# Spin in higher dimensions

1. Apr 22, 2010

### haael

Do you know of any papers about spin in dimensions>4? It seems that there are two independent spins in 4+1 dimensions, since you can replace spatial dimension 1 with 2 and 3 with 4, each pair not messing with the other.

I found only one paper on arxiv: <http://arxiv.org/abs/0908.2484> [Broken] on 5D black holes, where is stated that in 5D indeed there are two spins.

But what about more quantum physics? Did anyone have identified second spin with, say, isospin? Are there any works?

Last edited by a moderator: May 4, 2017
2. Apr 25, 2010

### Naty1

Unsure just what you are looking for but there should be a bunch of papers under String Theory.

3. Apr 25, 2010

### Spinnor

4. Apr 26, 2010

### tom.stoer

Spin in n dimensions is just the theory of representations of SO(n).

There are some surprises for small n, e.g. so(3) = su(2) and so(4) = su(2)+su(2), but for higher n it's just so(n).

5. Apr 26, 2010

### haael

I'm looking for a proof of Kaluza-Klein theory or like.

I'm a big fan of this theory and I would be very sad if the fifth dimension I dreamt of as a child simply didn't exist :). Yet, when I had finally understood what spin is, a simple test came to my mind: in 5d there shoul be two spins. So our particles should have some another degree of freedom. If they have it, Kaluza-Klein may be true. If not, farewell hyperspace.

Now I look for some papers that may have explained this. Do we really have such another spin? May isospin be our missing factor? Or maybe this another spin is somehow supressed or unobservable? Or maybe in Minkowski 4+1 spacetime there are no two spins at all? Curiosity is killing me.

I have read some papers on Kaluza-Klein and higher dimensions and they focus on "translation" degrees of freedom, but spin is kind of taboo topic.

However, if Kaluza-Klein was true, it would be simply beautiful. Unified gravity and electromagnetism (and weak interaction, if the second spin was isospin) in a pure geometric way. It can also explain CP violation. Supersymmetry may give us matter. I even read that the dilaton may be the Higgs itself.

So, if you know of any papers about physical consequences of five dimensions on spin, please point me some. Maths is appreciated, but I rather want to proove or disproove Kaluza-Klein.

6. Apr 26, 2010

### tom.stoer

I do not understand what you mean by "two spins". In higher dimensions spin is just the quantum number that corresponds to representations of SO(N). In higher dimensions the multiplets are more complicated, but it's still one physical quantity.

By the Coleman-Mandula no-go theorem you cannot combine isospin (or some other internal quantum number) with a spacetime symmetry, except for the "trivial" combination. In order to get a kind of intertwining between spin and isospin you need supersymmetry.

Can you please explain what you mean by "two spins"?

7. Apr 26, 2010

### haael

Two sets of spin quantum numbers, or a cartesian product of the two if you like.

Say, electron can have states (+1/2) and (-1/2). In 4 spatial dimensions it seems it should be (+1/2,+1/2), (+1/2, -1/2), (-1/2, +1/2), (-1/2, -1/2).

Also, you can define two spin operator vectors that will commute with each other:
$$S_i =\epsilon_{ijk} [S_j, S_k]$$
$$Z_i =\epsilon_{ijk} [Z_j, Z_k]$$

Isospin is the only thing that comes to my mind when I try to find interpretation of this another "spin". So, if you convince me that isospin can not be the manifestation of spin in another dimension, I will have to forget about Kaluza-Klein.

Note that this is only my limited knowlegde. I may be just plain wrong.

8. Apr 26, 2010

### tom.stoer

This is not how spin works in general. In 3-dim. space you have SO(3)~SU(2) which explains the half-integer values. In 4-dim. spacetime you have SO(3,1)~SU(2)*SU(2) which explains the bi-spinor structure of the Dirac equation. But in higher dimensions such a factorization is in general not available.

Spin and isospin are not intertwined. Isospin is invariant under Lorentz boosts and rotations (a rotated proton is still a proton) and vice versa (an iso-rotated proton becomes a neutron but the spin orientation is not affected).

If you want to get a non-trivial intertwining of spin and isospin (or any other internal symmetry) you have to find a loop-hole in the Coleman-Mandula no-go theorem - which could be supersymmetry ...

9. Jun 4, 2011

### ismaili

Dear guys,

I'm also interested in the definition of spin in various dimensions.
For the usual 4D QFT, the definition of spin comes from the other Casimir operator $$W_\mu \equiv \frac{1}{2}\epsilon_{\mu\nu\rho\sigma}P^\nu J^{\rho\sigma}$$However, in higher dimensions, we can have defined a lot of Casimir operators, each of them can derive a definition of "spin", so, I am wondering that what is the definition of spin in the usual literature concerning about theories in higher dimensions?

And, what's the general rule to find out all the Casimir operators in a giving symmetry? Thanks!