Plasma physics of extreme astrophysical environments

Astrophysical and space plasmas are complex [1]. To start with, they are almost always immersed in magnetic fields. In addition, they are often bathed in radiation fields with photons over a wide range of wavelengths across the electromagnetic spectrum. Both magnetic and radiation fields can be produced locally by the plasma itself or have an external origin (i.e., represent an external background).

Next, it is rather rare that one finds a completely quiescent, homogeneous plasma in complete mechanical and thermal equilibrium. Quite often, cosmic plasmas have some level of turbulence, i.e., a range of collective motions and structures, including eddies and various kinds of waves¹, on a broad range of scales, from the global system size down to dissipative scales (e.g., plasma microscopic scales, such as the ion Larmor radius ρ_i or the ion collisionless skin depth d_i). Turbulent motions may sometimes play an important dynamical role, i.e., exert pressure and carry energy.

Furthermore, in sufficiently collisionless plasmas, and especially if small-scale phenomena are involved, the electrons and ions (or positrons) do not have enough time to relax to a complete thermodynamic equilibrium, and hence they often have complex, multi-component distribution functions, e.g., a Maxwellian with a high-energy power-law tail. For example, in solar wind studies one often describes the plasma distribution function in terms of three components: the core, the halo, and the strahl [2]. In some extreme cases, e.g., in pulsar wind nebulae (PWNe), the thermal component is believed to be essentially absent completely and the plasma is characterized by a power-law distribution with a high-energy cutoff, e.g., [3]. Another important example of high-energy non-thermal particles present in the interstellar medium (ISM) are relativistic cosmic rays (CRs), characterized by a power-law energy distribution that extends up to 10²⁰ eV.

Finally, the chemical composition of cosmic plasmas can be quite diverse and complex. It can range from pure electron–positron plasmas at highest energies, e.g., in pulsar winds and nebulae, to usual electron–ion plasmas, to cold weakly ionized plasma with a whole host of ion and neutral species, e.g., in cold molecular clouds or in protostellar accretion disks. An important additional complication in the case of cold, partially ionized plasmas is that their proper treatment often requires taking into account a complex network of chemical and photo-chemical reactions. Furthermore, this chemistry and, especially, ionization–recombination balance, is often greatly affected by ubiquitous dust grains of different sizes. At the other end of the energy-density spectrum, nuclear chemistry and neutrino processes become important in dense and hot plasmas in central engines of supernovae (SNe) and gamma-ray bursts (GRBs).

These remarks lead us to conclude that one cannot adequately describe real space- and astrophysical plasmas with just two or three basic parameters, such as density and temperature, as it is often done in simplified textbook problems. Instead, one sees that these complex plasmas should be regarded as consisting of several separate but interpenetrating and actively interacting components or constituents. One can list them as follows: the thermal gas of electrons, ions, and perhaps positrons; a population of non-thermal (e.g., described by a truncated high-energy power-law tail) particles (e.g., CRs); a large-scale (background) magnetic field; electromagnetic radiation (photons); small-scale collective plasma waves or eddies; and finally, in certain cases, neutral atoms and molecules, and dust grains.

Importantly, all these constituents carry energy. The relative importance of the different plasma constituents in the overall energetics can, of course, vary from system to system. Interestingly, however, one often finds herself in a situation where at least several of the constituents are energetically non-negligible. The ISM in our own Galaxy is a good example (see, e.g., [4] for a review): here, remarkably, the energy densities (and hence pressures) of the thermal gas, of CRs, of the background magnetic field, of light, and of turbulence are all generally comparable to each other.

The plasma constituents exchange energy with each other through various physical processes, as illustrated in figure 1. For example, the process by which bulk kinetic energy is converted into magnetic energy is called the dynamo; the process that converts magnetic energy into the thermal, bulk kinetic, and non-thermal particle energies is magnetic reconnection; the process of conversion of large-scale bulk kinetic energy to heat and non-thermal particles is a shock. Other important energy-exchange processes include radiative processes (converting particle kinetic energy to radiation and vice versa) and turbulent dissipation. Some of these processes involve instabilities, for example: the Rayleigh–Taylor (RT) instability converts gravitational energy to bulk kinetic; the magneto-rotational instability (MRI) converts the energy associated with a large-scale differential rotation to small-scale kinetic and magnetic energy; the Kelvin–Helmholtz (KH) instability converts large-scale kinetic energy to turbulent kinetic and magnetic energy; the kink instability converts magnetic energy to bulk kinetic, etc.

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Studying the energy conversion/exchange processes between various constituents in complex astrophysical plasmas, i.e., of the various ways in which they interact with each other, is the main scope of Plasma Astrophysics.

