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.