Noncommutative standard model

Last updated

In theoretical particle physics, the non-commutative Standard Model (best known as Spectral Standard Model [1] [2] ), is a model based on noncommutative geometry that unifies a modified form of general relativity with the Standard Model (extended with right-handed neutrinos).

Contents

The model postulates that space-time is the product of a 4-dimensional compact spin manifold by a finite space . The full Lagrangian (in Euclidean signature) of the Standard model minimally coupled to gravity is obtained as pure gravity over that product space. It is therefore close in spirit to Kaluza–Klein theory but without the problem of massive tower of states.

The parameters of the model live at unification scale and physical predictions are obtained by running the parameters down through renormalization.

It is worth stressing that it is more than a simple reformation of the Standard Model. For example, the scalar sector and the fermions representations are more constrained than in effective field theory.

Motivation

Following ideas from Kaluza–Klein and Albert Einstein, the spectral approach seeks unification by expressing all forces as pure gravity on a space .

The group of invariance of such a space should combine the group of invariance of general relativity with , the group of maps from to the standard model gauge group .

acts on by permutations and the full group of symmetries of is the semi-direct product:

Note that the group of invariance of is not a simple group as it always contains the normal subgroup . It was proved by Mather [3] and Thurston [4] that for ordinary (commutative) manifolds, the connected component of the identity in is always a simple group, therefore no ordinary manifold can have this semi-direct product structure.

It is nevertheless possible to find such a space by enlarging the notion of space.

In noncommutative geometry, spaces are specified in algebraic terms. The algebraic object corresponding to a diffeomorphism is the automorphism of the algebra of coordinates. If the algebra is taken non-commutative it has trivial automorphisms (so-called inner automorphisms). These inner automorphisms form a normal subgroup of the group of automorphisms and provide the correct group structure.

Picking different algebras then give rise to different symmetries. The Spectral Standard Model takes as input the algebra where is the algebra of differentiable functions encoding the 4-dimensional manifold and is a finite dimensional algebra encoding the symmetries of the standard model.

History

First ideas to use noncommutative geometry to particle physics appeared in 1988-89, [5] [6] [7] [8] [9] and were formalized a couple of years later by Alain Connes and John Lott in what is known as the Connes-Lott model . [10] The Connes-Lott model did not incorporate the gravitational field.

In 1997, Ali Chamseddine and Alain Connes published a new action principle, the Spectral Action, [11] that made possible to incorporate the gravitational field into the model. Nevertheless, it was quickly noted that the model suffered from the notorious fermion-doubling problem (quadrupling of the fermions) [12] [13] and required neutrinos to be massless. One year later, experiments in Super-Kamiokande and Sudbury Neutrino Observatory began to show that solar and atmospheric neutrinos change flavors and therefore are massive, ruling out the Spectral Standard Model.

Only in 2006 a solution to the latter problem was proposed, independently by John W. Barrett [14] and Alain Connes, [15] almost at the same time. They show that massive neutrinos can be incorporated into the model by disentangling the KO-dimension (which is defined modulo 8) from the metric dimension (which is zero) for the finite space. By setting the KO-dimension to be 6, not only massive neutrinos were possible, but the see-saw mechanism was imposed by the formalism and the fermion doubling problem was also addressed.

The new version of the model was studied in [16] and under an additional assumption, known as the "big desert" hypothesis, computations were carried out to predict the Higgs boson mass around 170 GeV and postdict the Top quark mass.

In August 2008, Tevatron experiments [17] excluded a Higgs mass of 158 to 175 GeV at the 95% confidence level. Alain Connes acknowledged on a blog about non-commutative geometry that the prediction about the Higgs mass was invalidated. [18] In July 2012, CERN announced the discovery of the Higgs boson with a mass around 125 GeV/c2.

