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Strong field physics close to or above the Schwinger limit are typically studied with vacuum as initial condition or by considering test particle dynamics. However, with a plasma present initially, quantum relativistic mechanisms such as Schwinger pair creation are complemented by classical plasma nonlinearities. In this work we use the Dirac-Heisenberg-Wigner formalism to study the interplay between classical and quantum mechanical mechanisms in the regime of ultrastrong electric fields. In particular, the effects of initial density and temperature on the plasma oscillation dynamics are determined. Finally, comparisons with competing mechanisms such as radiation reaction and Breit-Wheeler pair production are made.

As is well known, for plasmas of high density and modest temperature, the classical kinetic theory needs to be extended. Such extensions can be based on the Schrödinger Hamiltonian, applying a Wigner transform of the density matrix, in which case the Vlasov equation is replaced by the celebrated Wigner–Moyal equation. Extending the treatment to more complicated models, we investigate aspects such as spin dynamics (based on the Pauli Hamiltonian), exchange effects (using the Hartree–Fock approximation), Landau quantization, and quantum relativistic theory. In the relativistic theory, we first study cases where the field strength is well-beyond Schwinger critical field. Both weakly relativistic theory (gamma factors close to unity) and strongly relativistic theory are investigated, using assumptions that allow for a separation of electron and positron states. Finally, we study the so-called Dirac–Heisenberg–Wigner (DHW) formalism, which is a fully quantum relativistic theory, allowing for field strengths of the order of the Schwinger critical field or even larger. As a result, the quantum kinetic theory is extended to cover phenomena such as Zitterbewegung and electron–positron pair creation. While the focus of this review is on the quantum kinetic models, we illustrate the theories with various applications throughout the manuscript.

We derive a system of coupled partial differential equations for the equal-time Wigner function in an arbitrary strong electromagnetic field using the Dirac-Heisenberg-Wigner formalism. In the electrostatic limit, we present a system of four coupled partial differential equations, which are completed by Ampères law. This electrostatic system is further studied for two different cases. In the first case, we consider linearized wave propagation in a plasma accounting for the nonzero vacuum expectation values. We then derive the dispersion relation and compare it with well-known limiting cases. In the second case, we consider Schwinger pair production using the local density approximation to allow for analytical treatment. The dependence of the pair production rate on the perpendicular momentum is investigated and it turns out that the spread of the produced pairs along with perpendicular momentum depends on the strength of the applied electric field.

We present a quantum kinetic theory for spin-1/2 particles, including the spin-orbit interaction, retaining particle dispersive effects to all orders in ℏ, based on a gauge-invariant Wigner transformation. Compared to previous works, the spin-orbit interaction leads to a new term in the kinetic equation, containing both the electric and magnetic fields. Like other models with spin-orbit interactions, our model features "hidden momentum". As an example application, we calculate the dispersion relation for linear electrostatic waves in a magnetized plasma, and electromagnetic waves in a unmagnetized plasma. In the former case, we compare the Landau damping due to spin-orbit interactions to that due to the free current. We also discuss our model in relation to previously published works. 

When the Zecman energy approaches the characteristic kinetic energy of electrons, Landau quantization becomes important. In the vicinity of magnetars, the Zeeman energy can even be relativistic. We start from the Dirac equation and derive a kinetic equation for electrons, focusing on the phenomenon of Landau quantization in such ultrastrong but constant magnetic fields, neglecting short-scale quantum phenomena. It turns out that the usual relativistic gamma factor of the Vlasov equation is replaced by an energy operator, depending on the spin state, and also containing momentum derivatives. Furthermore, we show that the energy eigenstates in a magnetic field can be computed as eigenfunctions of this operator. The dispersion relation for electrostatic waves in a plasma is computed, and the significance of our results is discussed.

