Quantum Geometry of Fundamental Interactions: From Algebraic Quantization to Modified Gravity

This paper presents a complete quantization scheme for an algebraic-geometric theory that unifiesgravity and gauge interactions through a torsion constraint. The fundamental structure is based onan extended Clifford algebra with orthogonal imaginary operators I, J, K, L that separate interactionsectors. The key constraint geometrically links spacetime torsion to gauge field strengths. We develop the functional integral formalism for this algebraic theory, showing factorization due to operator orthogonality: exp(iStotal) = Q X[cos SX ⊗ 1 + i sin SX ⊗ X]. At one-loop level, we derive quantum corrections to the gravitational potential. On cosmological scales, the Higgs sector L modifies Newton’s law toΦ(r) = −GM/r+ Λeffr^2/6, with Λeff = 2λHv^2R0 naturally yielding the observed dark energy density.This explains galactic rotation curves without dark matter. Photon scattering on black holes showspolarization conversion and modified Hawking radiation. The theory offers a path to ultravioletcomplete quantum gravity, with predictions testable through gravitational wave astronomy, galactic dynamics, and precision gravity measurements.本文提出一套完整的代数几何理论量化方案，该方案通过挠率约束（torsion constraint）统一引力与规范相互作用。其核心结构基于引入正交虚算子I、J、K、L的扩展克利福德代数（extended Clifford algebra），该算子可划分不同相互作用扇区。核心约束从几何层面将时空挠率与规范场强（gauge field strengths）关联起来。我们针对该代数理论构建了泛函积分形式（functional integral formalism），并证明了由算子正交性引发的因式分解关系：$e^{iS_{ ext{total}}} = Q otimes left[cos S_X otimes 1 + isin S_X otimes X ight]$。在单圈阶（one-loop level）近似下，我们推导了引力势的量子修正项。在宇宙学尺度下，希格斯扇区（Higgs sector）L将牛顿引力定律修正为$Phi(r) = -frac{GM}{r} + frac{Lambda_{ ext{eff}} r^2}{6}$，其中$Lambda_{ ext{eff}} = 2lambda_H v^2 R_0$可自然得出与观测结果一致的暗能量（dark energy）密度。该理论无需引入暗物质（dark matter）即可解释星系旋转曲线。光子与黑洞的散射过程展现出偏振转化与修正后的霍金辐射（Hawking radiation）现象。该理论为构建紫外完备（ultraviolet complete）的量子引力提供了可行路径，其相关预言可通过引力波天文学、星系动力学与高精度引力测量实验进行检验。