Doubling of the algebra and neutrino mixing within noncommutative spectral geometry

Abstract

We study physical implications of the doubling of the algebra, an essential element in the construction of the noncommutative spectral geometry model, proposed by Connes and his collaborators as offering a geometric explanation for the standard model of strong and electroweak interactions. Linking the algebra doubling to the deformed Hopf algebra, we build Bogogliubov transformations and show the emergence of neutrino mixing.PACS. 02.40.Gh Noncommutative geometry -14.60.Pq Neutrinos mass and mixing -12.10.Dm Unified theories and models of strong and electroweak interactions

Causality in noncommutative two-sheeted space-times

“…The choice of the C ⊕ C model is also motivated on physical grounds. The noncommutative Standard Model of particle physics, based on the algebra A SM = C ⊕ H ⊕ M 3 (C), is often described as a two-sheeted space-time [11]. Indeed, the space of pure states of the electroweak sector C ⊕ H consists of two points and although P (M 3 (C)) ∼ = CP 2 , all of its points are separated by an infinite distance as the Dirac operator D F commutes with the M 3 (C) part of the algebra [12,Remark 5.1].…”

Aspects of the Bosonic Spectral Action

“…Since deformed co-products are a basis of Bogogliubov transformations, one concludes that field mixing arises from the algebraic structure of the deformed co-product in the noncommutative Hopf algebra embedded in the algebra doubling of noncommutative spectral geometry. We can hence conclude that the SM derived from NCSG, includes neutrino mixing by construction [9].…”

“…The algebra doubling can also lead to neutrino oscillations. Linking the algebra doubling to the deformed Hopf algebra, one can build Bogogliubov operators as linear combinations of the co-product operators defined in terms of the deformation parameter obtained from the doubled algebra, and show the emergence of neutrino mixing [9]. In particular, one can write the mixing transformations connecting the flavour fields ψ f to the neutrino fields with nonvanishing masses ψ m as…”

“…The doubling of the algebra is intimately related to dissipation, gauge field structure (necessary to address the physics of the SM), as well as neutrino mixing, while it incorporates the seeds of quantisation [8,9]. Consider the classical Brownian motion of a particle of mass m with equation of motion…”

“…In the following, we will give a short description of the main elements of NCSG [4]- [7] , discuss its phenomenological particle physics consequences, the physical implications of the precise construction of the almost commutative manifold [8,9], and examine the cosmological consequences of the gravitational sector of the theory [10]- [18]. We will then address some issues regarding the traditional bosonic spectral action approach and highlight a new proposal [19].…”