Preserving Mass Shell Condition in the Stochastic Optimal Control Derivation of the Dirac Equation

The Dirac equation, central to relativistic quantum mechanics, governs spin-1/2 particles and their antiparticles, with each spinor component satisfying the Klein-Gordon equation - the quantum counterpart of the relativistic mass shell condition. Our prior work [V. Yordanov, Sci. Rep. 14, 6507 (2024)] derived Dirac equation using stochastic optimal control (SOC) theory by linearizing the Lagrangian's kinetic term and the Hamilton-Jacobi-Bellman equation, but failed to preserve the mass shell condition. Here, we introduce a novel SOC derivation that retains the nonlinear kinetic term and integrates spin-electromagnetic coupling into the potential, ensuring relativistic consistency. This approach not only addresses the limitations of the previous model but also deepens the link between stochastic mechanics and quantum theory, offering fresh insights into relativistic quantum phenomena.