Noncommutative standard model: Difference between revisions

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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.

Line 6 ⟶ 5:

| author1-link = Ali Chamseddine

| author2-link = Alain Connes

| journal = [[JHEPJournal 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><ref name="10.1007/JHEP11(2013)132">

}}

</ref>

<ref name="10.1007/JHEP11(2013)132">

{{cite journal | title = Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification

| last1 = Chamseddine | first1 = A.H.

Line 19 ⟶ 17:

| author1-link = Ali Chamseddine

| author2-link = Alain Connes

| journal = [[JHEPJournal 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).

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 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 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 Standard Model gauge group <math>G=\mathrm{SU}(3) \times \mathrm{SU}(2) \times U(1)</math>.

<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

| title = Simplicity of certain groups of diffeomorphisms

| last = Mather | first = John N.

| journal = Bulletin of the American Mathematical Society

| volume = 80

| issue = 2

| year = 1974

| pages = 271–273

| doi = 10.1090/S0002-9904-1974-13456-7 | doi-access = free

}}</ref>

and Thurston<ref name="10.1090/S0002-9904-1974-13475-0">

{{cite journal

| title = Foliations and groups of diffeomorphisms

| last = Thurston | first = William

| journal = Bulletin of the American Mathematical Society

| volume = 80

| year = 1974

| issue = 2 | pages = 304–307

| url = http://projecteuclid.org/euclid.bams/1183535407

| doi = 10.1090/S0002-9904-1974-13475-0 | doi-access = free

}}

</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.

), 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.

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

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.

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.

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.

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 Standard Model.

==History==

First ideas to use [[noncommutative geometry]] to particle physics appeared in 1988-89

== History ==

<ref name="connes_1998_essay">

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

{{cite book

| last = Connes | first = Alain

| last = Connes | first = Alain | author-link = Alain Connes

| year= 1990

| journal = The interface of mathematics and particle physics

| chapter = Essay on physics and noncommutative geometry

| year = 1988

| title = The Interface of Mathematics and Particle Physics (Oxford, 1988)

}}

| pages=9–48

</ref>

| series=Inst. Math. Appl. Conf. Ser., New Ser. |volume=24

<ref name="dv_1988_dcdnc">

| publisher=Oxford University Press

| ___location=New York

}}</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 - SeriesSérie I - Mathematics

| issue = 307

| pages = 403-408403–408

| 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

Line 60 ⟶ 96:

| 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">

</ref>

<ref name="10.1016/0370-2693(89)90083-X">

{{cite journal | title = Gauge bosons in a noncommutative geometry

| last1 = Dubois-Violette | first1 = Michel

Line 71 ⟶ 106:

| issue = 4

| year = 1989

| pages = 495-488495–488

| doi = 10.1016/0370-2693(89)90083-X

| bibcode = 1989PhLB..217..485D

}}

}}</ref><ref name="10.1063/1.528917">

</ref>

<ref name="10.1063/1.528917">

{{cite journal | title = Noncommutative differential geometry and new models of gauge theory

| last1 = Dubois-Violette | first1 = Michel

Line 84 ⟶ 118:

| issue = 31

| year = 1989

| pages = 495-488495–488

| 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 name="10.1016/0920-5632(91)90120-4">

</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 (mathematician)

| journal = Nuclear Physics B - Proceedings Supplements

| 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.

</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">

<ref name="10.1007/s002200050126">

{{cite journal | title = The Spectral Action Principle

| last1 = Chamseddine | first1 = Ali H.

Line 109 ⟶ 139:

| author1-link = Ali Chamseddine

| author2-link = Alain Connes

| journal = Communications in Mathematical Physics volume 186

| pages = 731–750

| year = 1997

| volume = 186 | issue = 3

| doi = 10.1007/s002200050126

| 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

Line 126 ⟶ 156:

| issue = 10

| year = 1997

| pages = 6357–6366 | doi = 10.1103/PhysRevD.55.6357

| arxiv = hep-th/9610035 | bibcode = 1997PhRvD..55.6357L | s2cid = 14692679

}}</ref>

</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

| pagesvolume = 123-128416

| pages = 123–128

| year = 1998

| issue = 1–2 | doi = 10.1016/S0370-2693(97)01310-5

| arxiv = hep-th/9709145

| bibcode = 1998PhLB..416..123G | s2cid = 15557600 }}

}}

</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 (physicist)|John W. Barrett]]<ref name="10.1063/1.2408400">

{{cite journal

<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)

| journal = Journal of Mathematical Physics, 48

| year volume= 200648

| 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]]

| title = Noncommutative Geometry and the standard model with neutrino mixing

<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

Line 161 ⟶ 193:

| 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">

<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.

Line 178 ⟶ 207:

| author3-link = Matilde Marcolli

| journal = Advances in Theoretical and Mathematical Physics

| volume = 11 | number = 6

| numberyear = 62007

| yearpages = 2006991–1089

| doi = 10.4310/ATMP.2007.v11.n6.a3 | arxiv = hep-th/0610241 | s2cid = 9042911

}}</ref> and under an additional assumption, known as the "big desert" hypothesis, computations were carried out to predict the [[Higgs boson]] mass around 170&nbsp;[[GeV]] and postdict the [[top quark]] mass.

| 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 book

<ref name="arxiv:0808.0534">

{{cite journal | titlechapter = Combined CDF and D0DØ Upper Limits on Standard Model Higgs Boson Production at High Mass (155−200−GeV/c2)(155-200-155–200&nbsp;GeV/''c^{''<sup>2)}(155−200−GeV</c2sup>) with 3 fb−1fb^{-1}fb−1&nbsp;fb⁻¹ of Datadata

| author = CDF and D0 Collaborations and Tevatron New Phenomena Higgs Working Group

| journaltitle = Proceedings, 34th International Conference on High Energy Physics (ICHEP 2008)

| year = 2008

| arxiv = 0808.0534

}}</ref> excluded a Higgs mass of 158 to 175&nbsp;GeV/''c''² at the 95% confidence level. Alain Connes acknowledged on a blog about non-commutative geometry that the prediction about the Higgs mass was invalidated.<ref>

}}

{{cite web

</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

| accessdatedate=4 August 2008

| access-date=4 August 2008

| url = http://noncommutativegeometry.blogspot.com/2008/08/irony.html

| url = https://noncommutativegeometry.blogspot.com/2008/08/irony.html

}}</ref>.

