Relativistic Effects in Spin Correlations Induced by QED Scattering and Wigner Rotations

Abstract: We study the relativistic nature of the interactions that, at tree level, generate spin correlations between two electrons in Møller scattering, as well as in an extended process involving a witness particle $C$. The corresponding processes, $e^{-}e^{-}\rightarrow e^{-}e^{-}$ and $e^{-}e^{-}C\rightarrow e^{-}e^{-}C$, are analyzed both in the center-of-mass frame and, for the former process, in a Lorentz-boosted frame where Wigner rotations arise. It is found that, through a nonrelativistic approximation of the scattering amplitudes, dipole-dipole and current-dipole interactions are responsible for the emergence of these correlations. This is evidenced by the variation of the von Neumann entropy of one electron for initially separable states, and of $C$ for an initially prepared three-particle entangled W-state. In Wigner rotations, the invariance of entropy under local unitary transformations is maintained at the expense of the emergence of quantum coherence in the density matrix at large rapidities. As a consequence, the final states of both particles are evaluated and shown to encode information about the scattering process through their spin expectation values. This framework is then used to comment on the correlations in the inelastic process $e^{-}e^{+}\rightarrowμ^{-}μ^{+}$, for which some research has reported differing results.

• The paper details how tree-level QED scattering and Wigner rotations generate spin correlations, with current-dipole interactions dominating Møller scattering.
• It uses reduced density matrices and von Neumann entropy to quantify variations in entanglement and spin expectation values under Lorentz boosts.
• The study finds that maximal initial entanglement remains unchanged, offering insights for precision measurement of spin correlations in particle collisions.

Relativistic Spin Correlations in QED: Scattering, Interactions, and Wigner Rotations

Context and Objectives

This paper provides a comprehensive analysis of the relativistic mechanisms underlying spin correlations induced by quantum electrodynamical (QED) scattering, with particular attention to Møller scattering, inelastic lepton-antilepton processes, and scenarios involving a spectator witness particle. The authors rigorously examine how spin entanglement and quantum coherence are generated, redistributed, or preserved under both non-relativistic and relativistic dynamics, and how these effects manifest in different inertial frames through the action of Wigner rotations.

Methodological Framework

The study is centered on the explicit calculation of reduced density matrices for relevant single-particle and subsystem states, both before and after the QED scattering events. Key ingredients include:

• Tree-level Scattering Amplitudes: Systematic expansion of Feynman amplitudes in terms of particle momenta enables identification of interaction types (Coulomb, current-current, dipole-dipole, current-dipole).
• Reduced Density Matrices: Construction and analysis of reduced density matrices for individual particles, with von Neumann entropy and spin expectation values serving as quantitative markers of correlations and mixedness.
• Wigner Rotations: Analysis of spin transformations under Lorentz boosts, particularly perpendicular to the motion of the particles, incorporates the full structure of the Poincaré group for characterizing changes in spin observables and coherence.
• Reference Frames: Calculations are performed both in the center-of-mass frame and in Lorentz-boosted frames, isolating the effects due to relativistic symmetry transformations.

Main Results

Spin Correlations in Møller Scattering

The analysis reveals that, at leading order, spin correlations arising in Møller scattering between electrons are exclusively due to current-dipole and dipole-dipole interactions. Notably, the magnitude of correlation generation is significantly larger for the current-dipole interaction—a consequence of the prohibition of spin-flip processes in the current term and the asymmetric structure of the associated amplitudes.

Numerical Findings:

• For initially separable spin states, scattering generates nonzero von Neumann entropy and measurable changes in spin expectation values, directly attributable to these interaction channels.
• For maximally entangled initial spin states, the final reduced density matrix remains maximally mixed and $\Delta S_{\text{Neumann}} = 0$. No additional correlations can be introduced into an already maximally entangled bipartite system via tree-level QED scattering dynamics.

Wigner Rotations and Lorentz Boosts

Under Lorentz boosts perpendicular to the collision axis, Wigner rotations induce nontrivial transformations of the spin basis:

• Quantum coherence emerges in the reduced density matrix, detectable through the presence of new off-diagonal components in the spin basis and the appearance of a spin expectation value orthogonal to the original quantization axis.
• Despite this basis-dependent redistribution, the total von Neumann entropy is rigorously preserved, verifying its Lorentz invariance at the level of local unitary transformations induced by boosts.

Three-Particle Correlations: Entangled Witness

When a third spectator particle is introduced (prepared in a tripartite W-state), the same interaction mechanisms—primarily current-dipole and dipole-dipole—mediate the creation and redistribution of correlations. The entropy and spin expectation value variations for the witness are found to be weaker than in the bipartite case due to the distribution of correlations among three subsystems.

Inelastic Processes: $e^- e^+ \to \mu^- \mu^+$

Analyzing the inelastic QED process with a lepton-antilepton initial state, the behavior parallels that observed in elastic e$^-$e$^-$ scattering:

• For initially separable spin states, entropy and spin expectation values of outgoing states increase following the scattering, consistent with correlation generation.
• For maximally entangled initial states, no further increase in von Neumann entropy is possible; consequently, $\Delta S_{\text{Neumann}} = 0$. This result is not compatible with certain prior claims in the literature that predict a nonzero entropy variation for such initial conditions.

Theoretical and Practical Implications

These findings clarify the origin of spin entanglement and decoherence in relativistic scattering processes, with direct implications for the interpretation of quantum information flow in quantum field theory. The demonstration of Lorentz invariance of von Neumann entropy under Wigner rotations rigorously confirms a key tenet of relativistic quantum information theory, and the identification of specific interaction channels responsible for correlation generation provides a microscopic foundation for analyzing entanglement production in scattering experiments.

Practically, these results are relevant for designing precision measurements of spin correlations in particle collisions, and for understanding the encoding of scattering process information in observable spin expectation values. The formalism also extends naturally to scenarios with more complex initial entanglement structures and multiple spectator degrees of freedom.

Perspectives and Future Directions

At tree level, the described mechanisms are exhaustive; however, the authors emphasize that at higher orders, loop corrections and real soft photon emission must be included to enforce full unitarity and physical consistency. This opens avenues for investigating the redistribution of quantum information in the presence of radiative corrections, the role of soft photons in entanglement dynamics, and the extension of these methods to non-Abelian gauge theories and gravitational scattering.

Experimental validation—involving measurement of spin correlations and entropy variations in relativistic scattering contexts—could sharpen or revise the theoretical predictions and elucidate the role of initial entanglement in quantum field dynamics.

Conclusion

This work establishes a detailed connection between specific QED interaction channels and the generation of spin correlations and quantum coherence in both elastic and inelastic scattering processes. It demonstrates the Lorentz-invariant nature of von Neumann entropy under Wigner rotations, specifies the mechanisms underlying entanglement redistribution, and addresses previously unresolved or contradictory findings in the literature. The study underscores the necessity of relativistic frameworks and group-theoretic methods for understanding quantum correlations in particle physics and sets the stage for further exploration of multi-particle entanglement dynamics in quantum field theories.

Reference: "Relativistic Effects in Spin Correlations Induced by QED Scattering and Wigner Rotations" (2606.08995)