A proposal to address the problem of the Higgs mass was published by Ali Chamseddine and Alain Connes in 2012 [1] by taking into account a real scalar field that was already present in the model but was neglected in previous analysis. Another solution to the Higgs mass problem was put forward by Christopher Estrada and Matilde Marcolli by studying renormalization group flow in presence of gravitational correction terms. [19]

See also

Notes

  1. 1 2 Chamseddine, A.H.; Connes, A. (2012). "Resilience of the Spectral Standard Model". Journal of High Energy Physics . 2012 (9): 104. arXiv: 1208.1030 . Bibcode:2012JHEP...09..104C. doi:10.1007/JHEP09(2012)104. S2CID   119254948.
  2. ↑ Chamseddine, A.H.; Connes, A.; van Suijlekom, W. D. (2013). "Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification". Journal of High Energy Physics . 2013 (11): 132. arXiv: 1304.8050 . Bibcode:2013JHEP...11..132C. doi:10.1007/JHEP11(2013)132. S2CID   18044831.
  3. ↑ Mather, John N. (1974). "Simplicity of certain groups of diffeomorphisms". Bulletin of the American Mathematical Society. 80 (2): 271–273. doi: 10.1090/S0002-9904-1974-13456-7 .
  4. ↑ Thurston, William (1974). "Foliations and groups of diffeomorphisms". Bulletin of the American Mathematical Society. 80 (2): 304–307. doi: 10.1090/S0002-9904-1974-13475-0 .
  5. ↑ Connes, Alain (1990). "Essay on physics and noncommutative geometry". The Interface of Mathematics and Particle Physics (Oxford, 1988). Inst. Math. Appl. Conf. Ser., New Ser. Vol. 24. New York: Oxford University Press. pp. 9–48.
  6. ↑ Dubois-Violette, Michel (1988). "Dérivations et calcul différentiel non commutatif". Comptes Rendus de l'Académie des Sciences, Série I (307): 403–408.
  7. ↑ Dubois-Violette, Michel; Kerner, Richard; Madore, John (1989). "Classical bosons in a non-commutative geometry". Classical and Quantum Gravity. 6 (11): 1709. Bibcode:1989CQGra...6.1709D. doi:10.1088/0264-9381/6/11/023. S2CID   250880966.
  8. ↑ Dubois-Violette, Michel; Kerner, Richard; Madore, John (1989). "Gauge bosons in a noncommutative geometry". Physics Letters B. 217 (4): 495–488. Bibcode:1989PhLB..217..485D. doi:10.1016/0370-2693(89)90083-X.
  9. ↑ Dubois-Violette, Michel; Kerner, Richard; Madore, John (1989). "Noncommutative differential geometry and new models of gauge theory". Journal of Mathematical Physics. 323 (31): 495–488. doi:10.1063/1.528917.
  10. ↑ Connes, Alain; Lott, John (1991). "Particle models and noncommutative geometry". Nuclear Physics B - Proceedings Supplements. 18 (2): 29–47. Bibcode:1991NuPhS..18...29C. doi:10.1016/0920-5632(91)90120-4. hdl: 2027.42/29524 .
  11. ↑ Chamseddine, Ali H.; Connes, Alain (1997). "The Spectral Action Principle". Communications in Mathematical Physics. 186 (3): 731–750. arXiv: hep-th/9606001 . Bibcode:1997CMaPh.186..731C. doi:10.1007/s002200050126. S2CID   12292414.
  12. ↑ Lizzi, Fedele; Mangano, Gianpiero; Miele, Gennaro; Sparano, Giovanni (1997). "Fermion Hilbert Space and Fermion Doubling in the Noncommutative Geometry Approach to Gauge Theories". Physical Review D. 55 (10): 6357–6366. arXiv: hep-th/9610035 . Bibcode:1997PhRvD..55.6357L. doi:10.1103/PhysRevD.55.6357. S2CID   14692679.
  13. ↑ Gracia-Bondía, Jose M.; Iochum, Bruno; Schücker, Thomas (1998). "The standard model in noncommutative geometry and fermion doubling". Physical Review B. 416 (1–2): 123–128. arXiv: hep-th/9709145 . Bibcode:1998PhLB..416..123G. doi:10.1016/S0370-2693(97)01310-5. S2CID   15557600.
  14. ↑ Barrett, John W. (2007). "A Lorentzian version of the non-commutative geometry of the standard model of particle physics". Journal of Mathematical Physics. 48 (1): 012303. arXiv: hep-th/0608221 . Bibcode:2007JMP....48a2303B. doi:10.1063/1.2408400. S2CID   11511575.
  15. ↑ Connes, Alain (2006). "Noncommutative Geometry and the standard model with neutrino mixing". Journal of High Energy Physics. 2006 (11): 081. arXiv: hep-th/0608226 . Bibcode:2006JHEP...11..081C. doi:10.1088/1126-6708/2006/11/081. S2CID   14419757.
  16. ↑ Chamseddine, Ali H.; Connes, Alain; Marcolli, Matilde (2007). "Gravity and the standard model with neutrino mixing". Advances in Theoretical and Mathematical Physics. 11 (6): 991–1089. arXiv: hep-th/0610241 . doi:10.4310/ATMP.2007.v11.n6.a3. S2CID   9042911.
  17. ↑ CDF and D0 Collaborations and Tevatron New Phenomena Higgs Working Group (2008). "Combined CDF and DØ Upper Limits on Standard Model Higgs Boson Production at High Mass (155–200 GeV/c2) with 3 fb−1 of data". Proceedings, 34th International Conference on High Energy Physics. arXiv: 0808.0534 .{{cite book}}: CS1 maint: numeric names: authors list (link)
  18. ↑ "Irony". 4 August 2008. Retrieved 4 August 2008.
  19. ↑ Estrada, Christopher; Marcolli, Matilde (2013). "Asymptotic safety, hypergeometric functions, and the Higgs mass in spectral action models". International Journal of Geometric Methods in Modern Physics. 10 (7): 1350036–68. arXiv: 1208.5023 . Bibcode:2013IJGMM..1050036E. doi:10.1142/S0219887813500369. S2CID   215930.

