# Conformal basis The idea of the conformal basis lies in the heart of [[0010 Celestial holography|celestial holography]]. Once one writes 4d scattering amplitudes in the conformal basis, they transform like correlators in a 2d CFT. This suggests that there might a better way to study and organise scattering amplitudes, and the duality between the conventional 4d physics and this 2d reformulation is called the celestial holography. ## What is the basis - primary functions with conformal weights on the principal continuous series ## Particle content of CCFT - for *stable* particles of spin $s$, a *complete* basis is given by celestial conformal [[0029 Primary operator|primaries]] with conformal weights $(h,\bar{h})=\left(\frac{\Delta+s}{2},\frac{\Delta-s}{2}\right)$ and $\Delta = 1+i \lambda$ (and positions $(z,\bar{z})$). - unstable particles decay and do not reach infinity - for *massless* particles, conformal fields operators are Mellin transforms of place wave functions in 4d Minkowski ## Wavefunctions for massless fields - for **massless** fields, just do [[0079 Mellin transform|Mellin transformation]] - massless spin-0 conformal primary - $f(\omega)=e^{\pm i \omega q \cdot X-\varepsilon \omega q^{0}}$ with $\varepsilon>0$ - -> $\phi^{\Delta, \pm}\left(X^{\mu} ; w ; \bar{w}\right)=\frac{\Gamma(\Delta)}{(\pm i)^{\Delta}} \frac{1}{\left(-q \cdot X \mp i \varepsilon q^{0}\right)^{\Delta}}$ with $\operatorname{Re}[\Delta]>0$ - n.b. $i\varepsilon$ needed - massless spin-1 and spin-2 conformal primary - see 3.2 of [[2020#Donnay, Pasterski, Puhm]] for precise forms and gauge choices - they are gauge equivalent to Mellin transforms of plane waves multiplied by polarisation vectors/tensors - graviton - [[LiuLowe2021]][](https://arxiv.org/pdf/2109.00037.pdf) ## Inner product - spin-1 $\left(A, A^{\prime}\right)_{\Sigma}=-i \int d \Sigma^{\rho}\left[A^{\nu} F_{\rho \nu}^{\prime *}-A^{\prime * \nu} F_{\rho \nu}\right]$ - spin-2$\left(h, h^{\prime}\right)_{\Sigma}=-i \int d \Sigma^{\rho}\left[h^{\mu \nu} \nabla_{\rho} h_{\mu \nu}^{\prime *}-2 h^{\mu \nu} \nabla_{\mu} h_{\rho \nu}^{\prime *}+h \nabla^{\mu} h_{\rho \mu}^{\prime *}-h \nabla_{\rho} h^{\prime *}+h_{\rho \mu} \nabla^{\mu} h^{\prime *}-\left(h \leftrightarrow h^{\prime *}\right)\right]$ ## Operators - mode expansion$\begin{aligned} &O^s\left(X^\mu\right)=\sum_{J=\pm s} \int d^2 w \int_{1-i \infty}^{1+i \infty}(-i d \Delta) & \\&\left[\mathcal{N}_{2-\Delta, s}^{+} \Phi_{2-\Delta,-J}\left(X_{+}^\mu ; w, \bar{w}\right) a_{\Delta, J}(w, \bar{w})+\mathcal{N}_{\Delta, s}^{-} \Phi_{\Delta, J}\left(X_{-}^\mu ; w, \bar{w}\right) a_{\Delta, J}(w, \bar{w})^{\dagger}\right] \end{aligned}$ - extracting the Fourier component $\mathcal{O}$ of a 4D operator $O$ using an inner product $\mathcal{O}_{\Delta, J}^{s, \pm}(w, \bar{w}) \equiv i\left(O^s\left(X^\mu\right), \Phi_{\Delta^*,-J}^s\left(X_{\mp}^\mu ; w, \bar{w}\right)\right)$ - commutators - $\left[a_{\Delta, J}(w, \bar{w}), a_{\Delta^{\prime}, J^{\prime}}\left(w^{\prime}, \bar{w}^{\prime}\right)^{\dagger}\right]=\delta_{J J^{\prime}} \delta^{(2)}\left(w-w^{\prime}\right) \boldsymbol{\delta}\left(i\left(\Delta+\Delta^{\prime *}-2\right)\right)$ - $\left[\mathcal{O}_{\Delta, J}^s(w, \bar{w}), O^s\left(X^\mu\right)\right]=i \Phi_{\Delta, J}\left(X_{-}^\mu ; w, \bar{w}\right)$ ## Refs - main - [[2017#Pasterski, Shao, Strominger (Jan)]] - [[0256 Massive particles in CCFT|massive]] scalar - [[2023#Hao, Taylor]] - generalisations - [[2017#Pasterski, Shao]]: Maxwell and linearised Einstein gravity - [[2020#Narayanan]] and [[2020#Iacobacci, Muck]] (independently): massive and massless fermions - [[2020#Pasterski, Puhm]]: all spins related by SUSY - [[2022#Donnay, Esmaeili, Heissenberg]]: $p$-forms - inner products and defining operators - [[2020#Donnay, Pasterski, Puhm]] - discrete basis - [[2022#Freidel, Pranzetti, Raclariu]] - [[2023#Cotler, Miller, Strominger]] - compact contour - [[2024#Mitra]]