Spin of a p-form

Demystifier

Spin of a 1-form (e.g. electromagnetic field) is 1.
What is the spin of a p-form?
(My first guess is p, but I know that higher p-forms appear in physically interesting supersymmetric theories, whereas spins higher than 2 do not appear there.)

Mr.Brown

I guess one important point here is that it doesn´t make much sense to talk abount spins higher than 2 in any elementary particle field as for example is pointed out in weinberg vol. 1 ( the chapter about infrared behaviour), as they seem to have any interactions only in the very high energy regiem.
But anyways i guess that your guess is right that for every spinor index you get 1/2 spin.
And for every lorentz index you get spin 1 as worked out in weinberg :)

The thing in susy theories is that your forms are not scalar valued, they are grassmann and scalar valued i guess that´s the reason isn´t it ?

Demystifier

Can you explain it in more detail? What does it mean that they are both grassmann AND scalar valued?

blechman

p forms do not have spin-p. I'm actually not sure what their spin is, but a spin-s field is represented by a SYMMETRIC tensor of rank-s. And forms are antisymmetric. So they don't have the right number of degrees of freedom.

Higher p-forms only exist in higher dimensions: in simply connected D=4, you can only write down 1-forms, which are equivalent to 1-tensors (EM field). You start to have higher p-form fields in D=10,11 Supergravity theories, for example. But as you say, these objects are not higher spin. I'm not an expert on these things, but I believe you can describe these higher p-form fields in terms of branes in string theory, where the usual "point particle" description fails, and all bets are off. But as I said, I'm not an expert...

Demystifier

Supersymmetric field theories in 10 or 11 dimensions exist even without string theory. In that sense, the usual point particle description does not fail.

blechman

in D=11 sugra there are solitonic (extended) solutions that couple to these form fields - they are given the name "branes" for whatever reason. This is independent of string theory.

However, it unfortunately doesn't answer your question - rather, these solitons are the things that COUPLE to the form-fields, not the form-fields themselves.

Let me try again: in higher dimensions (where nontrivial form fields exist), the reps of the Lorentz group are not given by a single half-integer, and so talking about "spin" of these objects is misleading. I think that's the right way to think about it - the little group allows for more complicated representations than just the usual ones in D=4. Again, I'm not sure I'd be willing to invest money in this, but maybe someone with more experience than me can work out the bugs. As far as susy is concerned: what is important is that the form fields are bosonic (all indices are vector-indices).

blechman

I was just reading John Terning's excellent text "Modern Supersymmetry" where he talks about supergravity in chapter 15. There, he looks at the sugra spectrum in D=10,11 dimensions and gets p-forms, as he should. But later he looks at what these fields like like to a four-dimensional observer (someone living on a 3-brane, for instance). He finds that the massless modes (lowest KK-states) coming from the p-form fields are all spin-1 or spin-0. There are several of them(!) but they all have these lower spins. This is nothing more than the KK-paradigm that usually happens.

So I stand by what I said above, that you can't talk about spin in the usual way for higher-dimension fields. But when looking at these things from the low-energy (4D) point of view, you see these objects as spin-1 and spin-0.

The only question left is what does the 4D picture look like for form fields. Once again, you can have 0-form fields (s=0) and 1-form fields (s=1). 3(4)-form fields are dual to 1(0)-form fields, so they're also s=1(0). The only question is: what about 2-form fields. My GUESS is that these are all exact - that is, they can be written as total exterior derivatives (coboundaries). This is true if the 2nd cohomology group of Minkowski space is trivial. Is it????

