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Solution of Dirichlet problem for scalar field in Ads

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I am reading "Anti de Sitter space and holography" by Witten. In this article he derives the two-point function for CFT from theADS/CFT correspondence for a massless scalar field living in the bulk.

To do so he introduces action of scalar field (with fixed ADS metric): $$I_\text{Ads}=\int_{B_{d+1}}\mathrm{d}^{d+1}x\sqrt{g}|d\phi|^{2}$$ For this action he solves the equation of motion (page 12) for a boundary value $\phi_{0}$ as Dirichlet boundary condition. The ADS metric is given by $$\mathrm{d}s^{2}=\frac{1}{x_{0}^{2}}(\sum_{i=0}^{d}(\mathrm{d}x^{i})^{2})$$ and he is looking for the Green's function $K$.

From what I know the general scheme of such an approach should look like this:

  1. $\partial^{2}\phi(x,t)=J(x,t)$ , where $ J$ is current.
  2. $\phi(x,t) = \phi_{free}(x,t)+\int{dy}G(x-y)J(y)$, where $\phi_{0}(x,t)$ solution of free wave equation.
  3. $\partial^{2}G(x)=\delta(x)$

I can see (page 13) that the final result is indeed given by the expression: $$\phi(x_{0},x_{i})=\int\mathrm{d}\textbf{x'}K(x_{0},\textbf{x},{x'})\phi_{0}(x'_{i})$$ I guess that $\phi_{free}(x,t)= 0$ because "there is no nonzero square-integrable solution of the Laplace equation (for Ads metric)"(page 7). Yet I do not understand why did he consider the equation for $K$ without the delta function (page 13): $$\frac{\mathrm{d}}{\mathrm{d}x_{0}}x_{0}^{-d+1}\frac{\mathrm{d}}{\mathrm{d}x_{0}}K(x_{0})=0$$ Is $K$ really a Green's function in the sense I understand it?

I think that the field equation is $D^{2}\phi(x_{i},x_{0})=\phi_{0}(x_{i},x_{0}= \infty,0)$. Is that right?


This post imported from StackExchange Physics at 2015-11-06 22:01 (UTC), posted by SE-user Yaroslav Shustrov

asked Oct 27, 2015 in Theoretical Physics by Yaroslav Shustrov (15 points) [ revision history ]
edited Nov 6, 2015 by Dilaton

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