# Insights Anyon demystified - Comments

1. Oct 28, 2016

### Demystifier

"Every quantum physicist knows that all particles are either bosons or fermions. And the standard textbook arguments that this is so do not depend on the number of dimensions.

On the other hand, you may have heard that in 2 dimensions particles can be anyons, which can have any statistics interpolating between bosons and fermions. And not only in theory, but even in reality. But how that can be compatible with the fact that all particles are either bosons or fermions? Where is the catch?"

https://www.physicsforums.com/insights/anyon-demystified/

Last edited by a moderator: Oct 28, 2016
2. Oct 28, 2016

### vanhees71

Hm, the only argument, why there are only bosons and fermions and no anyons I know about goes with the number of space dimensions, and indeed the anyons (i.e., anyonic quasiparticles in various condensed-matter contexts) live in 2 dimensions.

Also, indeed, if there is a non-symmetric interaction potential for two particles, then the two particles are in fact distinguishable, and you have no restriction concerning the symmetry whatsoever of the wave functions/quantum states under exchange of the particles. Also with two indistinguishable particles, you have only the bosonic and fermionic representation of the symmetric group of two elements. You need at least 3 indistinguishable particles to discuss anyons. So even on a semi-popular level there is a bit to demystify in your demystification :-).

3. Oct 28, 2016

### Greg Bernhardt

4. Oct 28, 2016

### Demystifier

Well, if you take e.g. the book by Streater and Wightman "PCT Spin Statistics and All That", which is a standard book with a rigorous derivation of spin-statistics theorem, they say nothing about the number of dimensions in the proof of the theorem. They assume Lorentz invariance, while quantum field theories with anyon statistics do not obey Lorentz invariance. (Attempts to construct 2+1 dimensional Lorentz invariant QFT's with intrinsic anyon statistics lead to problems.)

If you mean arguments based on non-relativistic QM (not QFT), then, in most general QM textbooks, the principle that only two statistics are possible is justified heuristically (not derived rigorously) by arguments which do not depend on number of dimensions. Of course, these general QM textbooks don't mention anyons.

5. Oct 28, 2016

### Demystifier

The anyon statistics is not based on the symmetric group, but on the braid group. It is an infinite group which has a finite symmetric group as a subgroup, even for 2 particles. In other words, anyons can be discussed even for 2 particles.

6. Oct 28, 2016

### vanhees71

What about the famous paper by Laidlaw and C. de Witt:

M. G. G. Laidlaw and C. M. DeWitt, Feynman Functional Integrals for Systems of Indistinguishable Particles, Phys. Rev. D, 3 (1970), p. 1375.

7. Oct 28, 2016

### vanhees71

This I don't understand. Perhaps it's worth to write an Insight with the sufficient amount of math!

8. Oct 28, 2016

### Demystifier

As I see from the Abstract, they rule out parastatistics, not anyons. These two exotic statistics should not be confused. Parastatistics is based on the symmetric group, anyons are based on the braid group.

9. Oct 28, 2016

### vanhees71

Okay, then I need some education on this. Any nice review(s)?

10. Oct 28, 2016

### Demystifier

11. Oct 28, 2016

### Demystifier

If the book above is not available to you (or is simply too big), try also
https://arxiv.org/abs/hep-th/9209066

12. Oct 28, 2016

### A. Neumaier

But the Lorentz group is based on 4-dimensional Minkowski space if nothing is said.

13. Oct 28, 2016

### A. Neumaier

14. Oct 28, 2016

### stevendaryl

Staff Emeritus
Hmm. John Baez in an old article claims that the spin-statistics theorem only applies for 4 or more spacetime dimensions:
http://math.ucr.edu/home/baez/braids/node2.html

"Now for the catch: the spin-statistics theorem only holds for spacetimes of dimension 4 and up."

15. Oct 28, 2016

### houlahound

How is this possible when reality is clearly not 2 dimensional?

16. Oct 29, 2016

### A. Neumaier

Because surfaces or thin films can be modeled in two space dimensions, and thin wires in one.

17. Oct 29, 2016

### houlahound

Then what's an atom, zero dimensions?

Obviously you wouldn't say that.

18. Oct 29, 2016

### A. Neumaier

It depends on the detailed level of modeling. A point particle has zero dimensions, indeed. For quantum chemistry, nuclei are treated as point particles. If one models a wire in full detail, it becomes 3-dimensional. But in mechanics one usually treats it as a 1-dimensional object. The same holds for much of the physics of nanowires. Fact is that these materials behave like predicted by lower-dimensional quantum field theory.

19. Oct 29, 2016

### houlahound

Well then I'm free to make up any particle thats true in some dimension. What's the basis for you saying this?

20. Oct 29, 2016

### A. Neumaier

The literature on the subject. Read some of the links given here: http://www.physicsoverflow.org/32114/
You are free to make up anything. The question is whether Nature will be described by your make-up.

People had studied low-dimensional quantum physics and their peculiarities for theoretical reasons, long before it was found that there are many interesting systems in Nature described by them.