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For a free scalar field of mass $m$ in Anti-de Sitter spacetime of dimension $d+1$ and radius $L$, use coordinates withThe conformal boundary is at $z=0$. The Klein-Gordon equation gives the radial leading balance $z^2{\varphi }_{zz}-(d-1)z{\varphi }_z-m^2{L}^2\varphi \simeq 0$. A power law $z^{\Delta }$ therefore obeys the AdS scalar mass--dimension relationFor distinct roots and generic nonresonant parameters,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. ^[ref]

The Breitenlohner-Freedman bound $m^2{L}^2\ge -d^2/4$ 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. ^[ref]

The holographic dictionary maps observables and parameters between a boundary quantum field theory and a bulk gravitational description. In the ordinary scalar Dirichlet prescription, one near-boundary coefficient acts as the source for a boundary local field operator, while another independent coefficient is related to its expectation value after holographic renormalization.

The AdS scalar mass--dimension relation connects the bulk mass to the boundary scaling dimension. Solving the bulk field equation with a specified boundary source then determines the response and hence boundary correlation functions. Source and response assignments depend on boundary conditions: alternative quantization, logarithmic terms and local counterterms require care. They cannot always be identified just by calling every leading coefficient a source and every subleading coefficient an expectation value.

Holographic renormalization subtracts local boundary counterterms from the regulated on-shell bulk action. It gives finite correlation functions and identifies the renormalized normalizable coefficient with an operator expectation value.

The divergence is local in the induced boundary metric and is cancelled by the leading holographic renormalization countertermOn the regulated cylinder this iswhich cancels the ${ϵ}^{-2}$ divergence of $I_{\mathrm{GR}}$. An overall sign changes if one uses the oppositely signed Euclidean generating functional; the local counterterm changes with it.

When $J(x)$ varies, the radial wave equation recursively generates local derivative terms before the normalizable response. SubstitutingintogivesBecause the boundary dimension is odd, no logarithmic term is required. Holographic renormalization subtracts these source-dependent pieces, leavingin the normalization $\mathcal{O}=A$. Equivalently,