mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

(especially twistor string theory), an amplituhedron is a geometric structure introduced in 2013 by

Nima Arkani-Hamed Nima Arkani-Hamed ( fa, نیما ارکانی حامد; born April 5, 1972) is an American-Canadian

and Jaroslav Trnka. It enables simplified calculation of

particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from ...

interactions in some

quantum field In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...

theories. In planar ''N'' = 4 supersymmetric Yang–Mills theory, also equivalent to the perturbative topological B model string theory in

twistor space In mathematics and theoretical physics (especially twistor theory), twistor space is the complex vector space of solutions of the twistor equation \nabla_^\Omega_^=0 . It was described in the 1960s by Roger Penrose and Malcolm MacCallum. Accor ...

, an amplituhedron is defined as a mathematical space known as the positive

Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...

. Amplituhedron theory challenges the notion that

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...

locality Locality may refer to: * Locality (association), an association of community regeneration organizations in England * Locality (linguistics) * Locality (settlement) * Suburbs and localities (Australia), in which a locality is a geographic subdivis ...

and

unitarity In quantum physics, unitarity is the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of qua ...

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 a conjecture that has passed many non-trivial checks, including an understanding of how locality and unitarity arise as consequences of positivity. Research has been led by

Nima Arkani-Hamed Nima Arkani-Hamed ( fa, نیما ارکانی حامد; born April 5, 1972) is an American-Canadian

Edward Witten Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...

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".

Description

When

subatomic particle In physical sciences, a subatomic particle is a particle that composes an atom. According to the Standard Model of particle physics, a subatomic particle can be either a composite particle, which is composed of other particles (for example, a p ...

s 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 In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.unitarity In quantum physics, unitarity is the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of qua ...

, the sum of the probabilities for every possible outcome is 1. The

on-shell In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called "on the mass shell" or simply more often on shell; while those that do not are called "off the mass shell", ...

scattering process "tree" may be described by a positive

, a structure in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

analogous to a

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the w ...

, that generalizes the idea of a

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

. A

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

is the ''n''-dimensional analogue of a 3-dimensional

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...

, the values being calculated in this case are scattering amplitudes, and so the object is called an ''amplituhedron''. Using

twistor theory In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a branch of theoretical and mathematical physics. Penrose proposed that twistor space should be the basic ar ...

, Britto–Cachazo–Feng–Witten recursion (

BCFW recursion The Britto–Cachazo–Feng–Witten recursion relations are a set of on-shell recursion relations in quantum field theory. They are named for their creators, Ruth Britto, Freddy Cachazo, Bo Feng and Edward Witten. The BCFW recursion method ...

) 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 amplitude In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process. The amplituhedron thus provides a more intuitive geometric model for calculations with highly abstract underlying principles. 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. 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.Anil Ananthaswamy; "The New Shape of Reality", ''New Scientist'', 29 July 2017, pages 28-31.

Implications

The twistor approach simplifies calculations of particle interactions. In a conventional

perturbative In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for w ...

approach to quantum field theory, such interactions may require the calculation of thousands of

Feynman diagrams In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduce ...

, most describing off-shell "virtual" particles which have no directly observable existence. In contrast,

provides an approach in which scattering amplitudes can be computed in a way that yields much simpler expressions. 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. The geometric nature of the theory suggests in turn that the nature of the universe, in both classical relativistic

and

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

, may be described with

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

. Calculations can be done without assuming the quantum mechanical properties of

and

. 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. 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. 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 still unknown, but it provides promising directions for research into theories about the real world.

References

External links

Grassmannian Geometry of Scattering Amplitudes Workshop, December 8–12, 2014
* *
''N'' = 4 ''D'' = 4 super Yang–Mills theory
from nLab * {{Cite arXiv , last=Postnikov, first=Alexander, date=2006-09-27, title=Total positivity, Grassmannians, and networks , eprint=math/0609764 2013 in science Gauge theories Geometry Quantum gravity Scattering theory

Description

Implications

See also

References

External links