Operator content of free, conformal field theories in low dimensions

Two of the most pressing phenomenological questions of our era ask can we get a quantitative handle on (1) QCD and (2) multi-particle scattering amplitudes relevant for background and new physics searches at the LHC and future colliders? In this talk, I will suggest that an improved understanding of spacetime kinematics could be of relevance for both of these pursuits. In particular, I look at how spacetime symmetry—either Poincaré or its enhancement to conformal symmetry—organize the Hilbert space of free field theories. Working in momentum space with spinors in d=3 and 4 dimensions, we explicitly decompose the Hilbert space into conformal representations, essentially computing the operator product expansion. For N-particle states, there is an action of U(N) symmetry group in d=4, with a remarkable pairing with the conformal group: the two groups dually control the decomposition. A geometric picture emerges, where observables—in the form of conformal primaries—are identified with the coset space U(N)/U(N-2). A similar story holds in d=3, replacing U(N) with O(N). These states have practical use to the problems raised earlier. They provide necessary input ingredients for a recently revived technique called Hamiltonian truncation, which may offer a numerical window into strongly coupled systems. Looking towards colliders, scattering states are identified with scalar primaries, and correspond to a non-redundant operator basis for effective field theories. We point out applications to the Standard Model effective field theory as well as the chiral Lagrangian.