We shall now transition from the general discussion of plasma astrophysics given in the previous section to a more focused discussion of a special sub-area that we, in this review, will call the Extreme Plasma Astrophysics. By this we shall mean plasma physics relevant to astrophysical environments with ultra-strong magnetic fields—magnetic fields approaching and exceeding the critical quantum-electrodynamical (QED) magnetic field strength

$$\begin{equation} B_* \equiv {{m_{\rm e}^2 c^3}\over{e\hbar}} \simeq 4.4\times 10^{13}\,{\rm G}. \end{equation} \tag{ 1 }$$

This critical field has a clear physical interpretation that the corresponding electron cyclotron energy ℏΩ_e is equal to the electron rest-mass energy m_ec² = 0.51 MeV. We will call magnetic fields exceeding this value, 'ultra-strong' or 'super-critical' (see section 3.1).

However, since the general spirit of this review is to look at plasma astrophysics as a study of energy exchange processes, what we here are really interested in is, in general, astrophysical environments with energy densities corresponding to that of the critical field—i.e., energy densities exceeding 10²⁶ erg cm⁻³; this also corresponds to blackbody radiation energy density (aT⁴, where a ≃ 7.57 × 10⁻¹⁵ erg cm⁻³ K⁻⁴ is the radiation constant) for a temperature of order the electron rest mass, k_BT ∼ m_ec². For the purposes of this review, we will call astrophysical environments with this level of energy densities 'extreme astrophysical environments'. Our main focus will be on plasma-physical aspects of such ultra-magnetized, and often relativistically hot and radiation-dominated plasmas. The character of the energy exchange processes discussed in the previous section may be significantly altered in these environments.

We would like to remark that, in addition to its direct astrophysical applications described below, this new frontier of plasma astrophysics is also interesting from the fundamental-physics point of view: it lies at the intersection of plasma physics, quantum field theory (namely, QED), high-energy-density (HED) physics, and high-energy astrophysics. A rich variety of exotic physical effects come into play in these extreme plasmas, making this inter-disciplinary new direction intellectually challenging and attractive.

This review is structured as follows. In section 2 we will describe various astrophysical contexts in which extreme, ultra-magnetized plasma environments are encountered: (1) interiors of neutron stars (NSs) (section 2.1); (2) magnetospheres of magnetars (section 2.2); (3) central engines of SNe and GRBs (section 2.3). The basic theoretical framework for various models of extreme plasmas is described in section 3. This section is primarily devoted to an exposition of quantum plasma physics. In particular, in section 3.1 we discuss the basic physics of super-critical magnetic fields. In section 3.2 we discuss non-relativistic quantum plasma physics, including the fundamental theoretical formalism for quantum kinetic theory (section 3.2.1), its application to linear waves and instabilities (section 3.2.2), quantum fluid theory (section 3.2.3), and spin quantum plasmas (section 3.2.4). In section 3.3 we describe the recent advances in relativistic quantum plasma (RQP) physics, namely, for spinless Klein–Gordon (KG) plasmas (section 3.3.1), for spin-1/2 Dirac fermion plasmas (section 3.3.2), and finally, for relativistic quantum hydrodynamics (QHD) (section 3.3.3). In section 4 we turn to astrophysical applications of QED plasma physics and discuss several examples: propagation of high-energy radiation through magnetar magnetospheres (section 4.1), large-scale magnetohydrodynamic (MHD) processes (section 4.2), thermodynamics of extreme astrophysical environments (section 4.3), interaction between magnetospheric currents and a magnetar surface (section 4.4), and reconnection of ultra-strong magnetic fields in the context of magnetar flares and GRB central engines (section 4.5). A summary of the review and concluding remarks are presented in section 5.

NSs are one of the three types, along with white dwarfs (WDs) and black holes (BHs), of compact objects that remain in the aftermath of the death of main-sequence stars (see [5] for review). Like BHs, NSs are born in SN explosions marking the death of massive stars (i.e., stars with initial zero-age main sequence (ZAMS) masses greater than roughly 8 solar masses).

NSs typically have masses between 1 and 2 solar masses, with most of the masses clustered around the Chandrasekhar mass of M_Ch ≃ 1.4 M_⊙, where M_⊙ ≃ 2 × 10³³ g is the solar mass. The most striking characteristics of NSs are perhaps their very small radii, of order 10 km, just a few times larger than the corresponding Schwarzschild radius, and, correspondingly, their extremely high central densities, comparable to the density of the nuclear matter, ρ_nuc ≃ 3 × 10¹⁴ g cm⁻³. In essence, the interior of a NS can be seen as just one giant nucleus. As the NS name suggests, the composition of the interior is dominated by neutrons; they comprise more than 90% of NS mass and their degeneracy pressure supports the star against gravitational collapse. In addition, there is a small admixture (a few percent) of protons and electrons. NSs are essentially macroscopic quantum objects: their interiors are believed to be superconducting and superfluid. Another unique feature that distinguishes NSs from all other stellar-mass celestial objects is that, with the exception of newly born NSs, they are believed to be covered with a solid crust, about a kilometer thick, composed mainly of iron with a density of order 10⁶ g cm⁻³.

