Bell Correlations from Prepared Coherence in Entangled Dirac Wavepackets

Bell correlations are usually formulated for an ideal spin singlet, for which the Bell–CHSH combination reaches the maximal quantum value $B=-2\sqrt{2}$, independent of detector separation. Here we derive Bell correlations from a more general physical state: an antisymmetrized pair of entangled Dirac wavepackets with source-prepared amplitude and phase coherence. The propagated branches are sampled locally by spatially separated endpoint detectors, yielding a separation-dependent CHSH value $B(Z)$. For a fixed CHSH analyzer geometry, the zero-separation, full-overlap limit gives $$B(0)=-2\sqrt{2},$$ independent of the preparation parameters. At large detector separation, once the direct branch-overlap contribution is suppressed, the surviving Bell–CHSH value approaches the prepared-coherence kernel $$B(\mathrm{\infty })={\mathcal{K}}_{\mathrm{c}\mathrm{o}\mathrm{h}}=-\sqrt{2}[1+\sin (2\theta )\cos \chi ]\mathrm{.}$$ Thus the asymptotic Bell value is controlled by the coherence fixed at the source through the amplitude balance $\theta$ and relative phase $\chi$. Bell violation is therefore a phase-sensitive local readout of prepared nonseparable Dirac-wave coherence: it rules out separable classical probability, but does not by itself require superluminal causation. In this wave-realist account, Bell correlations retain their full quantum content while remaining compatible with relativistic causal locality.