What is the explanation for the existence of anyons in 2 dimensions?

[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?"

Read more here
https://www.physicsforums.com/insights/anyon-demystified/

 

[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 :-).

 

[Greg Bernhardt]

Congrats on your first Insight!

 

[Demystifier]

Hm, the only argument, why there are only bosons and fermions and no anyons I know about goes with the number of space dimensions

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.

 

[Demystifier]

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.

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.

 

[vanhees71]

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.

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

 

[vanhees71]

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.

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

 

[Demystifier]

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

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.

 

[vanhees71]

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

 

[Demystifier]

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

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/dp/9812561609/?tag=pfamazon01-20
is quite good.

 

[Demystifier]

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

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

 

[A. Neumaier]

they say nothing about the number of dimensions in the proof of the theorem. They assume Lorentz invariance,

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

 

[A. Neumaier]

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

http://www.physicsoverflow.org/32114/
http://www.physicsoverflow.org/14488/
http://www.physicsoverflow.org/16826/

 

[stevendaryl]

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.

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."

 

[houlahound]

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.

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

 

[A. Neumaier]

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

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

 

[houlahound]

Then what's an atom, zero dimensions?

Obviously you wouldn't say that.

 

[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.

 

[houlahound]

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

 

[A. Neumaier]

What's the basis for you saying this?

The literature on the subject. Read some of the links given here: http://www.physicsoverflow.org/32114/

Well then I'm free to make up any particle that's true in some dimension.

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.

 

[houlahound]

OK that caliber of paper is above my skill level.

 

[A. Neumaier]

OK that caliber of paper is above my skill level.

Is http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/advanced-physicsprize2016.pdf about this year's Nobel price in physics (which was given for theoretical achievements in lower-dimensional physics) more understandable?

 

[houlahound]

Scan read it on phone, definitely be printing it off for a proper read. Thanks.

 

[Demystifier]

Is http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/advanced-physicsprize2016.pdf about this year's Nobel price in physics (which was given for theoretical achievements in lower-dimensional physics) more understandable?

In fact, it was this year's Nobel prize that motivated me to read more about anyons.

 

[Demystifier]

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

Yes, but you can repeat their proof step by step in any number of dimensions, including 2+1. Indeed, as I already said, there is no consistent local Lorentz invariant QFT in 2+1 dimensions with intrinsic anyon statistics.

 

[Demystifier]

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."

See my post above. So the spin-statistics theorem, with all the explicit assumptions of the theorem, is valid even in 2+1. In 2+1, anyons are OK as non-relativistic (first-quantized) QM, but not as relativistic local (second-quantized) QFT.

 

[stevendaryl]

See my post above. So the spin-statistics theorem, with all the explicit assumptions of the theorem, is valid even in 2+1. In 2+1, anyons are OK as non-relativistic (first-quantized) QM, but not as relativistic local (second-quantized) QFT.

I'm still trying to reconcile your claim that the theorem is valid in 2+1 dimensions with John Baez' claim that it is only valid in 4 (and higher) dimensions. Are you saying that there is a general theorem that doesn't assume Lorentz invariance, which is valid in 4 or more dimensions, and then a specialization that does assume Lorentz invariance, which is valid in any number of dimensions?

 

[stevendaryl]

I'm still trying to reconcile your claim that the theorem is valid in 2+1 dimensions with John Baez' claim that it is only valid in 4 (and higher) dimensions. Are you saying that there is a general theorem that doesn't assume Lorentz invariance, which is valid in 4 or more dimensions, and then a specialization that does assume Lorentz invariance, which is valid in any number of dimensions?

Or are you saying that Baez is just wrong about the loophole in 2+1 dimensions?

 

[stevendaryl]

Or are you saying that Baez is just wrong about the loophole in 2+1 dimensions?

The argument I heard, which I didn't really understand, is that if you consider paths in configuration space, the path where two identical particles exchange places is continuously deformable to a path were one of the particles rotates through 360 degrees and nothing else happens. So somehow there must be a relationship between particle exchange and rotations. But the deformation requires at least 3 dimensions, so there is a 2-D loophole. Something like that.