Demystifier

I think that the notion of spin is a physical (not purely mathematical) property NOT uniquely determined by its kinematic geometric properties. (For example, the electromagnetic field F=dA is a 2-form, but its spin is 1. A less familiar example is a symmetric second rank tensor which, in general, contains components of spin 0, 1, and 2. If this tensor is further restricted to satisfy the Einstein equation, then only spin 2 components remain.) Instead, spin is determined by dynamics, i.e. by the Lagrangian. More precisely, spin of a field is determined by its contribution to the total conserved angular momentum, which is determined by the Lagrangian. Since p-forms in supersymmetric theories couple in the same way as the electromagnetic field (the Lagrangian is quadratic in curvature F), which is very different from that of gravitational field (linear in curvature R), I conjecture that all p-forms in supersymmetric theories have spin 1.

CarlB

One of the interesting things about spin-1/2 and massless spin-1 is that both have 2 degrees of freedom.

And I believe that they both have the same geometric phase (or Berry phase); that is, when you send them through a series of direction changes and back to their original orientation, in such a way that the path encloses a spherical area of S ster radians, they both pick up a complex phase of exp( i S / 2 ).

This is one of the things that makes me wonder if they really are that different from each other.

Demystifier

This is true for the spin-2 as well. In fact, this is valid for any spin. But this is a property of 4 dimensions of spacetime. In other spacetime dimensions the number of massless degrees of freedom depends on the spin.

Demystifier

I have just found a paper that says something more concrete about that. See page 76 of
http://lanl.arxiv.org/abs/hep-th/9912164
It is true that dynamics is important, but it is not true that all p-forms in supersymmetric theories have spin 1. Instead, their spin may be 1 or 0, or they may carry no dynamical degrees of freedom at all. In a specific supersymmetric 4-dimensional theory discussed there we have:
3-form: no dynamical degrees of freedom
2-form: s=0
1-form: s=1
The 4-form does not appear in this supersymmetric theory, but a 4-form in 4 dimensions is dual to a scalar, which means that in any 4-dimensional theory a 4-form is either nondynamical or has spin s=0.

Demystifier

If so, then what is the meaning of the following frequently stated claim?
"D=11 is the maximal number of dimensions of a supersymmetric theory that does not contain massless fields with spins higher than 2."
Does the word "spin" in that sentence referres only to the 4-dimensional view of the higher dimensional theory?
It makes sense if one visualizes spin as a rotation in the 2-plane normal to the spatial direction of particle motion. Only in 3 space dimensions such a plane is unique.

blechman

I believe you are correct. A massless field in D=11 has its rotation properties ("spin") determined by its representation of SO(9) (the "liittle group") - When we talk about its "spin", we must be referring to a SO(2) subgroup of this larger group. I can't see how it makes sense otherwise.

But I'm not sure what you meant by your earlier post of "physical" vs "mathematical" spin. Spin is the quantum number that tells you what representation of the rotation group you are in. It is "physical" in the sense that ANY quantum number is physical! But that quantum number is only defined in D=4, since in higher dimensions, the representation is no longer simply a half-integer.

This thought shouldn't bother anyone too much: in 3 dimensions, you can have "anyons" which have "spin" any real number!

To CarlB's post: D=4 is a funny world, indeed! It is where massless degrees of freedom always have only 2 components, irregardless of their spin. It is also the "critical dimension" where the Wilson-Fisher fixed point of condensed matter becomes the trivial fixed point. Maybe G-d just loves the number 4!

Comments??? Complaints???

Demystifier

Is there an attempt to use these facts to explain why we live in (or see) 4 dimensions?

blechman

Eh, sounds like mystical numerology to me! It's a funny coincidence, but I don't think there's any more to it than that.

AlphaNumeric2

That is only true if your metric is Lorentzian. If your metric has 2 time-like directions then you can go up to D=12, as is done in F theory. I think beyond that you cannot get spin=<2 only, no matter how many time-like dimensions you have.
Can you please be more specific about which section? I looked in the 15th chapter about infrared effects but nothing popped out as relating to it.

Which page is it? Thanks.

gel

Doesn't the space of p-forms on Minkowski space define a repesentation of the Poincare group, in which case it can be decomposed into spin-k representations for some set of values for k (regardless of whether it occurs in field theory or not)?