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Notional amplituhedron visualization.

In mathematics and theoretical physics (especially twistor string theory), 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.[1][2]

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.[3][4]

The connection between the amplituhedron and scattering amplitudes is at present[when?] 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 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".[5]


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.[3] 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.[1]

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.[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 the planar limit of N = 4 D = 4 supersymmetric Yang–Mills theory, it describes the scattering amplitudes of particles described by this theory.[1] The amplituhedron thus provides a more intuitive geometric model for calculations with highly-abstract underlying principles.[6]

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.[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.[7]


The twistor approach simplifies calculations of particle interactions. In a conventional perturbative approach to quantum field theory, such interactions may require the calculation of thousands of Feynman 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.[8] 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.[9][7]

The geometric nature of the theory suggests in turn that the nature of the universe, in both classical relativistic spacetime and quantum mechanics, may be described with geometry.[7]

Calculations can be done without assuming the quantum mechanical properties of locality and unitarity. In amplituhedron theory, locality and unitarity arise as a direct consequence of positivity.[clarification needed] 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.[10]

Since the planar limit of the N = 4 supersymmetric Yang–Mills theory is a toy theory that does not describe the real world, the relevance of this technique for more realistic quantum field theories is currently[when?] unknown, but it provides promising directions for research into theories about the real world.[citation needed]

See also[edit]


  1. ^ a b c d e f g Arkani-Hamed, Nima; Trnka, Jaroslav (2014). "The Amplituhedron". Journal of High Energy Physics. 2014 (10): 30. arXiv:1312.2007. Bibcode:2014JHEP...10..030A. doi:10.1007/JHEP10(2014)030. S2CID 7717260.
  2. ^ Witten, Edward (2004). "Perturbative Gauge Theory As A String Theory In Twistor Space". Communications in Mathematical Physics. 1. 252 (1): 189–258. arXiv:hep-th/0312171. Bibcode:2004CMaPh.252..189W. doi:10.1007/s00220-004-1187-3. S2CID 14300396.
  3. ^ a b c Arkani-Hamed, Nima; Bourjaily, Jacob L.; Cachazo, Freddy; Goncharov, Alexander B.; Postnikov, Alexander; Trnka, Jaroslav (2012). "Scattering Amplitudes and the Positive Grassmannian". arXiv:1212.5605 [hep-th].
  4. ^ Ryan O'Hanlon (September 19, 2013). "How to Feel About Space and Time Maybe Not Existing". Pacific Standard.
  5. ^ Natalie Wolchover (September 17, 2013). "A Jewel at the Heart of Quantum Physics". Quanta Magazine.
  6. ^ "The Amplituhedron and Other Excellently Silly Words". 4 gravitons and a grad student. September 20, 2013. Archived from the original on September 25, 2013.
  7. ^ a b c Anil Ananthaswamy; "The New Shape of Reality", New Scientist, 29 July 2017, pages 28-31.
  8. ^ Kevin Drum (September 18, 2013). "Maybe Space-Time Is Just an Illusion". Mother Jones.
  9. ^ GraduatePhysics (2016-07-23), Nima Arkani-Hamed - Physics and Mathematics for the End of Spacetime, retrieved 2017-05-28
  10. ^ Musser, George (2015). Spooky Action at a Distance. New York: Farrar, Straus and Giroux. pp. 40–41. ISBN 978-0-374-53661-9.

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