Related Research Articles

<span class="mw-page-title-main">Alain Connes</span> French mathematician (born 1947)

Alain Connes is a French mathematician, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the Collège de France, Institut des Hautes Études Scientifiques, Ohio State University and Vanderbilt University. He was awarded the Fields Medal in 1982.

M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten's announcement initiated a flurry of research activity known as the second superstring revolution. Prior to Witten's announcement, string theorists had identified five versions of superstring theory. Although these theories initially appeared to be very different, work by many physicists showed that the theories were related in intricate and nontrivial ways. Physicists found that apparently distinct theories could be unified by mathematical transformations called S-duality and T-duality. Witten's conjecture was based in part on the existence of these dualities and in part on the relationship of the string theories to a field theory called eleven-dimensional supergravity.

<span class="mw-page-title-main">Standard Model</span> Theory of forces and subatomic particles

The Standard Model of particle physics is the theory describing three of the four known fundamental forces in the universe and classifying all known elementary particles. It was developed in stages throughout the latter half of the 20th century, through the work of many scientists worldwide, with the current formulation being finalized in the mid-1970s upon experimental confirmation of the existence of quarks. Since then, proof of the top quark (1995), the tau neutrino (2000), and the Higgs boson (2012) have added further credence to the Standard Model. In addition, the Standard Model has predicted various properties of weak neutral currents and the W and Z bosons with great accuracy.

In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a widely studied branch of theoretical and mathematical physics. Penrose's idea was that twistor space should be the basic arena for physics from which space-time itself should emerge. It has led to powerful mathematical tools that have applications to differential and integral geometry, nonlinear differential equations and representation theory, and in physics to general relativity, quantum field theory, and the theory of scattering amplitudes. Twistor theory arose in the context of the rapidly expanding mathematical developments in Einstein's theory of general relativity in the late 1950s and in the 1960s and carries a number of influences from that period. In particular, Roger Penrose has credited Ivor Robinson as an important early influence in the development of twistor theory, through his construction of so-called Robinson congruences.

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.

In theoretical physics, the matrix theory is a quantum mechanical model proposed in 1997 by Tom Banks, Willy Fischler, Stephen Shenker, and Leonard Susskind; it is also known as BFSS matrix model, after the authors' initials.

In mathematics, noncommutative topology is a term used for the relationship between topological and C*-algebraic concepts. The term has its origins in the Gelfand–Naimark theorem, which implies the duality of the category of locally compact Hausdorff spaces and the category of commutative C*-algebras. Noncommutative topology is related to analytic noncommutative geometry.

