Amplituhedron

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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]

Description[]

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]

Implications[]

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[]

• Associahedron
• Wilson loop

References[]

External links[]

Wikiquote has quotations related to: Amplituhedron

• Grassmannian Geometry of Scattering Amplitudes Workshop, December 8–12, 2014
• Nima Arkani-Hamed (2013-08-30). "The Amplituhedron" (video). SUSY 2013 Conference Video Archive.
• Scattering Without Space-Time Subrahmanyan Chandrasekhar Lecture, 25 September 2012 on YouTube
• N = 4 D = 4 super Yang–Mills theory from nLab
• Postnikov, Alexander (2006-09-27). "Total positivity, Grassmannians, and networks". arXiv:math/0609764.