# Confusion between Green function and solution of equation of motion in Witten's paper on holography and AdS

I was going through Witten's paper on AdS and holography , and am confused in section 2.4. He starts by considering a massless scalar action in Euclidean AdS spacetime, with a boundary value $$\phi_0$$. He then looks for a "Green function" $$K$$, but says that it satisfies the following condition:

$$L K = 0,$$

where $$L$$ is the Laplacian for the scalar field. But in general say when we have a differential equation $$L \phi = f,$$

in order to solve it we construct a Green function $$G$$ for the differential operator $$L$$, it has to satisfy

$$L_x G_{xy} = \delta(x-y)$$

with proper boundary conditions and the solutions to $$L$$ are given by $$\phi$$ such that $$\phi = \int_M G f$$. The solution and not the Green function satisfies

$$L \phi =0.$$

Is there a deeper reason behind calling it a Green function? Does this have any relation to the fact that he is working in Euclidean AdS and not Lorentzian AdS?



• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. Mar 1, 2020 at 11:39

Here $$x'$$ denotes a point on the boundary, and $$x$$ denotes a point in the bulk. The problem he is looking at is given some boundary value $$\phi_0(x')$$ of the scalar field $$\phi(x)$$, how do you solve the wave equation.

$$L \phi(x) = \phi_0(x')$$

In order to do that he is solving

$$L_x K_{xx'} = \delta(x-x').$$

One way to solve this is to just consider the solutions to the equation $$L K = 0$$, and see which component of this equation blows up at the boundary point $$x'$$, which is here given by $$x' = \infty$$, and show that this blowing up corresponds to a delta function. Here the blowing up is given by

$$K(x_0) = C x_0^d.$$

In equation (2.18) he shows that this blowing up corresponds to a delta function, by implementing an isometry transformation

$$x^i \to \frac{x^i}{x_0^2 + \sum_{i=1}^d x_i^2}$$

which maps $$x_0'=\infty$$ to the origin. He then shows that the Green function

$$K(x) = C \left( \frac{x^i}{x_0^2 + \sum_{i=1}^d x_i^2} \right)^d$$

satisfies all properties of a delta function

$$L_x K(x-x') = \delta(x-x')$$ in the limit $$x_0\to 0, x_i =0$$. Therefore the solution of the equation of motion is given by

$$\phi(x_0,x_i) = \int_M \phi_0(x') K(x-x')$$