New hopes for a noncommutative dual UV/IR phenomenological manifestation ?
In this paper we have proven that the real φ4-model on (Euclidean) noncommutative R4 is renormalisable to all orders in perturbation theory. The bare action of relevant and marginal couplings of the model is parametrised by four (divergent) quantities which require normalisation to the experimental data at a physical renormalisation scale. The corresponding physical parameters which determine the model are the mass, the ﬁeld amplitude (to be normalised to 1), the coupling constant and (in addition to the commutative version) the frequency of an harmonic oscillator potential. The appearance of the oscillator potential is not a bad trick but a true physical effect. It is the self-consistent solution of the UV/IR-mixing problem found in the traditional noncommutative φ4-model in momentum space. It implements the duality (see also ) that noncommutativity relevant at short distances goes hand in hand with a modiﬁed structure of space relevant at large distances. Such a modiﬁed structure of space at very large distances seems to be in contradiction with experimental data. But this is not true. Neither position space nor momentum space are the adapted frames to interpret the model. An invariant characterisation of the model is the spectrum of the Laplace-like operator which deﬁnes the free theory. Due to the link to Meixner polynomials, the spectrum is discrete. ... we see that the spectrum of the squared momentum variable has an equidistant spacing of 4Ω/θ . Thus, √(4Ω/θ) is the minimal (non-vanishing) momentum of the scalar ﬁeld which is allowed in the noncommutative universe. We can thus identify the parameter √Ω with the ratio of the Planck length to the size of the (ﬁnite!) universe. Thus, for typical momenta on earth, the discretisation is not visible.
(Submitted on 20 Jan 2004 (v1), last revised 4 Oct 2004 (this version, v2))
... to spectral noncommutative geometry
The spectral action delivers a huge mass term [of the Higgs] and one can check that it is consistent with the sign and order of magnitude of the quadratic divergence of the self-energy of this scalar field. However though this shows compatibility with a small low energy value it does by no means allow one to justify such a small value. Giving the term − 2H2 at unification scale and hoping to get a small value when running the theory down to low energies by applying the renormalization group, one is facing a huge fine tuning problem. Thus one should rather try to find a physical principle to explain why one obtains such a small value at low scale. In the noncommutative geometry model M × F of space-time the size of the finite space F is governed by the inverse of the Higgs mass. Thus the above problem has a simple geometric interpretation: Why is the space F so large (by a factor of 1016) in Planck units? There is a striking similarity between this problem and the problem of the large size of space in Planck units. This suggests that it would be very worthwhile to develop cosmology in the context of the noncommutative geometry model of space-time, with in particular the preliminary step of the Lorentzian formulation of the spectral action.This also brings us to the important role played by the dilaton field which determines the scale in the theory. The spectral action is taken to be a function of the twisted Dirac operator so that D2 is replaced with e−φD2e−φ. In  we have shown that the spectral action is scale invariant, except for the dilaton kinetic energy. Moreover, one can show that after rescaling the physical fields, the scalar potential of the theory will be independent of the dilaton at the classical level. At the quantum level, the dilaton acquires a Coleman-Weinberg potential  and will have a vev of the order of the Planck mass . The fact that the Higgs mass is damped by a factor of e−2φ, can be the basis of an explanation of the hierarchy problem.
(Submitted on 30 Nov 2008)
"It's very important to have a mental picture of what it's going on now ... it's very surprising why spacetime is macroscopic because when you write any type of equation in [quantum] physics you find something with a Planck size... and the idea is that eventually when we do cosmology the picture that is emerging [from our most recent work] is that the ...[spacetime] manifold M is like a butterfly ... folded in the product of two spheres of [Planckian] size but itself it has a macroscopic size and it unfolds... and one amazing thing is when you write down these maps [from M] to the spheres with their degrees [or winding numbers, related to the number of quanta of geometry] you find out there are natural operations in mathematics which are coming from self maps of the sphere. Now what is the effect of these operations on the butterfly ? They multiply the volume [of spacetime] by a constant ... and of course this is extremely reminiscent of e-folding in inflation..."
Alain Connes (17.12.2014) (oral transcript by the blogger)