Amplituhedron

Last updated
Notional amplituhedron visualization. Amplituhedron-0c.png
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]

Contents

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 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, Britto–Cachazo–Feng–Witten recursion (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 still unknown, but it provides promising directions for research into theories about the real world.[ citation needed ]

See also

Related Research Articles

<span class="mw-page-title-main">Quantum field theory</span> Theoretical framework combining classical field theory, special relativity, and quantum mechanics

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 and in condensed matter physics to construct models of quasiparticles.

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force. Thus, string theory is a theory of quantum gravity.

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 arena for physics from which space-time itself should emerge. It leads to a powerful set of mathematical tools that have applications to differential and integral geometry, nonlinear differential equations and representation theory and in physics to general relativity and quantum field theory, in particular to scattering amplitudes.

In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter spaces (AdS) which are used in theories of quantum gravity, formulated in terms of string theory or M-theory. On the other side of the correspondence are conformal field theories (CFT) which are quantum field theories, including theories similar to the Yang–Mills theories that describe elementary particles.

<span class="mw-page-title-main">Nima Arkani-Hamed</span> Iranian physicist (born 1972)

Nima Arkani-Hamed is an American-Canadian theoretical physicist of Iranian descent, with interests in high-energy physics, quantum field theory, string theory, cosmology and collider physics. Arkani-Hamed is a member of the permanent faculty at the Institute for Advanced Study in Princeton, New Jersey. He is also Director of the Carl P. Feinberg Cross-Disciplinary Program in Innovation at the Institute and director of The Center for Future High Energy Physics (CFHEP) in Beijing, China.

In mathematical physics, noncommutative quantum field theory is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functions are noncommutative. One commonly studied version of such theories has the "canonical" commutation relation:

The Coleman–Mandula theorem is a no-go theorem in theoretical physics. It states that "space-time and internal symmetries cannot be combined in any but a trivial way". Since "realistic" theories contain a mass gap, the only conserved quantities, apart from the generators of the Poincaré group, must be Lorentz scalars.

In theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists, such as Edward Witten and Cumrun Vafa, by analogy with Witten's earlier idea of topological quantum field theory.

In theoretical physics, Seiberg–Witten theory is a theory that determines an exact low-energy effective action of a supersymmetric gauge theory—namely the metric of the moduli space of vacua.

The J. J. Sakurai Prize for Theoretical Particle Physics, is presented by the American Physical Society at its annual April Meeting, and honors outstanding achievement in particle physics theory. The prize consists of a monetary award, a certificate citing the contributions recognized by the award, and a travel allowance for the recipient to attend the presentation. The award is endowed by the family and friends of particle physicist J. J. Sakurai. The prize has been awarded annually since 1985.

The term "bootstrap model" is used for a class of theories that use very general consistency criteria to determine the form of a quantum theory from some assumptions on the spectrum of particles. It is a form of S-matrix theory.

Asım Orhan Barut was a Turkish-American theoretical physicist.

Stanley Mandelstam was a South African theoretical physicist. He introduced the relativistically invariant Mandelstam variables into particle physics in 1958 as a convenient coordinate system for formulating his double dispersion relations. The double dispersion relations were a central tool in the bootstrap program which sought to formulate a consistent theory of infinitely many particle types of increasing spin.

The history of string theory spans several decades of intense research including two superstring revolutions. Through the combined efforts of many researchers, string theory has developed into a broad and varied subject with connections to quantum gravity, particle and condensed matter physics, cosmology, and pure mathematics.

<span class="mw-page-title-main">MHV amplitudes</span> Maximally helicity violating amplitudes

In theoretical particle physics, maximally helicity violating amplitudes (MHV) are amplitudes with massless external gauge bosons, where gauge bosons have a particular helicity and the other two have the opposite helicity. These amplitudes are called MHV amplitudes, because at tree level, they violate helicity conservation to the maximum extent possible. The tree amplitudes in which all gauge bosons have the same helicity or all but one have the same helicity vanish.

This page is a glossary of terms in string theory, including related areas such as supergravity, supersymmetry, and high energy physics.

N = 4 supersymmetric Yang–Mills (SYM) theory is a mathematical and physical model created to study particles through a simple system, similar to string theory, with conformal symmetry. It is a simplified toy theory based on Yang–Mills theory that does not describe the real world, but is useful because it can act as a proving ground for approaches for attacking problems in more complex theories. It describes a universe containing boson fields and fermion fields which are related by four supersymmetries. It is one of the simplest and one of the few finite quantum field theories in 4 dimensions. It can be thought of as the most symmetric field theory that does not involve gravity.

Twistor string theory is an equivalence between N = 4 supersymmetric Yang–Mills theory and the perturbative topological B model string theory in twistor space.

Freddy Alexander Cachazo is a Venezuelan-born theoretical physicist who holds the Gluskin Sheff Freeman Dyson Chair in Theoretical Physics at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada.

Zvi Bern is an American theoretical particle physicist. He is a professor at University of California, Los Angeles (UCLA).

References

  1. 1 2 3 4 5 6 7 Arkani-Hamed, Nima; Trnka, Jaroslav (2014). "The Amplituhedron". Journal of High Energy Physics . 2014 (10): 30. arXiv: 1312.2007 . Bibcode:2014JHEP...10..030A. doi:10.1007/JHEP10(2014)030. S2CID   7717260.
  2. ↑ Witten, Edward (2004). "Perturbative Gauge Theory As A String Theory In Twistor Space". Communications in Mathematical Physics. 1. 252 (1): 189–258. arXiv: hep-th/0312171 . Bibcode:2004CMaPh.252..189W. doi:10.1007/s00220-004-1187-3. S2CID   14300396.
  3. 1 2 3 Arkani-Hamed, Nima; Bourjaily, Jacob L.; Cachazo, Freddy; Goncharov, Alexander B.; Postnikov, Alexander; Trnka, Jaroslav (2012). "Scattering Amplitudes and the Positive Grassmannian". arXiv: 1212.5605 [hep-th].
  4. ↑ Ryan O'Hanlon (September 19, 2013). "How to Feel About Space and Time Maybe Not Existing". Pacific Standard.
  5. ↑ Natalie Wolchover (September 17, 2013). "A Jewel at the Heart of Quantum Physics". Quanta Magazine .
  6. ↑ "The Amplituhedron and Other Excellently Silly Words". 4 gravitons and a grad student. September 20, 2013. Archived from the original on September 25, 2013.
  7. 1 2 3 Anil Ananthaswamy; "The New Shape of Reality", New Scientist, 29 July 2017, pages 28-31.
  8. ↑ Kevin Drum (September 18, 2013). "Maybe Space-Time Is Just an Illusion". Mother Jones.
  9. ↑ GraduatePhysics (2016-07-23), Nima Arkani-Hamed - Physics and Mathematics for the End of Spacetime , retrieved 2017-05-28
  10. ↑ Musser, George (2015). Spooky Action at a Distance . New York: Farrar, Straus and Giroux. pp.  40–41. ISBN   978-0-374-53661-9.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.