vendredi 24 février 2017

(The inception of spectral) noncommutative geometry, its calculus and functional action principle (to model spacetime ?)

This is the second fragment of my Lover's dictionary of Spectral Physics, the entry is of course:


Noncommutative Geometry

Plato derives the knowledge of ideas from body by abstraction and cutting away, leading us by various steps in mathematical discipline from arithmetic to geometry, thence to astronomy, and setting harmony above them all. For things become geometrical by the accession of magnitude to quantity; solid, by the accession of profundity to magnitude; astronomical, by the accession of motion to solidity; harmonical, by the accession of sound to motion. 

Plato alleges that God forever geometrizes... meanwhile Connes and Chamseddine are computing what other children's of Archimedes have not finished to measure.
 Folklore


The geometric concepts have first been formulated and exploited in the Framework of Euclidean geometry. This framework is best described using Euclid’s axioms (in their modern form by Hilbert’). These axioms involve the set X of points pX of the geometric space as well as families of subsets: the lines and the planes for 3-dimensional geometry. Besides incidence and order axioms one assumes that an equivalence relation (called congruence) is given between segments, i.e., pairs of points (p,q),p,q X and also between angles, i.e., triples of points (a,O,b);a,O,b X. These relations eventually allow us to define the length |(p.q)| of a segment and the size (a,O,b) of an angle. The geometry is uniquely specified once these two congruence relations are given. They of course have to satisfy a compatibility axiom: up to congruence a triangle with vertices a,O,b X is uniquely specified by the angle (a,O,b) and the lengths of (a,O) and (0,b) ... Besides the completeness or continuity axiom, the crucial one is the axiom of unique parallel. The efforts of many mathematicians trying to deduce this last axiom from the others led to the discovery of non-Euclidean geometry...  
The introduction by Descartes of coordinates in geometry was at first an act of violence (cf. Ref. 2). In the hands of Gauss and Riemann it allowed one to extend considerably the domain of validity of geometric ideas. In Riemannian geometry the space Xn is an n-dimensional manifold. Locally in X a point p is uniquely specified by giving n real numbers x1,...,xn which are the coordinates of p. The various coordinate patches are related by diffeomorphisms. The geometric structure on X is prescribed by a (positive definite) quadratic form, gµν dxµdxν, (1.4) which specifies the length of tangent vectors... This allows, using integration, to define the length of a path γ... The analog of the lines of Euclidean or non-Euclidean geometry are the geodesics. The analog of the distance between two points p,q X is given by the formula, d(p,q)=Inf Length(γ)... where γ varies among all paths with γ(0)=p, γ(l)=q ... The obtained notion of “Riemannian space” has been so successful that it has become the paradigm of geometric spaceThere are two main reasons behind this success. On the one hand this notion of Riemannian space is general enough to cover the above examples of Euclidean and non-Euclidean geometries and also the fundamental example given by space-time in general relativity (relaxing the positivity condition Of (1.4)). On the other hand it is special enough to still deserve the name of geometry, the point being that through the use of local coordinates all the tools of the differential and integral calculus can be brought to bear ...   
Besides its success in physics as a model of space-time, Riemannian geometry plays a key role in the understanding of the topology of manifolds, starting with the Gauss Bonnet theorem, the theory of characteristic classes, index theory, and the Yang Mills theory. 
Thanks to the recent experimental confirmations of general relativity from the data given by binary pulsars there is little doubt that Riemannian geometry provides the right framework to understand the large scale structure of space-time. 
The situation is quite different if one wants to consider the short scale structure of space-time. We refer to Refs. 5 and 6 for an analysis of the problem of the coordinates of an event when the scale is below the Planck length. In particular there is no good reason to presume that the texture of space-time will still be the 4-dimensional continuum at such scales.  
In this paper we shall propose a new paradigm of geometric space which allows us to incorporate completely different small scale structures. It will be clear from the start that our framework is general enough. It will of course include ordinary Riemannian spaces but it will treat the discrete spaces on the same footing as the continuum, thus allowing for a mixture of the two. It also will allow for the possibility of noncommuting coordinates. Finally it is quite different from the geometry arising in string theory but is not incompatible with the latter since supersymmetric conformal field theory gives a geometric structure in our sense whose low energy part can be defined in our framework and compared to the target space geometry. 
It will require the most work to show that our new paradigm still deserves the name of geometry. We shall need for that purpose to adapt the tools of the differential and integral calculus to our new framework. This will be done by building a long dictionary which relates the usual calculus (done with local differentiation of functions) with the new calculus which will be done with operators in Hilbert space and spectral analysis, commutators.... The first two lines of the dictionary give the usual interpretation of variable quantities in quantum mechanics as operators in Hilbert space. For this reason and many others (which include integrality results) the new calculus can be called the quantized calculus’ but the reader who has seen the word “quantized” overused so many times may as well drop it and use “spectral calculus” instead. 
Alain Connes 
Received 4 April 1995; accepted for publication 7 June 1995

