How Do Green Functions Relate to Boundary Conditions in Quantum Field Theory?

[robousy]

Hey folks,

I'm trying to get a handle on my old Nemesis, Green functions. I have a massless scalar field confined between two parallel plates separated by a distance a (in the z dimension) and the field satisfies Dirichel BC's. Basically I'm trying to work from line 1 of a book to line 2 (K. Miltons the Casimir Effect p23).

'The Green function satisfies'

-\partial^2G(x,x')=\delta(x-x')

"We introduce a reduced Green function g(z,z) according to the Fourier Transform"

G(x,x')=\int\frac{d^dk}{(2\pi)^d}e^{i\vec{k}.(x-x')}\int\frac{d\omega}{2\pi}e^{-i\omega(t-t')}g(z,z')

This is all the book says so sorry of that's not much info. I'm fairly sure that \partial=\nabla+\frac{d}{dt}.

What I want to understand (and see the math for) is how to get from line 1 to line 2. I'm pretty sure that it involves Fourier transforms, but I would like to see it. Also, I don't understand the concept of a reduced green function. Can anyone either point me to a good reference, or better still explain how and why it is used.

I hope someone can walk me through this.

:)

 

[Ben Niehoff]

I don't know all the details of that particular problem, but what I remember from electrostatics is that the "reduced" Green's function is essentially a separation-of-variables technique. If we have, for some linear operator L,

LG(\vec x, \vec x') = \delta^3(\vec x - \vec x')

then we can write

\delta^3(\vec x - \vec x') = \delta(x - x') \, \delta(y - y') \, \delta(z - z')

(or we could use some other coordinate system, such as r, \phi, \theta, transforming to it with the proper Jacobian).

Given the boundary conditions, x and y are free, but z is bounded by the planes z=0 and z=a. So we can write

\delta(x - x') = \frac{1}{2\pi} \int dk_x \, e^{ik_x(x-x')}

which is just an identity using Fourier transforms (the integral is over the entire real line). We can do likewise in y, because y also has no boundaries. This leaves us:

LG = \frac{1}{(2\pi)^2} \int dk_x \, e^{ik_x(x-x')} \, \int dk_y \, e^{ik_y(x-x')} \, \delta(z - z')

My memory gets shaky at this point, but you might be able to see where to go. For L = \partial^2, L is pretty simple to invert for the two Fourier transforms. What's left is a function g(z, z') that needs to be solved for.

 

[robousy]

Thanks Ben. I think I actually realize what I don't understand now. Let's say our Geometry is in x,z and t. We can use techniques from Separation of variables to write our GF as:

G(x^\mu,x^\mu')=g(x,x')g(t,t')g(z,z')

Where the \musuperscript runs over x,z,t.

Ok up to here, but then the second line of my first post implies that

g(x,x')=\int\frac{dk}{2\pi}e^{ik.(x-x')}=\delta(x-x')
and
g(t,t')=\int\frac{d\omega}{2\pi}e^{i\omega.(t-t')}=\delta(t-t')

Ok, so here lies my problem. Why can I just assume that the reduced green function in x and t is just a delta function?

 

[Ben Niehoff]

I may have made a slight mistake; I'm not sure. But you should have

\partial^2 G = \delta(x - x') \, \delta(y - y') \, \delta(z - z') \, \delta(t - t')

rather than just G on the left hand side. Once you have that, you can choose to represent some of those delta functions by their Fourier transforms (or Bessel function series, or what-have-you) in the frequency domain.

The reduced Green's function in a particular dimension is not a delta function; it is \partial^2g that is a delta function.

 

[robousy]

ok, I think it should be:



G(x,xs')=\int\frac{d^dk}{(2\pi)^d}e^{i\vec{k}.(x-x')}\int\frac{d\omega}{2\pi}e^{-i\omega(t-')}g(z,z')

because the physics is contained in the z direction so we just expect plane wave solutions in the x and t so we can replace them with delta functions. Then we put this expression for G into the first equation G''.

Thanks for your insight! :)