A Projection Theory of Four-Dimensional Scalar Direction: A Geometric Interpretation for the Unification of Quantum Mechanics and Relativity

The fundamental incompatibility between general relativity and quantum mechanics remains a central challenge in theoretical physics. This paper proposes a unified geometric framework: introducing a four-dimensional scalar direction C as the cosmic background velocity, whose projection into three-dimensional space necessarily manifests as points, surfaces, or spheres. Based on this postulate, a stationary object carries momentum mC—a manifestation of participating in the overall cosmic motion. In local measurements, since all measuring instruments carry the same background momentum, the mC terms cancel out, making Newtonian mechanics P = mV a valid approximation that neglects the background. This framework provides unified geometric interpretations for several long-standing puzzles: quantum entanglement as multiple three-dimensional projections of the same four-dimensional entity; wave-particle duality as the three-dimensional slicing effect of a complete four-dimensional wave; black holes and wormholes as three-dimensional projections of four-dimensional vortices and folds; inertia, dark matter, and dark energy as manifestations of the background momentum mC and its projection effects. The framework satisfies the correspondence principle: it naturally reduces to Newtonian mechanics at low speeds, aligns with special and general relativity in their domains, and reproduces quantum effects at microscopic scales. This paper presents a qualitative geometric picture, laying a conceptual foundation for future mathematical formalization and experimental testing.