Intrinsic Nonlocality of Spin- and Polarization-Resolved Probabilities in Strong-Field Quantum Electrodynamics

Abstract

Spin and polarization are central to precision tests of fundamental physics and for interpreting radiation from astrophysical sources and ultraintense laser-matter experiments. Predictive modeling therefore requires not only energy spectra, but also angle-, spin-, and polarization-resolved particle distributions. Here, we demonstrate that a key assumption underlying current strong-field quantum electrodynamics (QED) models, i.e., that emission can be treated as an instantaneous random event sampled from a local differential rate, breaks down once emission angles, electron spin, and/or photon polarization are resolved. Namely, the resulting fully differential distribution can deviate strongly from the true result and can even yield inconsistent probabilities that take negative values. The physical reason is simple: a photon emission probability builds up over a finite length of the electron trajectory, the formation region, during which the electron direction changes by roughly the same small angle that defines the radiation cone. We therefore integrate over this formation region analytically to obtain a physically consistent electron spin and photon polarization model whose implementation is compatible with existing Monte Carlo and particle-in-cell (PIC) workflows. Simulations of a GeV-class electron-laser collision and of emission in a pulsar-like magnetic field reveal spin and polarization patterns that differ even qualitatively from state-of-the-art local models. In particular, our new model predicts substantial angle-dependent circular photon polarization where the standard approach yields none, and a pronounced helicity bias in the recoiling electrons absent from current predictions. These findings have direct implications for upcoming strong-field QED experiments and for interpreting polarized radiation from extreme astrophysical environments.

Figures (9)

• Figure 1: Panel a: Graphical illustration of the inequality in Eq. \ref{['diseq_3']}. The blue curve shows $z\,\,\text{Ai}(z)/\,\text{Ai}^\prime(z)$, while the orange horizontal line indicates a representative value of the constant $\sigma^\prime C_3$. The condition in Eq. \ref{['diseq_3']} is satisfied where the blue curve lies below the orange line. Panel b: $\bar{T}^{-1,+1}(\phi_+)$ (blue) and $\bar{T}^{+1,+1}(\phi_+)$ (orange) as functions of $\phi_+$, each normalized to its respective maximum. Both functions become negative over finite intervals of $\phi_+$, illustrating that $d P^{(\sigma,\sigma^\prime)}_{\mathrm{NCS}}/(d\phi_+\, d\omega\, d\theta\, d\varphi)$ is not positive definite. See the text for details.
• Figure 2: Angular coordinate system used to specify the photon emission direction $\bm{n}$ with respect to the electron velocity unit vector $\hat{\bm{v}}$. The polar angle $\theta$ is defined as the angle between $\bm{n}$ and $\hat{\bm{v}}$. Let $\bm{s}$ be the unit vector along the electron’s transverse (with respect to $\hat{\bm{v}}$) acceleration, and define $\bm{b}=\hat{\bm{v}}\times\bm{s}$. The azimuthal angle $\varphi$ is measured in the plane orthogonal to $\hat{\bm{v}}$, spanned by $(\bm{s},\bm{b})$, from $\bm{s}$ to the projection of $\bm{n}$ onto that plane.
• Figure 3: Distribution of admissible (green) and inadmissible (red) absolute values of the mean outgoing electron spin $\langle\bm{\eta}^{\,\prime}\rangle$ (panels a, c) and of the photon Stokes vector $\langle\bm{\xi}\rangle$ (panels b, d) versus the photon-to-electron energy ratio $\omega/\varepsilon$ and the photon emission angles $\theta$ and $\varphi$. An incoming electron has energy $\varepsilon=1~\mathrm{GeV}$ in a magnetic field $\bm{B}$ with velocity perpendicular to $\bm{B}$ and with spin vector $\bm{\eta}$ anti-aligned with $\bm{B}$ and quantum parameter $\chi_e=2$. We simulate $10^7$ events by sampling $\omega$ uniformly in $(0,\varepsilon)$ and $(\theta,\varphi)$ from the distribution $dP^{(\eta)}_{\mathrm{NCS}}/(dt\,d\omega\,d\theta\,d\varphi)$ obtained from Eq. \ref{['spa_res_2']} after summing over the final spin and polarization states. For each event, $\langle\bm{\eta}^{\,\prime}\rangle$ and $\langle\bm{\xi}\rangle$ are computed from Eqs. \ref{['final_spin_angles_def']} and \ref{['final_polarization_angles_def']}. Markers are green when $|\langle\bm{\eta}^{\,\prime}\rangle|\le 1$ (panels a, c) or $|\langle\bm{\xi}\rangle|\le 1$ (panels b, d), and red otherwise; red markers thus indicate events for which the local model implies a negative inferred probability, in direct violation of its probabilistic interpretation.
• Figure 4: Spin and polarization curves for NCS of an ultrarelativistic electron ($\varepsilon = 1$ GeV, $\chi_e = 2$) for five different initial spin states (purple: $1$; red: $0.5$; green: $0$; orange: $-0.5$; blue: $-1$) in a uniform magnetic field $\bm{B} = (0, B_y, 0)$ versus the photon-to-electron energy ratio $\omega/\varepsilon$. Panels a, b: mean outgoing electron spin component $\langle \eta^{\prime 2}\rangle$ (along $\bm{B}$). Panels c, d: mean photon Stokes parameter $\langle \xi^{3}\rangle$. In panel a and c colored curves are obtained by averaging the spin and polarization inferred from Eqs. \ref{['final_spin_angles_def']} and \ref{['final_polarization_angles_def']}, respectively, over $10^4$ sampled emission angles at each $\omega/\varepsilon$. In panels b, d colored curves show the same averages after discarding events with $|\bm{\eta}^{\,\prime}|>1$ or $|\bm{\xi}|>1$. Black dashed curves show the corresponding angle-integrated prediction from Eq. \ref{['final_spin_angles_def_u']} (panels a, b) and Eq. \ref{['final_polarization_angles_def_u']} (panels c, d).
• Figure 5: Spin and polarization curves for NCS of an ultrarelativistic electron ($\varepsilon = 1$ GeV, $\chi_e = 2$) for five different initial spin states (purple: $1$; red: $0.5$; green: $0$; orange: $-0.5$; blue: $-1$) in a uniform magnetic field $\bm{B} = (0, B_y, 0)$ versus the photon-to-electron energy ratio $\omega/\varepsilon$. Panel a: mean outgoing electron spin component $\langle \eta^{\prime 2}\rangle$ (along $\bm{B}$) according to Eq. \ref{['legitimate_SPA']}. Panel b: mean photon Stokes parameter $\langle \xi^{3}\rangle$ according to Eq. \ref{['legitimate_PPA']}. Black dashed curves show the corresponding local- angle-integrated prediction according to Eq. \ref{['final_spin_angles_def_u']} (panel a) and Eq. \ref{['final_polarization_angles_def_u']} (panel b).