Brainwashed by Pythagoras and Descartes?

Are extra metric dimensions really an illusion coming from Pythagoras theorem?

The idea to write this post comes from the following sentence :

Illusion of extra-dimension come from Pythagoras theorem, which is a particular case of Pythagoras^-1.

 Pierre Martinetti, (The standard model from the metric point of view: how noncommutative geometry provides extra-dimensions from Pythagoras theorem)

This sentence is extracted from the following slide...

and has to be associated with this other slide...

The standard model from the metric point of view
 How noncommutative geometry provides extra-dimensions from Pythagoras theorem

P. Martinetti, May 2008

... where the mysterious Pythagoras^-1 mentioned in the last sentence of the first slide refers in fact to a formula from noncommutative geometry, the mathematical framework envisioned by Alain Connes where a Riemannian-like line element is the inverse of a generalized Dirac Operator D.  In this context a simple relation between three squared Dirac operators ∂, D_I and D - defined respectively on a continuous 4D space, a discrete 0-dimensional one and the noncommutative tensor product of both - appears naturally as an "inverse Pythagorean equality" (third line of the first slide). This kind of relation has been first discussed by Connes here (in French) and more informally in a post there. It was extended in another article by F. d'Andrea and P. Martinetti in 2012.

Is the Higgs doublet the fifth column which undermines the Descartes commutative geometry?

To get a full understanding of the claim presented above one has to read an older article by Martinetti and Wulkenhaar which reads in part:

In the noncommutative approach to the standard model of elementary particles [10], spacetime appears as the product (in the sense of fibre bundles) of a continuous manifold by a discrete space... within the framework of noncommutative geometry, we investigate how the distance in the continuum evolves when the space-time of euclidean general relativity is tensorised by an internal space. We find that in many cases the relevant picture is the two sheets model [8]... Indeed, under precise conditions, the metric aspect of ”continuum × discrete” spaces reduces to the simple picture of two copies of the manifold. It was known [11,5] that the distance on each copy is the geodesic distance while the distance between the copies – the distance on the fibre – is a constant. But this does not give a complete description of the geometry, in particular the distance between different points on different copies. In this paper we show that this distance coincides with the geodesic distance within a (4+1)-dimensional manifold whose fifth component comes from the internal part of the geometry. This component is a constant in the simplest cases and becomes a function of the manifold when the metric fluctuates. Restricting ourselves to scalar fluctuations of the metric, which correspond to the Higgs sector in the standard model, it appears that the Higgs field describes the internal part of the metric in terms of a discrete Kaluza-Klein model...

The finite part of the geometry of the standard model with scalar fluctuations of the metric consists of a two-sheets model labelled by the single states of C and H. Each of the sheets is a copy of the Riemannian four-dimensional space-time endowed with its metric. The fifth component of the metric, corresponding to the discrete dimension, is

where (h₁,h₂) is the Higgs doublet and m_t the mass of the quark top... 

Noncommutative geometry intrinsically links the Higgs field with the metric structure of space-time. We have not considered the gauge field A_µ so it is not clear whether or not the interpretation of the Higgs as an extra metric component has a direct physical meaning. It is important to study the influence of the gauge fluctuation and, particularly, how it probably makes the metric of the strong interaction part finite.

Discrete Kaluza-Klein from scalar fluctuations in noncommutative geometry,

P. Martinetti, R. Wulkenhaar (last revised 17 Apr 2001 (this version, v2))

Replacing the Pythagorean knotted rope by a spin-half fermionic Dirac propagator for the quantum surveyor

Let us now recap the information already gained on the notion of dimension in a generalized spacetime viewed as a "product" of a continuous and a discrete space with the abstract distance concept forged by Connes in his real spectral triple paradigm:

• Notion of dimension is subtle: from the distance point of view, illusion of extra-dimension, that comes from the line elements satisfying Pythagore relation.
• But the metric dimension (defined as the rate of decrease of the eigenvalues of D) is still m=dim M.
• Still another dimension (KO dimension), important for massive neutrinos (see Chamseddine, Connes, Marcolli and Barrett)

The standard model from the metric point of view
 How noncommutative geometry provides extra-dimensions from Pythagoras theorem

P. Martinetti, May 2008

Disclaimer 

Of course, no disrespect is implied by the title of the post. It is a pale remake of the title of an informative article by Z. K. Silagadze "Brainwashed by Newton" which was already a remake by Anderson’s famous article ”Brainwashed by Feynman?” (do not miss Feynman Brainwashed? either by Nathan Isgur ;-)