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 noncommutativity
parameter 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 link
between quantum gravity and the mass spectrum.

6. Consistency with Chamseddine’s framework: Our approach does not replace but rather complements the Chamseddine construction, providing a
geometric 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.