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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]] 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>.
<math>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>Diff(\mathcal{X}) = \mathcal{G} \rtimes 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
}}
</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
| pages = 304-307
| doi = 10.1090/S0002-9904-1974-13475-0
}}
</ref>
that for ordinary (commutative) manifolds, the connected component of the identity in <math>Diff(\mathcal{M})</math> is always a simple group, therefore no ordinary manifold can have this semi-direct product structure.
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 continuous 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==
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