Quaternion Vacuum Mechanics: When Mathematical Symmetry Meets the Standard Model of Particle Physics

This article introduces a groundbreaking theoretical physics study that uses quaternion algebra to reinterpret the Standard Model of particle physics. Based on the quaternion structure of the vacuum, this study provides a new perspective for resolving dilemmas such as the parameter fine-tuning feel and fermion mass hierarchy in the Standard Model. It also offers explanations for issues like dark matter and the cosmological constant, and proposes testable experimental predictions.

Although the Standard Model of particle physics successfully describes three fundamental interactions and predicts many particles, it has problems such as a large number of parameters and a patchwork structure. Quaternions were discovered by Hamilton in 1843 and are a four-dimensional extension of complex numbers (q=a+bi+cj+dk, satisfying i²=j²=k²=ijk=-1), with advantages in describing three-dimensional rotations. The new study assumes that the vacuum is filled with quantum fluctuations of quaternion symmetry, providing a geometric basis for the parameters of the Standard Model.

The study found that constants such as the fine-structure constant and coupling constants in the Standard Model can be explained through the dimensional properties of quaternion algebra: the dimension of the SU(2) gauge group (3) corresponds to the tangent space dimension of the quaternion unit sphere S³, and the dimension of SU(3) (8) is related to the complex quaternion structure. If the vacuum has a specific quaternion geometry, the fermion mass matrix automatically exhibits the observed hierarchical structure, and the mass ratios of the three generations of fermions can be derived through the quaternion eigenvalue problem.

The form of the neutrino mass matrix predicted by quaternion vacuum mechanics deviates from DESI and Planck observation data by less than 1%; after correcting the coupling constant evolution equation, the three gauge interactions tend to unify at the Planck scale (≈1e19 GeV), and the value of the unification point can be calculated via the quaternion Casimir operator, which is consistent with the precise measurement data from LHC and LEP.

The excited state of the quaternion vacuum predicts a new neutral particle (mass ≈2.7e18 GeV), which matches the properties of dark matter candidates, and its relic density is consistent with the Planck-measured Ω_DM h²=0.12; combined with the holographic principle, there is a cancellation mechanism for vacuum energy contributions, and the effective cosmological constant is consistent with the observed value Λ≈1e-47 GeV⁴.

The theory proposes testable predictions: next-generation neutrino oscillation experiments (DUNE, Hyper-K) to verify the neutrino mass matrix; correction of the branching ratio of rare decays (μ→eγ); direct dark matter detection experiments (XENONnT, LZ) to search for the predicted particles; and DESI experiments to test the time evolution of the dark energy equation of state parameter w.

Quaternion vacuum mechanics opens a new direction for theoretical physics, indicating that deep physical laws are rooted in elegant mathematical structures. Although it is in the early stage and requires more verification, it reminds us of the unity of mathematical beauty and physical truth. As Einstein said, the intelligibility of the universe may be further revealed through such mathematical frameworks.