Guidance in solving Scalar Field with BC's

1. Oct 11, 2005

robousy

Hi all,

I am having some problems understanding the steps in a paper.

I've looked in books and asked other grad students but they have all not been of too much help and I am still stuck.

I have a massive scalar field mass $$\mu$$ interacting with two delta function potentials with lagrangian density:

$${\cal L}_{int}=- \frac{1}{2} \frac{\lambda}{a} {\delta (x) \phi(x)^2 - \frac{1}{2} \frac{\lambda^'}{a} {\delta (x-a) \phi(x)^2$$

with conditions:

$$\lambda, \lambda^{'} \rightarrow \infty \: \: \: \phi(0), \phi(a) \rightarrow 0$$

I am interested in calculating the Casimir energy between the plates which can be computed in terms of a Green function which is just the Time ordered product of the expectation of the fields with a Fourier Transform:

$$G(x,x^{'} )=\int \frac{d\omega}{2\pi}e^{i\omega (t-t^{'})}g(x,x^{'};\omega^{'} )$$

The reduced Green function satisfies:

$$\left -\frac{\partial^2}{\partial x^2}+\kappa^2+\frac{\lambda}{a}\delta(x) + \frac{\lambda^{'}}{a}\delta(x-a) \right g(x,x') = \delta(x-x^{'} )$$

where, $$\kappa^2=\mu^2-\omega^2$$

...they then go on to solve for g(x,x')

I am stuck on several parts of this.

1) What 'is' the 3rd equation. I am not sure if its a general wave equation that the field must satisfy, or if its the equations of motion of the field or what....and why is it multiplied by the reduced Greens function.

2) I am sure that noone can really be bothered to type out the solution of g(x,x') for me but if you could recommend a good source so that I can see the steps I would appreciate it.

ANY help whatsoever here would be good - physical insights...mathematical processes, just say anything you like! :)

Last edited: Oct 11, 2005
2. Oct 12, 2005

vanesch

Staff Emeritus
The way it is formulated, I have the impression it is simply the defining equation for the Greens function of the wave equation you'd get from the Lagrangian.
See it this way: the Lagrangian codes for a wave equation, right ? Like the Lagrangian for the free Dirac field codes for the Dirac equation. Now the Dirac equation has a Green's function attached to it, which is essentially the Dirac equation with a delta function on the right (and a prescription for how to circumvent the poles).
I'm GUESSING that that is what the 3rd equation stands for.
If I talk nonsense I hope that someone will point that out (gently )
cheers,
Patrick.

3. Oct 17, 2005

Zygotic Embryo

Sorry

4. Oct 17, 2005

robousy

ok, thanks to both of you. I'm still working on this (after a week!!) and have been looking at it every day. If anyone else can provide some tips then they will be appreciated.

5. Oct 20, 2005

robousy

Ok,

I think I am figuring out how to solve this.

Take the Fourier Transform of both sides, divide out term in brackets on left hand side to get g(k,x') by itself then take the inverse FT and solve using the residue theorem.