robousy
Oct11-05, 04:39 PM
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! :)
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! :)