Equation of motion Chern-Simons

In summary, the equation of motion for a particle in a mathematical space is L = aμ∂νaλ. The 2 in front comes from the derivative in front of one a. To get the equation of motion for a particle in ##a##, you must differentiate with respect to ##\partial_\nu a_\lambda##.
  • #1
Lapidus
344
11
The Lagrangian (Maxwell Chern-Simons in Zee QFT Nutshell, p.318)
Bildschirmfoto 2019-03-17 um 17.52.27.png

has as equation of motion:
Bildschirmfoto 2019-03-17 um 17.52.35.png


Where does the 2 in front come from?

Thank you very much
 

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  • #2
You have two ##a## in the ##\gamma## term.
 
  • #3
Right. And a derivative in front of one a.

Do I get one term from the RHS and one from the LHS of equation of motion and then I add them together?
Bildschirmfoto 2019-03-17 um 21.43.17.png
 

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  • #4
Lapidus said:
Right. And a derivative in front of one a.

Do I get one term from the RHS and one from the LHS of equation of motion and then I add them together?
View attachment 240456
Why don’t you try it and see what you get?
 
  • #5
I get the equation but without the 2 in front. I do not see how the 2 comes about. How to sum over the indices. I find the indices confusing. Hence my question.
 
  • #6
Lapidus said:
I get the equation but without the 2 in front. I do not see how the 2 comes about. How to sum over the indices. I find the indices confusing. Hence my question.
We cannot help you unless you show what you actually did. Otherwise we have no way of knowing where you went wrong.
 
  • #7
I differentiate γεμνλaμνaλ w.r.t. aμ and I get γεμνλνaλ

and γεμνλaμνaλ w.r.t. ∂νaλ which gives γεμνλaμ.

Thus γεμνλνaλ - γεμνλaμ = 0 , which is not 2γεμνλνaμ = 0.
 
  • #8
You forgot to take the derivative with respect to ##x^\nu## of the derivative with respect to ##\partial_\nu a_\lambda##.

Edit: Also, if you want the equation of motion for ##a_\mu##, you must take the derivative with respect to ##\partial_\nu a_\mu##, not ##\partial_\nu a_\lambda##.
 
  • #9
L = aμνaλ

∂L/∂aμ - ∂ν (∂L/∂(∂νaμ)) = ∂vaμ - ∂v ?

I do not know what and how to differentiate in the second term. Also, I need to add two identical terms to get the factor two. But there is a minus sign.
 
  • #11
Orodruin, I really appreciate your time and effort. I read carefully all your posts in this thread and the link you gave. But unfortunately, I still can not answer my initial question. Maybe I try somewhere else. Thank you!
 
  • #12
I am sorry you don't feel you have enough. I think you would benefit significantly from showing your computations in more detail instead of just stating what you get and by thinking of each step in terms of what I said in the Insight. I could of course just give you the derivation, but I doubt you would learn as much from that as you would if you present it, clarify exactly in which steps your confusions lie, and get help in seeing how to resolve them.

In particular, I think you are guilty of #8 in the Insight and that this is causing you trouble. However, it is difficult to tell since you have not given us your explicit stepwise computations, just what you think each term should be.
 
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  • #13
Lapidus said:
Where does the 2 in front come from?
Thank you very much
Do not differentiate with respect to repeated indices. Write [tex]\mathcal{L} = \epsilon^{\sigma\rho\nu} \ A_{\sigma} \ \partial_{\rho}A_{\nu} + A_{\sigma}J^{\sigma}.[/tex] Now, differentiate with respect to [itex]A_{\mu}[/itex], and use [itex]\frac{\partial A_{\eta}}{\partial A_{\mu}} = \delta^{\mu}_{\eta}[/itex] to get [tex]\frac{\partial \mathcal{L}}{\partial A_{\mu}} = \epsilon^{\mu\rho\nu} \ \partial_{\rho} A_{\nu} + J^{\mu} . \ \ \ \ \ (1)[/tex] Next, differentiate [itex]\mathcal{L}[/itex] with respect to [itex](\partial_{\tau}A_{\mu})[/itex] and use the identity [tex]\frac{\partial (\partial_{\rho}A_{\eta})}{\partial (\partial_{\tau}A_{\mu})} = \delta^{\tau}_{\rho} \ \delta^{\mu}_{\eta} ,[/tex] to obtain [tex]\frac{\partial \mathcal{L}}{\partial (\partial_{\tau}A_{\mu})} = \epsilon^{\sigma\tau\mu} \ A_{\sigma} = - \epsilon^{\mu\tau\sigma} \ A_{\sigma}.[/tex] Thus [tex]\partial_{\tau} \left( \frac{\partial \mathcal{L}}{\partial (\partial_{\tau}A_{\mu})} \right) = - \epsilon^{\mu\tau\sigma} \ \partial_{\tau}A_{\sigma} = - \epsilon^{\mu\rho\nu} \ \partial_{\rho}A_{\nu} . \ \ \ (2)[/tex] Now [itex](1) – (2) = 0[/itex] is the E-L equation. It gives you [tex]2 \epsilon^{\mu\rho\nu} \ \partial_{\rho}A_{\nu} + J^{\mu} = 0.[/tex]
 
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  • #14
Fantastic! Thanks Samalkhaiat! Kronecker Deltas from derivatives and indices swapping in the Levi-Civita. Got it.

I must study a little tensor calculus..
 

1. What is the Equation of Motion Chern-Simons?

The Equation of Motion Chern-Simons is a mathematical equation used in theoretical physics to describe the dynamics of particles in a three-dimensional space. It was developed by physicists Shiing-Shen Chern and James Harris Simons in the 1970s.

2. How does the Equation of Motion Chern-Simons differ from other equations of motion?

The Equation of Motion Chern-Simons differs from other equations of motion, such as Newton's Second Law, in that it takes into account the effects of topology and gauge symmetry. This makes it particularly useful for describing the behavior of particles in systems with non-trivial topology.

3. What is the significance of the Chern-Simons term in the equation?

The Chern-Simons term in the equation represents a topological invariant, which means that it remains unchanged even when the system undergoes continuous deformations. This makes it a powerful tool for studying systems with topological properties.

4. How is the Equation of Motion Chern-Simons used in physics?

The Equation of Motion Chern-Simons is used in a variety of fields in physics, including condensed matter physics, quantum field theory, and string theory. It has been particularly useful in studying the behavior of particles in topological insulators and superconductors.

5. Are there any current research developments related to the Equation of Motion Chern-Simons?

Yes, there is ongoing research related to the Equation of Motion Chern-Simons in various fields of physics. Some recent developments include its application in studying topological phases of matter and its connection to quantum information theory.

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