Positive Energy Conditions in 4D Conformal Field Theory

I will discuss a local energy condition, the spacetime averaged weak energy condition (SAWEC), in the context of conformal field theory (CFT). This condition is a bound on the energy density with spacetime averaging over a region of length scale $L$, $\langle T^{00}\rangle \geq -C/L^4$, where $C$ is a positive theory-dependent constant. We motivate this condition as a fundamental consistency requirement for any $\text{4D}$ quantum field theory. We argue that violation of this statement would have serious undesirable consequences for a theory. In particular, the theory would contain states indistinguishable from states of negative total energy by any local measurement, which would lead to unphysical instabilities. We apply the condition to $\text{4D}$ and $\text{3D}$ CFTs and derive bounds on the OPE coefficients of these theories. Interestingly, these conditions imply the positivity of the 2-point function of the energy-momentum tensor. Our $\text{4D}$ bounds are weaker than the ``conformal collider" constraints of Hofman and Maldacena, which were recently rigorously established. All calculations have been carried out in momentum-space using Wightman correlation functions. These methods may also be interesting on their own.