... we shall build our notion of geometry, in a very similar but somehow dual manner [to the Riemann's concept], on the pair (A, ds) of the algebra A of coordinates and the infinitesimal length element ds. For the start we only consider ds as a symbol, which together with A generates an algebra (A, ds). The length element ds does not commute with the coordinates, i.e. with the functions f on our space, f ∈ A. But it does satisfy non trivial relations. 
... we shall write down the axioms of geometry as the presentation of the algebraic relations between A and ds and the representation of those relations in Hilbert space. In order to compare different geometries, i.e. different representations of the algebra (A, ds) generated by A and ds, we shall use the following action functional,
(14) Trace(ϕ(ds/p))
where is the Planck length and ϕ is a suitable cutoff function which will cut off all eigenvalues of ds larger than p. We shall show in [CC] that for a suitable choice of the algebra A, the above action will give Einstein gravity coupled with the Lagrangian of the standard U(1)×SU(2)×SU(3) model of Glashow Weinberg Salam. The algebra will not be C(M) with M a (compact) 4-manifold but a non commutative refinement of it which has to do with the quantum group refinement of the Spin covering of SO(4). 
1 → Z/2 → Spin(4) → SO(4) → 1.  
Amazingly, in this description the group of gauge transformations of the matter fields arises spontaneously as a normal subgroup of the generalized diffeomorphism group Aut(A). It is the non commutativity of the algebra A which gives for free the group of gauge transformations of matter fields as a (normal) subgroup of the group of diffeomorphisms.
What the present paper shows is that one should consider the internal gauge symmetries as part of the diffeomorphism group of the non commutative geometry, and the gauge bosons as the internal fluctuations of the metric. It follows then that the action functional should be of purely gravitational nature. We state the principle of spectral invariance, stronger than the invariance under diffeomorphisms, which requires that the action functional only depends on the spectral properties of D=ds-1 in H. This is verified by the action,
I =Trace (ϕ(ds/ℓp))+<Dψ, ψ>
for any nice function ϕ from R*+ to R. We shall show in [CC] that this action gives the SM Lagrangian coupled with gravity. It would seem at first sight that the algebra A has disappeared from the scene when one writes down the above action, the point is that it is still there because it imposes the constraints [[D, a], b0]=0 ∀ a, b ∈ A and Σa0i[D, a1i]...[D, a4i]= γ coming from axioms [required to provide with the spectral calculus and the volume form]. It is important at this point to note that the integrality, n ∈ N of the dimension of a non commutative geometry appears to be essential to define the [algebraic formulation of a differential form called a] Hochschild cycle c∈Zn and in turns the chirality γ. This is very similar to the obstruction which appears when one tries to apply dimensional regularization to chiral gauge theories.
(Submitted on 8 Mar 1996)

This leads us to the postulate that: 
The symmetry principle in noncommutative geometry is invariance under the group Aut(A). 
We now apply these ideas to derive a noncommutative geometric action unifying gravity with the standard model. The algebra is taken to be A=C(M)⊗AF where the algebra AF is finite dimensional, AF=M3() and ℍ ⊂ M2() is the algebra of quaternions, ℍ ...  
A is a tensor product which geometrically corresponds to a product space, an instance of spectral geometry for A is given by the product rule
H = L2(M, S)⊗ HF , D = M ⊗ 1 + γ5 ⊗ DF  
where (HFDF) is a spectral geometry on AF, while both L2(M, S) and the Dirac operator M on M are as above. The group Aut(A) of diffeomorphisms falls in equivalence classes under the normal subgroup Int(A) of inner automorphisms. In the same way the space of metrics has a natural foliation into equivalence classes. The internal fluctuations of a given metric are given by the formula, 
D = D0 + A + JA-1,   A = Σai[D0, bi] , aibi ∈ A and A = A*... 
For Riemannian geometry these fluctuations are trivial. 
The hypothesis which we shall test in this letter is that there exist an energy scale Λ in the range 1015−1019 Gev at which we have a geometric action given by the spectral action...   
We now describe the internal geometry. The choice of the Dirac operator and the action of AF in HF comes from the restrictions that these must satisfy: 
J2 = 1 , [J, D] = 0,        [a, Jb*-1]=0 ,        [[D, a], Jb*-1]=0 ∀ a, b. (4) 
We can now compute the inner fluctuations of the metric and thus operators of the form: A = Σai[D, bi]. This with the self-adjointness condition A = A* gives a U(1), SU(2) and U(3) gauge fields as well as a Higgs field... 
It is a simple exercise to compute the square of the Dirac operator ... This can be cast into the elliptic operator form [7]: 
P = D2 = −(gµνµν · 1I + Aγµµ + B) 
where 1I, A µ and B are matrices of the same dimensions as D. Using the heat kernel expansion for Tr(e-tP) ... we can show that ... a very lengthy but straightforward calculation ... gives for the bosonic action ... {the standard model action coupled to Einstein and Weyl gravity} plus higher order non-renormalizable interactions suppressed by powers of the inverse of the mass scale in the theory}... 
We ... adopt Wilson’s view point of the renormalization group approach to field theory [9] where the spectral action is taken to give the bare action with bare quantities ... at a cutoff scale Λ which regularizes the action the theory is assumed to take a geometrical form.
The renormalized action receives counterterms of the same form as the bare action but with physical parameters  ... The renormalization group equations ... yield relations between the bare quantities and the physical quantities with the addition of the cutoff scale Λ. Conditions on the bare quantities would translate into conditions on the physical quantities. The renormalization group equations of this system were studied by Fradkin and Tseytlin [10] and is known to be renormalizable, but non-unitary [11] due to the presence of spin-two ghost (tachyon) pole near the Planck mass. We shall not worry about non-unitarity (see, however, reference 12), because in our view at the Planck energy the manifold structure of space-time will break down and must be replaced with a genuienly noncommutative structure.
Relations between the bare gauge coupling constants as well as equations (3.19) have to be imposed as boundary conditions on the renormalization group equations [9]. The bare mass of the Higgs field is related to the bare value of Newton’s constant, and both have quadratic divergences in the limit of infinite cutoff Λ... 
There are some relations between the bare quantities. The renormalized action will have the same form as the bare action but with physical quantities replacing the bare ones. The relations among the bare quantities must be taken as boundary conditions on the renormalization group equations governing the scale dependence of the physical quantities. These boundary condition imply that the cutoff scale is of order ∼ 1015 Gev and sin2θw∼0.21 which is off by ten percent from the true value. We also have a prediction of the Higgs mass in the interval 170 − 180 Gev. There is ... a stronger disagreement where Newton’s constant comes out to be too large... Incidentally the problem that Newton’s constant is coming out to be too large is also present in string theory where also has unification of gauge couplings and Newton’s constant occurs [15]. These results must be taken as an indication that the spectrum of the standard model has to be altered as we climb up in energy. The change may happen at low energies (just as in supersymmetry ...) or at some intermediate scale. This also could be taken as an indication that the the concept of space-time as a manifold breaks down and the noncommutativity of the algebra must be extended to include the manifold part.
(Submitted on 11 Jun 1996)