Since NSs represent the end-state of stellar evolution for a substantial fraction of the stellar population (namely, stars with ZAMS masses between 8 and about 20–25 M_⊙), there should be literally billions of old NSs flying around in our Galaxy and its halo². However, because they are so compact, they are very dim in the optical and so are very difficult to detect. Observationally, the main classes of NSs are the following:

• (1)  
  classical radio-pulsars, the first kind of NSs discovered, usually found by their pulsed periodic radio emission; currently there are almost 2000 of them known;
• (2)  
  accreting NSs in galactic x-ray binaries (XRBs), usually discovered as powerful x-ray sources (about 5% of the total known NS population), including x-ray pulsars and x-ray bursters;
• (3)  
  magnetars—NSs with magnetic fields of 10¹⁴–10¹⁵ G, identified observationally as anomalous x-ray pulsars (AXPs) and soft gamma repeaters (SGRs), see section 2.2.

There are just a few known magnetars in our galaxy and the magellanic clouds, but they are believed to comprise about 10% of the entire NS population at birth, e.g., [6–8].

An outstanding plasma astrophysics problem in the context of NSs has been the origin and evolution of their magnetic fields. It is remarkable that, despite the very narrow spread of NS masses and sizes, they seem to exhibit a huge spread in their (dipole) magnetic field strengths. The typical magnetic fields inferred in most regular NSs, e.g., radio- and x-ray pulsars, are of the order of 10¹² G, but there are also a large number of NSs with much weaker fields, perhaps as low as 10⁹ G and lower, in old systems such as recycled millisecond radio-pulsars and type-I x-ray bursters in low-mass XRBs (LMXBs). On the strong-field side, there are magnetars—NS with estimated magnetic fields of order 10¹⁵ G. Explaining this great diversity of NS magnetic fields is an important problem in modern astrophysics.

Some of this diversity may reflect a possible wide range of the field strengths that NSs are born with, but some probably may be attributed to magnetic field decay later in the NS's life. It is important to understand the role of both of these stages. Thus, the overall problem of explaining magnetic field distribution in NSs naturally splits into two rather independent sub-problems. The first one is the problem of establishing the initial distribution of NSs with respect to their magnetic fields, i.e., the initial NS magnetic field function [9]. The second problem is that of long-term evolution of the magnetic field in old NSs.

The first problem can be called the NS magnetogenesis problem: it aims at understanding the processes generating the initial magnetic field in NSs just as they are born in a core-collapse SN explosion. This problem is inextricably related to the problem of understanding the mechanisms of SN explosions (see section 2.3). There is a large number of astrophysically important questions that one would like to address: why are some (perhaps 10%) NSs (namely, magnetars) endowed with ultra-strong (10¹⁴–10¹⁵ G) magnetic fields at birth, whereas most NSs have much weaker (10¹² G) fields? Are there NSs with very weak initial magnetic fields, e.g., of order 10⁹–10¹⁰ G? What basic parameters of the progenitor massive star (e.g., its mass, rotation rate, metallicity, etc.) influence or control the large-scale magnetic field with which a newly born NS emerges after the SN explosion clears? How strong is this influence, i.e., is the resulting initial magnetic field essentially predetermined by the above progenitor's parameters or is it largely a random quantity? Is there any connection between NS fields and natal kicks?

