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In theoretical [[particle physics]], the '''non-commutative Standard Model''' (best known as '''Spectral Standard Model'''<ref name="10.1007/JHEP09(2012)104">
<ref name="10.1007/JHEP09(2012)104">▼
{{cite journal | title = Resilience of the Spectral Standard Model
| last1 = Chamseddine | first1 = A.H.
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| journal = [[Journal of High Energy Physics]]
| year = 2012
| volume = 2012 | issue = 9 | page = 104
| doi = 10.1007/JHEP09(2012)104 | arxiv = 1208.1030 | bibcode = 2012JHEP...09..104C | s2cid = 119254948
</ref>▼
{{cite journal | title = Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification
| last1 = Chamseddine | first1 = A.H.
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| journal = [[Journal of High Energy Physics]]
| year = 2013
| volume = 2013 | issue = 11 | page = 132
| doi = 10.1007/JHEP11(2013)132 | arxiv = 1304.8050 | bibcode = 2013JHEP...11..132C | s2cid = 18044831 }}
</ref>), is a model based on [[noncommutative geometry]] that unifies a modified form of [[general relativity]] with the [[Standard Model]] (extended with right-handed neutrinos).▼
</ref>▼
▲), is a model based on [[noncommutative geometry]] that unifies a modified form of [[general relativity]] with the [[Standard Model]] (extended with right-handed neutrinos).
The model postulates that space-time is the product of a 4-dimensional compact spin manifold <math>\mathcal{M}</math> by a finite space <math>\mathcal{F}</math>. The full Lagrangian (in Euclidean signature) of the [[Standard
The parameters of the model live at unification scale and physical predictions are obtained by running the parameters down through [[renormalization]].
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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 theory|Kaluza–Klein]] and [[Albert Einstein]], the spectral approach seeks unification by expressing all forces as pure gravity on a space <math>\mathcal{X}</math>.
The group of invariance of such a space should combine the group of invariance of [[general relativity]] <math>\text{Diff}(\mathcal{M})</math> with <math>\mathcal{G} = \text{Map}(\mathcal{M}, G)</math>, the group of maps from <math>\mathcal{M}</math> to the
<math>\text{Diff}(\mathcal{M})</math> acts on <math>\mathcal{G}</math> by permutations and the full group of symmetries of <math>\mathcal{X}</math> is the semi-direct product:
<math>\text{Diff}(\mathcal{X}) = \mathcal{G} \rtimes \text{Diff}(\mathcal{M})</math>
Note that the group of invariance of <math>\mathcal{X}</math> is not a simple group as it always contains the normal subgroup <math>\mathcal{G}</math>. It was proved by Mather<ref name="10.1090/S0002-9904-1974-13456-7">
{{cite journal
<ref name="10.1090/S0002-9904-1974-13456-7">▼
| last = Mather | first = John N.
| journal = Bulletin of the American Mathematical Society
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| year = 1974
| pages = 271–273
| doi = 10.1090/S0002-9904-1974-13456-7 | doi-access = free
▲}}</ref>
{{cite journal
▲{{cite journal | title = Foliations and groups of diffeomorphisms
| last = Thurston | first = William
| journal = Bulletin of the American Mathematical Society
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| year = 1974
| issue = 2 | pages = 304–307
| url = http://projecteuclid.org/euclid.bams/1183535407
|
</ref>
that for ordinary (commutative) manifolds, the connected component of the identity in <math>\text{Diff}(\mathcal{M})</math> is always a simple group, therefore no ordinary manifold can have this semi-direct product structure.
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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 <math>A = C^{\infty}(M) \otimes A_F </math> where <math>C^{\infty}(M)</math> is the algebra of differentiable functions encoding the 4-dimensional manifold and <math>A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})</math> is a finite dimensional algebra encoding the symmetries of the
== History ==
First ideas to use noncommutative geometry to particle physics appeared in 1988-89, <ref name="connes_1998_essay">
{{cite book
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| pages = 403–408
| year = 1988
}}</ref><ref name="DVKM_1989_CBNG">▼
▲</ref><ref name="DVKM_1989_CBNG">
{{cite journal | title = Classical bosons in a non-commutative geometry
| last1 = Dubois-Violette | first1 = Michel
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| number = 11
| year = 1989
| page = 1709 | doi = 10.1088/0264-9381/6/11/023 | bibcode = 1989CQGra...6.1709D | s2cid = 250880966
}}</ref><ref name="10.1016/0370-2693(89)90083-X">
{{cite journal | title = Gauge bosons in a noncommutative geometry
| last1 = Dubois-Violette | first1 = Michel
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| pages = 495–488
| doi = 10.1016/0370-2693(89)90083-X
| bibcode = 1989PhLB..217..485D
}}</ref><ref name="10.1063/1.528917">
{{cite journal | title = Noncommutative differential geometry and new models of gauge theory
| last1 = Dubois-Violette | first1 = Michel
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| pages = 495–488
| doi = 10.1063/1.528917
}}</ref> and were formalized a couple of years later by [[Alain Connes]] and [[John Lott (mathematician)|John Lott]] in what is known as the Connes-Lott model▼
▲</ref> and were formalized a couple of years later by [[Alain Connes]] and [[John Lott (mathematician)|John Lott]] in what is known as the Connes-Lott model
.<ref name="10.1016/0920-5632(91)90120-4">
{{cite journal | title = Particle models and noncommutative geometry
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| year = 1991
| volume = 18 | issue = 2 | pages = 29–47 | doi = 10.1016/0920-5632(91)90120-4
| bibcode = 1991NuPhS..18...29C | hdl = 2027.42/29524 | hdl-access = free
}}</ref> 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, <ref name="10.1007/s002200050126">
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| pages = 731–750
| year = 1997
| volume = 186 | issue = 3
| doi = 10.1007/s002200050126 | arxiv = hep-th/9606001 | bibcode = 1997CMaPh.186..731C | s2cid = 12292414
}}</ref> 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) ▼
▲</ref> 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)
<ref name="10.1103/PhysRevD.55.6357">
{{cite journal
| title = Fermion Hilbert Space and Fermion Doubling in the Noncommutative Geometry Approach to Gauge Theories | last1 = Lizzi | first1 = Fedele
| last2 = Mangano | first2 = Gianpiero
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| year = 1997
| pages = 6357–6366 | doi = 10.1103/PhysRevD.55.6357
| arxiv = hep-th/9610035 | bibcode = 1997PhRvD..55.6357L | s2cid = 14692679
▲}}</ref>
<ref name="10.1016/S0370-2693(97)01310-5">
{{cite journal
| title = The standard model in noncommutative geometry and fermion doubling | last1 = Gracia-Bondía | first1 = Jose M.
