FRACTIONAL DIRAC OPERATORS AND LEFT-RIGHT FRACTIONAL CHAMSEDDINE–CONNES SPECTRAL BOSONIC ACTION PRINCIPLE IN NONCOMMUTATIVE GEOMETRY

摘要

The generalization of the Chamseddine–Connes spectral triples action to its (left andnright) fractional counterpart is constructed within the context of the Riemann–Liouvillenand Erdelyi–Kober (left and right) fractional operators. In the fractional approach, thenDirac operators u0002D is approximated by u0002D(α)n(a,−) + u0002D(α)n(b,+)∀t ∈ [a, b] and the spectral triplen(C∞(M), L2(M, S), u0002D) is replaced by its fractional equivalent (C∞(M), L2(M, S),nu0002D(2α)n(a,−)) + (C∞(M), L2(M, S), u0002D(2α)n(b,+)), 0 < α < 1. When the (left) fractional actionnis applied to the noncommutative space defined by the spectrum of the Standard Model,none obtains many attractive characteristics including time-dependent gauge couplingsnconstants (g2n1, g2n2, g2n3), a time-dependent cosmological constant (Λcos), a time-dependentnscalar Ricci curvature (R), a time-dependent Newton’s coupling constant, and a timedependentnHiggs square mass m2nH1. Furthermore, (g2n1, g2n2, g2n3), Λcos, R, and m2nH1 werenfound to be nonsingulars at the Planck’s time. When the (left and right) fractionalnbosonic action is taken into account, all the previous functions are found to be complexified,nincluding gravity. Many additional interesting features are discussed and explorednin some details.