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 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]

For a scalar field of mass $m$ on ${\mathrm{AdS}}_{d+1}$ of radius $L$, the scaling dimension of the dual scalar primary obeys $m^2{L}^2=\Delta (\Delta -d)$. The two roots correspond to the two independent near-boundary falloffs.

When both scalar falloffs in anti-de Sitter spacetime are normalizable, alternative quantization exchanges which near-boundary coefficient is treated as the source and which as the expectation value. In AdS4, a field with $m^2=-2$ may be dual to a scalar operator of dimension one.