RS 1, massless scalar field, and sep of vars.

In summary, a person is seeking help with understanding a specific equation (2.1) in a paper about the Casimir force in Randall-Sundrum models. They are trying to get to equation 2.2, which involves Bessel functions, but have encountered a problem with their logic and are asking for assistance. Another person suggests adding boundary terms and using a different separation of variables to get the correct result.
  • #1
robousy
334
1
Hey, this is rather involved but I hope someone can help me out.

I am reading http://arxiv.org/abs/0704.3626 ( the casimir force in randall sundrum models) and am trying to get from equation 2.1 :

[tex]g^{\mu\nu}\partial_\mu\partial_\nu\Phi+e^{2ky}\partial_y(e^{-4ky}\partial_y\Phi)=0[/tex]

to equation 2.2 ...'The general solution for the non-zero modes can be eppressed in terms of Bessel functions of the 1st and 2nd kind as:

[tex]\chi(y)=e^{2ky}(a_1 J_2(\frac{m_Ne^{ky}}{k}) +a_2Y_2(\frac{m_Ne^{ky}}{k}) )[/tex]

I'll show you my attempts and if anyone has the patience can maybe help me.

First express field [tex]\Phi[/tex] as:
[tex]\Phi(x_\mu,y)=\phi(x_\mu)\chi(y)[/tex]

Plug this into the original differential equation

[tex](g^{\mu\nu}\partial_\mu\partial_\nu\phi)\chi+e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)\phi=0[/tex]

Divide by [tex]\phi\chi[/tex] to obtain


[tex] \frac{ (g^{\mu\nu}\partial_\mu\partial_\nu\phi) }{ \phi}+


\frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=0[/tex]

As the first term only depends on [tex]x_\mu[/tex] and the second on y, each term in the equation must be a constant, so for the [tex]\chi(y)[/tex] term we can write:

[tex]\frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=n^2[/tex]

where [tex]n^2[/tex] is some constant.

Tidying this up a bit we get:

[tex] e^{-2ky}\partial^2_y\chi} -4ke^{-2ky}\partial_y\chi-n^2\chi=0[/tex]

Make this easier to compare to Bessels equation by writing

[tex]e^{2ky} \rightarrow x^2, \chi\rightarrow y[/tex]

My final equation now looks like:

[tex]x^2y''-4kx^2y'-n^2y=0[/tex]

Whereas Bessels equation is

[tex]x^2y''+xy'+(x^2-n^2)y=0[/tex]

So this is my problem, I obtain an expression that is not quite bessels equation. Can anyone see the error of my logic??

Thank you so much.
 
Last edited:
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  • #2
robousy said:
Hey, this is rather involved but I hope someone can help me out.

I am reading http://arxiv.org/abs/0704.3626 ( the casimir force in randall sundrum models) and am trying to get from equation 2.1 :

[tex]g^{\mu\nu}\partial_\mu\partial_\nu\Phi+e^{2ky}\partial_y(e^{-4ky}\partial_y\Phi)=0[/tex]

to equation 2.2 ...'The general solution for the non-zero modes can be eppressed in terms of Bessel functions of the 1st and 2nd kind as:

[tex]\chi(y)=e^{2ky}(a_1 J_2(\frac{m_Ne^{ky}}{k}) +a_2Y_2(\frac{m_Ne^{ky}}{k}) )[/tex]

I'll show you my attempts and if anyone has the patience can maybe help me.

First express field [tex]\Phi[/tex] as:
[tex]\Phi(x_\mu,y)=\phi(x_\mu)\chi(y)[/tex]

Plug this into the original differential equation

[tex](g^{\mu\nu}\partial_\mu\partial_\nu\phi)\chi+e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)\phi=0[/tex]

Divide by [tex]\phi\chi[/tex] to obtain


[tex] \frac{ (g^{\mu\nu}\partial_\mu\partial_\nu\phi) }{ \phi}+


\frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=0[/tex]

As the first term only depends on [tex]x_\mu[/tex] and the second on y, each term in the equation must be a constant, so for the [tex]\chi(y)[/tex] term we can write:

[tex]\frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=n^2[/tex]

where [tex]n^2[/tex] is some constant.

