Effective-action approach to wave propagation in scalar QED plasmas

Yuan Shi, Nathaniel J. Fisch, and Hong Qin

Phys. Rev. A 94, 012124 – Published 29 July 2016

Abstract

A relativistic quantum field theory with nontrivial background fields is developed and applied to study waves in plasmas. The effective action of the electromagnetic 4-potential is calculated ab initio from the standard action of scalar QED using path integrals. The resultant effective action is gauge invariant and contains nonlocal interactions, from which gauge bosons acquire masses without breaking the local gauge symmetry. To demonstrate how the general theory can be applied, we give two examples: a cold unmagnetized plasma and a cold uniformly magnetized plasma. Using these two examples, we show that all linear waves well known in classical plasma physics can be recovered from relativistic quantum results when taking the classical limit. In the opposite limit, classical wave dispersion relations are modified substantially. In unmagnetized plasmas, longitudinal waves propagate with nonzero group velocities even when plasmas are cold. In magnetized plasmas, anharmonically spaced Bernstein waves persist even when plasmas are cold. These waves account for cyclotron absorption features observed in spectra of x-ray pulsars. Moreover, cutoff frequencies of the two nondegenerate electromagnetic waves are red-shifted by different amounts. These corrections need to be taken into account in order to correctly interpret diagnostic results in laser plasma experiments.

Received 13 March 2016

DOI:https://doi.org/10.1103/PhysRevA.94.012124

Physics Subject Headings (PhySH)

Bernstein plasma waves Bose gases Electromagnetic waves & oscillations Electron-cyclotron plasma waves Electrostatic waves & oscillations Gauge theories Laser-induced magnetic fields in plasmas Plasma waves Quantum field theory Strong-field-induced spectra X ray astronomy

Plasma PhysicsInterdisciplinary PhysicsGeneral PhysicsParticles & FieldsAtomic, Molecular & OpticalGravitation, Cosmology & Astrophysics

Authors & Affiliations

Yuan Shi^1,2,*, Nathaniel J. Fisch^1,2, and Hong Qin^1,2,3

¹Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544, USA
²Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA
³School of Nuclear Science and Technology, University of Science and Technology of China, Hefei, Anhui 230026, China

^*Corresponding author: yshi@pppl.gov

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Vol. 94, Iss. 1 — July 2016

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Images

Figure 1
Wave dispersion relations in a cold, unmagnetized, quasineutral, spinless, electron-ion plasma. The wave vector and the wave frequency are normalized to the plasma frequency. For various effects to be visible on the scale of this figure, parameters used for making this plot are $2 e^{2} / 3 {(4 π)}^{2} ε_{0} ℏ c = 20, ω_{p e} ℏ / m_{e} c^{2} = 0.2$ , and $m_{i} / m_{e} = 3$ . The solid black curves are the one-loop dispersion relations. The solid red curves are the dispersion relations that ignore the vacuum polarization. The dashed black curves are wave dispersion relations in a classical plasma. The upper curves are the electromagnetic waves, the middle curves are the Langmuir waves, the bottom curves are the ion acoustic waves, and the dashed gray line across the diagonal represents the light cone. Notice that near the light cone, wave dispersion relations in the relativistic quantum plasma asymptote to those in the classical plasma. While away from the light cone, they can differ substantially.
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Figure 2
Perpendicular wave dispersion relations in a magnetized, cold, spinless electron gas with immobile ions as neutralizing background. The wave vector and the wave frequency are normalized to the electron gyrofrequency. Parameters used for making this plot are $ω_{p e} / | Ω_{e} | = 0.7$ and $| Ω_{e} | ℏ / m_{e} c^{2} = 0.1$ . The solid curves are waves in a relativistic quantum plasma and the dotted curves are corresponding waves in a classical plasma. The black curves are the ordinary electromagnetic waves. The blue curves are the extraordinary electromagnetic waves hybridized with cyclotron resonances. The dashed gray line across the diagonal represents the light cone. Notice that near the light cone, wave dispersion relations in the relativistic quantum plasma asymptote to those in the classical plasma. While the classical dispersion relations only capture the upper-hybrid resonance $ω_{UH}$ , the relativistic quantum dispersion relations capture all cyclotron resonances. Notice that gaps between Bernstein waves remain open even when the relativistic quantum plasma is cold. Also notice that cyclotron resonances are not harmonically spaced. The fifth resonance occurs near $4 Ω$ instead of $5 Ω$ in this example.
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Figure 3
Parallel wave dispersion relations in a cold, magnetized, quasineutral, spinless electron-ion plasma. The wave vector and the wave frequency are normalized to the electron gyrofrequency. For various effects to be visible on the scale of this figure, parameters used for making this plot are $ω_{p e} / | Ω_{e} | = 0.7, | Ω_{e} | ℏ / m_{e} c^{2} = 0.1$ , and $m_{i} / m_{e} = 3$ . The solid curves are waves in a relativistic quantum plasma, and the dotted curves are corresponding waves in a classical plasma. The black and blue curves are the right- and left-handed circularly polarized electromagnetic waves, respectively. The red curves are nearly horizontal and they are the longitudinal electrostatic waves, which include a gapped Langmuir wave (upper) and a gapless acoustic wave (lower). The dashed gray line across the diagonal represents the light cone. Notice that near the light cone, wave dispersion relations in the relativistic quantum plasma asymptote to those in the classical plasma. While away from the light cone, they can differ substantially.
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Figure 4
(a) Faraday rotation per vacuum wavelength in a cold, magnetized, spinless electron gas. The wave frequency is normalized to the electron gyrofrequency. The solid and dashed black curves are Faraday rotations in a relativistic quantum plasma and a classical plasma, respectively. The red curve is their relative difference. Parameters used for making this plot are $ω_{p e} / | Ω_{e} | = 0.7$ and $| Ω_{e} | ℏ / m_{e} c^{2} = 0.1$ . Notice that near the classical cutoff $ω_{R 0}$ , Faraday rotations in the relativistic quantum plasma and the classical plasma differ significantly. (b) Region in the $n_{e} - B_{0}$ space where relativistic quantum corrections are important. The regions above the solid, dashed, and dotted black contours are regions where $δ > 100 %, 10 %$ , and $1 %$ , respectively. The classical cutoff frequency $ω_{R 0} = 10$ , 1, and $0.1$ eV along the blue, red, and gray contours, respectively. These two sets of contours combined can be used to determine how important relativistic quantum corrections are in a given situation. See text for further details.
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