The notion of spectral geometry has deep roots in pure mathematics. They have to do with the understanding of the notion of (smooth) manifold. While this notion is simple to define in terms of local charts i.e. by glueing together open pieces of finite dimensional vector spaces, it is much more difficult and instructive to arrive at a global understanding ... What one does is to detect global properties of the underlying space with the goal of characterizing manifolds... At the beginning of the 80’s, motivated by numerous examples of noncommutative spaces arising naturally in geometry from foliations or in physics from the Brillouin zone in the work of Bellissard on the quantum Hall effect, I realized that specifying an unbounded representative of the Fredholm operator was giving the right framework for spectral geometry ...
Over the years this new [noncommutative geometric paradigm of spectral nature] has been considerably refined ... The noncommutative geometry dictated by physics is the product of the ordinary 4-dimensional continuum by a finite noncommutative geometry which appears naturally from the classification of finite geometries of KO-dimension equal to 6 modulo 8 (cf. [15, 18]). The compatibility of the model with the measured value of the Higgs mass was demonstrated in [20] due to the role in the renormalization of the scalar field already present in [19].
In [21, 22], with Chamseddine and Mukhanov we gave the conceptual explanation of the finite noncommutative geometry from Clifford algebras and obtained a higher form of the Heisenberg commutation relations between p and q, whose irreducible Hilbert space representations correspond to 4-dimensional spin geometries. The role of p is played by the Dirac operator and the role of q by the Feynman slash of coordinates using Clifford algebras. The proof that all spin geometries are obtained relies on deep results of immersion theory and ramified coverings of the sphere. The volume of the 4-dimensional geometry is automatically quantized by the index theorem and the spectral model, taking into account the inner automorphisms due to the noncommutative nature of the Clifford algebras, gives Einstein gravity coupled with the slight extension of the standard model which is a Pati-Salam model. This model was shown in our joint work with A. Chamseddine and W. van Suijlekom [24, 25] to yield unification of coupling constants.
Th{e} quantization of the volume implies that the bothering cosmological leading term of the spectral action is now quantized and thus it no longer appears in the variation of the spectral action. Thus provided one understands how to reinstate all the ne details of the nite geometry (the one encoded by the Clifford algebras) such as the nuance on the grading and the number of generations, the variation of the spectral action will reproduce the Einstein equations coupled with matter.
Alain Connes 
Draft version from February 21, 2017

mercredi 22 février 2017

(How and when) Spectral Physics (was born) or first fragment of a Lover's Dictionary on...

Spectral Physics (draft for the core entry)



... in the beginning of the year 1666 (at which time I applyed my self to the grinding of Optick glasses of other figures then Sphericall) I procured me a triangular glasse Prisme to try therewith the celebrated phænomena of colours. And in order thereto having darkned my chamber & made a small hole in my window-shuts to let in a convenient quantity of the sun's light, I placed my Prism at its entrance that it might be thereby refracted to the opposite wall. It was at first a very pleasing divertisement to view the vivid & intense colours produced thereby; but after a while applying my selfe to consider them more circumspectly, I became surprized to see them in an oblong form, which according to the received lawes of refraction I expected should have been circular. 
They were terminated at the sides with streight lines, but at the ends the decay of light was so graduall that it was difficult to determine justly what was their figure, yet they seemed semicircular. 
Comparing the length of this Coloured Spectrum with its bredth I found it about five times greater, a disproportion soe extravagant that it excited me to a more then ordinary curiosity of examining from whence it might proceed; I could scarce think that the various thicknesse of the glasse, or the termination with shaddow or darknesse could have any influence on light to produce such an effect, yet I thought it not amisse to examine first those circumstances, & soe tryed what would happen by transmitting light through parts of the glasse of divers thicknesses, or through holes in the window of divers bignesses, or by setting the Prism without, so that the light might passe through it & bee refracted before it was terminated by the hole: but I found none of those circumstances materiall. The fashion of the colours was in all these cases the same.
Isaac Newton
Trinity Coll Cambridge. Feb. 6. 1671/2


I envision spectral physics as a scientific endeavour based on a set of experimental and conceptual spectroscopes to scrutinise and merge in a coherent picture the macroscopic and microscopic features of the phenomenological world physicists probe thanks to telescopes, high energy accelerators, extremely low temperature devices, very high magnetic fields ... etc, and confront with heuristic tools like quantum mechanics, thermal physics or general relativity while mathematicians formalise the computations confirmed by nature with theories like Euclidean geometry, calculus, Fourier analysis, Riemannian manifolds and their noncommutative extensions with a proper spectral calculus.


dimanche 19 février 2017

(Even more?) Simple GUTs come along with longer proton lifetime and hint of normal neutrino mass hierarchy

From hidden simplicities to a conjectured quantum gravity condition on matter content
Here are ideas and speculations from a more than twenty year old article by Giovanni AMELINO-CAMELIA which resonates interestingly - in my ears at least - with grand unification in the Spectral Pati-Salam Model retrospectively (underlining and bold emphasis as well as {...} are mine as usual):
It is probably worth emphasizing, at least for the benefit of the students ... that we have nothing (from the conceptual viewpoint) to assure us that nature should be describable in terms of simple laws. Still, most of us do expect this simplicity, probably extrapolating from the history of physics, which has proceeded through a series of steps of deeper understanding and simplification (such as the description of the baryon spectrum in terms of the quark model). From the point of view of this expected simplicity, the SM is quite unsatisfactory, since it leaves unanswered several questions; in particular,  
(Qa) Particle physics is described by a gauge theory with the peculiar gauge group GSM ≡SU(3)c⊗SU(2)L⊗ U(1)Y .
(Qb) The corresponding three coupling constants, αs, α2, and αY , are free parameters of the model. (Qc) A peculiar bunch of IRREPs (irreducible representations) of SU(3)c⊗SU(2)L hosts the quarks and leptons.
(Qd) The hypercharge assignments to the quark and lepton IRREPs are arbitrary. (Qe) The Yukawa couplings are arbitrary.
(Qf) The entries of the Cabibbo-Kobayashi-Maskawa matrix are arbitrary.
(Qg) Each of the quarks and leptons of the model is present in triplicate copy (family structure).
(Qh) A peculiar (bunch of) IRREP(s) hosts the Higgs particles.
(Qi) The parameters of the Higgs potential, which determine the Higgs mass(es) and all the aspects of symmetry breaking, are arbitrary
.
(Qj) The anomaly cancellation is a (apparently accidental) result of the structure of the (arbitrarily selected) matter content of the model.  
GUTs [models with a (grand)unified description of particle interactions] have been investigated primarily because, as illustrated in the discussion of SO(10) GUTs given in the next section, they address/simplify (Qa)-(Qf) and, in some cases, (Qj), and are therefore good candidates for the description of particle physics if the trend of incremental simplifications of this description is to continue. However, it should be noted that not only GUTs bring no improvement in relation to (Qg), but they actually increase the complexity associated to (Qh) and (Qi). Therefore, from the “aesthetic” viewpoint, GUTs have merits and faults (with the merits outnumbering, but not necessarily outweighing, the faults). 
Phenomenological encouragement for the GUT idea comes from the observed low-energy values of αs, α2, and αY , which appear to be arranged just as needed for unification. Indeed, (although the simplest GUT candidate, minimal SU(5), does not pass this test[1]) there are several examples of GUTs which reproduce these data on the coupling constants while being consistent with the ... experimental lower limit on proton decay τp→e+π0 ≥ 9×1032 years {in 1996 and τp→e+π0 >1.6×1034 years at 90% confidence using Super-Kamiokandedata from April 1996 to March 2015}. One important feature of phenomenologically consistent GUTs is that they must involve at least one extra scale, besides the unification scale MX, at which the RGEs (renormalization group equations) of the SM couplings are modified... 
Perhaps the simplest GUTs meeting the minimum requirement of agreement with the data on the coupling constants and with the experimental limit on proton decay are some SO(10) models, which are reviewed in the next section. They naturally (see next section) predict a two-scale breaking to GSM; in fact, a typical SO(10) breaking chain is given by  
 
