarily all of them do.

It also isn't manifestly obvious that even if gravity is mediated via a Standard Model-like boson force carrier that it is really a single unitary force transmitted by a single kind of force carrier.  The weak nuclear force is transmitted by three kinds of spin-1 particles (the W+, W- and Z).  The strong nuclear force is transmitted by eight varieties of gluons.  There could be, for example, a whole family of gravitons that combined act like a massless spin-2 boson, broken up by the nature of the particles that emit them, or chirally, or in some other respect.

Why spins higher than two are not fine for elementary particles

What about j=3 or higher? In that case, we would produce an even larger number of wrong-sign polarizations of the one-particle states created by the creation operators transforming as j greater than or equal to 3
tensors. The corresponding conserved charges would have to transform as j≥2 tensors. And they

 have too many components in d≥4 . In fact, if this high number of tensor components were conserved, one could prove that interactions are so constrained that they de facto vanish. Any momentum exchange between the lowest-mass scalar particles would violate the conservation laws.

This is the essence of the Coleman-Mandula theorem. There can't be conserved charges with spin greater than one. It follows – through our negative-norm-based arguments – that there can't be any

fundamental fields with j≥3 in your theory.

Well, string theory – and also its currenly fashionable "toy model", the Vasiliev higher-spin theory – circumvents this ban but the ability of these theories to avoid the conclusion critically depends on

 their having an infinite number of excitations with arbitrarily high spins j and their subtle interplay.

Let me mention that fields with spin j=5/2 would have to come with conserved charges with spin





 j= 3/2which is already too high and prohibits interesting interactions. So j=2 is indeed the highest spin of "ordinary" fundamental fields.

Comment: As Lubos notes, the Coleman-Mandula theorem has at least one loophole that string theory and SUSY attempt to exploit.  But, it does provide suggestive evidence, at least, that there are good theoretical reasons why there might not be higher spin-2 fundamental bosons.

One important loophole is the word "fundamental" in this context.  Some very important forces, such as the nuclear binding force that holds atomic nuclei together, are not fundamental and may be mediated through composite particles such as pi-mesons whose behavior is rooted more fundamentally in Standard Model QCD.  Coleman-Mandula does not bar these kind of emergent composite force carriers from having an intrinsic spin j that is greater than 2.

Context

What makes all of this news is that the Standard Model has particles of spins 0, 1/2 and 1 which have been observed, but not of spins 3/2 and 2. 

SUSY theories have spin 3/2 particles, but no spin 3/2 fundamental particles have been observed.  (Some exotic hadrons, made up of three quarks bound by gluons, with spin 3/2 have been observed, so we know how to experimentally identify such particles if they are out there.)

An observation of a single fundamental spin 3/2 particle (which would be a fermionic superpartner of a Standard Model boson such as a photon, weak force boson or gluon) would definitively shift the balance in favor of SUSY.  But, while we can theoretically describe them, just as we can theoretically describe all sorts of mythical animals (e.g. unicorns), we have yet to see a single such particle.

Spin 3/2 particles could just be too heavy for current experiments to spawn and to unstable to continue to exist for more than a moment once they come into being.  But, they also simply might not exist.