Abstract
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.
Highlights
Quantum relativistic treatment of plasmas are of interest in several different contexts [1,2,3]
We have studied the DHW formalism in the 1D electrostatic limit
For this case, the 16 scalar equations of the general theory can be reduced to four scalar equations given in (25), which only needs to be complemented by Ampere’s law (26)
Summary
Quantum relativistic treatment of plasmas are of interest in several different contexts [1,2,3]. A particular phenomena of much interest is electron-positron pair production [12,13,14,15,16,17,18,19], that has received much attention since this interesting process might eventually be viable in the laboratory. Simplified quantum relativistic models of plasmas have been presented by, e.g., Refs. The simplified system is used to derive a dispersion relation for Langmuir waves, demonstrating that wave-particle interaction with the quantum vacuum is possible, leading to electron-positron pair creation. The reduced electrostatic equations are used to study the influence of perpendicular momentum (perpendicular referring to the direction of the electric field) on the process of pair production in vacuum. We present our main conclusions and provide an outlook for future work
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