Giving new momentum to the quantization of the spectral noncommutative standard model...

...removing the independent "masslessness of the photon" axiom ?

Noncommutative geometry (NCG) provides a particularly elegant way to derive and describe the structure and the Lagrangian of the Standard Model in curved spacetime and its coupling to gravitation [1]. The main ingredients of this approach is an algebra A=C^∞(M)⊗A_F (where M is a Riemann spin manifold M and A_F=C⊕H⊕M₃(C)) [1] a Hilbert space H=L²(M,S)⊗H_F (where H_F is 96-dimensional), and a Dirac operator D. The elements a of the algebra are represented by bounded operators π(a) over H. In this approach, the gauge bosons are described by gauge potentials (i.e. non- commutative one-forms) in Ω¹_DA, where Ω_DA is the differential graded algebra (DGA) constructed from A, whose differential is calculated by using the commutator with D. 

From the physical point of view, a striking success of the noncommutative geometric approach is that the algebra, the Hilbert space and the Dirac operator of the Standard Model can be derived from a few simple axioms, including the condition of order zero, the condition of order one and the condition of massless photon [2,3,4]. Then, the Lagrangian of the Standard Model coupled to (Riemannian) gravity is obtained by counting the eigenvalues of the Dirac operator D [1]. 

Still, this approach is not completely physical because it is formulated in the Riemannian (instead of Lorentzian) signature and is not quantized. Therefore, the original NCG approach was variously modified, by using Lie algebras [5], twisted spectral triples [6] or Lie algebroids and derivation-based NCG [7], to deal with models that do not enter into the standard NCG framework (e.g. quantum groups or Grand Symmetry)[8] ... 

Boyle and Farnsworth [9] recently used Eilenberg’s algebra extension method to build an algebra E where the universal differential graded algebra (DGA) Ω built on A (see section III) is extended by the Hilbert space H. This is physically more satisfactory because the gauge field, the field intensity, the curvature and the Lagrangian densities are noncommutative differential forms, which belong to Ω up to an ideal described below. They observed that the associativity of the algebra E imposes a new condition (of order two) which is satisfied by the finite part A_F of the Standard Model and removes a somewhat arbitrary axiom in Chamseddine and Connes’ derivation [4]. This axiom requires the Dirac operator D_F of the finite algebra to commute with a specific family of elements of A_F. It is called the condition of massless photon because it ensures that the photon has no mass. 

However, the approach proposed by Boyle and Farnsworth has two serious drawbacks: i) it is not valid for a spin manifold (i.e. the canonical spectral triple (C^∞(M),L²(M,S),D_M) does not satisfy the condition of order two); (ii) it uses the DGA algebra Ω in which gauge fields with vanishing representation (i.e. A ∈ Ω such that π(A)=0) can have non-zero field intensity (i.e. π(dA)≠0). This makes the Yang-Mills action ill defined [1]. A consistent substitute for Ω is the space Ω_D of noncommutative differential forms which is a DGA built as the quotient of Ω by a differential ideal J usually called the junk. To solve both problems, we define an extension E of the physically meaningful algebra Ω_D of noncommutative differential forms by a representation space M_D that we build explicitly. Since the algebra Ω_D is a DGA, we require the extension E to be also a DGA and we obtain that M_D must be a differential graded bimodule over Ω_D (see below). The most conspicuous consequence of this construction is a modification of the condition of order two proposed by Boyle and Farnsworth, which provides exactly the same constraints on the finite part of the spectral triple of the Standard Model, but which is now consistent with the spectral triple of a spin manifold. As a consequence, the full spectral triple of the Standard Model (and not only its finite part) now satisfies the condition of order two and enables us to remove the condition of massless photon. 

...

To take into account the differential graded structure of Ω_D, we built a differential graded bimodule that takes the junk into account. The grading transforms the Boyle and Farnsworth condition on the commutator [π(δa),π(δb)◦] = 0 into a condition on the anticommutator {π(δa),π(δb)◦} ∈ K, which is now satisfied for the full Standard Model and not only for its finite part. 

This indicates that, in a reinterpretation of the noncommutative geometric approach to field theory, the differential graded structure of the boson fields must be accounted for. This is a good news for any future quantization and renormalization of NCG because the differential graded structure is also an essential ingredient of the Becchi-Rouet-Stora-Tyutinand Batalin-Vilkovisky approaches. 

Our differential graded bimodule retains the advantages of the Boyle and Farnsworth approach: (i) it unifies the conditions of order zero and one and the condition of massless photon into a single bimodule condition; (ii) it can be adapted to non-associative or Lie algebras. 

We hope to use our construction for the quantization of a noncommutative geometric description of the Standard Model coupled with gravity.

The Standard Model as an extension of the noncommutative algebra of forms

Christian Brouder (IMPMC), Nadir Bizi (IMPMC), Fabien Besnard

(Submitted on 15 Apr 2015)

Voilà une belle illustration de ce que la physique théorique/mathématique (ne) vit (que) des hypothèses /axiomes dont elle se libère un peu comme l'art ne vit que des contraintes qu'il s'impose pour paraphraser Albert Camus (commentant André Gide).

Mise à jour : l'article de Brouder, Bizi et Besnard a reçu récemment un ajout très intéressant dans une seconde version datée du 5 juin que j'évoquerai bientôt.