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): IAds=∫Bd+1dd+1x√g|dϕ|2
For this action he solves the equation of motion (page 12) for a boundary value
ϕ0 as Dirichlet boundary condition. The ADS metric is given by
ds2=1x20(d∑i=0(dxi)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:
- ∂2ϕ(x,t)=J(x,t) , where J is current.
- ϕ(x,t)=ϕfree(x,t)+∫dyG(x−y)J(y), where ϕ0(x,t) solution of free wave equation.
- ∂2G(x)=δ(x)
I can see (page 13) that the final result is indeed given by the expression: ϕ(x0,xi)=∫dx'K(x0,x,x′)ϕ0(x′i)
I guess that
ϕfree(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):
ddx0x−d+10ddx0K(x0)=0
Is
K really a Green's function in the sense I understand it?
I think that the field equation is D2ϕ(xi,x0)=ϕ0(xi,x0=∞,0). Is that right?
This post imported from StackExchange Physics at 2015-11-06 22:01 (UTC), posted by SE-user Yaroslav Shustrov