Causality constraints in quantum field theory

Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In conformal field theory, I will show how causality is encoded in crossing symmetry and reflection positivity of Euclidean correlators, and derive constraints on the interactions of low-lying operators directly from the conformal bootstrap. I will explain the connection between the causality constraints in the lightcone limit and the averaged null energy condition. In particular, causality implies that the averaged null energy must be positive in interacting quantum field theory in flat spacetime. Whereas, for large-N CFTs, causality constraints in the Regge limit leads to a stronger condition which is the CFT version of the gravity condition proposed by Engelhardt and Fischetti. This causality condition leads to strong constraints on CFT 3-point couplings. In particular, I will also show that in four-dimensions, conformal field theories with large-N and a sparse spectrum exhibit universal, gravity-like behavior and the anomaly coefficients satisfy a=c.