Quantum foundations
Where the Born rule and the guidance equation come from, and when Born-looking statistics can hide a departure from them.
Three works in four archived versions · latest release · single-author works by Nidhal Mghirbi
The question
Answered When can restricted measurements hide a conserved nonequilibrium?
Bounds in terms of mixing and record sensitivity.
The papersin reading order, with what each settled
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Galilean covariance almost fixes the quantum probability current
Nidhal Mghirbi · Zenodo,
The continuity equation fixes only the divergence of the quantum probability current. Imposing Galilean boost covariance pointwise, for any one-body current, local or nonlocal, pure, mixed or many-body, leaves exactly the Schrödinger current plus a linear functional of the density; rotational covariance with conservation removes that term in three or more dimensions, and conservation with the momentum–translation relation removes it in every dimension. The same framework sorts the known alternatives, among them Nelson’s, Doebner–Goldin, chiral and Deotto–Ghirardi currents.
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Observable quantum equilibrium under controlled measurements
Nidhal Mghirbi · Zenodo,
A microscopic departure from the Born distribution can be conserved while every permitted measurement becomes insensitive to it. Under a quantified mixing hypothesis, the paper bounds how well any complete experimental record, including apparatus readings, clocks and adaptive controls, can tell such an ensemble from equilibrium; a driven Bohmian oscillator and a square-well calculation with 131,072 trajectories make the bounds concrete.
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The relativistic necessity of the Born rule: uniqueness from Poincaré symmetry and dynamical preservation
Nidhal Mghirbi · International Journal of Quantum Foundations 12 (2026) 1–28 · Zenodo, ; first version
From the symmetries of the Klein–Gordon and Dirac equations and the representation theory of the Poincaré group, the paper derives their conserved currents and argues that, for spin-1/2 particles, the probability density is the unique admissible form, reducing to in the non-relativistic limit; within de Broglie–Bohm theory it proves that this distribution is preserved by the dynamics.