Preserving mass shell condition in the stochastic optimal control derivation of the Dirac equation

Abstract The Dirac equation, central to relativistic quantum mechanics, governs spin-$\frac{1}{2}$ particles and their antiparticles, with each spinor component satisfying a Klein-Gordon-type equation -- the quantum counterpart of the relativistic mass-shell condition.
In our prior work [V. Yordanov, Sci. Rep. 14, 6507 (2024)], we derived the Dirac equation using stochastic optimal control (SOC) theory by linearizing both the Lagrangian's kinetic term and the Hamilton-Jacobi-Bellman (HJB) equation, but that earlier linearized derivation did not preserve the relativistic mass-shell condition. 
Here, we formulate the SOC problem starting from a specified covariant Lagrangian for a single relativistic charged particle in an external electromagnetic field, in which the standard relativistic square-root kinetic term and the minimal electromagnetic coupling are retained in their original form.
In addition, the Lagrangian is supplemented by a covariant spin-field coupling term describing the interaction of the particle's intrinsic spin with the electromagnetic field.
Because this spin-field term is matrix-valued, it is incorporated through scalarization so that the HJB problem remains scalar.
This yields a relativistically consistent SOC derivation in which the $\hbar \to 0$ limit of the HJB equation recovers the proper-time relativistic Hamilton--Jacobi equation, whose stationary form yields the classical mass-shell condition, while the stationary HJB equation yields the quantum-corrected mass-shell relation.
We illustrate the theory with stochastic simulations of the Dirac-Landau problem (electron in a uniform magnetic field): at the analytic optimal drift the average stochastic action attains a local minimum, and the average action components are consistent with the Dirac-Landau values within statistical uncertainty.