Noncommutative standard model: Difference between revisions

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Revision as of 17:18, 29 March 2021

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), 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-KleinKaluza–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]].

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==Motivation==

Following ideas from [[Kaluza-KleinKaluza–Klein theory|Kaluza–Klein]] and [[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>Diff(\mathcal{M})</math> with <math>\mathcal{G} = Map(\mathcal{M}, G)</math>, the group of maps from <math>\mathcal{M}</math> to the standard model gauge group <math>G=SU(3) \times SU(2) \times U(1)</math>.

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</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">

<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.

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| arxiv = hep-th/0608221

| bibcode = 2007JMP....48a2303B | s2cid = 11511575 }}

</ref> and [[Alain Connes]],<ref name="10.1088/1126-6708/2006/11/081">

,<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

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| 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

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</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">

<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 GeV/''c''²) with 3 fb⁻¹ of data

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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">

.<ref name="10.1142/S0219887813500369">

{{cite journal | title = Asymptotic safety, hypergeometric functions, and the Higgs mass in spectral action models

| last1 = Estrada | first1 =Christopher

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==See also==

* [[Noncommutative geometry]]

* [[Noncommutative quantum field theory]]

* [[Timeline of atomic and subatomic physics]]

==Notes==

{{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 |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=http://cds.cern.ch/record/285273 }}