Understanding AdS and quantization in AdS

maverick280857

In section 6.3 of D'Hoker and Freedman's paper, they give the following result for the geodesic distance in AdS:

[tex]d(z,w) = \int_{w}^{z}ds = \ln\left(\frac{1+\sqrt{1-\zeta^2}}{\zeta}\right)[/tex]

with

[tex]\zeta \equiv \frac{2 z_0 w_0}{z_0^2 + w_0^2 + (\vec{z}-\vec{w})^2}[/tex]

How does one derive this?

Ben Niehoff

First you have to solve the geodesic equation. It's not hard if you use some tricks. AdS is a homogeneous space. In general, homogeneous spaces include spheres, hyperbolic spaces, dS and AdS spaces. Every homogeneous space embeds isometrically as a quadric surface in a flat space of one dimension higher; this flat space may turn out to have mixed signature (++..+ for spheres, -+..+ for hyperbolic spaces and dS, --+..+ for AdS).

So think about the 2-sphere embedded in R^3. Can you see a very simple way to determine the 2-sphere's geodesics from this embedding? There should be a fairly obvious way to use symmetry to your advantage.

The same technique applies in AdS. Then, once you have the geodesics of AdS, you simply integrate the line element along them.

It would also help to read about hyperbolic spaces, particularly the Poincare upper-half-plane model, because it applies very well to AdS spaces, and the UHP model corresponds to the coordinates you are using mostly in this thread.

maverick280857

Thanks Ben, but could you pleas esuggest references where this might be worked out or something? I need to figure this out reasonably quickly, in order to move on to the solution of the free wave equation in AdS. and I am hard pressed for time.