| last2 = Iochum | first2 = Bruno
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Only in 2006 a solution to the latter problem was proposed, independently by [[John W. Barrett (physicist)|John W. Barrett]]<ref name="10.1063/1.2408400">
{{cite journal
| title = A Lorentzian version of the non-commutative geometry of the standard model of particle physics | last = Barrett | first = John W.
| author-link=John W. Barrett (physicist)
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| volume= 48
| year = 2007
| issue = 1 | page = 012303
| doi = 10.1063/1.2408400 | arxiv = hep-th/0608221 | bibcode = 2007JMP....48a2303B | s2cid = 11511575
}}</ref> and Alain Connes,<ref name="10.1088/1126-6708/2006/11/081">▼
{{cite journal
▲</ref> and Alain Connes,<ref name="10.1088/1126-6708/2006/11/081">
| last = Connes | first = Alain
| author-link=Alain Connes
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| volume = 2006
| year = 2006
| issue = 11 | page = 081
| doi = 10.1088/1126-6708/2006/11/081 | arxiv = hep-th/0608226 | bibcode = 2006JHEP...11..081C | s2cid = 14419757
}}</ref> 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.▼
▲</ref> 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,<ref name="10.4310/ATMP.2007.v11.n6.a3">
{{cite journal | title = Gravity and the standard model with neutrino mixing
| last1 = Chamseddine | first1 = Ali H.
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| author3-link = Matilde Marcolli
| journal = Advances in Theoretical and Mathematical Physics
| volume = 11 | number = 6
| year = 2007
| pages = 991–1089
| doi = 10.4310/ATMP.2007.v11.n6.a3 | arxiv = hep-th/0610241 | s2cid = 9042911
}}</ref>
▲</ref>, 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<ref name="arxiv:0808.0534">
{{cite book
| chapter = Combined CDF and DØ Upper Limits on Standard Model Higgs Boson Production at High Mass (155–200
| author = CDF and D0 Collaborations and Tevatron New Phenomena Higgs Working Group
| title = Proceedings, 34th International Conference on High Energy Physics
| year = 2008
| arxiv = 0808.0534
}}</ref> excluded a Higgs mass of 158 to 175
{{cite web
| title = Irony
| date=4 August 2008
| access-date=4 August 2008
| url =
}}</ref> In July 2012, CERN announced the discovery of the [[Higgs boson]] with a mass around 125
A proposal to address the problem of the Higgs mass was published by [[Ali Chamseddine]] and Alain Connes in 2012
<ref name="10.1007/JHEP09(2012)104"/> 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.<ref name="10.1142/S0219887813500369">
{{cite journal
| title = Asymptotic safety, hypergeometric functions, and the Higgs mass in spectral action models | last1 = Estrada | first1 =Christopher
| last2 = Marcolli | first2 = Matilde
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| number = 7
| year = 2013
| pages = 1350036–68
| doi = 10.1142/S0219887813500369 | arxiv = 1208.5023 | bibcode = 2013IJGMM..1050036E | s2cid = 215930 }}
</ref>
== See also ==
* [[Noncommutative geometry]]
* [[Noncommutative algebraic geometry]]
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* [[Timeline of atomic and subatomic physics]]
== Notes ==
{{reflist}}
== References ==
* {{cite book |last1=Connes |first1=Alain |author-link=Alain Connes |year=1994 |url=http://www.alainconnes.org/docs/book94bigpdf.pdf |title=Noncommutative Geometry |publisher=Academic Press |isbn=0-12-185860-X}}
* {{cite journal |last1=Connes |first1=Alain |author-mask=1 |year=1995 |title=Noncommutative geometry and reality |journal=Journal of Mathematical Physics |volume=36 |issue=11 |pages=6194–6231|doi=10.1063/1.531241 |bibcode=1995JMP....36.6194C |url=https://cds.cern.ch/record/285273 }}
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* {{cite book |doi=10.1007/978-3-540-31532-2_6 | arxiv=hep-th/0111236 | last1=Schucker | first1=Thomas | title=Topology and Geometry in Physics | chapter=Forces from Connes' Geometry | series=Lecture Notes in Physics | year=2005 | volume=659 | pages=285–350 | bibcode=2005LNP...659..285S| isbn=978-3-540-23125-7 | s2cid=16354019 }}
== External links ==
* [http://www.alainconnes.org/ Alain Connes's official website] with [http://www.alainconnes.org/en/downloads.php downloadable papers.]
* [
{{DEFAULTSORT:Noncommutative Standard Model}}
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