A Time-Symmetric Variational Formulation of Quantum Mechanics with Emergent Schrödinger Dynamics and Objective Boundary Randomness

Abstract: We present a time-symmetric variational formulation of nonrelativistic quantum mechanics in which Schrödinger dynamics and a Bohm-type guidance law arise as emergent Euler-Lagrange optimality conditions rather than postulates. The formulation is expressed in terms of probability density and current fields subject to a continuity constraint and two-time boundary conditions. A Fisher-information regularization term generates the quantum potential, yielding the Schrödinger equation when the optimality system is expressed in complex form. Unlike standard Bohmian mechanics, which requires an auxiliary Quantum Equilibrium Hypothesis ($P = |ψ|^2$), our primal-dual formulation satisfies the Born rule by construction. The trajectories emerge not from an external guidance wave, but as the unique hydrodynamic flow minimizing the Fisher-regularized action between two-time boundary constraints. Deterministic trajectories thus emerge only as effective, coarse-grained descriptions, with randomness entering objectively at the interface of boundary constraints.