Importantly, in SO(10) the hypercharge Y is the combination of two generators belonging to the Cartan, Y = T3R + (B − L)/ 2 , (2) where B − L and T3R belong respectively to the SU(4)PS (the SU(4) containing SU(3)c and U(1)B−L, which was first considered by Pati and Salam) and the SU(2)R subgroups of SO(10). This leads to a high unification point MX if the intermediate symmetry group G′ contains SU(2)R and/or SU(4)PS, since then, between MZ and MR, the Abelian evolution of αY (predicted by SM) is replaced by the non-Abelian one of either component of Y . MX is connected with the masses of the lepto-quarks that can mediate proton decay, and this SO(10) mechanism for a higher unification point turns out to be useful in allowing to meet the condition  
MX ≥ 3.2×1015 GeV , (3)  
which is necessary... for agreement with the present experimental limit on proton decay. Although (relatively) simple GUTs, such as SO(10), can work, they are affected by the hierarchy problem, and this causes many to prefer SUSY (supersymmetric) GUTs... In this paper (but not necessarily elsewhere) I take the position that for the GUTs (whether they are SUSY or not) the aesthetic advantages and the consistency with the observed low-energy values of the gauge coupling constants outweigh the damage done in regard to (Qh) and (Qi). This motivates me to look for possible ways to associate hidden simplicities to the apparently complicated GUT structures affecting (Qh) and (Qi) (and (Qg)); the reader is warned of the fact that the resulting discussion is accordingly quite speculative.

From ... properties of the smallest IRREPs of SO(10) one can easily see that the typical pattern of SSB of SO(10) to GSM has two steps; indeed, with the exception of the singlet in the 144, the little group of all the above mentioned GSM singlets is larger than GSM. Actually, either for phenomenological or for technical reasons, some of the above mentioned GSM singlets, cannot be used for the first SSB step...The previous considerations lead to four scenarios, in which the first steps of breaking are: 
(Ia) SO(10) → SU(3)c⊗SU(2)L⊗SU(2)R⊗U(1)B−L×D 
(Ib) SO(10) → SU(4)PS⊗SU(2)L⊗SU(2)R 
(Ic) SO(10) → SU(3)c⊗SU(2)L⊗SU(2)R⊗U(1)B−L 
(II) SO(10) → SU(4)PS⊗SU(2)L⊗SU(2)R×D,  
The type-I SO(10) models require that an appropriate vector3 in the space of GSM singlets of the 210 acquires a v.e.v. (vacuum expectation value) at the GUT scale. Analogously the type-II model requires that the GSM singlet of the 54 acquires a v.e.v. at the GUT scale. An appealing[10] possibility for the completion of the models of type-Ia,b,c and type-II is the one of realizing the second SSB step, at a scale MR, with the GSM singlet of the 126⊕126 representation, and the third SSB step with a combination of the SU(3)c⊗U(1)e.m.-invariant vectors of two 10’s, in such a way to avoid the unwanted relation mt = mb ... Through the see-saw mechanism, the scale MR is related to the masses of the (almost) left-handed neutrinos... 
The type-Ia,b,c and type-II models have been studied... and they have been found to be consistent with the unification of couplings and the experimental bound on the proton lifetime, although in the case of the type-II model the consistence with the experimental bound on the proton lifetime is only marginal... Within the see-saw mechanism, one also finds that these models predict masses for the (almost) left-handed neutrinos in an interesting range.  
This phenomenology depends however on the values of the parameters of the Higgs potential, which are free inputs of the model. Most importantly, as mentioned above, the parameters of the Higgs potential must be chosen so that the desired SSB pattern is realized. Although this does not involve a particular fine tuning[10, 11, 12, 13], it does introduce an element of undesirable arbitrariness in the models. Similarly, the “matter ingredients” of the models (e.g., in the type-I models, 16⊕16⊕16 for the fermions plus 10⊕10⊕126⊕126⊕210 for the Higgs bosons) is selected with the only constraint of reproducing observation (i.e. the matter content is not constrained by any requirement of internal consistency of the models)... 
One way to render a GUT more predictive would be the discovery of a dynamical mechanism (quasi) fixing the values of the parameters of the Higgs potential at the GUT scale MX. In this section I discuss one such mechanism which might be available when looking at the GUT as an effective low-energy description of a more fundamental theory (possibly including gravity).  
I observe that, besides leading to the possibility of an increased predictivity (in the sense clarified above) of the SSB pattern, viewing GUTs as effective low-energy descriptions of a more fundamental theory, with the associated RG implications, requires a modification of the conventional tests of the naturalness of a GUT. These conventional tests typically assign a “low grade” to GUTs in which a fine tuning of the Higgs parameters is needed for a phenomenological SSB pattern; however, the effective-theory viewpoint on GUTs would require to check whether the phenomenological SSB pattern corresponds to fine tuning of the Higgs parameters at the scale M∗. It is plausible that a scenario requiring no fine tuning of the Higgs parameters at M∗ might correspond via the RG running (for example in presence of an appropriate infra-red fixed point) to a narrow region (apparent fine tuning) of the Higgs parameter space at MX , where the SSB is decided. On the other hand, it is also plausible that a SSB pattern corresponding to a significant portion of the Higgs parameter space actually requires some level of fine tuning at M∗ (for example, the considered portion of Higgs parameter space might be “disfavored” by the RG running). 
Concerning the scale M∗ at which the GUT becomes relevant as an effective low-energy theory, it should be noticed that, while  any scenario with M∗ >MX is plausible, the present (however limited) understanding of physics beyond the GUT scale MX suggests that M∗ could be within a few orders of magnitude of the Planck scale MP . In fact, it is reasonable to expect that beyond the GUT there is a theory incorporating gravity (a quantum gravity), and MP is the scale believed to characterize this more fundamental theory. 
It is also important to realize that the type of RG naturalness that I am requiring for GUTs is really a minimal requirement once the GUT is seen as an effective low-energy description of a more fundamental theory. In order to get a consistent GUT from this viewpoint one would also want that “nothing goes wrong” in going from the scale M∗ to the scale MX . For example, SSB should not occur “prematurely” at a scale µSSB such that MX <µSSB <M∗ , and the running of the masses involved in the RGEs should be taken into account. In relation to this point, it is interesting that the investigation of the RG naturalness of the parameters of the Higgs potential might ultimately help understanding also the emergence of the GUT scale. At present this scale is just a phenomenological  input of a GUT, resulting from the observed (low-energy) values of the GSM coupling constants, but it would be interesting to see it emerging as a scale within the GUT. By studying the RGEs for the parameters of the Higgs potential one might find such a scale; for example, assuming not-too-special initial conditions for the parameters at the scale M∗, one might find that the running of the parameters is such that SSB of the GUT occurs typically in the neighborhood of a certain scale hopefully a phenomenologically reasonable one).  

