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.
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.
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.
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The twistor approach simplifies calculations of particle interactions. In a perturbative approach to quantum field theory, such interactions may require the calculation of hundreds of Feynman diagrams. In contrast, twistor theory provides an approach in which scattering amplitudes can be computed in a way that yields much simpler expressions.
The twistor approach was relatively abstract. The amplituhedron provides an underlying model. Its geometric nature suggests the possibility that the nature of the universe, both classical relativistic spacetime and quantum mechanics, can be described with geometry. 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. They are encoded in the positive geometry of the amplituhedron, via the singularity structure of the integrand for scattering amplitudes.
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 unknown, but it provides promising directions for research into theories about the real world.
- Arkani-Hamed & Trnka 2013.
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