Quantum Fractal Dynamics from Complex Relativistic Kinematics

We establish an exact mathematical isomorphism mapping special relativistic kinematics in complex velocity space directly to microscopic quantum fractal trajectories (Δ x ~ √Δ t) and local spacetime geometry. Guided by the principle that scale transformations of measurement standards must map onto geometric symmetries, we extend standard kinematics using Nottale's complex velocity field V^μ = v^μ - i u^μ. We demonstrate that in the sub-Compton observational limit (Δ tobs τC ≡ /mc²), the inverse complex Lorentz boost factor γ(V)⁻¹ scales proportionally with √Δ tobs. Applying this inverse Lorentz time dilation transforms classical linear geodesics into the characteristic quantum diffusion relation (Δ x ∝ √Δ t). Crucially, we prove that local conformal symmetry in metric space (gμν = Ω² ημν) is physically equivalent to continuous Lorentz boost rescalings in complex velocity space. By identifying the local Conformal Scale Factor Ω(x) directly with the inverse complex Lorentz factor (Ω(x) = γ(V(x))⁻¹), we derive a scale-covariant metric representation gμν(x) = γ(V(x))⁻² ημν. This proves that quantum fractal dynamics, relativistic time dilation, and local spacetime curvature originate from a unified kinematic transformation.