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Inmathematics andtheoretical physics (especiallytwistor string theory), anamplituhedron is a geometric structure introduced in 2013 byNima Arkani-Hamed and Jaroslav Trnka. It enables simplified calculation ofparticle interactions in somequantum field theories. InplanarN = 4 supersymmetric Yang–Mills theory, also equivalent to the perturbativetopological B model string theory intwistor space, an amplituhedron is defined as a mathematical space known as the positiveGrassmannian.[1][2]
Amplituhedron theory challenges the notion thatspacetimelocality andunitarity are necessary components of a model of particle interactions. Instead, they are treated as properties that emerge from an underlying phenomenon.[3][4]
The connection between the amplituhedron and scattering amplitudes is a conjecture that has passed many non-trivial checks, including an understanding of how locality and unitarity arise as consequences of positivity.[1] Research has been led byNima 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".[5]
Whensubatomic particles interact, different outcomes are possible. The evolution of the various possibilities is called a "tree", and theprobability amplitude of a given outcome is called itsscattering amplitude. According to the principle ofunitarity, the sum of the probabilities (the squared moduli of the probability amplitudes) for every possible outcome is 1.
Theon-shell scattering process "tree" may be described by a positiveGrassmannian, a structure inalgebraic geometry analogous to aconvex polytope, that generalizes the idea of asimplex inprojective space.[3] Apolytope is then-dimensional analogue of a 3-dimensionalpolyhedron, the values being calculated in this case are scattering amplitudes, and so the object is called anamplituhedron.[1]
Usingtwistor theory, Britto–Cachazo–Feng–Witten recursion (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.[3] The scattering amplitude can thus be thought of as the volume of a certain polytope, the positive Grassmannian, in momentum twistor space.[1]
When the volume of the amplituhedron is calculated in theplanar limit ofN = 4D = 4 supersymmetric Yang–Mills theory, it describes thescattering amplitudes of particles described by this theory.[1]
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 specificcell 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.[1]
The twistor approach is relatively abstract. While amplituhedron theory provides an underlying geometric model, the geometrical space is not physical spacetime and is also best understood as abstract.[6]
The twistor approach simplifies calculations of particle interactions. In a conventionalperturbative approach to quantum field theory, such interactions may require the calculation of thousands ofFeynman diagrams, most describing off-shell "virtual" particles which have no directly observable existence. In contrast,twistor theory provides an approach in which scattering amplitudes can be computed in a way that yields much simpler expressions.[7] Amplituhedron theory calculates scattering amplitudes without referring to such virtual particles. This undermines the case for even a transient, unobservable existence for such virtual particles.[6]
The geometric nature of the theory suggests in turn that the nature of the universe, in both classicalrelativisticspacetime andquantum mechanics, may be described withgeometry.[6]
Calculations can be done without assuming the quantum mechanical properties oflocality andunitarity. In amplituhedron theory, locality and unitarity arise as a direct consequence of positivity.[4] They are encoded in the positive geometry of the amplituhedron, via the singularity structure of the integrand for scattering amplitudes.[1] Arkani-Hamed suggests this is why amplituhedron theory simplifies scattering-amplitude calculations: in the Feynman-diagrams approach, locality is manifest, whereas in the amplituhedron approach, it is implicit.[8]
