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

Abstract

Lagrangian for a single relativistic charged particle in an external electromagnetic field, retaining both the standard relativistic square-root kinetic term and the minimal electromagnetic coupling in their original forms. The Lagrangian is supplemented by a covariant spin--field coupling describing the interaction between the particle's intrinsic spin and the electromagnetic field. Because this term is matrix-valued, it is incorporated through a scalarization procedure so that the HJB problem remains scalar. Accordingly, the present construction is restricted to electromagnetic backgrounds for which $\sigma^{\mu\nu}F_{\mu\nu}$ admits a spacetime-independent eigenspinor, and the Dirac equation is obtained within the corresponding fixed spin sector. The resulting SOC formulation is relativistically consistent: in the limit $\hbar\to0$, the HJB equation reduces to the proper-time relativistic Hamilton--Jacobi equation, whose stationary form yields the classical mass-shell condition, whereas the stationary HJB equation yields a quantum-corrected mass-shell relation. The theory is illustrated through stochastic simulations of the Dirac--Landau problem for an electron in a uniform magnetic field. At the analytic optimal drift, the average stochastic action attains a local minimum, and its individual components agree with the corresponding Dirac--Landau values within statistical uncertainty.