The NS magnetogenesis problem is very difficult to address observationally—we simply do not have observational tools to measure the initial NS magnetic field distribution function. However, there have been a significant number of theoretical papers devoted to this subject in the past couple of decades, in particular, in relation to the origin of magnetars. It is generally believed, to a large degree by analogy with the solar magnetic field, that a NS's magnetic field is produced by a large-scale α Ω MHD dynamo in a rapidly rotating convective proto-NS in the first few seconds of its evolution [9–11]. Turbulent dynamo action is usually thought to require two essential ingredients: turbulence (driven by, e.g., thermal convection or by some other instability) and a large-scale (differential) rotation. Both of these conditions are likely to be present in young proto-NSs. First, in the first few seconds of their evolution, NSs are indeed believed to experience vigorous convection driven by neutrino cooling (e.g., [12]). In addition, they may become turbulent as a result of some active MHD instability, such as the MRI [13–15] or the Tayler–Spruit process [16–18]. As for rotation—the second important ingredient needed for dynamo—proto-NSs can indeed be rapid rotators: since the gravitational collapse of a WD-like progenitor core is very rapid (a couple hundred milliseconds), angular momentum is conserved during this process and so the emerging NS rotation rate is basically determined by the rotation rate of the core. For example, as was suggested by Duncan and Thompson [10], for a newly-born NS to develop a large-scale magnetic field of order 10¹⁵ G, it needs to have an initial rotation period of order a millisecond³. This in turn requires, by angular momentum conservation, the progenitor core to have a rotation period of just a few seconds. This argument illustrates an important link that ties the NS rotation, and hence, presumably, its large-scale magnetic field and its potential as a GRB central engine (see section 2.3), to the parameters of the progenitor star. As was suggested by Heger et al [19], such a rapid progenitor core rotation rate is actually very difficult to maintain during the preceding evolution phase of the progenitor, due to angular momentum losses to the outer stellar envelope. This implies that the birth of a NS with a millisecond spin period must be a rare event. (Note that there is a class of NSs that have millisecond periods (so-called recycled millisecond pulsars, see below) but these are probably not their initial periods—these are old, weak magnetic field NSs in binary systems that have been spun up by accretion.)

One important factor that hampers our understanding of magnetic field generation in proto-NSs is that we still do not have a reliable predictive theory of MHD dynamo. While significant progress in reproducing key features of the solar dynamo has been achieved with the help of numerical simulations in the past few years (e.g., [20, 21]), this success is yet to be extended to the NS magnetogenesis problem. Thus, the question of how the neutrino-convection-driven large-scale MHD dynamo operates at extreme densities and magnetic fields in a rapidly cooling (by neutrino emission) hot proto-NS during the first few seconds of its life remains an open frontier of NS research.

The second important aspect of the NS magnetic field distribution problem is the problem of long-term evolution of the magnetic field, in particular, their decay over time. Two important astrophysical contexts in which this problem arises are: (1) the decay of magnetic field in accreting NSs in LMXBs and the origin of recycled millisecond pulsars, and (2) the decay of magnetic field in magnetars on 10²–10⁴ year time scale and the resulting cessation of magnetar activity.

Long-term magnetic field decay in normal isolated NSs due to processes such as ohmic dissipation, ambipolar diffusion, and Hall drift, and its effect on the NS thermal evolution has been considered in a number of studies [22–29]. In addition, the problem of magnetic field decay/burial in accreting NSs has attracted a lot of attention and has been the subject of a significant research effort in recent years due to its importance for understanding the origin of recycled millisecond radio-pulsars (see [30, 31] for reviews). These are relatively old NSs found in some binary systems, which have presumably been spun up to millisecond periods by accretion sometime in the past. Maintaining such a rapid rotation rate over an extended period of time, however, implies that the standard magnetospheric spin-down mechanism is very weak in these systems. In particular, the observed very slow spin-down rates indicate large-scale dipolar magnetic fields as low as 10⁸–10⁹ G, significantly lower than the standard 10¹² G fields of 'normal' isolated pulsars. One popular explanation for this is an effective 'burial' of the NS magnetic field during the accretion stage (e.g., [23, 32, 33]). This is consistent with the observation that older actively accreting NSs in LMXBs typically have weaker inferred magnetic fields (10⁹–10¹⁰ G) than younger accreting NSs in HMXB, detected as x-ray pulsars (∼10¹² G).

In the context of magnetars, magnetic field decay on the time scale of 10⁴ years and its effect on the star's thermal evolution and on its potential for powering the x-ray emission in AXPs has been investigated by several authors, e.g., [29, 34–42]. We will discuss some of these processes in section 2.2.

Finally, there is a question of the structure of the magnetic field inside NSs, including magnetars, once they are fully formed. Since NSs stop being convective fairly soon, just a few seconds after their birth, the usual convective MHD dynamo processes no longer operate in such NSs. Differential rotation also decays quickly and hence the Tayler–Spruit [16, 18] dynamo action also ceases. The magnetic field inside a NS then eventually relaxes to a stable lowest-energy magnetostatic equilibrium configuration, consistent with global magnetic helicity constraints and with field-line pinning in the solid crust. There has been a significant amount of theoretical work in recent years on calculating such equilibrium configurations and assessing their stability [43–51].

Magnetars are isolated young NSs with large-scale magnetic fields of order 10¹⁵ G—much stronger than the 'mere' ∼10¹² G fields typical for normal NSs (e.g., radio-pulsars or x-ray pulsars) and also stronger than the critical quantum magnetic field B_* ∼ 4 × 10¹³ G. Since the focus of this review is on physical processes, rather than on observational phenomenology, it will be convenient for us to define magnetars as a class of NSs with large-scale magnetic fields in excess of B_*.

