= AdS scalar-field boundary asymptotics
For a free <scalar field> of <mass> $m$ in <Anti-de Sitter spacetime> of dimension $d+1$ and radius $L$, use coordinates with
$$
ds^2=\frac{L^2}{z^2}(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu).
$$
The <conformal boundary> is at $z=0$. The <Klein-Gordon equation> gives the radial leading balance $z^2\phi_{zz}-(d-1)z\phi_z-m^2L^2\phi\simeq0$. A power law $z^\Delta$ therefore obeys the <AdS scalar mass--dimension relation>
$$
\Delta_\pm=\frac d2\pm\nu,\qquad\nu=\sqrt{\frac{d^2}{4}+m^2L^2}.
$$
For distinct roots and generic nonresonant parameters,
$$
\phi(z,x)=z^{\Delta_-}(J(x)+\cdots)+z^{\Delta_+}(A(x)+\cdots).
$$
In standard quantization, $J$ is the source of the dual <local field operator>; $A$ determines its <expectation value> after <holographic renormalization>, with a <normalization> fixed by the bulk <action>. <Holographic duality with a view toward many-body physics by John McGreevy>[Sections 4.1–4.4, equations 4.25 and 4.35–4.40, source and response prescription]{ref}
The <Breitenlohner-Freedman bound> $m^2L^2\geq-d^2/4$ makes $\nu$ real. At $\nu=0$ the roots coincide and a second solution involves $z^{d/2}\log z$. Resonant expansions can also contain logarithms. In the appropriate <mass> window, <Alternative quantization of an AdS scalar field> changes which coefficient acts as source. <Holographic duality with a view toward many-body physics by John McGreevy>[Sections 4.3 and 4.7, logarithmic terms and changed boundary prescription]{ref}
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