Amplituhedron

The amplituhedron is a high-dimensional geometric structure which, in areas of quantum field theory , leads to a greatly simplified calculation via its geometry. It has been researched since 2013 by a group led by Nima Arkani-Hamed .

background

Projection of a (simplified) 7-dimensional amplituhedron structure

Mathematically, it is a generalization of a positive Graßmann manifold - this represents a -dimensional space area in a superordinate, -dimensional space. The basic structure of the amplitude hedron cannot be based on triangles, but on any polygons. The term amplituhedron denotes that probability amplitudes can also be represented spatially, namely polyhedron- shaped (English poly hedron ). Due to its facet structure and possible meaning, it is also popularly compared to a jewel . ${\displaystyle Gr(k,n)}$ ${\displaystyle k}$ ${\displaystyle n}$

In the case of particle collisions, as they occur e.g. B. take place in a particle accelerator , numerous combinations of new particles can arise. Their probabilities are called scatter amplitudes . The calculation turns out to be extremely time-consuming as the number of particles increases. Even a normal gluon collision can only be solved using several hundred Feynman diagrams or thousands of calculation terms. A surprising finding is that the probabilities of a scattering process are equivalent to the comparatively easily determinable ( -dimensional) volume of a certain amplitude hedron. The dimensionality and shape of the specific amplitude hedra correspond to the number and properties of the particles involved (such as their helicity ). As a result, the amplituhedron as a geometric body represents a dual formulation of quantum field theoretical processes. So far this has only been valid in an idealized theory, the supersymmetric Yang-Mills theory . ${\displaystyle k}$

Consequences

A generalized confirmation of the amplituhedron theory would have profound consequences for the worldview of physics. The concepts of locality and unitarity , which are viewed as fundamental, are therefore merely emergent phenomena. That means, on a deeper level, spacetime would not be necessary, nor would probabilities have to add up to 1. This could pave the way for the development of quantum gravity . In theory, an infinite-sided amplituhedron represents in its subspaces the totality of all amplitudes of all physical processes.

literature

• New Scientist, No. 3136: The geometry that could reveal the true nature of space-time. (Cover story), July 29, 2017
• Nima Arkani-Hamed, Jacob Bourjaily, Freddy Cachazo , Alexander Goncharov , Alexander Postnikov, Jaroslav Trnka: Scattering Amplitudes and the Positive Grassmannian , 2012. arxiv : 1212.5605 (158 p.)
• Nima Arkani-Hamed, Jaroslav Trnka: The Amplituhedron , 2013. arxiv : 1312.2007 (36 pp.)
• Sebastian Franco, Daniele Galloni, Alberto Mariotti, Jaroslav Trnka: Anatomy of the Amplituhedron. 2014. arxiv : 1408.3410 (74 pp.)
• Nima Arkani-Hamed, Hugh Thomas, Jaroslav Trnka: Unwinding the Amplituhedron in Binary , 2017. arxiv : 1704.05069 (43 pp.)
• Jacob Bourjaily, Hughes Thomas: What is an amplituhedron?, Notices AMS, 2018, No. 2, pdf
• N. Arkani-Hamed, J. Bourjaily, F. Cachazo, AB Goncharov, A. Postnikov, J. Trnka: Grassmannian geometry of scattering amplitudes , Cambridge UP 2016

Web links

• Video: The Amplituhedron - Lecture by Nima Arkani-Hamed at the SUSY 2013 Conference (111 MB, 59 min, English)
• Items to Amplituhedron in the magazine Wired (English)

Individual evidence

1. ↑ Scientists Discover a Jewel at the Heart of Quantum Physics in Wired magazine
2. ↑ ^{a b c} Natalie Wolchover: A Jewel at the Heart of Quantum Physics. In: quantamagazine.org. September 17, 2013, accessed August 1, 2017 .  
3. ↑ Ryan O'Hanlon: How to Feel About Space and Time Maybe Not Existing (interview with Jacob Bourjaily). In: Pacific Standard. September 20, 2013, accessed August 1, 2017 .