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Vacuum polarization and pair production in time-dependent electric fields: A quantum-kinetic-equation approach

I. A. Aleksandrov, V. A. Bokhan, A. I. Baksheev, A. Kudlis

TL;DR

This work develops a nonperturbative quantum-kinetic-equation framework to study vacuum polarization and $e^+e^-$ pair production in spatially homogeneous, time-dependent electric fields of arbitrary polarization. By constructing an adiabatic-basis, the authors derive a closed set of ten real QKEs for $f$, $\mathbf f$, $\mathbf u$, and $\mathbf v$ that track particle densities, spin, and vacuum polarization, and they relate these to the Dirac-Heisenberg-Wigner (DHW) formalism. They compute momentum-resolved spectra and vacuum-induced expectation values such as the electron-positron current, the energy-momentum tensor, and the angular-momentum tensor, and address ultraviolet logarithmic divergences through two consistent renormalization schemes—the one-potential subtraction and Pauli-Villars large-mass subtraction—ensuring gauge invariance and compatibility with DHW results. The framework yields finite, gauge-invariant expressions for observables, with explicit reduction to linear polarization, providing a robust basis for quantitative studies of nonperturbative QED in strong, time-dependent fields and enabling applications to realistic laser-driven scenarios.

Abstract

The evolution of the vacuum state in a time-dependent external electric field of arbitrary polarization is investigated within a nonperturbative framework of quantum kinetic equations (QKEs). In our previous work [Phys. Rev. Res. 6, 043009 (2024)], a revised version of the QKEs was derived by using an adiabatic basis constructed from one-particle Hamiltonian eigenfunctions in a spatially homogeneous electric field. In this study, we present an extensive analysis of these equations with particular emphasis on observable quantities. Specifically, we compute momentum-resolved particle yields, the induced electron-positron current, the energy-momentum tensor, and the angular-momentum tensor. We also discuss in detail the charge-renormalization procedure required to remove logarithmic divergences. It is shown that our results are consistent with the previous findings obtained via the Dirac-Heisenberg-Wigner formalism. Our analysis provides a firmer theoretical basis for investigations of nonperturbative effects in strong electric fields.

Vacuum polarization and pair production in time-dependent electric fields: A quantum-kinetic-equation approach

TL;DR

This work develops a nonperturbative quantum-kinetic-equation framework to study vacuum polarization and pair production in spatially homogeneous, time-dependent electric fields of arbitrary polarization. By constructing an adiabatic-basis, the authors derive a closed set of ten real QKEs for , , , and that track particle densities, spin, and vacuum polarization, and they relate these to the Dirac-Heisenberg-Wigner (DHW) formalism. They compute momentum-resolved spectra and vacuum-induced expectation values such as the electron-positron current, the energy-momentum tensor, and the angular-momentum tensor, and address ultraviolet logarithmic divergences through two consistent renormalization schemes—the one-potential subtraction and Pauli-Villars large-mass subtraction—ensuring gauge invariance and compatibility with DHW results. The framework yields finite, gauge-invariant expressions for observables, with explicit reduction to linear polarization, providing a robust basis for quantitative studies of nonperturbative QED in strong, time-dependent fields and enabling applications to realistic laser-driven scenarios.

Abstract

The evolution of the vacuum state in a time-dependent external electric field of arbitrary polarization is investigated within a nonperturbative framework of quantum kinetic equations (QKEs). In our previous work [Phys. Rev. Res. 6, 043009 (2024)], a revised version of the QKEs was derived by using an adiabatic basis constructed from one-particle Hamiltonian eigenfunctions in a spatially homogeneous electric field. In this study, we present an extensive analysis of these equations with particular emphasis on observable quantities. Specifically, we compute momentum-resolved particle yields, the induced electron-positron current, the energy-momentum tensor, and the angular-momentum tensor. We also discuss in detail the charge-renormalization procedure required to remove logarithmic divergences. It is shown that our results are consistent with the previous findings obtained via the Dirac-Heisenberg-Wigner formalism. Our analysis provides a firmer theoretical basis for investigations of nonperturbative effects in strong electric fields.
Paper Structure (21 sections, 127 equations, 1 figure)

This paper contains 21 sections, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Derivation of the QKE system
  3. General framework and notation
  4. Adiabatic correlation functions
  5. Quantum kinetic equations
  6. Perturbation theory
  7. Unrenormalized current, energy-momentum tensor, and angular-momentum tensor
  8. Electron-positron current
  9. Energy-momentum tensor
  10. Angular-momentum tensor
  11. Summary. Physical interpretation of the QKE functions
  12. Large-momentum behavior and ultraviolet divergences
  13. Renormalization
  14. One-potential contribution to the current density and subtraction term
  15. Current density via large-mass subtraction
  16. ...and 6 more sections

Figures (1)

  • Figure 1: Series representation of the one-loop current density \ref{['eq:j_mean']}. The interaction with the external background is viewed as a one-photon exchange between the vacuum fermionic loop (black) and the classical external current $\mathbf{j}_\text{ext} (t) = -\dot{\mathbf{E}} (t)$ (red small circles). The first diagram in the right-hand side encodes the one-potential contribution $\mathbf{j}^{(\text{1P})} (t) = \mathbf{j}^{(\text{1P})}_\text{pol} (t)$. The diagrams with an even number of the external-field legs vanish according to Furry's theorem.