In Witten’s approach one can relate expectation values of local operators in the CFT to the normalizable boundary behavior of fields in AdS space [14, 15].

For certain masses there are inequivalent sets of normalizable modes, a complication analyzed in [15, 17] which we will not consider further here.

To see this, use the AdS invariance to set ${\tau }^{\prime }={\rho }^{\prime }=0$, work in light-front coordinates $x^{\pm }=\tau \pm \rho$, and write the step function as $\theta (x^{+})\theta (-x^{-})+\theta (x^{-})\theta (-x^{+})$.

We could have chosen the left part of the light cone. In either case, this step is not possible in higher dimensions.

The smearing function defined in (17) does not transform nicely under AdS isometries. One can see this, for example, by doing the sum explicitly for a massless field. Fortunately the nonuniqueness of the smearing function allowed us to construct an AdS-covariant expression.

It is perhaps not so surprising that the Poincaré mode sum automatically vanishes at timelike separation: the Poincaré boundary field involves a complete set of functions $e^{-i\omega T}$ with $\omega \in \mathbb{R}$, so the expression (25) for the Poincaré smearing function is unique.

One blob if the bulk point is inside the Poincaré patch or any of its images under $\tau \to \tau +2\pi$, two blobs otherwise.

In fact, it is the dimensional reduction of a BTZ black hole [27].

In fact, this expression is valid for any $P$. However, if $P$ is in the left Rindler wedge it is simpler to work with a smearing function that is nonzero only at spacelike separation on the left boundary.

For a bulk point in the right Rindler wedge or any of its images under $\tau \to \tau +2\pi$ one gets a single blob on the right boundary. For a bulk point in the left wedge or any of its images it is more natural to use a smearing function with a single blob on the left boundary.

Despite appearances, the Hartle-Hawking state is independent of $M$. In fact, the global vacuum, the Poincaré vacuum and the Hartle-Hawking state are identical [23, 24, 32].

A similar property held in the Poincaré patch: the signal of the Poincaré horizon was that the smearing function extended to Poincaré time $T\to \pm \infty$.

Equivalently one needs an operator that acts both at real time $t$ and at complex time $t+i\beta /2$.

Positive frequency only in the sense that they multiply annihilation operators in the mode expansion of the field. They do not transform by a definite phase under a Kruskal time translation.

This is the same prescription used in Rindler space [45] or, more generally, on a bifurcate Killing horizon [30].