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Consider $AdS_{d+1}/ CFT_d$ duality. Suppose the bulk metric in Poincare coordinates is $$ds^2=\frac{l^2}{z^2}\left(dz^2+h_{ij}(x)dx^idx^j\right)$$ near the boundary, i.e. as $z\to 0$. Let $\sigma_1,\sigma_2$ be Cauchy surfaces for the boundary spacetime $Mink_d$, and let $\Sigma_i$ be an asymptotically spacelike bulk surface anchored to $\sigma_i$ at the boundary. (Really, I'd like to demand that the $\Sigma_i$ are Cauchy surfaces for the bulk, but given the metric is quantised, I don't know what this would mean).

The CFT gives transition amplitudes $\langle\phi_2 |U_{\sigma_1\to \sigma_2}|\phi_1\rangle$ between configurations $\phi_1,\phi_2$ on $\sigma_1,\sigma_2$ (resp.). Likewise, the bulk quantum gravity theory ought to give some transition amplitudes $\langle \text{geom}_2 |U_{\Sigma_1\to \Sigma_2}|\text{geom}_1\rangle$ between geometries on $\Sigma_1$ and geometries on $\Sigma_2$. (I am being loose with the word "geometries", but I imagine this data should roughly consist of the 3-metric on each slice, as well as the extrinsic curvature, modulo gauge freedom).

Question: Does the $AdS/CFT$ duality give us a way to relate the bulk and the boundary transition amplitudes?

For example, suppose we wish to calculate some bulk amplitude $\langle \text{geom}_2 |U_{\Sigma_1\to \Sigma_2}|\text{geom}_1\rangle$. What I have in mind is a method where we somehow cleverly map the bulk BCs (i.e. the geom$_i$) to some BCs at $\sigma_i$ for the boundary CFT, so that the bulk amplitude can be recast as a corresponding CFT amplitude (again with source term $h_{ij}$). Does such a prescription exist?

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  • $\begingroup$ The bulk partition function (with boundary conditions) is equal to the boundary partition function (with sources) with the identification boundary condition = source. You can use this to figure out what a transition amplitude corresponds to in the bulk. $\endgroup$
    – Prahar
    Feb 6, 2022 at 18:51
  • $\begingroup$ The bulk partition function which appears in the GPKW dictionary only has boundary conditions specified at large radius (i.e. the corresponding sources in the CFT only depend on the asymptotic bulk geometry). Im asking whether theres a stronger duality, telling us about transition amplitudes for the whole bulk geometry, not just at infinity. $\endgroup$
    – nodumbquestions
    Feb 6, 2022 at 19:35
  • $\begingroup$ What do you mean by transition amplitudes for the whole bulk geometry? In a diffeomorphism invariant theory, the data on a single Cauchy slice is entirely determined by the fields on the boundary of that Cauchy slice. This is a consequence of the constraint equations in GR. More precisely, the bulk $in$ and $out$ states are completely fixed by boundary conditions at infinity on a Cauchy slice. This same data is then used to evaluate the transition amplitude in the CFT. $\endgroup$
    – Prahar
    Feb 6, 2022 at 19:41
  • $\begingroup$ Note that this is quite distinct from what happens in quantum field theory for which the data is given by the fields on the entire bulk Cauchy slice. $\endgroup$
    – Prahar
    Feb 6, 2022 at 19:41
  • $\begingroup$ "The data on a single Cauchy slice is entirely determined by the fields on the boundary of that Cauchy slice." I don't understand this. Couldn't one imagine different geometries on a slice which have the same values at infinity? For instance, one geometry could be flat, whereas the other contains a star/planet... $\endgroup$
    – nodumbquestions
    Feb 7, 2022 at 0:01

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There is no unique answer to this question. It is possible to map certain bulk geometries to certain CFT boundary geometries, and thereby calculate certain bulk transition amplitudes in terms of CFT transition amplitudes. However, it is also possible to map different bulk geometries to different CFT boundary geometries, in which case the bulk transition amplitudes cannot be simply calculated in terms of CFT transition amplitudes.

It is also worth noting that there are many different CFTs, each of which may give rise to different bulk-CFT correspondences. So, even if a given bulk-CFT correspondence can be calculated, it may be difficult to determine which CFT is the most appropriate to use.

In general, the AdS/CFT correspondence provides a powerful tool for calculating certain bulk transition amplitudes in terms of CFT transition amplitudes. However, it is not always possible to use this correspondence to calculate every possible bulk transition amplitude.

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