Quantum Fractal Dynamics from Complex Relativistic Kinematics

Abstract

We establish an exact mathematical isomorphism mapping special relativistic kinematics in complex velocity space directly to microscopic quantum fractal trajectories ($\Delta x \sim \sqrt{\Delta t}$) 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 $\mathcal{V}^\mu = v^\mu - i u^\mu$. We demonstrate that in the sub-Compton observational limit ($\Delta t_{\text{obs}} \ll \tau_C \equiv \hbar/mc^2$), the inverse complex Lorentz boost factor $\gamma(\mathcal{V})^{-1}$ scales proportionally with $\sqrt{\Delta t_{\text{obs}}}$. Applying this inverse Lorentz time dilation transforms classical linear geodesics into the characteristic quantum diffusion relation ($\Delta x \propto \sqrt{\Delta t}$). Crucially, we prove that local conformal symmetry in metric space ($g_{\mu\nu} = \Omega^2 \eta_{\mu\nu}$) is physically equivalent to continuous Lorentz boost rescalings in complex velocity space. By identifying the local Conformal Scale Factor $\Omega(x)$ directly with the inverse complex Lorentz factor ($\Omega(x) = \gamma(\mathcal{V}(x))^{-1}$), we derive a scale-covariant metric representation $g_{\mu\nu}(x) = \gamma(\mathcal{V}(x))^{-2} \eta_{\mu\nu}$. This proves that quantum fractal dynamics, relativistic time dilation, and local spacetime curvature originate from a unified kinematic transformation.