A note on the two point function on the boundary of AdS spacetime

TL;DR

This paper introduces a new method to compute the two-point function on the boundary of AdS spacetime for a massless scalar field, confirming agreement with established holographic techniques and exploring state inductions related to the BTZ black hole.

Contribution

A novel approach using conformal techniques and Minkowski space limits to calculate boundary correlators in AdS, aligning with existing holographic prescriptions.

Findings

The new method matches results from Boundary-limit Holography and Witten recipe.

States in AdS induce consistent boundary states for the BTZ black hole.

Normalizable modes contribute similarly to non-normalizable modes in the holographic correspondence.

Abstract

We calculate by a new way the two point function on the boundary of AdS spacetime in 1+2 dimensions for the massless conformal real scalar field. The result agrees with the answer provided by the Boundary-limit Holography and Witten recipe. This is done in Poincar\'{e} coordinates. The basic ingredients of this new method are conformal techniques, quantum fields defined on a half of Minkowski spacetime and a limit inspired by the Boundary-limit Holography. We also show that a state in AdS, the global vacuum, in three dimensions induces a state on the conformal boundary of AdS spacetime, which in turn induces a state on the BTZ black hole. On the other hand the same state in AdS induces a state on the BTZ black hole which in turn induces a state on its conformal boundary. The two ways of getting the state on the conformal boundary of the BTZ black hole coincide for the massless conformal real scalar field. We point out that the normalizable modes in the AdS/CFT correspondence for the BTZ black hole give a similar contribution as the non-normalizable modes used in the Witten prescription. We also give some clues on why the Witten and the Boundary-limit Holography prescription coincide.