Spin & Polarization-Resolved QED Processes

• Spin- and polarization-resolved QED processes are a framework that resolves particle spin and photon polarization in quantum electrodynamics to analyze transition rates and interference effects.
• They employ methodologies such as Dirac spinors, density matrices, LCFA techniques, and Monte Carlo simulations to predict outcomes in both strong-field and collider settings.
• These processes provide key insights into entanglement and nonlocality, influencing experimental designs in high-intensity laser experiments and condensed matter physics.

Spin- and polarization-resolved QED processes constitute a central domain in high-energy, strong-field, and condensed matter physics, where the quantum properties of both matter and radiation—spin and polarization—profoundly influence transition rates, correlations, and observables. In such processes, the quantum electrodynamics (QED) S-matrix is resolved not only in momenta and numbers but in initial and final particle spin states (typically described by Stokes vectors or spin-density matrices) and photon polarization (in a linear or circular basis). This resolution is indispensable in ultraintense laser-matter interactions, collider experiments, quantum kinetic and transport modeling, and in understanding Bell-type entanglement and quantum nonlocality in relativistic regimes.

1. Fundamental Formalism and Operator Structure

At the fully quantum level, evaluation of QED amplitudes for processes such as nonlinear Compton scattering and Breit–Wheeler pair production requires both incoming and outgoing particle states (e.g., electrons, positrons, photons) to be individually resolved in spin (for fermions) or polarization (for photons). The underlying formalism uses Dirac spinors for electrons/positrons and explicit polarization four-vectors or Stokes parameters for photons, embedded in exact S-matrix elements or, for background fields, in Furry-picture Volkov or Landau states (Seipt et al., 2020, Chen et al., 2022, Kholodov et al., 2023).

For a generic process, the fully differential rate is expressed as

$$d\Gamma = \mathrm{Tr}\left[\rho_{\text{in}} \,\mathcal{M}^\dagger\,\rho_{\text{out}}\,\mathcal{M}\right]\,d\Phi,$$

where $\rho_{\text{in}}$, $\rho_{\text{out}}$ are initial/final spin/polarization density operators, $\mathcal{M}$ the QED amplitude, and $d\Phi$ the phase-space element.

Spin–polarization structure enters via:

• Spin density matrices for electrons/positrons: $\rho_e = (1+\boldsymbol{\sigma}\cdot \mathbf{n})/2$ with polarization vector $\mathbf{n}$ (Qian et al., 12 Nov 2025).
• Photon polarization density matrices: $$\rho_\gamma = (1+\mathbf{\sigma}\cdot\mathbf{\xi})/2$$ with Stokes vector $\mathbf{\xi}$, components corresponding to linear and circular polarization.
• Stokes/Mueller matrix formalism: Process probabilities are decomposed as $$P = \frac12 \mathbf{N}_{\text{out}}^\top\,\mathbf{M}\,\mathbf{N}_{\text{in}}$$, where $\mathbf{M}$ is a $4\times 4$ Mueller matrix and $\mathbf{N}$ is the Stokes (spin) 4-vector (Torgrimsson, 2020).

Loop corrections (self-energy, polarization operator) are incorporated as additional elements in the Mueller matrix, modifying both diagonal and off-diagonal (spin-transfer) elements.

2. Polarization-Resolved Processes in Strong and Weak Fields

A. Strong-Field QED

In ultraintense laser environments ($a_0\gg 1$, $I\gtrsim 10^{23}~\text{W/cm}^2$), processes are dominated by nonlinear Compton and nonlinear Breit–Wheeler mechanisms. The Locally Constant Field Approximation (LCFA) enables rendering spin- and polarization-resolved rates in analytic or semi-analytic forms, crucial for implementation in QED-PIC and kinetic codes (Seipt et al., 2020, Chen et al., 2022, Wan et al., 2023, Qian et al., 12 Nov 2025).

The LCFA double-differential rates for Compton scattering are: $$\frac{d^2 W_{fi}}{du\,dt} = \frac{W_0}{2}\left[F_0 + \xi_1 F_1 + \xi_2 F_2 + \xi_3 F_3\right],$$ where $F_i$ are Bessel/Airy function combinations, $\xi_j$ the photon Stokes parameters, and $S_{i,f}$ the electron spin vectors (Chen et al., 2022). Analogous structures hold for pair production.

Correlation structure: The final spin–photon polarization distribution is not separable; joint probabilities for electron spin and photon polarization involve interference terms (e.g., $F_2$ encodes spin–circular polarization helicity transfer).

Spin–flip and non-flip transitions are resolved via decomposition of rates, enabling predictions for final-state spin and polarization as functions of laser field geometry, pulse profile, and initial state.

B. Weak/Moderate Fields and Collider Regimes

In QED processes at moderate energies or in vacuum/weak background, such as electron-nucleus elastic scattering, photoabsorption, or $e^+e^-$ pair production, the spin- and polarization-resolved amplitudes acquire additional structure:

• Elastic lepton-nucleus scattering: The "beam-normal spin asymmetry" $A_n$ (Sherman function) quantifies parity-conserving spin-flip interference, sensitive to two-photon exchange, radiative loop corrections (vacuum polarization and vertex + self-energy), and nuclear response.

$$A_n = A_n^{\rm Born} + \Delta A_n^{\rm disp} + \Delta A_n^{\rm VP} + \Delta A_n^{\rm SE+V},$$

with explicit nonperturbative Dirac equations used for partial-wave expansions (Jakubassa-Amundsen, 2024).

