There might be more hopes in spectral noncommutative geometry, Wimplitio, than are thought of in our present phenomenology

In memoriam Daniel Kastler***, probably the first physicist prophet of noncommutative geometry

Hope(s) encouraged by science retrodiction?

We feel that non-commutative geometry is as fundamental to physics as Minkowskian and Riemannian geometry. Let us try to explain this by comparing the standard model of particle physics and general relativity. From a chronological point of view, this comparison is difficult, because Riemannian geometry existed well before general relativity. However, the field theoretic approach allows to introduce general relativity in close analogy to classical electrodynamics without use of Riemannian geometry. Therefore this approach is well suited for our comparison. 

So let us imagine a world ignoring Riemannian geometry where physicists try to describe
gravity. They are inspired by Maxwell who takes a field A of spin 1, a second order differential operator D_Max and writes down his field equation

D_Max A = (1/c²ε₀) j,

where j is the source, charge density and currents, and ǫ0 is the proportionality constant from Coulomb’s law. After many ingenious and expensive experiments and theoretical trials and errors, the physicists agree on the standard model of gravity. It starts from a particular spin 2 field g, and a second order differential operator D_Ein. The field equation is

D_Ein g = −(8πG/c⁴)T,

where the source T is energy-momentum density and currents, and G is the proportionality constant from Newton’s universal law of gravity. Although in perfect agreement with experiment, this standard model has draw backs: who ordered spin 2? Maxwell’s differential operator D_Max contains 8 summands, the gravitational one D_Ein results from brute force and contains roughly 80 000 summands. Some of these summands still are inaccessible to experiment. At this stage, Riemannian geometry is discovered, the spin 2 field is recognized as the metric and the differential operator D_Ein is recognized as the curvature if the unknown summands are chosen properly. Most physicists say: so what, just fancy mathematics. Some dream of a geometric unification of all forces. Later, even more expensive experiments will test the predictions of Riemannian geometry coming from the unknown summands. 

If, in the real world, we qualify general relativity as revolution, we have several criteria.

• Postdiction: the theory correctly reproduces experimental data, that remain unexplained in the old theories, e.g. the precession of perihelia of Mercury.

• Prediction: the theory can be in contradiction with future experimental data, e.g. deflection of light.

• New concepts, e.g. curved spacetimes, absence of universal time.

• Reticence of the majority.  

Our purpose is to explain that for non-commutative geometry the analogue of g in the imaginary world is the Higgs field, the analogue of D_Ein  is the Lagrangian of the standard model of electro-weak and strong interactions. Postdictions of the theory are that fermions sit in fundamental representations, that weak interactions violate parity that strong interactions are vector like ... There is also a prediction, the mass of the Higgs, accessible to experiment in about ten years. New concepts are fuzzy spacetimes — that is spacetimes with an uncertainty relation — and discrete spacetimes.

Riemannian and Non-commutative Geometry in Physics

Bruno Iochum, Daniel Kastler, Thomas Schucker (Marseille)

(Submitted on 2 Nov 1995 (v1), last revised 9 Nov 1995 (this version, v3))

With the benefit from history and (h)in(d)sight

A spectral triplet consists of an associative algebra A, a representation on a Hilbert space H classifying the fermions and a Dirac operator D. The invariance group is simply the automorphism group of A. The later is chosen to be a tensor product of the infinite dimensional, commutative algebra of differentiable functions on spacetime M by a matrix algebra A describing an internal space, A = C^∞(M)⊗A. Then the automorphism group is the semi-direct product of diffeomorphisms and gauge transformations. The latter are inner automorphisms. In the commutative case, A=ℂ, there are only diffeomorphisms and the Dirac operator simply encodes the metric. If A is noncommutative, for instance A=ℂ⊕ℍ⊕M₃(ℂ) for the standard model, then the metric 'fluctuates', that is, it picks up additional degrees of freedom from the internal space, the Yang-Mills connection and the Higgs scalar. In physicist's language, the spectral triplet is the Dirac action of a multiplet of dynamical fermions in a background field. This background field is a fluctuating metric, consisting of so far adynamical bosons of spin 0, 1 and 2. The remaining two action pieces are obtained exclusively from the spectrum of the covariant Dirac operator DA indexed by the quantum one-form A. These two pieces together are simply the number of eigenvalues of |DA| that are smaller than Λ, i.e. tr F(|DA|/Λ) with F the characteristic function of the unit interval. This function of Λ can be calculated conveniently from the heat kernel expansion [6] and if F was the logarithm then we would have an old physical interpretation of this action formula, the dynamics of the bosons would be induced from one-loop quantum corrections with fermions circulating in the loop. With the characteristic function instead, this action is essentially Klein-Gordon together with spontaneous symmetry breaking for spin 0, Yang-Mills for spin 1 and Einstein-Hilbert for spin 2. In this approach all coupling constants are fixed leading to the well known numerical problems, the Planck mass sets the scale. The hope is that these evaporate once the fuzziness of spacetime is properly taken into account. This hope is encouraged from history. Let us recall that Maxwell's relation c²=(ε₀µ₀)^-1 relates a velocity to static coupling constants. At his time the speed of light was believed to be frame dependent. It was only by accepting the revolution of Minkowskian geometry on spacetime that this problem evaporated.