There is a number of interesting phenomena exhibited by magnetars, and understanding them often requires a certain degree of plasma physics. Observationally, magnetars are detected through their powerful x-ray and γ-ray emission (see, e.g., [52] for a review). Based on their high-energy behavior, magnetars are usually subdivided into two classes—AXPs and SGRs. There are only a handful of objects of each class known. Both AXPs and SGRs are characterized by a powerful persistent x-ray emission (L_X = 10³⁴–10³⁶ erg s⁻¹), including a non-thermal power-law hard x-ray component observed in some AXPs (extending to at least 100 keV). However, in addition to this x-ray emission, SGRs also produce occasional very intense outbursts of gamma-ray activity (see below).

Both classes of magnetars have relatively slow rotation P = 5–12 s, detected through the modulation of their persistent x-ray emission, but relatively high spin-down rates $(\dot{P} \sim 10^{-10}{\hbox{--}}10^{-12})$ and hence young spin-down ages $(P/\dot{P} \lesssim 10^3{\hbox{--}}10^{5}$ years, consistent with the ages of their associated SN remnants). This provides a key evidence for high magnetic fields in these objects. SGRs have somewhat higher spin-down rates than AXPs and hence are believed to be younger (10³ years instead of 10⁴–10⁵ for AXPs). This has led to the suggestion that after a few thousand years SGRs lose their γ-ray activity and evolve into AXPs. In any case, an important implication of the very young ages of magnetars is that, even though the number of known magnetars is very small compared with the number of known regular NSs such as radio-pulsars (which are, on average, much older, with typical spin-down ages of millions of years), a rather significant fraction of all NSs, perhaps as high as ∼10%, are born as magnetars [6–8].

2.2.1. Persistent x-ray emission of AXPs and SGRs.

2.2.2. SGR giant γ-ray flares.

2.2.3. Propagation of electromagnetic radiation through a magnetar magnetosphere.

Another important example of extreme astrophysical environments is central engines and inner parts of magnetically driven jets in SNe and GRBs.

Both SNe and GRBs are very important and still not fully understood celestial phenomena. These spectacular cosmic explosions, marking the death of massive stars, have fascinated both astrophysicists and the general public alike for decades. Trying to understand their progenitors and the explosion mechanisms has been the subject of extensive research efforts. Still, however, there is no complete consensus among scientists on some of the key principal questions, such as the actual explosion mechanism or the nature of the central engine (for GRBs). These systems are very complex, rich in physics that is quite different from that encountered under any other circumstances. Of relevance to our review, among the various uncertainties impeding our progress, there are many that have to do with extreme plasma physics, as we shall describe in this subsection.

Before we proceed, we note that, although many basic physical characteristics of magnetar magnetospheres and SN/GRB central engines are similar (sizes, time-scales, typical magnetic field strengths, etc.), there are also many important differences. The physical conditions in SN and GRB central engines are quite exceptional and are not encountered anywhere else in the Universe since the Big Bang. In particular, the temperatures, reaching 10⁹ K, and the matter densities, of the order of the nuclear density of ∼10¹⁴ g cm⁻³, and hence the plasma pressures, are much higher here than around magnetars. Under these extreme conditions, the nuclear composition undergoes rapid changes as the material transforms from predominantly iron in the progenitor core to neutron-rich matter of the resulting NS. Also, neutrinos play a very important role in both the dynamics (exerting pressure) and, especially, the energetics of these explosions.

Before we turn to the discussion of the plasma-physical problems important to the operation of SN and GRB central engines, we shall give a brief basic introduction to each of them.

2.3.1. Supernovae.

2.3.2. Gamma-ray bursts.

Magnetic fields approaching B_* in magnitude introduce a number of novel features to a plasma. In this section we first briefly review the motions of single particles in an ultra-strong magnetic field, and then the basic properties of a thermal-equilibrium photo-leptonic plasma with an energy density corresponding to that of the critical magnetic field. This is followed by a discussion of light propagation through ultra-magnetized vacuum, which is a non-trivial issue due to the effect of vacuum polarization. Finally, we discuss equilibrium statistical mechanics of a quantum plasma in an ultra-strong magnetic field.

The discussion is purposefully succinct as the material is reviewed thoroughly elsewhere. Single-particle motions and solutions of the Dirac equation are discussed in detail by Strange [183]. The cold and warm plasma dielectric tensor for quantizing magnetic fields and radiative processes are reviewed in depth by Harding and Lai [84].

