Modified vacuum polarization in the presence of a plasma

TL;DR

The study addresses how vacuum polarization behaves in a plasma under strong fields by unifying vacuum and matter quantum effects within the Dirac-Heisenberg-Wigner (DHW) framework for a 1D electrostatic field. By applying weak-field and slow-time expansions, it derives a quasi-classical description of electron and positron distributions (the WCEP system) that are coupled through vacuum polarization, while recovering known vacuum responses from Euler-Heisenberg theory, including derivative corrections. The analysis shows that vacuum polarization dominates over matter-induced quantum corrections in the ultra-relativistic regime (approximately when $\gamma \gtrsim 300$), whereas matter effects are more important at lower relativistic factors, highlighting when a classical current plus vacuum polarization suffices. The results provide a quantitative criterion for when vacuum contributions must be included and lay out a framework extendable to more general field geometries and wave-particle interaction studies.

Abstract

We study vacuum polarization due to strong fields, in the presence of an electron-positron plasma. For this purpose, we expand quantum kinetic equations using weak fields and slow temporal scales as expansion parameters. It is demonstrated that the evolution of the Dirac field can be described by classical-like distribution functions for electrons and positrons, which are weakly coupled through quantum interactions. Furthermore, we deduce that these coupling terms give rise to well-known expressions for vacuum polarization, in addition to quantum modifications proportional to the content of real particles. Depending on the initial plasma density, the dominant quantum corrections to classical evolution may arise from real particle couplings or from the vacuum polarization associated with virtual particles. The implications of our results are discussed.

Figures (3)

• Figure 1: The evolution of $E(t)$ (upper panel) and $A(t)$ (lower panel) based on \ref{['Esolved']} for the initial conditions $E(t=0)=-0.01$, $A(t=0)=0$, and $(dE/dt)(t=0)=0$. $E_p=0.01$ and $\gamma_p\approx10$. The plasma density, $n_0=\int \tilde{f}(t=-T_B) \, d^2q$, was chosen as $n_0=0.0024$.
• Figure 2: The quantity $R^{vp}_T/E_p^{2}$ (black, dots) compared with the constant value $C_1$ (grey, dashed line). Upper panel: dependence on $E_p$ for fixed $\gamma_p = 5000$. Lower panel: dependence on $\gamma_p$ for fixed $E_p = 0.3$.
• Figure 3: The quantity $R^{mat}_T/n_0$ (black, dots) compared with the constant value $C_2$ (grey, dashed line). Upper panel: dependence on $E_p$ for fixed $\gamma_p = 5000$. Lower panel: dependence on $\gamma_p$ for fixed $E_p = 0.3$.