# Lorentzian CFT by Rychkov (TASI 2019) [YouTube Lecture 1](https://www.youtube.com/watch?v=btzDT-HAYAM) [YouTube Lecture 2](https://www.youtube.com/watch?v=DJfHAFKKibU) [YouTube Lecture 3](https://www.youtube.com/watch?v=vRO57qEqREc) [YouTube Lecture 4](https://www.youtube.com/watch?v=ddjKtIvnWY0) ## Applications of Lorentzian - conformal collider - lightcone and bootstrap - causality and ANEC - Lorentzian OPE inversion formula ## Why is this permitted? - Lattice -> CFT in IR ## Euclidean CFT in d > 2 - [[0028 Conformal symmetry]] - specify transformation rules for primaries - no need to specify the rules for descendants independently - $O(x)\rightarrow \pi_f(O)(x)$, $f\in$ Conf - this is a representation of - conformal invariance: - $\langle O(x_1)\dots\rangle=\langle\pi_f(O)(x_1)\dots\rangle$ - **scalar field** - $\pi_f(O)(x)=\Omega(x^\prime)^{-\Delta}O(x^\prime)$, where $x^\prime=f^{-1}x$ - for a rigid transformation $x\rightarrow \lambda x$, $\Omega=\lambda$ (just a number), and we get $\langle O_1(x_1)O_2(x_2)\dots\rangle=\lambda^{-\Delta_1-\dots-\Delta_n}\langle O_1(x_1/\lambda)\dots\rangle$ - in most physical application, $\Delta$ is a positive real number - **fields with spin** - $\pi_f(O)(x)=\Omega(x^\prime)^{-\Delta}\rho(R(x^\prime))O(x^\prime)$ where $R$ is the rotation matrix - where $J^{\mu}{}_\nu=\frac{\partial f^\mu}{\partial x^\nu}=\Omega(x)R^\mu{}_\nu(x)$ - **two point functions** - scalar $\langle O(x)O(0)\rangle=\frac{N}{|x|^{2\Delta}}$ - $N=1$ by convention - tensorial (we will focus on bosonic so we neglect spinorial representations) - $\langle O(x)_{\mu_1\dots\mu_l}O(0)_{\nu_1\dots\nu_l}\rangle=\frac{N}{|x|^{2\Delta}}T^{(2)}_{\mu_1\dots\mu_l\nu_1\dots\nu_l}$ - where $T$ needs to be conformally invariant - turns out there are very few such structures, and in this case of 2-pt. function, there is only one: $I_{\mu_1\nu_1}\dots I_{\mu_l\nu_l}$ with $T_{\mu\nu}(x)=\delta_{\mu\nu}-\frac{2x_\mu x_\nu}{x^2}$ - **three-point functions** - in general a finite number of conformally invariant tensor structures $T^{(3)}$ - scalar-scalar-scalar - $\langle O_1(x_1)O_2(x_2)O_3(x_3)\rangle=\frac{C_{123}}{|x_{12}|^{h_{123}}|x_{23}|^{h_{231}}|x_{13}|^{h_{132}}}$ where $h_{ijk}=\Delta_i+\Delta_j-\Delta_k$ - scalar-scalar-spin-$l$ - multiply by $T^{(3)}_{\mu_1\dots\mu_l}=V_{\mu_1}\dots V_{\mu_l}-trace$, where $V_{\mu}=\frac{1}{|x_{12}||x_{13}||x_{23}|}(x^\mu_{13}x^2_{23}-x^\mu_{23}x^2_{13})$ - can derive using embedding formalism - **4-point functions** - now kinematics does not fix the function uniquely - but two point functions and three point functions completely fix higher point functions - the technique that allows this is called [[0030 Operator product expansion]] - divergent configurations - four points on a circle in a certain order (special case: a line) - OPE does not converge - the 4-point function itself is divergent - for convergent configurations, the rate of convergence depends on how far it is from these special configurations - **convergence of OPE** - can separate a PI into three parts by a surface: integration over field inside, outside, and on the surface in three steps - can treat as an inner product of the in-state and the out-state - can add insertions insides and outside -> this computes correlation functions - in order for this correlation function to be a vector inner product, we want the states to be in a Hilbert space: positive inner product - for **unitary** CFT (in Euclidean this means **reflection positivity**), we do have this - then we can decompose states on a sphere into eigenstates of the dilatation operator - since a correlator is just an inner product between two states, and since OPE just rewrites each state as in the basis of dilatation eigenstates, we know the OPE must converge because rewriting in a different basis does not make a finite inner product divergent - => OPE must converge ## QFT Wightman Axioms - [[0165 Wightman axioms]] - formulated in $\mathbb{R}^{1,d-1}$ - assume Hilbert space $\mathcal{H}$ - Poincare group element $g\in(a,\Lambda)$ = translation + Lorentz; and a unitary transformation operator corresponding to it, $U_g$ - assume that the spectrum of the translation generator belongs to the forward lightcone $\text{spec } P^\mu \in \overline{V}_+$ - i.e. momentum is forward and at most null - unique vacuum, $\Omega$, invariant under Poincare, $U_g\Omega=\Omega$ - local operators, $\phi(x)$, transforms like $U_g^{-1}\phi(x)U_g=\rho(\Lambda)\phi(g^{-1}x)$, where $\rho$ is an irrep. of Lorentz - **micro-causality** (R3) - Wightman functions are **tempered distributions** (R0): $(\Omega, \phi(x_1)\dots\phi(x_n)\Omega)\equiv \mathcal{W}(x_1,\dots,x_n)$ - $\int \mathcal{W}(x_1,\dots,x_n) f_1(x_1)\dots f_n(x_n)dx_1\dots dx_n<\infty$ with $f_i\in S$ (Schwartz function) - Schwartz function: infinitely differentiable; the function itself and all its derivatives decay at infinity faster than any polynomial -> behave nicely under Fourier transforms (Schwartz remain Schwartz) - consequences / properties - **spectral property** (R5) (can show from the axioms) - now let $W(\xi_1,...,\xi_{n-1})\leftarrow\mathcal{W}(x_1,...,x_n)$ with $\xi_i=x_{i+1}-x_i$ - Fourier transform $\hat{W}(q_1,...,q_{n-1})$ has support only in the product of forward null cones $\overline{V}_+\times...\times\overline{V}_+$ - n.b. due to previous condition, the Fourier transform exists and is also a tempered function - **positivity property** $(\psi,\psi)\ge0$ (R2) - $\psi=\sum_{n=0}^N\int\phi(x_1)...\phi(x_n)f_n(x_1,...,x_n)dx_1...dx_n\Omega$ - write the condition in terms of the functions $f_n$: will get $\sum_{n,m}\int\mathcal{W}_{n+m}f_n^*f_mdx_1...dx_{n+m}\ge0$ ### Wightman reconstruction theorem - it means we can forget about the Hilbert space - **Wightman reconstruction theorem**: given a sequence of Wightman functions $\mathcal{W}_n$, one can construct everything else: $\mathcal{H}, U_g, \phi(x)$ - n.b. even if the sequence does not contain all Wightman functions, one can construct enough of the Hilbert space to reproduce these Wightman functions ### Analytic continuation to Euclidean - $W(\xi_1,...,\xi_{n})=\int\hat{W}(q_1,...,q_n)e^{i\sum q_k.\xi_k}dq_1...dq_n$ - can analytically continue $\xi_i$ from $\mathbb{R}^d$ to part of $\mathbb{C}^d$, provided $\Im[\xi_i]\in V_+$ - rough idea for showing **why the analytic continuation exists**: this makes the exponent decaying (we are using mostly negative signature) in the forward light cone so that the Fourier transform is well-defined; i.e. the RHS can be analytically continued and still converge - property: Lorentz invariance is preserved, and in fact complexified Lorentz invariance is also true - $W(\Lambda\xi)=W(\xi)$ for $\Lambda$ in the *complexified* Lorentz group: any complex matrix satisfying $\Lambda^T\eta\Lambda=\eta$ - **Schwinger functions**: ==$\mathcal{S}((\tau_1,\vec x_1),(\tau_2,\vec x_2),...)=\mathcal{W}((-i\tau_1,\vec x_1),(-i\tau_2,\vec x_2),...)$== - suppose $\tau_1>\tau_2>\tau_3...$ - $\xi_k=x_{k+1}-x_k$ -> $\Im[\xi_k]=(\tau_k-\tau_{k+1},\vec 0) \in V_+$ - => $\mathcal{S}$ is analytic at the points specified by its arguments (non-coincidental points) - also, can show this is rotation invariant - **reflection positivity** (E2) - what does positivity property of the Wightman functions $\mathcal{W}$ translate to in the Euclidean - first write $\mathcal{W}(x_1,x_2,x_3,x_4)$=(\Omega,\phi(0,\vec x_1)e^{-iH(t_1-t_2)}\phi(0,\vec x_2)e^{-iH(t_2-t_3)}\phi(0,\vec x_3)e^{-iH(t_3-t_4)}\phi(0,\vec x_4)\Omega)$ - then $\mathcal{S}(x_1,x_2,x_3,x_4)$=(\Omega,\phi(0,\vec x_1)e^{-H(\tau_1-\tau_2)}\phi(0,\vec x_2)e^{-H(\tau_2-\tau_3)}\phi(0,\vec x_3)e^{-H(\tau_3-\tau_4)}\phi(0,\vec x_4)\Omega)$ - consider $\vec x_2=\vec x_3$ and $\vec x_1=\vec x_4$, also consider $\tau_1>\tau_2>0>\tau_3=-\tau_2>\tau_4=-\tau_1$, then symmetric w.r.t. $\tau=0$ - for this particular configuration, $\mathcal{S}=(\psi,\psi)>0$, $\psi=e^{-H\tau_2}\phi(0,\vec