# Mellin transform The Mellin transform is a convenient technique in AdS/CFT. Mellin amplitudes are nice functions. The Mellin transform is also useful in [[0010 Celestial holography|celestial holography]], where bulk amplitudes get transformed into boundary correlators. ## Refs - Mack 2009 ## Transforming the amplitudes - massless particles - in [[2019#Fan, Fotopoulos, Taylor]]: $\mathcal{A}_{J_{1} \ldots J_{n}}\left(\Delta_{i}, z_{i}, \bar{z}_{i}\right)=\left(\prod_{i=1}^{n} g\left(\lambda_{i}\right) \int d \omega_{i} \omega_{i}^{i \lambda_{i}}\right) \delta^{(4)}\left(\sum_{i} \epsilon_{i} \omega_{i} q_{i}\right) \mathcal{M}_{\ell_{1} \ldots \ell_{n}}\left(\omega_{i}, z_{i}, \bar{z}_{i}\right)$ - in [[2019#Pate, Raclariu, Strominger, Yuan]]: $\widetilde{\mathcal{A}}_{s_{1} \cdots s_{n}}\left(\Delta_{1}, z_{1}, \bar{z}_{1}, \cdots, \Delta_{n}, z_{n}, \bar{z}_{n}\right)=\left(\prod_{k=1}^{n} \int_{0}^{\infty} d \omega_{k} \omega_{k}^{\Delta_{k}-1}\right) \mathcal{A}_{s_{1} \cdots s_{n}}\left(\epsilon_{1} \omega_{1}, z_{1}, \bar{z}_{1}, \cdots, \epsilon_{n} \omega_{n}, z_{n}, \bar{z}_{n}\right)$ ## General conditions on functions - $\mathcal{M}[f](\Delta)=\int_{0}^{\infty} d \omega \omega^{\Delta-1} f(\omega) \equiv \varphi(\Delta)$ - inverse $\mathcal{M}^{-1}[\varphi](\omega)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} d \Delta \omega^{-\Delta} \varphi(\Delta)=f(\omega)$ - Mellin transform is well-defined if $\int_{0}^{\infty} d \omega \omega^{k-1}|f(\omega)|<\infty$ for some $k>0$ - inverse exists if $c>k$ - if both conditioned satisfied: $\varphi(\Delta)=\mathcal{M}\left[\mathcal{M}^{-1}[\varphi]\right](\Delta)$ ## Generalisations - [[2021#Brandhuber, Brown, Gowdy, Spence, Travaglini]]