The Origin of Mass: Yukawa Couplings from Berezin Quantization in Noncommutative Spectral Geometry

We present a comprehensive geometric theory of mass generation within the framework of noncommutative spectral geometry. Building on the foundational work of Chamseddine and Connes [1, 2], we provide a geometric interpretation ofthe abstract algebraic structures in terms of tori T 2 (fermions) and spheres S2 (bosons). The bilinear functional T (I) µν (a, b), previously identified as the Berezin operator kernel [7], is shown to encode all mass parameters of the Standard Model.Key results include:1. Spectrum on the noncommutative torus: We derive the exact eigenvalues of the generalized Dirac operator on T 2 θ , incorporating the noncommutativityparameter θ into the mode expansion.2. Higgs mechanism geometrically: The Higgs field emerges from off-diagonal components of the finite Dirac operator DF. Its vacuum expectation value,acting through the Berezin operator B⟨Φ⟩, generates masses for the W and Z bosons, yielding m2 W = g2v2/4.3. Yukawa couplings from the bilinear functional: We prove that the Yukawa matrix elements are precisely the matrix elements of the bilinear functional projected onto the Higgs sector:yff′ = 1v⟨f|T (H)µν |f′⟩. (2)4. Unified mass formula: All particles satisfy m = ℏ/Rc ·F(topology), linking mass to the fundamental geometric scale Rc and topological quantum numbers(m, n). The mass hierarchy follows naturally from excited torus states: the top quark corresponds to m, n ∼ 10^3.5. Gravity-mass connection: The same bilinear functional Tµν that yields the graviton propagator also encodes Yukawa couplings, establishing a direct linkbetween quantum gravity and the mass spectrum. 6. Consistency with Chamseddine’s framework: Our approach does not replace but rather complements the Chamseddine construction, providing ageometric interpretation of the abstract algebraic structures while preserving all quantitative predictions.All modifications are controlled by the noncommutativity scale ΛNC ∼ 1016 GeV,ensuring consistency with all current experiments while making testable predictions for future collider and astrophysical observations.