3.1.1. Single-particle motion.

When describing the motion of an electron in a super-critical magnetic field, the necessity of a quantum-relativistic treatment can be seen by considering the Landau quantization of electron motion. For B ≳ B_*, the Schrödinger equation does not suffice and one must solve the Dirac equation [γ^μ(∂_μ + ieA_μ/ℏc) + im_ec/ℏ]Ψ = 0 to find the proper Landau orbital energies. For a uniform magnetic field, the solution of the Dirac equation reveals energy eigenstates with energies

$$\begin{equation} \left(\frac{E_{r,n,p_{z}}}{m_{\rm e} c^{2}}\right)^{2}=1+\frac{p_{z}^{2}}{m_{\rm e}^{2}c^{2}}+2b(n+r-1), \end{equation} \tag{ 2 }$$

where r = 1, 2 differentiates between particle spin states, n = 0, 1, 2... is the energy quantum number, p_z is the momentum component parallel to the magnetic field, and b ≡ B/B_* (see, e.g., [184]). Thus, even an electron in the ground state has a kinetic energy on the order of its rest mass for b ≃ 1. Equation (2) also implies that, unless the plasma temperature exceeds the electron mass by a factor several times unity, most electrons and positrons are confined to the lowest Landau levels. The interaction of particle spins with the magnetic field is also seen to be significant. For particles with spin magnetic moment μ_m, if μ_m · B ∼ k_BT, the spins may not be randomized and collective spin interactions may introduce novel plasma features.

For relativistic electrons in a magnetic field, the characteristic length scale is the size of the ground-state wave function [84],

$$\begin{equation} \rho_{L}=\sqrt{\frac{\hbar c}{eB}}=6.44\times10^{-4}\, (B/{1\, \rm G})^{-1/2}\,{\rm cm}, \end{equation} \tag{ 3 }$$

which represents the quantum equivalent of the Larmor radius. For B = B_* this becomes equal to the (reduced) electron Compton wavelength,

$$\begin{eqnarray} &&\fl \rho_{\rm L*}\equiv \rho_{\rm L}(B=B_*) = l_{\rm C} = \frac{\lambda_{\rm C}}{2\pi} \equiv \frac{\hbar}{m_{\rm e} c} = 3.87 \times10^{-11}\,{\rm cm}.\nonumber\\ \end{eqnarray} \tag{ 4 }$$

The fact that ρ_L approaches the fundamental quantum length scale l_C calls for the plasma description to be carried into the quantum domain.

3.1.2. Basic properties of pair plasmas with extreme energy densities corresponding to $B_*^2/8\pi$.

Critical quantum field introduces a characteristic energy density $u_{*}=B_{*}^{2}/8\pi=7.8\times10^{25}\,{\rm erg\,cm}^{-3}$. This high level of energy density indicates the potential for plasmas embedded in such a field to become relativistically hot and dense if a substantial portion of the magnetic energy is converted into thermal energy. For example, a thermal gas of radiation and pairs in thermal equilibrium at zero chemical potential with an energy density u_* has a temperature of k_BT = 1.33 m_ec², electron and photon number densities of n_e = 1.34 × 10³¹ cm⁻³ and n_γ = 9.9 × 10³⁰ cm⁻³ as found by using

$$\begin{eqnarray} \fl n_{\rm pair}\left(\theta\right) &=& 2\int_{mc^{2}}^{\infty}\rmd \epsilon\, a\left(\epsilon\right) \frac{1}{\rme^{\epsilon/k_{\rm B} T}+1} \nonumber\\ \fl & =& 16\pi\lambda_{\rm C}^{-3}\theta^{3} \int_{\theta^{-1}}^{\infty} \rmd x\sqrt{x^{2}- \theta^{-2}}\, \frac{x}{\rme^{x}+1}\nonumber\\ \fl &\equiv& 16\pi\lambda_{\rm C}^{-3}\theta^{3}I_{1}\left(\theta\right) , \end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray} \fl u_{\rm pair}\left(\theta\right) &=& 2 \int_{mc^{2}}^{\infty} \rmd \epsilon\, a_{\rm e} \left(\epsilon\right)\frac{\epsilon}{\rme^{\epsilon/k_{\rm B} T} +1} \nonumber\\ \fl &=& 16\pi\frac{mc^{2}}{\lambda_{\rm C}^{3}}\theta^{4} \int_{\theta^{-1}}^{\infty} \rmd x \sqrt{x^{2} -\theta^{-2}}\, \frac{x^{2}}{\rme^{x}+1}\nonumber\\ \fl &=& 16\pi mc^{2}\lambda_{\rm C}^{-3}\theta^{4} I_{2}\left(\theta\right) , \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &&\fl u_{\gamma}\left(\theta\right)=\frac{8\pi^{5}}{15}mc^{2}\lambda_{\rm C}^{-3}\theta^{4} , \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} &&\fl n_{\gamma}\left(\theta\right)=16\pi\zeta\left(3\right)\lambda_{\rm C}^{-3}\theta^{3} , \end{eqnarray} \tag{ 8 }$$

where

$$\begin{equation} a_{e}\left(\epsilon\right)=8\pi\lambda_{\rm C}^{-3}\frac{\epsilon\, \sqrt{\epsilon^{2}-\left(mc^{2}\right)^{2}}}{\left(mc^{2}\right)^{3}} \end{equation} \tag{ 9 }$$

the energy density of states for massive spin 1/2 fermions [185], θ ≡ k_BT/m_ec² and x ≡ /k_BT, d = k_BTdx.

