This work extends the classical hidden variables theory proposed in Ref. [1] to the relativistic regime by considering the Lorentz-invariant Complex Sine-Gordon Equation (CSGE) as a fundamental model. In this framework, soliton solutions of the CSGE are interpreted as classical particles moving within a chaotic background that induces fluctuations in their position and momentum. We show that these fluctuations satisfy an exact uncertainty principle in the non-relativistic limit, leading to an emergent Schrödinger equation governing the ensemble dynamics of solitons. An explicit relation between the background noise intensity and the Planck constant is obtained. Furthermore, the simulations confirm that the probability density of solitons evolves in agreement with the Schrödinger equation for small velocities, while deviations at higher velocities suggest the emergence of a relativistic generalization of the Schrödinger equation. These results support the hypothesis that quantum mechanics may arise as an effective statistical description of an underlying deterministic field theory.
[1] D. Gomila, “Solitons, chaos, and quantum phenomena: A deterministic approach to the Schrödinger equation”, Physical Review Research 7, 033276 (2025).