Power-series dynamics equation of fields based on geometric algebra and high-order covariant-derivative schemes

Based on previous work, we explore in the context of quantum field theory and cosmology a geometric algebra (GA)-based approach to field dynamics through a multi-derivative power-series equation. By decomposing the power series into a partial sum of order four and a residual power series, we derive two separate equations involving high-order covariant derivatives, suggesting an emergent scale invariance. A key result is the formulation of a quartic differential operator, leading to Dirac-like equations that predict four family generations of particles. The residual power series describes field modes dominated by hyper-accelerations. Small coefficients in the residual power series reduce it to an equation involving just the fourth-power covariant derivative. This equation reduces the partial sum to order three, thereby suppressing the fourth generation and simplifying the spectrum of the field modes. This theoretical framework offers insights into the Standard Model, positing that weakening of hyper-acceleration terms in the early universe led to the decay of the fourth generation, leaving the three family generations observed at present. We identify two mechanisms for mass generation, which may explain the presence of left- and right-handed fields and the absence of right-handed neutrinos. The implications of this field theory extend to early cosmology, suggesting potential observational imprints in the cosmic microwave background, nucleosynthesis, and large-scale structure. This study invites further exploration into how these advanced field equations challenge established paradigms in both quantum field theory and general relativity, offering a new perspective on the dynamics of fields, particle generations, and cosmological evolution.