Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

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 [tex]\mu[/tex] interacting with two delta function potentials with lagrangian density:

[tex]{\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

[/tex]

with conditions:

[tex] \lambda, \lambda^{'} \rightarrow \infty \: \: \: \phi(0), \phi(a) \rightarrow 0 [/tex]

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:

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

The reduced Green function satisfies:

[tex]\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^{'} )[/tex]

where, [tex]\kappa^2=\mu^2-\omega^2[/tex]

...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! :)

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# Guidance in solving Scalar Field with BC's

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