Published November 9, 2025 | Version v2

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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 emergence of the quantum equilibrium. This establishes a solid basis for transforming the Quantum Equilibrium Hypothesis from a postulate into a dynamical attractor or potentially a mathematical necessity.

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2025-11-09

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