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The AdS scalar-field boundary asymptotics areThus the holographic dictionary becomes especially simple for the flat field $\psi =\varphi /z$:up to the overall normalization chosen for the bulk action and possible local counterterms. The leading mode is the source and the normalizable subleading mode is the response.

For a bulk scalar with $m^2=\Delta (\Delta -d)$ in unit-radius Poincaré AdS, the AdS scalar-field boundary asymptotics areThus $J$ and $\mathcal{O}$ are, up to normalization and local counterterms, the nonnormalizable and normalizable boundary coefficients respectively.

For $d=3$, $\Delta =1$, and $J=0$, alternative quantization givesThe bulk Klein-Gordon equation issoAt separated points in any state,This contains only radial bulk derivatives, as required.

For a boundary-independent source, the AdS scalar-field boundary asymptotics reduce toOne may therefore extract the source and response directly:The operator expectation value is proportional to $A$; choosing the boundary-operator normalization $\mathcal{O}=A$ gives the requested expressions. A conventional action normalization can instead multiply this relation by the fixed factor $2\Delta -d=5$.