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In theoretical [[particle physics]], the
<ref name="
{{cite journal | title = Resilience of the Spectral Standard Model
| last1 = Chamseddine | first1 = A.H.
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}}
</ref>
<ref name="
{{cite journal | title = Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification
| last1 = Chamseddine | first1 = A.H.
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| journal = [[JHEP]]
| year = 2013
| doi =
| arxiv = 1304.8050
}}
</ref>
), is an extension of the [[Standard Model]] minimally coupled to a modified form of [[general relativity]] expressed in the framework of [[noncommutative geometry]]. In that sense, it unifies gravity and particle physics in a common mathematical framework.
The model postulates that space-time is mildly non-commutative by tensoring the continuous 4-dimensional space by a finite non-commutative space (a matrix algebra). It is therefore close in spirit to [[Kaluza-Klein theory]] but without the problem of massive tower of states.
The [[Lagrangian]] of the full [[Standard Model]] minimally coupled to gravity is obtained by the action of pure gravity over that tensored space.
It is worth stressing that it is more than a simple reformation of the [[Standard Model]]. This unification implies a few constraints on the parameters of the Standard Model. For example, unlike [[Quantum Field Theory]], in [[noncommutative geometry]] the scalar sector is strongly constrained.
==History==
First ideas to use [[noncommutative geometry]] to particle physics appeared in 1988-89
<ref name="connes_1998_essay">
{{cite journal | title = Essay on physics and noncommutative geometry
| last = Connes | first = Alain
| author-link = Alain Connes
| journal = The interface of mathematics and particle physics
| year = 1988
}}
</ref>
<ref name="dv_1988_dcdnc">
{{cite journal | title = Dérivations et calcul différentiel non commutatif
| last = Dubois-Violette | first = Michel
| journal = Comptes Rendus de l'Académie des Sciences Paris - Series I - Mathematics
| issue = 307
| pages = 403-408
| year = 1988
}}
</ref>
<ref name="DVKM_1989_CBNG">
{{cite journal | title = Classical bosons in a non-commutative geometry
| last1 = Dubois-Violette | first1 = Michel
| last2 = Kerner | first2 = Richard
| last3 = Madore | first3 = John
| journal = Classical and Quantum Gravity
| volume = 6
| number = 11
| year = 1989
}}
</ref>
<ref name="10.1016/0370-2693(89)90083-X">
{{cite journal | title = Gauge bosons in a noncommutative geometry
| last1 = Dubois-Violette | first1 = Michel
| last2 = Kerner | first2 = Richard
| last3 = Madore | first3 = John
| journal = Physics Letters B
| volume = 217
| issue = 4
| year = 1989
| pages = 495-488
| doi = 10.1016/0370-2693(89)90083-X
}}
</ref>
<ref name="10.1063/1.528917">
{{cite journal | title = Noncommutative differential geometry and new models of gauge theory
| last1 = Dubois-Violette | first1 = Michel
| last2 = Kerner | first2 = Richard
| last3 = Madore | first3 = John
| journal = Journal of Mathematical Physics
| volume = 323
| issue = 31
| year = 1989
| pages = 495-488
| doi = 10.1063/1.528917
}}
</ref>
, and were formalized a couple of years later by [[Alain Connes]] and [[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
| last1 = Connes | first1 = Alain
| last2 = Lott | first2 = John
| author1-link = Alain Connes
| author2-link = John Lott
| journal = Nuclear Physics B - Proceedings Supplements
| year = 1991
| doi = 10.1016/0920-5632(91)90120-4
}}
</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">
{{cite journal | title = The Spectral Action Principle
| last1 = Chamseddine | first1 = Ali H.
| last2 = Connes | first2 = Alain
| author1-link = Ali Chamseddine
| author2-link = Alain Connes
| journal = Communications in Mathematical Physics volume 186
| pages = 731–750
| year = 1997
| doi = 10.1007/s002200050126
| arxiv = hep-th/9606001
}}
</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
| last3 = Miele | first3 = Gennaro
| last4 = Sparano | first4 = Giovanni
| journal = Physical Review D
| volume = 55
| issue = 10
| year = 1997
| doi = 10.1103/PhysRevD.55.6357
| arxiv = hep-th/9610035
}}
</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
| last3 = Schücker | first3 = Thomas
| journal = Physical Review B, 416
| pages = 123-128
| year = 1998
| doi = 10.1016/S0370-2693(97)01310-5
| arxiv = hep-th/9709145
}}
</ref> 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]]
<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
| journal = Journal of Mathematical Physics, 48
| year = 2006
| doi = 10.1063/1.2408400
| arxiv = hep-th/0608221
}}
</ref> and [[Alain Connes]]
<ref name="10.1088/1126-6708/2006/11/081">
{{cite journal | title = Noncommutative Geometry and the standard model with neutrino mixing
| last = Connes | first = Alain
| author-link=Alain Connes
| journal = Journal of High Energy Physics
| volume = 2006
| year = 2006
| doi = 10.1088/1126-6708/2006/11/081
| arxiv = hep-th/0608226
}}
</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.
| last2 = Connes | first2 = Alain
| last3 = Marcolli | first3 = Matilde
| author1-link = Ali Chamseddine
| author2-link = Alain Connes
| author3-link = Matilde Marcolli
| journal = Advances in Theoretical and Mathematical Physics
| volume = 11
| number = 6
| year = 2006
| doi = 10.4310/ATMP.2007.v11.n6.a3
| arxiv = hep-th/0610241
}}
</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 journal | title = Combined CDF and D0 Upper Limits on Standard Model Higgs Boson Production at High Mass (155−200−GeV/c2)(155-200-GeV/c^{2)}(155−200−GeV/c2) with 3 fb−1fb^{-1}fb−1 of Data
| author = CDF and D0 Collaborations and Tevatron New Phenomena Higgs Working Group
| journal = Proceedings, 34th International Conference on High Energy Physics (ICHEP 2008)
| year = 2008
| arxiv = 0808.0534
}}
</ref>
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 falsified
<ref>
{{cite_web
| title = Irony
| accessdate=4 August 2008
| url = http://noncommutativegeometry.blogspot.com/2008/08/irony.html
}}</ref>.
In July 2012, CERN announced the discovery of the [[Higgs boson]] with a mass around 125 Gev.
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
| author2-link = Matilde Marcolli
| journal = International Journal of Geometric Methods in Modern Physics
| volume = 10
| number = 7
| year = 2013
| doi = 10.1142/S0219887813500369
| arxiv = 1208.5023
}}
</ref>.
==See also==
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