Amplituhedron

An amplituhedron is a geometric structure introduced in 2013 by Nima Arkani-Hamed and Jaroslav Trnka. It enables simplified calculation of particle interactions in some quantum field theories. In planar N = 4 supersymmetric Yang–Mills theory, also equivalent to the perturbative topological B model string theory in twistor space, an amplituhedron is defined as a mathematical space known as the positive Grassmannian.

Amplituhedron theory challenges the notion that space-time locality and unitarity are necessary components of a model of particle interactions. Instead, they are treated as properties that emerge from an underlying phenomenon.

The connection between the amplituhedron and scattering amplitudes is at present a conjecture that has passed many non-trivial checks, including an understanding of how locality and unitarity arise as consequences of positivity.

Research has been led by Nima Arkani-Hamed. Edward Witten described the work as “very unexpected" and said that "it is difficult to guess what will happen or what the lessons will turn out to be."

In the approach, the on-shell scattering process "tree" is described by a positive Grassmannian, a structure in algebraic geometry analogous to a convex polytope, that generalizes the idea of a simplex in projective space. A polytope is the n-dimensional analogue of 3-dimensional polyhedrons, and the values being calculated in this case are scattering amplitudes, and so the object is called an amplituhedron.

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