Perturbative scattering amplitudes in gauge theories have remarkable
simplicity and hidden infinite dimensional symmetries that are completely
obscured in the conventional formulation of field theory using Feynman
diagrams. This suggests the existence of a new understanding for scattering
amplitudes where locality and unitarity do not play a central role but are
derived consequences from a different starting point. In this note we provide
such an understanding for N=4 SYM scattering amplitudes in the planar limit,
which we identify as ``the volume" of a new mathematical object--the
Amplituhedron--generalizing the positive Grassmannian. Locality and unitarity
emerge hand-in-hand from positive geometry.Comment: 36 pages, 14 figure
