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. 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 spacetime 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'. When subatomic particles interact, different outcomes are possible. The evolution of the various possibilities is called a 'tree' and the probability of a given outcome is called its scattering amplitude. According to the principle of unitarity, the sum of the probabilities for every possible outcome is 1. The on-shell scattering process 'tree' may be 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 a 3-dimensional polyhedron, the values being calculated in this case are scattering amplitudes, and so the object is called an amplituhedron. Using twistor theory, BCFW recursion relations involved in the scattering process may be represented as a small number of twistor diagrams. These diagrams effectively provide the recipe for constructing the positive Grassmannian, i.e. the amplituhedron, which may be captured in a single equation. The scattering amplitude can thus be thought of as the volume of a certain polytope, the positive Grassmannian, in momentum twistor space. When the volume of the amplituhedron is calculated in the planar limit of N = 4 D = 4 supersymmetric Yang–Mills theory, it describes the scattering amplitudes of subatomic particles. The amplituhedron thus provides a more intuitive geometric model for calculations whose underlying principles were until then highly abstract. The twistor-based representation provides a recipe for constructing specific cells in the Grassmannian which assemble to form a positive Grassmannian, i.e. the representation describes a specific cell decomposition of the positive Grassmannian. The recursion relations can be resolved in many different ways, each giving rise to a different representation, with the final amplitude expressed as a sum of on-shell processes in different ways as well. Therefore, any given on-shell representation of scattering amplitudes is not unique, but all such representations of a given interaction yield the same amplituhedron.