Complex Stochastic Optimal Control Foundation of Quantum Mechanics

Recent studies have expanded the use of the stochastic Hamilton Jacobi Bellman (HJB) equation to include complex variables for deriving quantum mechanical equations. However, these studies typically assume that it is valid to apply the HJB equation directly to complex numbers, an approach that overlooks the fundamental problem of comparing complex numbers to find optimal controls. This paper addresses how to properly apply the HJB equation in the context of complex variables. Our findings significantly reevaluate the stochastic movement of quantum particles, directly influenced by the Cauchy Riemann theorem. These insights not only deepen our understanding of quantum dynamics but also enhance the mathematical rigor of the framework for applying stochastic optimal control in quantum mechanics.