Positive Moments for Scattering Amplitudes

What does define the space of theories that are found at the endpoints of the renormalization group evolution subject to the fundamental principles of quantum mechanics and relativity? 

While this question is still too hard to be addressed in generality, I will present in this talk the complete set of positivity conditions that 2-to-2 scattering amplitudes have to satisfy at low energy in order to descend from a unitary and causal microscopic theory. Positivity of the amplitudes translates into bounds of the low-energy Wilson coefficients that go beyond the implementation of symmetries, ruling out candidate effective theories that scale very quickly with energy in the infrared. I will discuss the connection between these physical questions and the mathematics of the multidimensional moment problem, and its challenges to further progress.