I also want to stress that one could consider additional consistency/naturalness conditions in order to render the GUT consistent with a working cosmological (early universe evolution) scenario. Such conditions should be properly formulated in the language of finite temperature field theory, and should take into account the fact that (contrary to the expectations often expressed in the literature) the dependence of couplings on the renormalization scale is different from their temperature dependence... 
As illustrated by the review of SO(10) GUTs ... GUTs typically involve a remarkably complex matter content. Most notably, the Higgs sector consists of several carefully selected IRREPs of the GUT group, and, like in the SM, the fermionic sector of leptons and quarks is arbitrary and is plagued by the family triplication. This complexity might well be telling us that the GUT idea needs drastic revisions; however, in this paper I take the point of view that the complexity of the matter content might be only apparent. I therefore want to mention a few appealing scenarios in which this complexity might arise from a fundamental simplicity. For continuity with the line of argument advocated in the previous section, let me start by mentioning the possibility that as an effective low-energy description of a more fundamental theory, the (effective) matter content of the GUT at the scale MX might be fixed by the RG running. It is in fact plausible that some IRREPs tend to get heavy masses via RG running, whereas the masses of other IRREPs (the ones relevant for symmetry breaking and low-energy phenomenology) might tend to be light (i.e. order MX or less). This type of running of masses (or other parameters) might even be responsible for the cancellation of anomalies at low energies; indeed, the RG running is known to be easily driven by symmetries... 
Another hypothesis which has been gaining some momentum in the literature on low-dimensional Quantum-Gravity toy models is that Quantum Gravity might be quite selective concerning the type of matter “it likes to deal with”, i.e. the requirement of overall consistency of Quantum Gravity might fix the matter content. Results pointing (however faintly) in this direction can be found for example in certain studies of discretized two-dimensional Quantum Gravities[25], and studies of the Dirac quantization of certain two-dimensional Quantum Gravities in the continuum [26 {26'}]... 

Perhaps the only robust concept discussed in this paper is the one concerning the way in which the conventional tests of the naturalness of a GUT need to be modified if the GUT is seen as a low-energy effective description of a more fundamental theory. 

On a more speculative side, I also articulated the hope that the correct GUT (if there is one) could be such that its SSB pattern is essentially predicted (in the sense of the RG naturalness I discussed) rather than being a free input; this would fit well the general trend of increased predictivity at each new stage of our understanding of particle physics. 
I have also looked at the complexity of the matter content of GUTs, and explored the possibility that this might be an apparent complexity, hiding a fundamental simplicity... I have speculated on a few appealing candidates for this simplicity; however, it is reasonable to expect that real progress in this direction will require dramatic new developments. 
(Submitted on 9 Oct 1996)



Two challenges for some courageous graduate students in particle physics phenomenology and cosmology:
  • Explore the phenomenological consequences of the spectral Pati-Salam model(s) in particular those concerning neutrino masses and mixing* and check its compatibility with current low-energy data and potential falsifiability with future planned experiments. Investigate the possible "Renormalisation Group naturalness" of this simple(r?) GUT(s).  
  • Build a cosmological model with a scope from a 1012 GeV leptogenesis horizon to the current 0.23 eV scale, compatible with current astrophysical data and based only on the matter content of the spectral Pati-Salam model and the phenomenology of mimetic gravity consistent with the spectral action principle. Explore the phenomenological implications in particular concerning the different cosmological backgrounds (gravitational, electromagnetic and neutrino sectors).


* Neutrino may sing "from their GUTs"
From a particle physics phenomenologist perspective, spectral models may appear not very appealing as the matter content does not offer great perspective regarding the discovery of an exotic particle at a man made accelerator or even detector: no sparticle or wimp for instance. Nevertheless the recent article summarized below shows how some minimal grand unified models can help us to stay tuned on the faint neutrino song:
Minimal SO(10) grand unified models provide phenomenological predictions for neutrino mass patterns and mixing. These are the outcome of the interplay of several features, namely: i) the seesaw mechanism; ii) the presence of an intermediate scale where B-L gauge symmetry is broken and the right-handed neutrinos acquire a Majorana mass; iii) a symmetric Dirac neutrino mass matrix whose pattern is close to the up-type quark one. In this framework two natural characteristics emerge. Normal neutrino mass hierarchy is the only allowed, and there is an approximate relation involving both light-neutrino masses and mixing parameters. This differs from what occurring when horizontal flavour symmetries are invoked. In this case, in fact, neutrino mixing or mass relations have been separately obtained in literature. In this paper we discuss an example of such comprehensive mixing-mass relation in a specific realization of SO(10) and, in particular, analyse its impact on the expected neutrinoless double beta decay effective mass parameter hmeei, and on the neutrino mass scale. Remarkably a lower limit for the lightest neutrino mass is obtained (mlighest  ≥ 7.5×10-4 eV, at 3 σ level)