We have extended previous quantum kinetic results to compute the exchange correction to the electrostatic electron susceptibility for arbitrary frequencies and wavenumbers in the low temperature limit. This has allowed us to make a general comparison with a much used hydrodynamic expression, based on density functional theory, for exchange effects. For low phase velocities, as for ion-acoustic waves, wave-particle interaction leads to a strong enhancement of the exchange correction and the hydrodynamic result is smaller by an order of magnitude. The hydrodynamic expression gives a useful approximation when the phase velocity is 2.5 times the Fermi velocity. If this condition is not fulfilled, the hydrodynamical theory gives misleading results. We discuss the implications of our results for the model choice for quantum plasmas, especially regarding particle dispersive effects.

We study a recently derived fully relativistic kinetic model for spin-1/2 particles. First, the full set of conservation laws for energy, momentum, and angular momentum are given together with an expression for the (nonsymmetric) stress-energy tensor. Next, the thermodynamic equilibrium distribution is given in different limiting cases. Furthermore, we address the analytical complexity that arises when the spin and momentum eigenfunctions are coupled in linear theory by calculating the linear dispersion relation for such a case. Finally, we discuss the model and give some context by comparing with potentially relevant phenomena that are not included, such as radiation reaction and vacuum polarization.

For short wavelengths, it is well known that the linearized Wigner-Moyal equation predicts wave damping due to wave-particle interaction, where the resonant velocity shifted from the phase velocity by a velocity v(q) = hk/2m. Here h is the reduced Planck constant, k is the wavenumber and m is the electron mass. Going beyond linear theory, we find additional resonances with velocity shifts nv(q), n= 2,3, ..., giving rise to a new wave-damping mechanism that we term multi-plasmon damping, as it can be seen as the simultaneous absorption (or emission) of multiple plasmon quanta. Naturally this wave damping is not present in classical plasmas. For a temperature well below the Fermi temperature, if the linear (n = 1) resonant velocity is outside the Fermi sphere, the number of linearly resonant particles is exponentially small, while the multi-plasmon resonances can be located in the bulk of the distribution. We derive sets of evolution equations for the case of two-plasmon and three-plasmon resonances for Langmuir waves in the simplest case of a fully degenerate plasma. By solving these equations numerically for a range of wave-numbers we find the corresponding damping rates, and we compare them to results from linear theory to estimate the applicability. Finally, we discuss the effects due to a finite temperature.

In this review we give an overview of the recent work on quantum kinetic theories of plasmas. We focus, in particular, on the case where the electrons are fully degenerate. For such systems, perturbation methods using the distribution function can be problematic. Instead we present a model that considers the dynamics of the Fermi surface. The advantage of this model is that, even though the value of the distribution function can be greatly perturbed outside the equilibrium Fermi surface, deformation of the Fermi surface is small up to very large amplitudes. Next, we investigate the short-scale dynamics for which the Wigner-Moyal equation replaces the Vlasov equation. In particular, we study wave-particle interaction, and deduce that new types of wave damping can occur due to the simultaneous absorption (or emission) of multiple wave quanta. Finally, we consider exchange effects within a quantum kinetic formalism to find a model that is more accurate than those using exchange potentials from density functional theory. We deduce the exchange corrections to the dispersion relations for Langmuir and ion-acoustic waves. In comparison to results based on exchange potentials deduced from density functional theory we find that the latter models are reasonably accurate for Langmuir waves, but rather inaccurate for ion acoustic waves.

In this paper, we derive a fully relativistic kinetic theory for spin-1/2 particles and its coupling to Maxwell's equations, valid in the long-scale-length limit, where the fields vary on a scale much longer than the localization of the particles; we work to first order in (h) over bar. Our starting point is a Foldy-Wouthuysen (FW) transformation, applicable to this regime, of the Dirac Hamiltonian. We derive the corresponding evolution equation for the Wigner quasidistribution in an external electromagnetic field. Using a Lagrangian method we find expressions for the charge and current densities, expressed as free and bound parts. It is furthermore found that the velocity is nontrivially related to the momentum variable, with the difference depending on the spin and the external electromagnetic fields. This fact that has previously been discussed as "hidden momentum" and is due to that the FW transformation maps pointlike particles to particle clouds for which the prescription of minimal coupling is incorrect, as they have multipole moments. We express energy and momentum conservation for the system of particles and the electromagnetic field, and discuss our results in the context of the Abraham-Minkowski dilemma.