x_2)e^{-H(\tau_1-\tau_2)}\phi(0,\vec x_1)\Omega$ - **a possible statement of reflection positivity (not general)**: for a correlator with an even number of operators, when the points are arranged such that it becomes an inner product -> must be positive - **full reflection positivity** - we derive that by translating the positivity condition of Schwinger functions - define $\Theta (\tau,x)=(-\tau,x)$ - ==$\sum_{n,m}\mathcal{S}_{n+m}\Theta f_n^* f_m \ge0$== for any sequence of functions $f_0, ..., f_n$ satisfying the technical conditions below - technical: - need $f_0,...,f_n$ have support only at positive time half plane - need $f$ to vanish at coincident points: when we defined analytic continuation, we did not allow $\taus to become equal ## Osterwalder-Schrader theorem - [[0153 Analytic continuation of Euclidean QFT]] - [[0112 Osterwalder-Schrader reconstruction theorem]] - take Schwinger functions $\mathcal{S}$ at *non-coincidental points* satisfying conditions: - (E1) rotation invariance, translation invariance - (E3) permutation symmetry - (E2) reflection positivity - (E$0^\prime$) growth condition (subtle and introduce only in the second paper) - $|\int dx_1...dx_n\mathcal{S}_n(x_1,...,x_n)f_n(x_1,...,x_n)|\le\sigma_n||f||_{const\cdot n}$ where $\|f\|_{N}=\operatorname{max}_{x}\left(1+|x|^{2}\right)^{\frac{N}{2}} \max _{|\alpha| \leqslant N}\left|\partial^{\alpha} f(x)\right|$, $\sigma_n\le\alpha\cdot (n!)^\beta$ - then one recovers Wightman functions $\mathcal{W_n}$ satisfying all Wightman axioms - limitation - this works in general QFT, but it is very difficult to check in practice. it requires knowledge of *all* $n$-point function (even if you just want to analytically continue e.g. the 4-pt function) to check the growth condition - even for some interesting CFTs you cannot check this - can circumvent this in CFT by assuming a different set of axioms: bootstrap axioms ## Theorem for CFT - work with Kravchuk and Qiao - a first paper: [[2020#Kravchuk, Qiao, Rychkov]] - statement of the theorem - $\mathcal{S}_{n+1}(y_1,...,y_{n+1})=S_n(x_1,...,x_n)$, $x_1=y_1-y_2$, $\tau_i\rightarrow\epsilon_i +it_i$, $\epsilon_i\rightarrow0$ - will have to show ==$|S(x_1,...,x_n)|\le const\cdot P(\frac{1}{\epsilon_i},|\vec x_i|,|\tau_i|)$== - $P$: polynomial in its arguments - consequences - limit as $\forall\epsilon_i\rightarrow0$ exists and is a tempered distribution - can find $\hat W$ such that $S(x_1,...,x_n)=\int dq_1...dq_n\hat W(q_i)e^{-\sum_i q_i^0\tau_i+i \vec q_i.\vec x_i}$ and $\hat W$ has support only at $q_i^0\ge0$ - actually support of $\hat W$ is in $\bar{V}_+$ - applications - 2 pt function - $S(x)=\frac{1}{(x^2)^\Delta}=\frac{1}{(\tau^2+\vec x^2)^\Delta}=\frac{1}{((\epsilon+it)^2+\vec x^2)^\Delta}$ - $|S(x)|\le \frac{1}{\epsilon^{2\Delta}}$ so the limit exists - 3-pt function - $S(x_1,x_2)=\frac{1}{(x^2_1)^\#(x^2_2)^\#(x_1+x_2)^{2\#}}$ - $|S(x_1,x_2)|\le\frac{1}{(\epsilon_1)^\#(\epsilon_2)^\#(\epsilon_1+\epsilon_2)^{\#}}$ - 4-pt. or higher - non-trivial - there we do have such simple formulas - 4-pt. function - in QFT difficult to do the analytic continuation, but in CFT it is possible - details see lecture 3 video - higher point functions - the method no longer works - don't know how to do ## On the boundary of AdS - a **paradox**: finite conformal transformations in Minkowski do not preserve causal ordering - a point can reach infinity and emerge from the opposite null boundary, which then can be at the past of some other point - resolved by going to the covering space: just the statement that the cylinder is composed of many Poincare patches - Lüscher-Mack 1975 - [[LuscherMack1975]] - already defined in Lorentzian, but we want to have a CFT defined in Euclidean signature - there is some argument saying that if the correlation functions are causal on the Poincare patch then there are causal on the whole cylinder - no rigorous proof ## Open questions - higher point functions - the method for 4-pt function no longer works