# What's a spin-3 field?

1. Aug 15, 2010

### petergreat

Is a spin-3 field conceivable?
For a spin-1 field, you need gauge invariance to cancel negative-norm states.
For a spin-2 field, you need diffeomorphism to cancel negative-norm states.
What should be done to a spin-3 field then?

2. Aug 15, 2010

### bcrowell

Staff Emeritus
You can have a composite object like a nucleus that has spin 3. At low energies, you can't tell from its statistical behavior that it's not a fundamental particle.

3. Aug 15, 2010

### MTd2

He is asking for a field of a fundamental particle.

4. Aug 16, 2010

### Demystifier

Massive spin fields do not have gauge invariance (for spin 1) or diffeomorphism invariance (for spin 2). Thus, petergreat is probably interested in massless spin fields. I am not sure why, but I think it is impossible to construct a "consistent" massless field theory for spin larger than 2. There are some notes on it, with further references, in the Weinberg´s QFT textbook.

5. Aug 16, 2010

### chrispb

The reason why is the Coleman-Mandula theorem: http://en.wikipedia.org/wiki/Coleman%E2%80%93Mandula_theorem" [Broken]. The basic idea is that if you have a spin-3 field, you have a spin-3 conserved current with it, and therefore there's a conserved current with two Lorentz (vector) indeces. This theorem forbids that.

To take the theorem apart, if you were to try to conserve both your spin-3 current and momentum in a 2 -> 2 collision, you would find that the outgoing momentum must be the incoming momentum, up to various sign issues. This implies that they didn't interact in the first place.

Last edited by a moderator: May 4, 2017
6. Aug 16, 2010

### MTd2

There are always loop holes to that theorem, which is always linked to interesting theories. Two examples are supersymmetry the other and quantum grups in 2+1 dimensions, which is linked to non abelian anyons, with an application to quantum computation. Maybe spin 3 is related to the former?

7. Aug 16, 2010

### petergreat

I saw Lubos' post about http://motls.blogspot.com/2010/02/holography-vasilievs-higher-spin.html" [Broken].

Lubos says Coleman-Mandula theorem can be bypassed in a theory containing infinitely many particles with arbitrarily high spin. But the theory is not the same as perturbative string theory because in the latter case the higher-spin particles are not massless.

So what on earth is this "higher-spin gauge theory"? Is it a well-defined Lorentz invariant field theory? Or is it something lacking a complete definition like string theory? I would be surprised if this theory of massless higher-spin particles is already in good shape, since no one has really figured out massless spin-2 (graviton) yet!

Last edited by a moderator: May 4, 2017
8. Aug 17, 2010

### haushofer

But this spin-3 field must be a gauge field then, right?

How does String Theory circumvent this?

Last edited by a moderator: May 4, 2017
9. Aug 17, 2010

### Demystifier

String states with spin 3 are MASSIVE states, so they do NOT correspond to a gauge field.

10. Aug 17, 2010

### MTd2

He understood that. What he finds strange it is that theory is gauge and massive.

PS.: I thought he was quoting petergreat's last post. Now it is me that thinks it is strange.

11. Aug 17, 2010

### MTd2

I don't know. From what petergreat said, it is "higher-spin gauge theory". Does't make sense.

12. Aug 17, 2010

### chrispb

I've never heard of a spin-3 particle that wasn't composite in nature, either stringy or not. (I'm certainly no expert in all of this yet, so that doesn't mean it doesn't exist!) When I say spin, I'm referring to its Lorentz group structure, which would imply that the conserved charge is a Lorentz generator and not a gauge one. This'd restrict the charge to a tensor product of two ps or maybe the metric.

You COULD make a gauge group of SU(2) and then have "spin"-3 particles which just lived in the 7-dimensional rep of the gauge group. I don't see anything wrong with that. I will say that I don't know of any fundamental particle that has a quantum numbers that belong to anything but the fundamental, trivial or adjoint reps of their gauge groups, which I've always found a little strange.

My knowledge of string theory is incredibly limited. What goes into the CM theorem are locality, a mass gap, a restriction to an algebra (as opposed to a superalgebra) and perhaps a few other things. SUSY circumvents things by extending the Poincare algebra into the SUSY superalgebra. I'm under the impression that string theory has a mass gap, so I really don't know where subtle tricks would come up.

13. Aug 17, 2010

### Demystifier

String theory can be viewed as a gauge field theory with INFINITE number of massive fields. Perhaps the non-existence theorems refer only to theories with a finite number of fields?

14. Aug 17, 2010

### Demystifier

It seems he is referring to the Vasiliev's theory, of which I know nothing.

15. Aug 17, 2010

### MTd2

I found this recent review paper:

http://arxiv.org/abs/1007.0435

How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples

Xavier Bekaert, Nicolas Boulanger, Per Sundell
(Submitted on 2 Jul 2010)
Aiming at non-experts, we explain the key mechanisms of higher-spin extensions of ordinary gravity. We first overview various no-go theorems for low-energy scattering of massless particles in flat spacetime. In doing so we dress a dictionary between the S-matrix and the Lagrangian approaches, exhibiting their relative advantages and weaknesses, after which we high-light potential loop-holes for non-trivial massless dynamics. We then review positive yes-go results for nonabelian cubic higher-derivative vertices in constantly curved backgrounds. Finally we outline how higher-spin symmetry can be reconciled with the equivalence principle in the presence of a cosmological constant leading to the Fradkin--Vasiliev vertices and Vasiliev's higher-spin gravity with its double perturbative expansion (in terms of numbers of fields and derivatives).

16. Aug 18, 2010

### Demystifier

MTd2, thanks for the link. I allways wanted to see a non-expert review on that stuff.

17. Aug 20, 2010

### haushofer

Ah, ok, how silly of me. So these "higher spin states" are massive, do not correspond to gauge particles, and as such there is no problem in having them.