# Anyon Demystified

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?

This, of course, is discussed in many papers (and a few books) devoted to anyons. But my intention is not to present a summary of the standard literature. I want to explain it in my own way which, I believe, demystifies anyons in a way that cannot be found explicitly in the existing literature. I will do it in a conceptual non-technical way with a minimal number of explicitly written equations. Nevertheless, the things I will say can be viewed as a reinterpretation of more elaborated equations that can easily be found in the standard literature. In this sense, my explanation is meant to be complementary to the existing literature.

Consider a 2-particle wave function ##\psi({\bf x}_1,{\bf x}_2)##. The claim that it is either bosonic or fermionic means that it is either symmetric or antisymmetric, i.e.

$$\psi({\bf x}_1,{\bf x}_2)=\pm \psi({\bf x}_2,{\bf x}_1) ….. (1)$$

But suppose that the wave function satisfies a Schrodinger equation with a potential ##V({\bf x}_1,{\bf x}_2)##, which has a property of being asymmetric

$$V({\bf x}_1,{\bf x}_2) \neq V({\bf x}_2,{\bf x}_1) .$$

In general, with an asymmetric potential, the solutions of the Schrodinger equation will not satisfy (1). And yet, no physical principle forbids such asymmetric potentials. It looks as if it is very easy to violate the principle that wave function must be either bosonic or fermionic.

But that is not really so. The principle that wave function must be symmetric or antisymmetric refers only to identical particles, i.e. particles that cannot be distinguished. On the other hand, if the potential between the particles is not symmetric, then the particles are not identical, i.e. they can be distinguished. In that case, (1) does not apply.

Now assume that the asymmetric potential takes a very special form, so that the wave function of two non-identical bosons or fermions takes the form

$$\psi({\bf x}_1,{\bf x}_2)=e^{i\alpha} \psi({\bf x}_2,{\bf x}_1)$$

where ##\alpha## is an arbitrary real number. This is the anyon. And there is nothing strange about it, it is simply a consequence of the special interaction between two non-identical particles. The effect of interaction is to simulate an exotic statistics (exotic exchange factor ##e^{i\alpha}##), while the “intrinsic” statistics of particles (i.e. statistics in the absence of exotic interaction) is either bosonic or fermionic.

The only non-trivial question is, does such interaction exists? It turns out (the details of which can be found in standard literature) that mathematically such an interaction exists, provided that the particles live in 2 dimensions and that the potential is not really a scalar potential ##V({\bf x}_1,{\bf x}_2)## but a vector potential ##{\bf A}({\bf x}_1,{\bf x}_2)##. And physically, that is in the real world, such interaction does not exist for elementary particles such as electrons, but only for certain quasi-particles in condensed matter physics. These are the main conceptual ideas of anyons, while the rest are technical details that can be found in standard literature.

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 :-).

Congrats on your first Insight!

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. 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.

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.

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.

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.

http://link.aps.org/abstract/PRD/v3/i6/p1375

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

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

http://link.aps.org/abstract/PRD/v3/i6/p1375

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.

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

I don't think that I can explain it better that the standard literature, so I would rather refer you to some standard literature. For instance, the explanation in the book

https://www.amazon.com/Fractional-Statistics-Quantum-Theory-2nd/dp/9812561609

is quite good.

If the book above is not available to you (or is simply too big), try also

https://arxiv.org/abs/hep-th/9209066