 

[rubi]

The proof of spin-statistics in Streater & Wightman depends heavily on 4 dimensions through the theory of complexified Lorentz transformations and the analytic continuation of the Wightman functions to the extended tube. It's just that Streater & Wightman prove these facts in earlier chapters and the proof of spin-statistics just silently uses them.

 

[Demystifier]

The argument I heard, which I didn't really understand, is that if you consider paths in configuration space, the path where two identical particles exchange places is continuously deformable to a path were one of the particles rotates through 360 degrees and nothing else happens. So somehow there must be a relationship between particle exchange and rotations. But the deformation requires at least 3 dimensions, so there is a 2-D loophole. Something like that.

There are two ways to define quantum statistics. One is in terms of particle exchange. The other is in terms of algebra of field operators. The former is a definition for QM, where the fundamental degrees are particles. The later is a definition for QFT, where the fundamental degrees are fields. The two definitions are closely related, but not fully equivalent.

The argument above is a valid argument in QM. But it is not a valid argument in QFT. QM and QFT are different theories. Even if they are not so different in the case of bosons and fermions, they are very different in the case of anyons. The standard spin-statistics theorem is a theorem about relativistic QFT, not a theorem about QM. We have a consistent QM theory of anyons in 2+1, but not a consistent local relativistic QFT of anyons in 2+1. The standard spin-statistics theorem and the anyon theory cannot be directly compared because they talk about different objects; one is talking about fields and the other about particles.

It seems that Baez (and apparently many others) failed to realize that two different ways of defining statistics cannot be directly compared. The fact that anyons as particles can live in 2+1 QM does not contradict the other fact that anyons as fields can not live in 2+1 local relativistic QFT.

 

[Demystifier]

The proof of spin-statistics in Streater & Wightman depends heavily on 4 dimensions through the theory of complexified Lorentz transformations and the analytic continuation of the Wightman functions to the extended tube. It's just that Streater & Wightman prove these facts in earlier chapters and the proof of spin-statistics just silently uses them.

You may be right about that, but the original Pauli version of spin-statistics theorem does not depend on it.

For various derivations of spin-statistic relation see the book
https://www.amazon.com/dp/9810231148/?tag=pfamazon01-20

 

[stevendaryl]

There are two ways to define quantum statistics. One is in terms of particle exchange. The other is in terms of algebra of field operators. The former is a definition for QM, where the fundamental degrees are particles. The later is a definition for QFT, where the fundamental degrees are fields. The two definitions are closely related, but not fully equivalent.

This is a topic for another thread, but I am intrigued by the possibility of a pure-particle formulation of relativistic quantum mechanics that is equivalent to QFT. It would at first seem impossible, because of particle creation, but there are at least two hand-wavy approaches to getting around that: (1) the Dirac sea idea, and (2) viewing particle creation in terms of a particle that can travel back and forth in time. I don't know whether there has been any serious effort to make either of these work rigorously.

 

[Demystifier]

This is a topic for another thread, but I am intrigued by the possibility of a pure-particle formulation of relativistic quantum mechanics that is equivalent to QFT. It would at first seem impossible, because of particle creation, but there are at least two hand-wavy approaches to getting around that: (1) the Dirac sea idea, and (2) viewing particle creation in terms of a particle that can travel back and forth in time. I don't know whether there has been any serious effort to make either of these work rigorously.

I guess I could say something about that too, but not in this thread.

 

[rubi]

You may be right about that, but the original Pauli version of spin-statistics theorem does not depend on it.

Well, Pauli's proof also requires 4 dimensions, since it also makes use of the $SU(2)$ angular momentum theory.

This is a topic for another thread, but I am intrigued by the possibility of a pure-particle formulation of relativistic quantum mechanics that is equivalent to QFT. It would at first seem impossible, because of particle creation, but there are at least two hand-wavy approaches to getting around that: (1) the Dirac sea idea, and (2) viewing particle creation in terms of a particle that can travel back and forth in time. I don't know whether there has been any serious effort to make either of these work rigorously.

There is a nice theorem by Currie, Jordan & Sudarshan that says that (Hamiltonian) relativistic particle theories must be non-interacting, both classically and quantum mechanically.