Simple question regarding anyons

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In summary, the conversation discusses the flux-tube model for anyons and the introduction of a vector potential for a N particle system. The conversation also includes a question about a step in the calculation and a discussion about the use of polar coordinates and the inclusion of a factor of r in the calculation. The conversation ends with a request for a simple explanation of the topic.
  • #1
ReyChiquito
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I have a simple question regarding the flux-tube model for anyons. It may sound complicated but it isnt. So here we go.

Considering the interaction term [tex]L_{s}=\frac{\hbar\theta}{\pi}\dot{\phi}[/tex] where [tex]\frac{\theta}{\pi}=\alpha[/tex] is called "anyon parameter" (0 for bosons, 1 for fermions), and [tex]\phi[/tex] is the relative angle between particles.

I have proven that the Hamiltonian in relative coordinates for that kind of system can be written as
[tex]H_{r}=\frac{p_{r}^2}{m}+\frac{(p_{\phi}-\hbar\alpha)^2}{mr^2}.}[/tex]

In order to generalize the Hamiltonian for a N partices system, the book (Fractional Statistics and Quantum Theory by Khare) introduces the next vector potential:

[tex]a_{i}(\bold{r})=\frac{\Phi}{2\pi}\frac{\epsilon_{ij}r_{j}}{\bold{r^2}}[/tex] where [tex]\epsilon_{ij}[/tex] is the antisimetric tensor (i asume).

Then the book goes
[tex]\mbox{Thus }a_{x}=\frac{\Phi}{2\pi}\frac{y}{x^2+y^2}\mbox{, }a_{y}=\frac{\Phi}{2\pi}\frac{-x}{x^2+y^2}\mbox{, or in polar coordinates }[/tex]
[tex]a_{r}=0\mbox{, }a_{\phi}=\frac{\Phi}{2\pi}[/tex]

I know it seems simple to deduce this step but i don't get it, here is what I've done:
[tex]a_{i}(\bold{r})=\frac{\Phi}{2\pi\bold{r^2}}\left(\begin{array}{cc}y\\-x\end{array}\right)=\frac{\Phi}{2\pi\bold{r^2}}\left(\begin{array}{cc}rsin\phi\\-rcos\phi\end{array}\right)=-\frac{\Phi}{2\pi}\frac{1}{r}\bold{\hat{\phi}}[/tex]

What am i doing wrong??

I asked a friend of mine and he mentioned something about the metric. To tell you the truth, i don't know what he is talking about. Can anybody explain this to me please?
 
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  • #2
Yea that looks right up to the last step, you basically have it in front of you *the last equality is wrong tho*.. Keep in mind you pick up a factor of r from the transformation..

You know, ds^2 = dr^2 + r^2 d(theta)^2
 
  • #3
Thx.. i knew i was missing something.

Can you explain me from where this factor arrives?

Its directly from calculus (i.e. jacobian) or has to do something with tensors and metric?
 
  • #4
It's the radial coordinate of the polar coordinates. The formula Haelfix gave is the length differential in polar coordinates.
 
  • #5
I understand that, maybe i need to refrace my question.

Why this isn't an ordinary change of variables?

I don't see any rates involved so i don't understand why do i have to include that factor.

If this where a simple calculus problem the tangent vector can be described in polar coordinates as
[tex]\vec{T}=\left(\begin{array}{cc}-y\\x\end{array}\right)=r\hat{\phi}[/tex]

if i used
[tex]\vec{T}=r^2\hat{\phi}[/tex]

i would be describing the wrong point in space right?, plus, how to correct the minus sign? isn't supossed to be a right hand system?

I KNOW that what I am doing is wrong, and i understand that the r factor must be included and the book is right, but i don't see any reason for including that factor.

Am i missing something simple and i need to review my clac notes?

ps. my quantum mechanics course sucked big time, it was like half spetial functions course and half "learn the dirac algebra and the conmutator operator", i harldy saw any of the stuff u should see in this subject (ie Cohen)
 
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  • #6
Dumping here. Just to see if anybody can explain it to me like i was a 4 year old :P
 

1. What are anyons?

Anyons are a type of particle that falls under the category of quantum statistics. They have properties that are somewhere between those of fermions and bosons.

2. How are anyons different from fermions and bosons?

Anyons have exotic properties that are not found in fermions or bosons. They can have fractional spin, which means they can rotate by a fraction of a full rotation rather than just a full rotation like fermions and bosons. They also exhibit a unique type of quantum exchange known as braiding, where their paths are intertwined and affect each other's quantum states.

3. What is the significance of anyons in physics?

Anyons have been studied for their potential applications in quantum computing and topological quantum field theory. They also play a role in understanding the behavior of matter in certain states, such as in the fractional quantum Hall effect.

4. Can anyons be observed in experiments?

While anyons have not been directly observed, their effects have been observed and studied in experiments. For example, the fractional quantum Hall effect is a well-known physical phenomenon that is attributed to the behavior of anyons.

5. Are anyons a recent discovery?

Anyons were first proposed in the late 1970s by physicist Frank Wilczek, but they have gained more attention and research in recent years due to their potential applications in quantum computing and topological quantum field theory.

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