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Ultra-magnetized plasmas, where the magnetic field strength exceeds theSchwinger field of about $B_{Q}\approx4\times10^{13}$~gauss, become of greatscientific interest, thanks to the current advances in laser-plasma experimentsand astrophysical observations of magnetar emission. These advances demandbetter understanding of how quantum electrodynamics (QED) effects influencecollective plasma phenomena. In particular, Maxwell's equations becomenonlinear in the strong-QED regime. Here we present the `QED plasma framework'which will allow one to {\em systematically} explore collective phenomena in aQED-plasma with arbitrarily strong magnetic field. Further, we illustrate theframework by exploring low-frequency modes in the ultra-magnetized, cold,electron-positron plasmas. We demonstrate that the classical picture of fivebranches holds in the QED regime; no new eigenmodes appear. The dispersioncurves of all the modes are modified. The QED effects include the overallmodification to the plasma frequency, which becomes field-dependent. They alsomodify resonances and cutoffs of the modes, which become both field- andangle-dependent. The strongest effects are (i) the {\em field-inducedtransparency of plasma} for the O-mode via the dramatic reduction of thelow-frequency cutoff well below the plasma frequency, (ii) the {\em Alfven modesuppression} in the large-$k$ regime via the reduction of the Alfven moderesonance, and (iii) the {\em O-mode slowdown} via strong angle-dependentincrease of the index of refraction. These results should be important forunderstanding of a magnetospheric pair plasma of a magnetar and for laboratorylaser-plasma experiments in the QED regime.

Plasma modes in QED super-strong magnetic fields of magnetars and laser plasmas