In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize the de Rham (co)homology of manifolds. These notions were independently introduced by Boris Tsygan (homology) and Alain Connes (cohomology) in the 1980s. These invariants have many interesting relationships with several older branches of mathematics, including de Rham theory, Hochschild (co)homology, group cohomology, and the K-theory. Contributors to the development of the theory include Max Karoubi, Yuri L. Daletskii, Boris Feigin, Jean-Luc Brylinski, Mariusz Wodzicki, Jean-Louis Loday, Victor Nistor, Daniel Quillen, Joachim Cuntz, Ryszard Nest, Ralf Meyer, and Michael Puschnigg.

In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name "field with one element" and the notation F1 are only suggestive, as there is no field with one element in classical abstract algebra. Instead, F1 refers to the idea that there should be a way to replace sets and operations, the traditional building blocks for abstract algebra, with other, more flexible objects. Many theories of F1 have been proposed, but it is not clear which, if any, of them give F1 all the desired properties. While there is still no field with a single element in these theories, there is a field-like object whose characteristic is one.

In particle physics and string theory (M-theory), the ADD model, also known as the model with large extra dimensions (LED), is a model framework that attempts to solve the hierarchy problem. The model tries to explain this problem by postulating that our universe, with its four dimensions, exists on a membrane in a higher dimensional space. It is then suggested that the other forces of nature operate within this membrane and its four dimensions, while the hypothetical gravity-bearing particle graviton can propagate across the extra dimensions. This would explain why gravity is very weak compared to the other fundamental forces. The size of the dimensions in ADD is around the order of the TeV scale, which results in it being experimentally probeable by current colliders, unlike many exotic extra dimensional hypotheses that have the relevant size around the Planck scale.

In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries. Example 2 × 2 quasideterminants are as follows:

In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher by Hairer & Wanner (1974), is an infinite-dimensional Lie group first introduced in numerical analysis to study solutions of non-linear ordinary differential equations by the Runge–Kutta method. It arose from an algebraic formalism involving rooted trees that provides formal power series solutions of the differential equation modeling the flow of a vector field. It was Cayley (1857), prompted by the work of Sylvester on change of variables in differential calculus, who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics.

In noncommutative geometry and related branches of mathematics and mathematical physics, a spectral triple is a set of data which encodes a geometric phenomenon in an analytic way. The definition typically involves a Hilbert space, an algebra of operators on it and an unbounded self-adjoint operator, endowed with supplemental structures. It was conceived by Alain Connes who was motivated by the Atiyah-Singer index theorem and sought its extension to 'noncommutative' spaces. Some authors refer to this notion as unbounded K-cycles or as unbounded Fredholm modules.

In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory. As a result of this change some variables that are usually continuous may become discrete. Often only such discrete variables are called "quantized"; usage varies.

Geometry of quantum systems is mainly phrased in algebraic terms of modules and algebras. Connections on modules are generalization of a linear connection on a smooth vector bundle written as a Koszul connection on the -module of sections of .

Nikita Alexandrovich Nekrasov is a Russian mathematical and theoretical physicist at the Simons Center for Geometry and Physics and C.N.Yang Institute for Theoretical Physics at Stony Brook University in New York, and a Professor of the Russian Academy of Sciences.

<span class="mw-page-title-main">Matilde Marcolli</span> Italian mathematician and physicist

Matilde Marcolli is an Italian and American mathematical physicist. She has conducted research work in areas of mathematics and theoretical physics; obtained the Heinz Maier-Leibnitz-Preis of the Deutsche Forschungsgemeinschaft, and the Sofia Kovalevskaya Award of the Alexander von Humboldt Foundation. Marcolli has authored and edited numerous books in the field. She is currently the Robert F. Christy Professor of Mathematics and Computing and Mathematical Sciences at the California Institute of Technology.

In mathematics, a Bost–Connes system is a quantum statistical dynamical system related to an algebraic number field, whose partition function is related to the Dedekind zeta function of the number field. Bost & Connes (1995) introduced Bost–Connes systems by constructing one for the rational numbers. Connes, Marcolli & Ramachandran (2005) extended the construction to imaginary quadratic fields.

Ali H. Chamseddine is a Lebanese physicist known for his contributions to particle physics, general relativity and mathematical physics. As of 2013, Chamseddine is a physics Professor at the American University of Beirut and the Institut des hautes études scientifiques.

Henri Moscovici is a Romanian-American mathematician, specializing in non-commutative geometry and global analysis.

References

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.