Noncommutative standard model: Difference between revisions

Revision as of 08:20, 7 November 2023

Revision as of 18:24, 23 December 2023

Line 70:

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 <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, <ref name="connes_1998_essay">

,<ref name="connes_1998_essay">

{{cite book

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

Line 141 ⟶ 140:

</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 193 ⟶ 191:

| arxiv = hep-th/0608221

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

</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

Line 230 ⟶ 228:

| 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 invalidated.<ref>

{{cite web

| title = Irony

Line 238 ⟶ 236:

}}</ref> In July 2012, CERN announced the discovery of the [[Higgs boson]] with a mass around 125 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">