The solid (dashed) lines bound the allowed region in the mlighest –(mee) plane obtained by spanning the 3 σ ranges for the neutrino mixing parameters [9] in case of NH (IH). The dotted (dot-dashed) line is the prediction of eq. (9) on the effective mass, once the NH (IH) best-fit values of the neutrino mixing parameters are adopted [9]. The shaded region represents the 3 σ area obtained according to the neutrino mass-mixing dependent sum rule of eq. (9).
(Submitted on 2 Jan 2017 (v1), last revised 11 Jan 2017 (this version, v2))


//Last edit February 20, 2017

samedi 18 février 2017

All roads lead to Clifford algebra (and praise the spinors for that)!

More of algebra pfest...
... in a very recent geometric algebra perspective on the discrete parameters and symmetries of the standard model:
A simple geometric algebra is shown to contain automatically the leptons and quarks of a generation of the Standard Model (SM), and the electroweak and color gauge symmetries...{Any structure aiming to describe the particles and forces of the SM has to include two instances of the complex Clifford algebra Cℓ6. Since the two instances are isomorphic, the minimal solution is to identify them. This minimal algebraic structure is the Standard Model Algebra, ASM:=Cℓ(χ⊕ χ)≅ Cℓ6 with a preferred Witt decomposition that splits the algebra into minimal left ideals: χ⊕ χ where χ is a complex three-dimensional vector space.} The minimal left ideals determined by the Witt decomposition correspond naturally pairs of leptons or quarks whose left chiral components interact weakly. The Dirac algebra is a distinguished subalgebra acting on the ideals representing leptons and quarks. The resulting representations on the ideals are invariant to the electromagnetic and color symmetries, which are generated by the bivectors of the algebra. The electroweak symmetry is also present, and it is already broken by the geometry of the algebra. The model predicts a bare Weinberg angle θW given by sin2W)=0.25... {which} seems more encouraging that that of 0.375 predicted by the SU(5), SO(10), and other GUTs. But it is still not within the range estimated experimentally. Depending on the utilized scheme, the experimental values for sin2W), range between ≃ 0.223 and ≃ 0.24 (Erler and Freitas, 2015). In particular, CODATA gives a value of 0.23129(5) (Mohr and Newe, 2016). As in the case of the SU(5) prediction ... a correct comparison would require taking into account the running of the coupling constants due to higher order perturbative corrections... 
The most known GUT models based on the inclusion of the Standard Model group into a larger simple Lie group range from SU(5) (Georgi and Glashow, 1974) and Spin(10) (Georgi, 1975), and their supersymmetric versions, to much larger extensions. The model proposed in this article has something in common with them, by using representations of the gauge groups on exterior algebras. It differs by not predicting new bosons and proton decay, by explaining why not any Lie group and not any representation appear to be manifested as particles, and by a different value of the Weinberg angle... 
The proposed model, by unifying various aspects of the Standard Model, may also be a first step toward a simpler and more insightful Lagrangian. This model clearly cannot include gravity on equal footing with the other forces, but its geometric nature and the automatic inclusion of the Dirac algebra associated with the metric may allow finding new connections with general relativity and gravity. At this stage these prospects are speculative, but this is just the beginning. Another future step is to investigate the quantization within this framework. Since the model does not make changes to the SM, it may turn out that the Lagrangian and the quantization are almost the same as those we know. But the constraints introduced by the Standard Model Algebra may be helpful in these directions too. An interesting difference is the electroweak symmetry breaking induced purely by geometry... The Higgs boson is not forbidden by the model, being allowed to live in its usual space associated with the weak symmetry, and it is useful to generate the masses of the particles. But it gained a more geometric interpretation, which may find applications in future research. The proposed model does not make any assumptions about the neutrino, except that it is represented as a four-spinor. This includes the possibility that the neutrino is a Weyl spinor, refuted, and that it is a Dirac or Majorana spinor, which is still undecided. This again depends on the dynamics. It is not excluded that subsequent development of this model may decide the problem in one way or another, at theoretical level.
Discrete properties of leptons and quarks in the Standard Model Algebra.
(Submitted on 14 Feb 2017)
Comment
I find this article interesting for several reasons. First it shows on one side the powerfulness of Clifford algebras, one particular mathematical facet of noncommutativity, to describe in a compact way the discrete parameters of the Standard model. But on the other side it exemplifies the limited scope of this geometric algebra perspective: if one wants to add gravity and its continuous parameters in a unified picture then another mathematical aspect of noncommutativity provides invaluable services thanks to a powerful analytical machinery namely spectral geometry as envisioned byConnes and his collaborator Chamseddine: 
... the study of pure gravity for spectral geometries involving the algebra B=Mn(C(M)) instead of the usual commutative algebra C(M) of smooth functions, yields Einstein gravity on M minimally coupled with Yang-Mills theory for the gauge group SU(n). The Yang-Mills gauge potential appears as the inner part of the metric, in the same way as the group of gauge transformations (for the gauge group SU(n)) appears as the group of inner diffeomorphisms. This simple example shows that the noncommutative world incorporates the internal symmetries in a natural manner as a slight refinement of the algebraic rules on coordinates. There is a certain similarity between this refinement of the algebraic rules and what happens when one considers super-space in supersymmetry, but unlike in the latter case the algebraic rules are semi-simple rather than nilpotent. The effect is also somewhat similar to what happens in the Kaluza-Klein scenario since it is pure gravity on the new geometry that produces the mixture of gravity and gauge theory. But there is a fundamental difference since the construction does not alter the metric dimension and thus does not introduce the infinite number of new modes which automatically come up in the Kaluza-Klein model. In this manner one stays much closer to the original input from physics and does not have to argue that the new modes are made invisible because they are very massive.
Space-Time from the spectral point of viewAli H. Chamseddine, Alain Connes(Submitted on 5 Aug 2010)

Of course my last remark doesn't intend to underestimate the scientific work of neither O. Stoica (of whom I follow the interesting blog) nor other geometric algebra practitioners. Nevertheless most physicists have not yet appropriated themselves Clifford algebras despite the fact that quantum physics seems to provide naturally (some of?) them. Probably they need more experimental or conceptual incentives. Moreover history has proved that it happened to be misused in the past. Time will tell then if the first article highlighted in this post will help. 


