Electromagnetism and Quantum Chromodynamics in Noncommutative Spectral Geometry: From Waves and Solitons to Confinement

We present a comprehensive analysis of electromagnetic and strong interactions within the framework of noncommutative spectral geometry, where gauge fields are identified with spacetime contorsion. This geometric unification reveals that all elementary particles — leptons, quarks, photons, gluons — are different manifestations of the same fundamental object: the axial vortex.
Key results include:
1. Electromagnetic sector: Photons emerge as spherical torsion waves propagating along generalized geodesics. The fundamental relations E ⊥ B ⊥ k are preserved as geometric necessities, but the vacuum impedance becomes energy-dependent: Z0(ω) = Z (0) 0 [1 − α 2(ω/ΛNC)2]. The Coulomb potential acquires a contact term:

 ϕ(r) = 14πϵ0 q r[1 + θ2 2 δ3(r)], regularizing the classi-
cal singularity. Light cones become energy-dependent ("rainbow geometry"), yet microcausality is rigorously preserved. Like Born-Infeld theory, our model forbids electromagnetic shock waves.
2. Strong interaction sector: Quarks are described as tori T 2 with non-trivial SU(3) bundles classified by the first Chern class c1 ∈ Z3. Gluons are spherical torsion waves in the SU(3) sector, propagating with a modified dispersion relation. Quark-gluon interactions occur through topological resonance conditions analogous to the photoelectric effect, with color conservation enforced by ∆c1 = ±1.
3. Confinement: The energy required to separate two colored tori with total Chern class ctotal1 ̸= 0 mod 3 grows linearly with distance, producing a string
tension σ = g23 8π ln(Rcut/Rc) ≈ 0.18 GeV2, matching lattice QCD results. This is a topological necessity, not a dynamical accident.
4. Nucleons as bound states: A proton or neutron is the connected sum of three quark tori, Σ3 = T 2#T 2#T 2, a genus-3 surface whose topological structure explains confinement and proton stability. The color singlet condition c1 + c2 + c3 = 0 mod 3 ensures the total SU(3) bundle is trivial.
5. Unification: All particles emerge from the same spectral triple (A, H, D, J,γ) with finite algebra AF = C ⊕ H ⊕ M3(C). The differences between electrons, quarks, photons, and gluons are merely different projections in the gauge indices and the effective scale ϕeff0

All modifications are controlled by the noncommutativity scale ΛNC ∼ 1016 GeV,
ensuring consistency with all current experiments while making testable predictions for future high-energy and astrophysical observations.