• Entangled pair production and QED Bell tests: In $\gamma\gamma\to e^+e^-$, joint probabilities for spin-resolved final states depend nontrivially on the beam energy and initial photon polarization, with clear violations of Bell inequalities at all energies. The QED amplitude does not reduce to naive spin addition; "sum-angle" terms and energy dependence (e.g., in $P_{\rm lin}[\theta_+,\theta_-;\omega]$) are present (Yongram, 2011, Yongram et al., 2013).
• Resonances in strong $B$: In a strong magnetic field, spin–polarization resolved cross sections factorize in the Breit–Wigner form at cyclotron resonance, with spin-resolved numerator originating from first-order (synchrotron, one-photon pair-creation) spin-resolved rates (Kholodov et al., 2023).

3. Quantum Kinetic, Transport, and Cascade Phenomena

When systems are sufficiently dense or under non-equilibrium conditions, a quantum kinetic equation approach is required. The Wigner-function or density-matrix formalism is extended to vector/axial sectors (spin-resolved) and includes QED-type collision terms (Seipt et al., 2023, Fang et al., 2022). For instance:

• The axial (spin) distribution $f_A$ for massless electrons obeys: $$(p\cdot \partial) f_A + \hbar\,\partial_\mu [S^{\mu\nu}_{(u)}(p)]\,\partial_{p_\nu} f_V = \mathcal{C}_A^{\text{HTL}}[f_V, f_A],$$ with $\mathcal{C}_A$ incorporating leading-log quantum corrections and encoding spin polarization driven by spacetime gradients (vorticity, chemical potential, shear) (Fang et al., 2022).
• Radiative polarization, anomalous precession, and quantum RR: Quantum corrections—radiative emission, loop (self-energy) effects—feed into both the radiation-reaction force and spin torque equations. At $\mathcal{O}(\alpha)$, both direct spin-flip transitions and anomalous precession (QED $g-2$ effects in background fields) appear, requiring augmented kinetic equations (Seipt et al., 2023, Torgrimsson, 2020).
• Spin-resolved cascades: In QED cascades seeded in strong fields, inclusion of full spin and polarization effects leads to reduction of the total particle multiplication rate (due to spin-asymmetry in emission/pair creation probabilities) and leads to significant polarization in both leptons and photons—sometimes at odds in high-energy tails with Sokolov–Ternov expectations due to "spin-straggling" (Seipt et al., 2020).

4. Monte Carlo, PIC Simulation Realizations, and Algorithmic Structures

Spin- and polarization-resolved QED processes are implemented in QED-Monte-Carlo and QED-PIC codes using precomputed LCFA expressions and stochastic sampling algorithms:

• Sampling steps: (1) Draw emitted photon energy fraction, (2) conditional sampling of final state spin (via expectation axis method, aligning with $g/w$ vector), (3) sample photon polarization $\xi$ (Qian et al., 12 Nov 2025, Chen et al., 2022).
• Density matrix tracking: Each particle carries a polarization vector (electron/positron spin or photon Stokes vector), advanced between QED events via T-BMT precession, and updated stochastically on emission or absorption (Wan et al., 2023, Qian et al., 12 Nov 2025).
• Code validation: Match with analytic rates for fixed $\chi$ (K-function spectra), comparison against Geant4 (for bremsstrahlung), published cascade and helicity-transfer scenarios, and cross-code benchmarks (Qian et al., 12 Nov 2025).

Limitations derive from the LCFA's breakdown at small $\chi$ or short pulses, approximate two-step conditional sampling, and collinear emission assumptions; photon polarization memory beyond single emissions (e.g., quantum birefringence) is typically neglected, though extensions exist (Torgrimsson, 2020).

5. Entanglement, Bell Inequalities, and Quantum Information Aspects

Spin- and polarization-resolved QED processes naturally generate entanglement between different degrees of freedom, with measures such as concurrence and tangle computed from reduced density matrices (Tanaka et al., 26 Feb 2025, Yongram et al., 2013, Yongram, 2011). Angular geometry, field configuration, and electronic structure modify the degree and structure of entanglement:

• Explicit violation of classical bounds: For $\gamma\gamma\to e^+e^-$, the joint spin probability $P_{\mathrm{lin}}(\theta_+,\theta_-;\omega)$ departs from naive "singlet spin" addition, showing CHSH $|S|>2$ at all energies—an entirely relativistic QED prediction.
• Condensed matter and cavity systems: In photoemission and X-ray emission (e.g., XEPECS of Ti$_2$O$_3$), entanglement between electron spin and X-ray polarization is sensitive to local crystal field, hybridization, and geometry (Tanaka et al., 26 Feb 2025), while in two-mode cavity QED, spin–photon mapping provides high-fidelity projective readout and quantum network node capabilities (Eto et al., 2010).

6. Experimental Implications and Open Challenges

Quantitative, spin–polarization-resolved QED calculations underpin the design and interpretation of diverse high-intensity laser-plasma, relativistic beam, heavy-ion, and condensed matter experiments:

• Polarization- and spin-resolved signatures are central for predicting and analyzing strong-field QED phenomena at ELI, XFEL, LUXE, FACET II, and future colliders.
• Entanglement and nonlocality tests: Direct measurement of CHSH violations in relativistic QED, as well as the angular/energy dependence of polarization correlations, test quantum field theory foundations, and differentiate from local hidden variable models (Yongram, 2011, Yongram et al., 2013).
• Unresolved theory–experiment discrepancies (e.g., spin asymmetry in high-energy elastic scattering) indicate the importance of including higher-order QED effects and hadronic/nuclear excitations beyond the two-photon plus one-loop level (Jakubassa-Amundsen, 2024).

Further development of beyond-LCFA theoretical methods, improved sampling algorithms for higher-order correlation, and comprehensive treatments of multiparticle entanglement remain essential for the next generation of spin- and polarization-resolved QED studies.