# Reflection positivity (in CFT) Reflection positivity imposes reality conditions on the OPE coefficients in CFT. An example is the following. The four-point *scalar* correlator is constrained to take the form$S_4\left(x_1, x_2, x_3, x_4\right)=\left(\frac{x_{24}^2}{x_{14}^2}\right)^{\frac{1}{2} \Delta_{12}}\left(\frac{x_{14}^2}{x_{13}^2}\right)^{\frac{1}{2} \Delta_{34}} \frac{g_{34}^{21}(u, v)}{\left(x_{12}^2\right)^{\frac{1}{2}\left(\Delta_1+\Delta_2\right)}\left(x_{34}^2\right)^{\frac{1}{2}\left(\Delta_3+\Delta_4\right)}},$where$g_{21}^{21}(x, \bar{x})=\sum_pC_{12}{}^pC_{21}{}^p \mathcal{F}_{21}^{21}(p | x) \bar{\mathcal{F}}_{21}^{21}(p | \bar{x}).$It turns out that $\mathcal{F}\bar{\mathcal{F}}$ is positive for even spins and negative for odd spins, so reflection positivity requires that$C_{12}{}^pC_{21}{}^p(-1)^{s_p}$be positive. Conformal symmetry requires $C_{21}{}^p= (-1)^{s_p}C_{12}{}^p$, so $(C_{12}{}^p)^2$ must be positive, hence positivity of $C_{12}{}^p$ itself (for scalar 1 and 2). A similar derivation works for 1 and 2 with general spins I believe (I haven't found references for this). In the example above, we have chosen a basis such that $O_p^\dagger=O_p$. Reflection positivity of the two-point function says $\langle\mathcal{O}^\dagger_p O_p\rangle\propto C_{pp\mathbb{1}}>0$, and we have used this fact to normalise $C_{pp\mathbb{1}}=1$. Had we chosen $O_p^\dagger=-O_p$ (we can achieve this by multiplying $i$ in the definition of $O_p$), reflection positivity of the two-point function would require $\langle\mathcal{O}^\dagger_p O_p\rangle\propto -C_{pp\mathbb{1}}>0$, so we can only normalise $C_{pp\mathbb{1}}=-1$ rather than positive unity. In this case, reflection positivity of the four-point function states$C_{12}{}^pC_{12}{}^pC_{pp\mathbb{1}}=-C_{12}{}^pC_{12}{}^p>0,$so $C_{12}{}^{p}$ is pure imaginary as well as $C_{12p}\equiv C_{12}{}^pC_{pp\mathbb{1}}=-C_{12}{}^p$. ## Refs - [[2021#Kravchuk, Qiao, Rychkov]]: when all three operators are self-conjugate-reflected i.e. satisfy (2.27), the OPE coefficients $f^a_{ijk}$ must be real - [[2021#Lanosa, Leston, Passaglia]]: footnote 3 talks about spin dependence \[*Acknowledgement*: I thank Jiaxin Qiao for teaching me about this.\]