The Higgs boson became the first discovered spinless elementary particle, one with
and quarks have
Physics Nobel Prize) and because all energy at the frequency
If all expectation values etc.
are demanded to be periodic with period
up to an overall phase. But if
time dependence is
must return to the original value which means that
or particles, the particle(s) must have
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
tensors. The corresponding conserved charges would have to transform as
have too many components in
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
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
Let me mention that fields with spin
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.
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.