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, A_SM:=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 θ_W given by sin²(θ_W)=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 sin²(θ_W), 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.

The Standard Model Algebra

Ovidiu Cristinel Stoica

(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=M_n(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. 

As far as noncommutative geometry is concerned I think the mathematical demonstration that the two very specific Clifford algebra required for a Pati-Salam gauge unification model 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 is already a potential strong incentive for physicists!

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. 

L'espace-temps double

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