Note that the characteristic inter-particle spacing is then comparable to the reduced Compton wave length: $l_{\rm int} = n_{\rm e}^{-1/3} \simeq 4.2\times10^{-11}\,{\rm cm} = 1.1\, l_C$. Since the Compton wave length λ_C = 2πl_C is the de Broglie wavelength of a particle with kinetic energy equal to its rest energy, we see that the inter-particle spacing in the above-described plasma is comparable to the average de Broglie wavelength 〈λ_dB〉 = 〈2πℏ/p〉. This fact, once again, underscores the importance of quantum effects in such a plasma.

Finally, the inverse plasma parameter, written as the ratio of Coulomb repulsion energy to the average kinetic energy, is

$$\begin{eqnarray} &&\fl \Gamma_{\rm p}=\frac{e^{2}}{l_{\rm int}\theta m_{\rm e} c^{2}}=\frac{r_{\rm e}}{l_{\rm int}}\, \theta^{-1} = \frac{\alpha}{2\pi}\, \frac{\lambda_{\rm C}}{\theta n^{-1/3}}\nonumber\\ &&= \alpha\left(\frac{I_{1}\left(\theta\right)}{\pi^{2}}\right)^{1/3}\sim\alpha, \end{eqnarray} \tag{ 10 }$$

where α ≡ e²/ℏc ≃ 1/137 is the fine structure constant. For example, at θ = 1.33, Γ_p = 0.005; this justifies the treatment of electrons as a weakly coupled plasma. For high temperatures (θ ≫ 1) the integral I₁(θ) asymptotically approaches 3ζ(3)/2 = 1.80. As expected, in the ultrarelativistic limit the plasma is weakly coupled due to the smallness of the fine structure constant α.

3.1.3. Vacuum polarization and electromagnetic wave propagation in ultra-strong magnetic fields.

3.1.4. Equilibrium quantum statistical mechanics with a uniform magnetic field.

Electrons obey Fermi–Dirac statistics and thus, in a magnetic field with energy levels given by equation (2), have a distribution function proportional to the occupancy fraction

$$\begin{eqnarray} &&\fl f\left(p_{\Vert},n,r,\mu\right)\nonumber\\ &&=\frac{1}{{\rm exp}\left[\theta^{-1}\sqrt{1+\frac{p_{\Vert}^{2}} {m^{2}c^{2}}+2b(n+r-1)}-\tilde{\mu}\right]+1}, \end{eqnarray} \tag{ 11 }$$

where p_|| is the momentum component parallel to the background magnetic field and $\tilde{\mu}=\mu/k_{\rm B} T$ is the normalized chemical potential. The photons obey Bose–Einstein statistics,

$$\begin{equation} f_{\gamma}\left(\omega\right)=\frac{1}{\rme^{\hbar\omega/k_{\rm B} T}-1}. \end{equation} \tag{ 12 }$$

Canuto and Chiu [196] obtained the equation of state for an ideal electron gas in a magnetic field of arbitrary strength using solutions of the Dirac equation and the energy-momentum tensor with Lagrangian density ${{\mathscr{L}}}\equiv-mc^{2}\psi\bar{\psi} - (1/2)\,\hbar c(\bar{\psi}\gamma_{\mu}\partial^{\mu}\psi-\partial_{\mu} \bar{\psi}\gamma^{\mu}\psi)$, where ψ is the (electron) field and $\bar{\psi}$ is the adjoint (positron) field. Inserting the spinor solutions of the Dirac equation in a uniform magnetic field and assuming a statistical ensemble with weight given by (11), they obtained expressions for the perpendicular pressure

$$\begin{eqnarray} &&\fl P_{\bot} = 2\pi b^{2}\frac{mc^{2}}{\lambda_{\rm C}^{3}}\nonumber\\ &&\times \sum_{n=0}^{\infty} \sum_{r=1}^{2} \int_{-\infty}^{\infty} \frac{\rmd p_{\Vert}}{mc}\, f(p_{\Vert},n,r,\mu)\, \frac{mc^{2}\left(n+r-1\right)}{E\left(p_{\Vert},n,r\right)} , \end{eqnarray} \tag{ 13 }$$

