Time-Delayed Causality as a Classical Alternative to Quantum Mechanics: Hydrogen in Neutral Newtonian Mechanics

This study develops a strictly classical time-delayed model of the hydrogen atom, with retarded Coulomb forces and finite-difference inertia over a state-dependent delay, to reproduce key quantum features without quantum postulates. The delay law is anchored in the Abraham, Lorentz and Dirac self-time and fixed by matching the Rydberg constant, thereby determining a neutral delay scale and an effective energy frequency proportionality without inserting Planck’s constant into the equations of motion. With this single calibration, a neutral mean-power condition and a phase-locking rule between orbital motion and delay generate a discrete ladder of radii and energies reproducing the Rydberg spectrum without imposing angular-momentum quantisation. Neutral radial and azimuthal phases, constructed from the retarded trajectory, define two indices that act as classical analogues of principal and orbital quantum numbers and satisfy simple sum rules for elliptic orbits, mirroring Sommerfeld’s extension of the Bohr model. Interpreting the neutral delay as a minimal time window for inferring position and velocity yields a classical position–momentum error product of order Planck’s constant. In weakly driven settings, non-zero mean neutral power turns the delayed inertial channel into an energy reservoir, allowing deterministic level-to-level transitions between long-lived neutral orbits and preventing the catastrophic radiative collapse of the Rutherford atom.