Eigenmodes in an ultra-relativistic ultra-magnetized pair QED-plasma

TL;DR

This paper extends QED-plasma theory to ultrarelativistic, nonneutral electron–positron plasmas in ultra-strong magnetic fields, deriving dispersion relations for normal modes within a framework that includes vacuum nonlinearities and a relativistic temperature distribution. It shows that QED effects primarily renormalize the plasma frequency via $\omega_{p*}^2=\omega_p^2/(1-C_δ)$ and introduce field-dependent corrections through $\alpha_ε$ and $\alpha_μ$, with $\alpha_ε$ growing roughly linearly with $B$ for $B\gg B_Q$, while $\alpha_μ$ remains small. The analysis reveals B-field–induced transparency of the O-mode, temperature-driven shifts of low-frequency cutoffs, and a largely preserved mode structure (Alfvén, fast magnetosonic, X-, O-, Langmuir) with QED modifications that are especially relevant for neutron star magnetospheres and FRB propagation. These results advance understanding of wave propagation in magnetar environments and in high-field laser-plasma contexts by clarifying how strong-field QED and ultrarelativistic temperatures reshape plasma dispersion and refractive indices.

Abstract

Ultra-relativistic quantum-electrodynamic (QED) plasmas, characterized by magnetic field strengths approaching and even exceeding the Schwinger field of approximately $B_{Q} \approx 4 \times 10^{13}$ gauss, hold significant interest for laser-plasma experiments and astrophysical observations of neutron stars and magnetars. In this study, we investigate the joint modification of normal plasma modes in ultra-relativistic electron-positron plasmas, both charge neutral and non-neutral, by the super-strong magnetic field and plasma relativistic temperature. Our analysis shows that the most substantial modification concerns the reduction of the plasma frequency cutoff, resulting in relativistic and field-induced transparency. Additionally, we observe a temperature-independent modification of the index of refraction of electromagnetic waves, which coincides with the behavior observed in a cold QED plasma.

Figures (9)

• Figure 1: Regions approximately corresponding to Case I (light-blue) and Case II (orange) are shown for two values of temperature $\Theta$ and two propagation angles $\theta$. Above the dashed black line, waves are superluminal and decoupled from Landau damping. The damping can be significant in the unshaded region below the dashed line, in-between the blue and orange regions. Case I dominates waves' dispersion, especially in the ultra-relativistic plasma.
• Figure 2: QED modification of the plasma frequency (log-linear scale, left axis) and quantities $\alpha_{\epsilon}, \alpha_{\mu}$ (log-log scale, right axis) as a function of the magnetic field strength $B/B_{Q}$.
• Figure 3: The schematic representation the plasma dispersion curves $\omega\left(k\right)$ for electrically neutral ($\Delta n/n=0$) and non-neutral ($\Delta n/n=0.8$) relativistic magnetized plasma. The units are arbitrary, but we set the speed of light to $c=1$. We set the numerical values of the plasma and cyclotron frequencies to be$\omega_p=1$, $\Omega=3$. The temperature parameter is chosen to be $\Theta=3$ for illustrative purposes only as, formally, $\Theta\gg1$ in the ultrarelativistic plasma. Both standard (dashed blue curves) and QED-modified with $B/B_Q=100$ (orange curves) branches of plasma normal modes are shown. The wave branches are labeled as follows: “A”—Alfvén wave, “F”—fast magnetosonic wave, “X”—extraordinary electromagnetic wave, “O”—ordinary electromagnetic wave(in a neutral plasma, it consists of two branches split around the cyclotron frequency), “W”—whistler wave, “Z”—Z-mode (the lower-frequency branch of the extraordinary wave, also called the slow extraordinary mode), "L"-Langmuir mode.
• Figure 4: The schematic representation of the index of refraction squared $N^2\left(\omega\right)$ (top row) and the plasma dispersion curves $\omega\left(k\right)$ (bottom row) for electrically neutral and non-neutral classical plasmas. The units are arbitrary, but we set the speed of light to $c=1$. We set the numerical values of the plasma and cyclotron frequencies to be $\omega_p=1$, $\Omega=3$, and $\theta=\pi/3$. The cold plasma case is in green, while the thermal plasma case with $\Theta=10$ is in blue. Both plots are for the non-QED case with $B/B_Q\rightarrow0$. Solid lines depict propagating waves, i.e., with $N^2>0$, and dashed lines depict evanescent branches with $N^2<0$. The wave branches are labeled as follows: “A”—Alfvén wave, “F”—fast magnetosonic wave, “X”—extraordinary electromagnetic wave, “O”—ordinary electromagnetic wave (in a neutral plasma, it consists of two branches split around the cyclotron frequency), “W”—whistler wave, “Z”—Z-mode (the lower-frequency branch of the extraordinary wave, also called the slow extraordinary mode), "L"-Langmuir mode.
• Figure 5: The schematic representation of the index of refraction squared $N^2\left(\omega\right)$ (top row) and the plasma dispersion curves $\omega\left(k\right)$ (bottom row) for electrically neutral and non-neutral QED plasmas. The units are arbitrary, but we set the speed of light to $c=1$. We set the numerical values of the plasma and cyclotron frequencies to be $\omega_p=1$, $\Omega=3$, and $\theta=\pi/3$. The wave branches are labeled as in Fig. \ref{['fig:dispersion-compare-classical']}. The cold plasma case is in purple, while the thermal plasma case with $\Theta=10$ is in orange. Both plots are for the QED case with $B/B_Q=300$. Solid lines depict propagating waves, i.e., with $N^2>0$, and dashed lines depict evanescent branches with $N^2<0$.