Spin-3 Field: Gauge Invariance & Diffeomorphism?

[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?

 

[bcrowell]

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.

 

[MTd2]

He is asking for a field of a fundamental particle.

 

[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.

 

[chrispb]

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.

The reason why is the Coleman-Mandula theorem: http://en.wikipedia.org/wiki/Coleman%E2%80%93Mandula_theorem" . 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.

 

[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?

 

[petergreat]

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

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!

 

[haushofer]

The reason why is the Coleman-Mandula theorem: http://en.wikipedia.org/wiki/Coleman%E2%80%93Mandula_theorem" . 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.

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

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.

How does String Theory circumvent this?

 

[Demystifier]

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

How does String Theory circumvent this?

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

 

[MTd2]

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

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.

 

[MTd2]

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

 

[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.

 

[Demystifier]

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

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?

 

[Demystifier]

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

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

 

[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).

 

[Demystifier]

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

 

[haushofer]

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

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.