I would like to conclude with the following text by Claude Daviau that summarizes in an exemplary way I think some real benefits and potential dangers of clifford algebras in  physics:
L'univers qui nous entoure est encore très largement à découvrir et la physique est loin d'avoir fini d'en faire le tour. Nous devons nous souvenir qu'à la fin du XIXème siècle certains physiciens pensaient que l'essentiel était déjà compris. Il n'y avait plus que quelques problèmes irritants, comme celui du corps noir. Mais de ces quelques difficultés qui restaient sont sorties des choses aussi importantes que la physique quantique et la relativité... 
Entre la physique de 1860 et la physique de 2010, de nombreuses différences mineures cachent deux différences fondamentales. La première est en apparence purement physique, c'est l'existence de la constante de Planck. La seconde concerne aussi les mathématiques, c'est l'utilisation des nombres complexes. Ces deux différences sont liées: avant les quanta, les nombres complexes n'avaient pas pris pied en physique. Ils s'y sont introduits, à la marge, parce que l'exponentielle complexe permet d'écrire simplement la trigonométrie... C'est Schrödinger qui, le premier, s'est aperçu... que l'onde {de l'électron} était à valeur complexe et non pas réelle. Pourquoi en est-il ainsi ? La réponse que l'on peut faire du point de vue mathématique, est simple : l'espace physique étant de dimension 3, son algèbre de Clifford contient des objets de carré -1. Encore faut-il alors justifier la nécessité physique de l'utilisation des algèbres de Clifford. 
Cette nécessité physique vient de l'existence de particules de spin 1/2.  Nous avons expliqué plus haut comment l'invariance relativiste, pour une particule de spin 1/2, nécessite l'utilisation de l'algèbre de Clifford d'espace, donc entraîne l'utilisation des nombres complexes. La découverte du spin de l'électron remonte à 1926, elle n'avait pas été prévue avant par la théorie physique. Longtemps la physique a sous-estimé les nouveautés que cela implique... 
La ...  raison pour laquelle on n'a pas compris vraiment la nouveauté c'est la difficulté de l'outillage mathématique. Tant que l'on utilise des opérateurs infinitésimaux, c'est à dire que l'on confond groupe de Lie et algèbre de Lie du groupe, on ne peut pas faire la différence entre les groupes d'invariance en jeu : les groupes sont globalement différents, mais localement identiques ! Il faut être vraiment vigilant et pointilleux pour comprendre qu'il y a un problème entre l'axiomatique de la théorie quantique et l'équation de Dirac pour l'électron...  
... on a tout interprété à partir des seules équations d'onde non relativistes. C'était choquant pour Louis de Broglie, qui avait conçu l'idée de l'onde à partir de la cinématique relativiste... la théorie quantique axiomatisée postule pour le vecteur d'état qu'il doit suivre une équation de Schrödinger. Comme c'est l'équation de Dirac...qui fonctionne pour l'atome d'hydrogène, la théorie axiomatisée n'est pas en droit de contraindre tout modèle à suivre ses règles...   
En remettant au coeur de la théorie physique les groupes d'invariance qui sont réellement nécessaires, on peut apercevoir que le groupe d'invariance de l'électromagnétisme est plus vaste que ce que l'on avait cru jusqu'ici. En conséquence les invariants sont moins nombreux, les contraintes sont plus grandes. 
Claude Daviau, 
Je publie 2011

About this last author I must confess I am very respectful for his long march through Clifford algebras from the Dirac equation to his own spinor wave equation for all objects of the first generation of fermions (electron, neutrino, quarks u and d with three states of color each) which is form invariant under a greater group than the relativistic group. I am also intrigued by some of its consequences and I have heuristical inclination towards a research program to deepen the "complex" information stored in the wave function or rather density operator. I know of course it is ... dangerous, to attribute any additional "real" meaning to them but I believe about the possibility that some imaginary parameters [in quantum physics] possess a "hidden reality" endowed with the assumed power of exerting "gespenstische Fernwirkungen" (Einstein). And indeed I can't follow Daviau in his speculations about monopoles because I am utterly skeptical about the claimed experimental evidences they rely on.

//last edit on February 22, 2017

vendredi 17 février 2017

Deux quanta de géométrie vallent bien une maille de supersymétrie

For physicists
As an addendum to the former post I recommend warmly to watch the very lively lecture of Ali Chamseddine below. It is about the quanta of geometry the Higgs boson has "brought" us thanks to the noncommutative spectroscope. Chamseddine deals also with the physical consequences and answers nicely the many questions from the physicist hearing even the ones about supersymmetry or string theory and he does it in his own prejudice-free or balanced way (see the next paragraph).


Quanta of geometry Lecture by Ali Chamseddine
Future Prospects for Fundamental Particle Physics and Cosmology Workshop
 May 4 – 8, 2015



Just an anecdote
In 1973 I got a scholarship from government of Lebanon to pursue my graduate studies at Imperial College, London. Shortly after I arrived, I was walking through the corridor of the theoretical physics group, I saw the name Abdus Salam on a door. At that time my information about research in theoretical physics was zero, and since Salam is an Arabic name, and the prime minister in Lebanon at that time was also called Salam, I knocked at his door and asked him whether he is Lebanese. He laughed and explained to me that he is from Pakistan. He then asked me why I wanted to study theoretical physics. I said the reason is that I love mathematics. He smiled and told me that I am in the wrong department. In June 1974, having finished the Diploma exams I asked Salam to be my Ph.D. advisor and he immediately accepted and gave me two preprints to read and to chose one of them as my research topic. The first paper was with Strathdee [1] on the newly established field of supersymmetry (a word he coined), and the other is his paper with Pati [2] on the first Grand Unification model, now known as the Pati-Salam model. Few days later I came back and told Salam that I have chosen supersymmetry which I thought to be new and promising. Little I knew that the second project will come back to me forty years later from studying the geometric structure of space-time, as will be explained in what follows. In this respect, Salam was blessed with amazing foresight.
(Submitted on 3 Jun 2016)