Tidying this up a bit we get:

[tex] e^{-2ky}\partial^2_y\chi} -4ke^{-2ky}\partial_y\chi-n^2\chi=0[/tex]

Make this easier to compare to Bessels equation by writing

[tex]e^2ky \rightarrow x^2, \chi\rightarrow y[/tex]

My final equation now looks like:

[tex]x^2y''-4kx^2y'-n^2y=0[/tex]

Whereas Bessels equation is

[tex]x^2y''+xy'+(x^2-n^2)y=0[/tex]

So this is my problem, I obtain an expression that is not quite bessels equation. Can anyone see the error of my logic??

Thank you so much.

But I don't see how you can make that change of variable. In Bessel equation, the parameter "x" is the variable differentiated with respect to! So you must be more careful. I will look into it when I have some time tonight.
 
  • #3
Cool Kdv, any help appreciated, and I see what you are saying about the change of variable so thank you on that point.
 
  • #4
Hmmm I seem to remember working this out before. I think the problem is that you have to add boundary terms to make it work right, that is, to get a good definition of von Neuman or Dirichlet B.C.'s.

Check out the Lecture notes by Gherghetta : http://arxiv.org/abs/hep-ph/0601213. I think he works it out in detail, but I could be wrong.

Also you have to be careful. The separation of variables that you should use is

[tex]\Phi(x_{\mu},y) = \sum_n \phi_n(x_{\mu})\chi_n(y)[/tex]

up to some constant that gives you the correct mass dimension. This gives you the KK tower, etc.
 
  • #5
robousy said:
Cool Kdv, any help appreciated, and I see what you are saying about the change of variable so thank you on that point.

I can't get it to work out. But I suspect there is something more to the story. They mention treating separately the N=0 mode from the others and this does not show up at all in the steps you followed. See the post by BenTheMan too.
 
  • #6
I couldn't get it to work out either, so I glanced at the reference given in the original post's paper. This reference is the the link given in BenTheMan's post. As BenTheMan said, there's another term.
 
  • #7
Ok guys, I did have a look at the Les Houches Lectures last week but it didn't help much. I'll take another look this evening and see if I can make progress. Thanks all for having a look at this.

Cheers for the correction in my separation of vars Ben. Hope all is good with you!

Rich
 

Related to RS 1, massless scalar field, and sep of vars.

1. What is RS 1 and how does it relate to physics?

RS 1, also known as Randall-Sundrum 1, is a model proposed by physicists Lisa Randall and Raman Sundrum to explain the weakness of gravity compared to other fundamental forces. It introduces an extra dimension of spacetime, called the "bulk," in addition to the four dimensions we are familiar with. This model has been used to address questions about the hierarchy problem and the origin of dark energy.

2. What is a massless scalar field and why is it important?

A massless scalar field is a hypothetical field in particle physics that has no mass and interacts with other particles through the Higgs mechanism. It is important because it is used to describe the behavior of certain particles, such as the Higgs boson, and plays a crucial role in the Standard Model of particle physics.

3. How does separation of variables work in physics?

Separation of variables is a mathematical technique used to solve differential equations that are separable. In physics, it is often used to simplify complex equations and make them easier to solve. This technique involves isolating each variable in an equation and solving for it individually, while assuming that the other variables are constant.

4. What are some applications of separation of variables in physics?

Separation of variables is widely used in many areas of physics, including classical mechanics, electromagnetism, and quantum mechanics. It is used to solve problems involving oscillatory systems, heat transfer, and wave propagation. In quantum mechanics, separation of variables is used to solve the Schrödinger equation for systems with multiple particles.

5. Can you explain how RS 1, massless scalar field, and separation of variables are related?

RS 1 and the massless scalar field are both concepts used in theoretical physics to explain different phenomena. Separation of variables is a mathematical technique often used in physics to solve complex equations. These concepts are not directly related, but they can be used together to understand and model various physical systems, such as the behavior of particles in extra dimensions predicted by RS 1 models.

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