From scattering amplitudes to the dilatation generator in N=4 SYM

The complete spin chain representation of the planar SYM dilatation generator has long been known at one loop, where it involves leading nearest-neighbor 2 → 2 interactions. In this work, we use superconformal symmetry to derive the unique solution for the leading L → 2 interactions of the planar dilatation generator for arbitrarily large L . We then propose that these interactions are given by a scattering operator that has SYM tree-level scattering amplitudes as matrix elements. We provide compelling evidence for this proposal, including explicit checks for L = 2, 3 and a proof of consistency with superconformal symmetry.