# Two-point functions (Green's functions, propagators, etc) ## From Fourier space to real space - commutation functions are closed contours, while propagation functions are open contours extending to infinity - a good summary in this [pdf](https://ncatlab.org/nlab/files/KGPropagatorsOnMinkowskiTable.pdf) and more details at [NCatLab](https://ncatlab.org/nlab/show/causal+propagator) - [[2002#Son, Starinets]] defines several 2-point functions in Fourier space ## From Feynman propagator to Green's function The Feynman propagator is defined by $\Delta_{F}(x-y)=\langle 0|T \phi(x) \phi(y)| 0\rangle=\left\{\begin{array}{cc}D(x-y) & x^{0}>y^{0} \\ D(y-x) & y^{0}>x^{0}\end{array}\right.$where $D(x-y)\equiv\langle 0|\phi(x) \phi(y)| 0\rangle$. Now one can show (see e.g. David Tong's QFT notes) that $\Delta_{F}(x-y)=\int \frac{d^{4} p}{(2 \pi)^{4}} \frac{i e^{-i p \cdot(x-y)}}{p^{2}-m^{2}+i \epsilon}.$Direct computation then shows that the Feynman propagator *is* the Green's function: $\begin{aligned}\left(\partial_{t}^{2}-\nabla^{2}+m^{2}\right) \Delta_{F}(x-y) &=\int \frac{d^{4} p}{(2 \pi)^{4}} \frac{i}{p^{2}-m^{2}}\left(-p^{2}+m^{2}\right) e^{-i p \cdot(x-y)} \\ &=-i \int \frac{d^{4} p}{(2 \pi)^{4}} e^{-i p \cdot(x-y)} \\ &=-i \delta^{(4)}(x-y). \end{aligned}$ ## Propagators in AdS - embedding formalism - [[2014#Costa, Goncalves, Penedones]] ## Propagator v.s. path integral The usual path integral gives time-ordered correlators, whereas Wightman functions (or "contour" ordered correlators) need path integral on time-folds: $\begin{aligned}\frac{1}{i} \Delta(x-y)&=\langle 0|\mathcal{T} \hat{\phi}(x) \hat{\phi}(y)| 0\rangle \equiv \int \mathcal{D} \phi(x) \phi(x) \phi(y) e^{i S[\phi]}\\&=\left.\frac{\delta}{i \delta J(x)} \frac{\delta}{i \delta J(y)} \int \mathcal{D} \phi(x) e^{i\left(S[\phi]+\int d^{4} x J(x) \phi(x)\right)}\right|_{J=0}.\end{aligned}$See e.g. [stack exchange](https://physics.stackexchange.com/questions/133701/propagators-green-s-functions-path-integrals-and-transition-amplitudes-in-quan) for more details. ## Propagator on non-static backgrounds In principle, to obtain the two-point function, it seems that we can just solve the field equations on complicated curved spacetime. However, how do we connect to the Euclidean section (which is not well-defined) by matching as in [[2008#Skenderis, van Rees (May)]]? This is something I haven't understood. In practice, one usually take the **geodesic approximation**: - see e.g. [[2012#Balasubramanian, Bernamonti, Craps, Keranen, Keski-Vakkuri, Muller, Thorlacius, Vanhoof]] - for dS space (using complex geodesics), see - [[2022#Aalsma, Faruk, van der Schaar, Visser, de Witte]] - [[2022#Chapman, Galante, Harris, Sheorey, Vegh]] ## Examples - [[2023#He, Li]]: two point thermal correlator in Einstein-Maxwell - see also [[0473 Retarded Green's function]] ## Black hole singularity and boundary two point function - [[2023#Horowitz, Leung, Queimada, Zhao]]: in the presence of a [[0117 Shockwave|shockwave]] - [[2024#Ceplak, Liu, Parnachev, Valach]]: boundary [[0030 Operator product expansion|OPE]] ## Thermal two-point functions - [[2025#Buric, Gusev, Parnachev]]