On the universal Chamseddine-Connes action 1. Details of the action computation

Bruno Iochum, Daniel Kastler, Thomas Schucker

(Submitted on 18 Jul 1996)

One of the basic problems facing theoretical physics is to determine the nature of space-time. This is intimately related to the problem of unifying all the fundamental interactions including gravity, and thus is not independent of solving the problem of quantum gravity. In a series of papers we have made important understanding uncovering a first approximation of the hidden structure of space-time. Our assumption is that at energies below the planck scale, space-time can be approximated as a product of a continuous four-dimensional manifold by a finite space. We were able to show in [14] that finite spaces satisfying the axioms of noncommutative geometry are severely restricted, and the corresponding irreducible representations on Hilbert spaces can only have dimensions which are the square of integers, or the double of such a square. The second possibility is the only one allowed when the finite space has dimension 6 modulo 8 (in the sense of K-theory or more pragmatically of the periodicity of Clifford algebras) as imposed by the need to have the total dimension 2 = 4 + 6 modulo 8 in order to be able to write down the Fermionic part of the action. Together with the restriction of imposing a unitary–symplectic structure and grading on the finite noncommutative space, this singles out 42 = 16 as the number of physical fermions per generation. Then, in the same way as was shown in [13], this predicts the existence of right-handed neutrinos, and the see-saw mechanism. Our present framework using the classification of finite spaces is stronger and the symmetries of the standard model emerge, rather than [being] assumed and put in by hand**. This construction, using the spectral action principle, predicts certain relations between the coupling constants, that can only hold at very high energies of the order of the unification scale. The spectral action principle is the simple statement that the physical action is determined by the spectrum of the Dirac operator D. This has now been tested in many interesting models including Superstring theory [6], noncommutative tori [30], Moyal planes [34], 4D-Moyal space [37], manifolds with boundary [12], in the presence of dilatons [10], for supersymmetric models [5] and torsion cases [38]. The additivity of the action forces it to be of the form Trace f (D/Λ). In the approximation where the spectral function f is a cut-off function, the relations given by the spectral action are used as boundary conditions and the couplings are then allowed to run from unification scale to low energy using the renormalization group equations. The equations show, when fitted to the low energy boundary conditions, that the three gauge coupling constants and the Newton constant nearly meet (within few percent) at very high energies, two or three orders from the Planck scale. This might be a coincidence but it can also be an indication that a more fundamental theory exists at unification scale and manifests itself at low scale through integration of the intermediate modes, as in the Wilson understanding of renormalization.

...

The compatibility between the values at low energy (obtained by integration over the fluctuations in the intermediate scales) and observation is a basic test of the general idea but in case this test is passed, one needs to go much further and develop a theory that takes over at higher scales... In [27] an analogy was developed between the phase transitions which occur in the number theoretic context and a scenario of spontaneous symmetry breaking involving the full gravitational sector. If substantiated, this could show how geometry would emerge from the computation of the KMS states of an operator theoretic system, closely related to a matrix model with basic variable the Dirac operator D. It is worthwhile to note, at this point, that, at the conceptual level, the spectral action is closely related to an entropy since it can be written as the logarithm of a number of states in the second quantized Fermionic Hilbert space.

Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I

Ali H. Chamseddine, Alain Connes

(Submitted on 3 Apr 2010)

**see an important update here.

//addendum 12/09/2015

Reading the nice art-book Radioactive : Pierre & Marie Curie, a tale of love and fallout by Lauren Redniss I was struck by a coïncidence : Daniel Kastler passed away a 4th of July just like Marie Curie (81 years before) and the very same day when the Higgs boson discovery at LHC were officially publicized (three years ago).

//addendum 25/07/2022

The curious reader may appreciate the resilience & effectiveness of noncommutative geometry, particularly in its spectral incarnation as a tentative framework for unification of all fundamental interactions including gravity having a look on its achievements since the Higgs boson discovery in 2012 in this brief non technical review:

We give an overview of the applications of noncommutative geometry to physics. Our focus is entirely on the conceptual ideas, rather than on the underlying technicalities. Starting historically from the Heisenberg relations, we will explain how in general noncommutativity yields a canonical time evolution, while at the same time allowing for the coexistence of discrete and continuous variables. The spectral approach to geometry is then explained to encompass two natural ingredients: the line element and the algebra. The relation between these two is dictated by so-called higher Heisenberg relations, from which both spin geometry and non-abelian gauge theory emerges. Our exposition indicates some of the applications in physics, including Pati--Salam unification beyond the Standard Model, the criticality of dimension 4, second quantization and entropy.

Noncommutativity and Physics: A non-technical review

Ali H. Chamseddine, Alain Connes, Walter D. van Suijlekom