The Relativistic Necessity of the Born Rule: Uniqueness from Poincaré Symmetry and Dynamical Preservation
Year: 2025
Type: article
Abstract: This paper is a pedagogical and rigorous synthesis of prior foundational works. It demonstrates that the probability density ρ = |ψ|² is a natural and necessary feature of relativistic quantum mechanics, not an arbitrary postulate. The work systematically derives the conserved currents of the Klein-Gordon and Dirac equations from their U(1) symmetry via Noether's theorem. It grounds this derivation in the representation theory of the Poincaré group, arguing for the uniqueness of the ψ̄γ⁰ψ form for spin-1/2 particles. The analysis resolves the Klein-Gordon equation's historical problems by reinterpreting its current as a charge density that correctly reduces to |ψ|² in the non-relativistic limit. The framework is extended to many-body systems in configuration space. Within the de Broglie-Bohm interpretation, the paper proves the property of equivariance, showing quantum equilibrium is dynamically preserved. This synthesis provides a foundational bridge between quantum probability and special relativity's symmetry principles. It deliberately avoids claims of a first-principles derivation from nothing, acknowledging the assumed complex field structure. The work serves as a crucial foundation for upcoming, more challenging papers on dynamical relaxation to quantum equilibrium. This establishes a solid basis for transforming the Quantum Equilibrium Hypothesis from a postulate into a dynamical attractor.
Author: Mghirbi, Nidhal
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W7082157574
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