Demo · Surfaceology
In the simplest theory of colliding particles, the textbook recipe — sum over Feynman diagrams — drowns in its own bookkeeping. Surfaceology trades the diagrams for curves on a surface, and turns scattering into a counting problem.
Take the simplest theory of colliding particles — colored scalars with a cubic vertex, tr φ³. To get an amplitude you sum over Feynman diagrams. A handful at tree level; billions once you add loops.
Surfaceology's first move: forget the diagram. Draw the n particles as marked points on the boundary of a disk. The dynamics will live on this surface.
Every Feynman diagram is just a way of slicing the disk into triangles with non-crossing curves — a triangulation. The sum over diagrams becomes a sum over curves on the surface.
All triangulations are the corners of a single polytope — the associahedron. The amplitude is read straight off its shape: a positive geometry, the colored-scalar cousin of the amplituhedron.
Add a loop and the surface grows a hole. Now a curve can wind around it — once, twice, forever. Infinitely many curves appear where diagrams used to be.
And yet the whole all-loop amplitude is fixed by a beautifully simple count of these curves — one combinatorial rule per surface. The curve integral replaces the sum over diagrams entirely.
Written this way, the amplitude at every loop order is a single curve integral — no diagrams, no Feynman rules. Locality and unitarity are not assumed; they emerge from the geometry of curves on the surface. It is part of a program — sometimes called surfaceology — to recast scattering in a language where spacetime and quantum mechanics look like consequences of deeper combinatorics, much as the amplituhedron does for maximally supersymmetric theories.