Solid Theoretical Research In Natural Geometric Structures
Juan Maldacena , Geometry and Quantum Mechanics Vision Talks at Strings 2014, 27 juin 2014What is the spectral noncommutative programme for physics?
A written answer in 2007:
[Our] approach to physics can be summarized as a strategy to interpret the complicated input of the phenomenological Lagrangian of gravity coupled with matter as coming from a ﬁne structure (of the form [continuous ]M[anifold]×F[inite space]) in the geometry of space-time. Extrapolating this to uniﬁcation scale (i.e. assuming the big desert) gives predictions which can be compared with experiment. Of course we do not believe that the big desert is there and a key test when “new physics” will be observed is to decide whether it will be possible to interpret the new terms of the Lagrangian in the same manner from noncommutative spaces and the spectral action. This type of test already occurred with the new neutrino physics coming from the Kamiokande experiment and for quite some time I believed that the new terms would simply not ﬁt with the spectral action principle. It is only thanks to the simple idea of decoupling the KO-dimension from the metric dimension that the problem was resolved (this was also done independently by John Barrett  with a similar solution). At a more fundamental level the fact that the action functional can be obtained from spectral data suggests that instead of just looking at the inner ﬂuctuations of a product metric on M×F, one should view that as a special case of a fully uniﬁed theory at the operator theoretic level i.e. a kind of spectral random matrix theory where the [Dirac] operator varies in the symplectic ensemble...
Alain Connes, Noncommutative geometry and the spectral model of space-time, 2007
Probably the true geometry [to describe physics] is entirely finite...There is no no-go theorem known in quantum gravity that you cannot replace a [continuum] space by a discrete space...The idea is that whereas it is not possible to "cut" the riemannian space into small points by respecting the symmetries [like the Lorentz one] it is perfectly sensible to cut it [following the spectral action principle] to do it by cutting [the spectrum of the Dirac operator acting] in the Hilbert space of [spin 1/2] fermions [at high energy] and then you have all the symmetries that you want and this has been done in few examples but it's a general principle * [in the operator theoretic framework of non commutative geometry].The general principle is that probably the true physical theory is a theory of random matrices. The random matrices are the Dirac operators and they are finite dimensional, very high dimensional but finite... This is why you probably don't have any problem with the functional integral... What we are trying to do is we are trying to guess that gradually, from the shadow that we have which is partly riemannian and now from this strory you can see it is only partly Riemannian ... it's a little bit non commutative and probably when you go further in energy it will become more and more non commutative.
(transcription of) Alain Connes (' answer to a question raised by the blogger at séminaire Algèbres d'opérateurs) during a presentation by Ali Chamseddine entitled: Spectral geometric unification, 06/26/2014
There is one important advantage of the spectral point of view when compared to the old idea of a discrete space-time, which is that continuous symmetry groups survive the operation of truncating the Hilbert space H to the ﬁnite dimensional subspace H(Λ) corresponding to eigenvectors of the Dirac operator for eigenvalues ≤ Λ where Λ is a cutoff scale. Indeed, any unitary operator U commuting with D will automatically restrict to H(Λ). It could well be that the coherence of the spectral action principle indicates that our continuum picture of space-time is only an approximation to a completely ﬁnite spectral geometry whose underlying Hilbert space is ﬁnite dimensional. Of course the basic ingredients such as J and γ will still be present, but the algebra A itself will have no reason to remain commutative. In this scenario, once we go up in energy towards the uniﬁcation scale, the small amount of noncommutativity encoded in the ﬁnite geometry F to model the present scale, will gradually creep in and invade the whole algebra of coordinates which will become a huge matrix algebra at Planck scale. The noncommutativity of the algebra of coordinates means that the “internal” degrees of freedom have gradually replaced the external ones and that the notion of “point” has disappeared since a matrix algebra admits only one irreducible representation.
//last edition: July the 11th 2014