Transitional Bell Correlation from Dirac Wavepackets

TL;DR

This work revisits Bell correlations for entangled particles by treating each electron as a spatially extended Dirac wavepacket and modeling localized detectors, thereby restoring the physical spatial structure of measurements. By constructing Gaussian Dirac wavepackets, forming a singlet state, and evaluating a windowed two-particle correlator, the authors derive a closed-form expression for the transitional Bell parameter $\mathcal B(\zeta;\kappa)$ that interpolates between quantum and classical limits as the wavepacket overlap changes. A critical threshold $\kappa_\ast\approx0.618$ separates regimes where diffusion maintains overlap from those where directed propagation dominates, leading to either persistent quantum violation ($|\mathcal B|>2$) or rapid relaxation to the classical bound. The results highlight that Bell violations in realistic setups arise from local wavepacket overlap rather than nonlocal mechanisms, with measurable consequences dependent on the detected observable, and suggest analogous overlap-driven effects in photonic Bell tests and solid-state qubits.

Abstract

We derive a closed-form expression for the Bell--CHSH correlation of entangled, counter-propagating electrons using realistic Dirac wavepackets and localized detection. In contrast to the conventional distance-independent result, the Bell parameter evolves continuously from the quantum bound $2\sqrt{2}$ to the classical limit $2$ as the spatial overlap of the two waves decreases. The quantum enhancement arises entirely from transverse overlap, showing that the Bell violation reflects the local overlap of propagating Dirac waves rather than any action at a distance.

Figures (1)

• Figure 1: Transitional Bell parameter $|\mathcal{B}(\zeta;\kappa)|$ as a function of normalized separation $\zeta$. Blue and red lines mark the quantum bound ($|\mathcal{B}|=2\sqrt{2}$) and classical limit ($|\mathcal{B}|=2$). Cyan: $\kappa=0.5$, where diffusion maintains substantial overlap and the violation persists over a broad range of separations. Orange: $\kappa=1.0$, where directed propagation causes the overlap to decay more rapidly and $|\mathcal{B}|$ enters the classical regime once the separation exceeds the wavepacket width.