Brainwashed by Pythagoras and Descartes?
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...
The standard model from the metric point of view
How noncommutative geometry provides extra-dimensions from Pythagoras theoremP. Martinetti, May 2008
In the noncommutative approach to the standard model of elementary particles , 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 ... 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
Noncommutative geometry intrinsically links the Higgs field with the metric structure of space-time. We have not considered the gauge field Aµ 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.
Replacing the Pythagorean knotted rope by a spin-half fermionic Dirac propagator for the quantum surveyor
- 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)
How noncommutative geometry provides extra-dimensions from Pythagoras theorem