# Why just int/half-int spins are fundamental?

1. Apr 19, 2010

### MTd2

I just don't get it. Some people, deep down in their hearts, want to see computation as the most fundamental thing ever. This is the reasoning behind the holographic idea, at least from Verlinde. Variable spin makes it possible quantum computation through anyons, yet people still want spin like it is in the usual world, integer or half integer until plank scale and assume all kinds of weird things down there: strings, loops, several dimensions, variable dimensions, variable speed of light.

So, why variable spin gets ignored?

2. Apr 19, 2010

### atyy

3. Apr 19, 2010

### MTd2

Yeah, and John Baez liberated it from 3dimensions too, up to 4 :)

http://arxiv.org/abs/gr-qc/0603085

4. Apr 20, 2010

### atyy

But you can have quantum computation without anyons - just not topological.

5. Apr 20, 2010

### MTd2

But it is interesting that the fact that is topological, that is, precludes the existence of a metric, makes the thing more interesting to set up the condition for emerging space out of it?

6. Apr 20, 2010

### atyy

Last edited: Apr 20, 2010
7. Apr 20, 2010

### meopemuk

Some people do not make these weird assumptions.

Because (half-)integrer spin is a direct consequence of two fundamental things:

1. rotational symmetry
2. postulates of quantum mechanics

Eugene.

8. Apr 20, 2010

### genneth

Now now. Let's not be rash. You also need 3D and above for point like-particles. In 2D, point-like particles can have arbitrary phase relations. More generally, if you don't assume particles as your starting point, but more generalised, extended objects, then even more complex braiding is possible, even in high dimensions.

9. Apr 20, 2010

### MTd2

Not really. You can have those arbitrary relations in 4d too. John Baez found a way in the paper I pointed out above.

10. Apr 20, 2010

### arivero

There are two parts here:

a) what multiples and submultiples of unit of angular momentum are needed, or allowed, by mathematics. The integer version amounts to label the spherical harmonics (in any dimension, if you wish), and then the subinteger quantities due to different commutation rules and other needs of labeling.

b) why the unit of angular momentum is the same for all the problems. The other constant, "c", is more easily to be understood as universal as it related to geometry of the space-time, and it adjust the units of t and x in the 4-vector. But the role of h seems internal, and still it is a universal quantity.

11. Apr 20, 2010

### MTd2

I guess I should have clarified it better. I referred to the particle spin, only.

12. Apr 20, 2010

### dx

It's basically because the distance between two successive eigenvalues of an operator like Sz is always 1. So if there a state at 0, then you get the series {..., -1, 0, 1, ...}. If there is no state at 0, then there must be a state at some 'a' > 0, and by symmetry, a state at -a < 0. Since the difference between these must be 1, 'a' must be 1/2, and therefore you get the series {..., -3/2, -1/2, 0, 1/2, 3/2, ...}.

13. Apr 20, 2010

### MTd2

But you are assuming that angular moment is isotropic, that is, it doesn't change if you move it. The spin of anyons are not isotropic, but position dependent to the spin and position of other particles.

14. Apr 20, 2010

### Haelfix

It would help if you describe what exactly you are talking about. If we are talking about

1) D=4
2) Fundamental, point particles
3) Quantum mechanics is applicable + special relativity is applicable to all objects

Then it is a theorem that the only things you can have are half integer spins like 0, 1/2, 1, 3/2... It essentially drops out of the spin-statistics theorem.

15. Apr 20, 2010

### dx

What do you mean spin is not isotropic for anyons? The fact is that space is isotropic. Are you saying space is not isotropic for anyons? How can that be?

16. Apr 20, 2010

### MTd2

Hmm, I was not talking about point particles, you see as I gave the example of string theory or the paper from John Baez. My question was kind of generic, of why don't I see theories without int/ half-int theories.

17. Apr 20, 2010

### MTd2

Not the space, but the value of the spin depends on the position of the path described by the particle as well as their relative position.

18. Apr 20, 2010

### dx

I don't think it matters whether the particles are points or strings, since the angular momentum operator is basically a rotation of space, and it would have the same commutation relations etc.

Last edited: Apr 20, 2010
19. Apr 20, 2010

### MTd2

Int/ half int labels make sense if you are talking about point particles, because you can always make them the center of cylindrical and polar coordinates and do not worry about mass distribution. But if your object is extended, a line, there will be a couple between spin and the orbital angular momentum in relation to the center of coordinates.

In the case of string theory, I really cannot understand why anyons are not straightforwardly used because it is a 1+1 theory.

20. Apr 20, 2010

### suprised

It is because of locality, or single-valuedness of correlation functions.

In higher dimensions it is simply a consequence of the representation theory of the rotation group, it just has spinor reps but there are no fractional spin reps (together with basic rules of quantum mechanics, as others have already said here).