From Algebraic Mixing Matrices to Dynamical Field Equations: A Unified Spectral Deformation Framework for Flavor, Electroweak, and Color Interactions
Description
We present a rigorous mathematical framework that elevates four fundamental mixing matrices of the Standard Model — the Weinberg rotation matrix (2 × 2, electroweak mixing), the Gell-Mann matrices (3 × 3, SU(3) color generators), and the geometrically reconstructed CKM and PMNS matrices (3 × 3, real orthogonal non-symmetric bi-reflection matrices) — into fully dynamical field equations. For each matrix, we derive: (i) its characteristic polynomial with explicit invariants (trace, determinant, and geometric coefficients d), (ii) the corresponding partial differential equation (PDE) of order equal to the matrix dimension, with temporal derivatives on the left and spatial operators (Laplacian and gradient) on the right, fully equipped with all physical constants (ℏ, c, α, gs, θW , MZ , mt, mτ ), and (iii) the equivalent Volterra integral equation including initial conditions. All derivations are dimensionally consistent; we introduce a characteristic frequency ω0 to render the differential operator dimensionless. We prove that these PDEs yield dispersion relations that encode scale-dependent mixing, finite propagation speed, and spatially varying CP phases. For the Gell-Mann sector, the third-order equation naturally produces a confining potential with string tension σ ∝ qω30 /c2. For the PMNS sector, the two extra initial conditions of the third-order equation are rigorously identified with the Majorana phases. All theorems are accompanied by explicit proofs with full step counting, dimensional checks, and consistency verifications with known low-energy limits. This framework reduces the number of free parameters in the flavor sector and provides a dynamical origin for the running of CKM and PMNS parameters. Appendices F–I demonstrate the universality of the framework by applying it to Pauli matrices, Dirac matrices, and the traditional (unitary) CKM and PMNS matrices, yielding characteristic equations, field equations, and integral equations in each case. The framework is complete in the sense of Theorem 8.0 and contains no unresolved conjectures.
Files
from_matrix_to_equation.pdf
Additional details
Additional titles
- Alternative title (English)
- From Algebraic Mixing Matrices to Dynamical Field Equations
Dates
- Submitted
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2025-12-31
References
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