AdS scalar-field boundary asymptotics

A scalar field of mass $m$ in ${\mathrm{AdS}}_{d+1}$ behaves near the conformal boundary as $\varphi =z^{d-\Delta }J+z^{\Delta }A+\cdots$, where $m^2=\Delta (\Delta -d)$. The leading coefficient is the source and the normalizable coefficient determines the operator expectation value.

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.