Amplituhedron

In quantum field theory, a branch of theoretical physics, the amplituhedron is a theoretical geometric object defined within an infinite-dimensional space known as the Grassmannian that dramatically simplifies calculations of particle interactions of some quantum field theories and challenges the notion that space-time locality and unitarity are necessary components of a model of particle interactions, as opposed to properties that emerge from some underlying phenomenon. Research into the object has been led by Nima Arkani-Hamed.

Implications

In the approach, on-shell scattering processes are described by a positive Grassmanian, an object in algebraic geometry that generalizes the idea of a simplex in projective space. A simplex is a polytope, a kind of higher dimensional polyhedron, and the values being calculated are scattering amplitudes, and so the object is called an amplituhedron.

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. This 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, which could aid the investigation of theories of quantum gravity.

It also simplifies the 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, the amplituhedron is a non-perturbative approach in which scattering amplitudes can be computed with a single equation.

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.

See also

• Grassmannian
• Twistor space
• Wilson loop

References

• Natalie Wolchover (September 17, 2013). "A Jewel at the Heart of Quantum Physics". Quanta Magazine.
• "The Amplituhedron". SUSY 2013 Conference Video Archive. 2013-08-30.
• Nima Arkani-Hamed. "Scattering Without Space-Time"., Subramanyan Chandrasekhar Lecture, 25 September 2012.

External links

• N = 4 D = 4 super Yang–Mills theory from nLab
• Arxiv paper on Total positivity, Grassmannians, and networks (Sept 2006)
• Arxiv paper on Scattering Amplitudes and the Positive Grassmannian (Dec 2012)