}}</ref> In July 2012, CERN announced the discovery of the [[Higgs boson]] with a mass around 125 Gev&nbsp;GeV/''c''².

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

<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

Line 217 ⟶ 240:

| number = 7

| year = 2013

| pages = 1350036–68

| doi = 10.1142/S0219887813500369

| arxiv = 1208.5023 | bibcode = 2013IJGMM..1050036E | s2cid = 215930 }}

</ref>

}}

</ref>.

== See also ==

* [[Noncommutative geometry]]

* [[Noncommutative quantum fieldalgebraic theorygeometry]]

* [[Noncommutative quantum field theory]]

*[[Timeline of atomic and subatomic physics]]

* [[Timeline of atomic and subatomic physics]]

== Notes ==

{{reflist}}

{{Reflist}}<!--added under references heading by script-assisted edit-->

== References ==

* [[{{cite book |last1=Connes |first1=Alain |author-link=Alain Connes]] (|year=1994) ''[|url=http://www.alainconnes.org/docs/book94bigpdf.pdf |title=Noncommutative geometry.]''Geometry |publisher=Academic Press. {{ISBN|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 }}

* -------- (1995) "Noncommutative geometry and reality," ''J. Math. Phys.'' 36: 6194.

* --------{{cite (1996)journal "[https://|arxiv.org/abs/=hep-th/9603053 |doi=10.1007/BF02506388 |title=Gravity coupled with matter and the foundation of noncommutativenon-commutative geometry]," ''Comm.|year=1996 Math.|last1=Connes Phys.''|first1=Alain 155:|author-mask=1 109|journal=Communications in Mathematical Physics |volume=182 |issue=1 |pages=155–176 |bibcode=1996CMaPh.182..155C |s2cid=8499894}}

* {{cite web |last1=Connes |first1=Alain |author--------mask=1 (|year=2006) "[|url=http://www.alainconnes.org/docs/einsymp.pdf |title=Noncommutative geometry and physics]," }}

* --------{{cite andbook |last1=Connes [[Matilde|first1=Alain |author-mask=1 |last2=Marcolli |M.first2=Matilde |author2-link=Matilde Marcolli]], ''[|year=2007 |url=http://www.alainconnes.org/en/downloads.php |title=Noncommutative Geometry: Quantum Fields and Motives.]'' |publisher=American Mathematical Society (2007).}}

* {{cite journal |arxiv=hep-th/9606001 |doi=10.1007/s002200050126 |title=The Spectral Action Principle |year=1997 |last1=Chamseddine |first1=Ali H. |last2=Connes |first2=Alain |journal=Communications in Mathematical Physics |volume=186 |issue=3 |pages=731–750 |bibcode=1997CMaPh.186..731C |s2cid=12292414}}

* Chamseddine, A., A. Connes (1996) "[https://arxiv.org/abs/hep-th/9606001 The spectral action principle]," ''Comm. Math. Phys.'' 182: 155.

* {{cite journal |arxiv=hep-th/0610241 |doi=10.4310/ATMP.2007.v11.n6.a3 |title=Gravity and the standard model with neutrino mixing |year=2007 |last1=Chamseddine |first1=Ali H. |last2=Connes |first2=Alain |last3=Marcolli |first3=Matilde |journal=Advances in Theoretical and Mathematical Physics |volume=11 |issue=6 |pages=991–1089 |s2cid=9042911}}

* Chamseddine, A., A. Connes, [[Matilde Marcolli|M. Marcolli]] (2007) "[https://arxiv.org/abs/hep-th/0610241 Gravity and the Standard Model with neutrino mixing]," ''Adv. Theor. Math. Phys.'' 11: 991.

* {{cite journal |arxiv=0705.0489 |last1=Jureit, |first1=Jan-H., Thomas |last2=Krajewski, |first2=Thomas Schücker,|last3=Schucker and|first3=Thomas |last4=Stephan |first4=Christoph A. Stephan (2007) "[https://arxiv.org/abs/0705.0489 |title=On the noncommutative standard model]," ''|journal=Acta Phys. Polon.'' B38:B 3181-3202|year=2007 |volume=38 |issue=10 |pages=3181–3202 |bibcode=2007AcPPB..38.3181J}}

* {{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 }}

*Schücker, Thomas (2005) ''[https://arxiv.org/abs/hep-th/0111236 Forces from Connes's geometry.]'' Lecture Notes in Physics 659, Springer.

== External links ==

* [http://www.alainconnes.org/ Alain Connes's official website] with [http://www.alainconnes.org/en/downloads.php downloadable papers.]

* [httphttps://resonaances.blogspot.com/2007/02/alain-connes-standard-model.html Alain Connes's Standard Model.]

{{DEFAULTSORT:Noncommutative Standard Model}}

[[Category:ParticlePhysics physicsbeyond the Standard Model]]

[[Category:Noncommutative geometry]]