Why we could be very happy that a Higgs boson has been discovered

Possible insights from the 125 GeV Higgs boson in a spectral hypothesis perspective
After reading a recent post by Peter Woit on his blog Not Even Wrong, I left there an accepted comment that retrospectively appears to me as a bold attempt to put across the exact opposite of the following views:

The only place left to look for a way out of this swamp [the variety of electroweak symmetry breaking theories], it seems to me, is in strongly interacting chiral gauge theories. Many talented theorists have thought about this.... There are surely wonders hidden in the subject of strongly interacting chiral gauge theories. If we are forced to deal with them to deal by physics at the SU(2)×U(1) breaking scale, we may find them. If instead a Higgs is discovered and the physics at the SU(2)×U(1) breaking scale can be described by perturbation theory, we probably never will. This would be the real source of my sadness if a Higgs were discovered. It would mean that nature had missed a chance to teach us some new and interesting field theory. Personally, I don’t think that she would be so malicious.
Howard Georgi 
Perspective on Higgs Physics II, ed. G.L. Kane. World Scientific, 1997.

I do not think either that nature is malicious but subtle, just like the Lord ... or quantum field theory. 

Indeed as regular posts in this blog try to argue, nature or more specifically the 125 GeV Higgs boson has already not missed the opportunity to teach us - but in a barely audible voice - some new and interesting quanta of geometry that might provide insight about the ultra-high scale of seesaw mechanism and leptogenesis and hints to enlighten the adelic sectors* of astrophysics and cosmology namely black holes, dark matter and dark energy (*this neologism, beyond a wink to the suffix used to create adjectives imparting a specific form of verve, is inspired by the greek adjective ἄδηλος which means literally "not self-evident" and figuratively "obscure").

I talk about "a barely audible voice" for two reasons. The first one is anecdotal and has to do with the fact that contemporary high energy physics papers reporting about noncommutative spectral developments derived from the 125 GeV Higgs boson are relevant but rare while it appears to me that the spectral noncommutative geometrization of the Higgs could find a place in the theoretical physics ecosystem behind the discovery of the Higgs boson as it is nicely reviewed by James D. Wells (in an article from which I borrowed the above Georgi's quote). The second one is more substantial and refers to the fact that the insight physicists could gain from taking at face value the spectral noncommutative post-diction of the Higgs boson 125 GeV mass relies on the effectively very weak mixing (10-9) between the electroweak higgs doublet and a computed ultra heavy higgs standard model singlet responsible for the seesaw mechanism and the spontaneous breaking of a Pati-Salam gauge group.

One can criticize my argument about a "spectral higgs brother hypothesis" (not to mention the quanta of geometry) as immoderate speculation(s) of course. But with the hindsight from history J.D Wells writes : 
... the Higgs boson hypothesis was an immoderate speculation, and therefore faith in theory argumentation and speculation was mandatory for the discovery program to proceed and reach its fulfillment. The Higgs boson could not have been discovered experimentally by accident.  

Then I ask my reader : why do we not try to take advantage of a mathematically coherent formalism that provides effective computational tools to follow the breadcrumb trail from the standard model higgs at the TeV scale to his big brother at the ZeV one or even YeV, investigate the possibility that electroweak symmetry breaking is related to gravity through an almost commutative extension of spacetime and is triggered somehow by noncommutative fluctuations that impart a proper dynamics to geometry with consequences as mimetic dark matter, dark energy and a limiting spacetime curvature? 

Nature is not malicious but the human Hi(gg)story of the understanding of the electroweak symmetry breaking definitely proves to be devilish.

At least it was for spectral noncommutative geometers of whom it is worth reminding here the basic paradigm:
We thus view a given geometry as an irreducible representation of the algebraic relations between the coordinates and the line element, while the choice of such representations breaks the natural invariance group of the theory. The simplest instance of this view of geometry as a symmetry breaking phenomenon is what happens in the Higgs sector of the standard model.
Alain Connes 

These new geometers of physics have failed first to predict the correct mass of the standard model Higgs boson. But they learned from its very value and works it triggered among physicists worried about the stability of the Higgs potential at ultra heavy scales how to remove an incorrect assumption they made. To improve then the coherence of their framework they have subsequently understood how to give up an axiom of noncommutative geometry and uncovered new inner fluctuations of geometry that impact the structure of the Higgs fields and make them composite somehow. As a consequence they could postdict - following a more constrained theory - the correct SM Higgs boson mass, predicting a mixing with another scalar field responsible for the spontaneous symmetry breaking of a Pati-Salam gauge symmetry as I have already reported above. Last but not least, this new impetus has lead them to write an elaborate Heisenberg-like equation and they have established the mathematical demonstration that the two very specific Clifford algebra required for a Pati-Salam gauge unification model in the spectral point of view are exactly the pair required in the Feynman slash of the proper coordinates to recover with a generalized Dirac operator any 4 dimensional Riemannian manifold with a quantized volume! (19/02/2017 update: A. Connes has made available yesterday a new paper in English that covers specifically this point that is the core of the first six hours of last lecture at the Collège to reach an audience as broad as possible).

So to conclude let me come back to Howard Georgi and be bold once again to write him the following message:
Dear Sir, 
I hope you have welcomed too the discovery of the 125 GeV Higgs boson in 2012 even if you had some doubts or prejudices against its existence. I cannot say if the wonders lying in strongly interacting chiral gauge theories are realy hidden by this scalar boson but I notice you have not lost the expectation to connect the Higgs phase and the confining phase 
I wonder what would be your take on the possible spectral noncommutative world hidden behind the specific electroweak symmetry breaking nature has chosen for us. Insofar as this new geometric paradigm suggests an extension of space where the program of unification of the standard model gauge interactions that you initiated with a few others is the natural result of a subtle dynamic extending that of gravity, I do not doubt that you would be interested to learn how discrete extra dimensions or rather some fine structure of spacetime could be more than a metaphor thanks to the Higgs boson discovery.
Yours respectfully.

This post is dedicated to a spectral heroine researcher : Charlotte Dempière and a fantasized student of her dreaming of a subtle loose way to connect unparticle physicsscale invariance at low accelerations and mimetic matter in a new quantum world.

// Last edit 18 february 2017