the parallel pressure

$$\begin{eqnarray} &&\fl P_{\Vert} = 2\pi b\frac{mc^{2}}{\lambda_{\rm C}^{3}}\sum_{n=0}^{\infty}\sum_{r=1}^{2}\int_{-\infty}^{\infty}\frac{\rmd p_{\Vert}}{mc}\, f(p_{\Vert},n,r,\mu)\, \frac{p_{\Vert}^{2}/m}{E\left(p_{\Vert},n,r\right)} ,\nonumber\\ \end{eqnarray} \tag{ 14 }$$

the energy density

$$\begin{eqnarray} &&\fl U = 2\pi b\frac{mc^{2}}{\lambda_{\rm C}^{3}}\sum_{n=0}^{\infty}\sum_{r=1}^{2}\int_{-\infty}^{\infty}\frac{\rmd p_{\Vert}}{mc}\, f(p_{\Vert},n,r,\mu)\, \frac{E\left(p_{\Vert},n,r\right)}{mc^{2}} ,\nonumber\\ \end{eqnarray} \tag{ 15 }$$

and the number density

$$\begin{equation} n = 2\pi b\, \frac{mc^{2}}{\lambda_{\rm C}^{3}}\sum_{n=0}^{\infty}\sum_{r=1}^{2}\int_{-\infty}^{\infty}\frac{\rmd p_{\Vert}}{mc}\, f(p_{\Vert},n,r,\mu) . \end{equation} \tag{ 16 }$$

These expressions can be evaluated numerically for any value of θ and $\tilde{\mu}$.

The physical quantities of importance in determining the degree of degeneracy are the density n (or the Fermi temperature T_F), the temperature T, and the chemical potential μ. For reference, they are related to each other by the normalization condition (see [197])

$$\begin{eqnarray} &&\fl - {\rm Li}_{\frac{3}{2}}\left(-\rme^{\mu/k_{\rm B} T}\right) = \frac{4}{3\sqrt{\pi}}\left(\frac{T_{\rm F}}{T}\right)^{3/2}= \frac{4\pi^{3/2}\hbar^{3}n}{\left(2m k_{\rm B} T\right)^{3/2}}\nonumber\\ &&= \sqrt{2}\pi^{3/2}n\lambda_{\rm dB}^{3} , \end{eqnarray} \tag{ 17 }$$

where Li is the polylogarithm function defined as

$$\begin{equation} {\rm Li}_{\nu}\left(z\right) \equiv \sum_{k=1}^{\infty}\frac{z^{k}}{k^{\nu}} . \end{equation} \tag{ 18 }$$

As we saw from the discussion in the preceding section, when dealing with plasmas immersed in ultra-strong magnetic fields, one has to confront the question of how to do plasma physics while accounting for Landau quantization, quantum statistics, the finite width of the wavefunction, quantum tunneling and particles with spin. This leads into the formalism of quantum plasma physics. This section is intended as a brief overview of the key ideas of non-RQP physics. It also aims at serving as a handy reference to various fundamental results in the theory of quantum plasmas. The generalization to RQPs, applicable to extreme plasma astrophysics, is discussed in section 3.3.

The study of collective effects in many-body quantum systems has its roots in condensed matter physics, and much early knowledge in non-RQP physics comes from that discipline. Recently, however, quantum plasmas have become popular theoretical and experimental subjects in their own right.

There has been a significant amount of work on non-RQP physics. The general procedure employed in modern quantum plasma theory can be outlined as follows. The equations of quantum plasma physics typically take the same form as their classical counterparts, but have additional ℏ-containing terms. The basis of modern quantum plasma theory is the Wigner quasi-probability distribution function (WF), through which a kinetic theory is derived. Defined through the density matrix, the WF is a quantum analog of phase-space density, which facilitates comparison with the classical kinetic theory. Differentiating the Wigner function with respect to time and using the Schrödinger equation, one can obtain the so-called Moyal kinetic equation [198] that governs the evolution of quantum plasmas. For the case of particles interacting with a self-consistent mean electromagnetic field, the Moyal equation can be interpreted as a quantum analog of the Vlasov equation. The evolution of the self-consistent electromagnetic field is then governed by the Maxwell equations as in the classical case. Having built this general theoretical framework, one can then apply it to study various plasma processes. This, however, has been done only for a few problems, mostly linear waves. Next, just as in the classical plasma physics, one can transition from kinetic to fluid description by taking the moments of the WF and the Moyal kinetic equation (or its appropriate analog). The resulting quantum fluid equations are very similar to the classical fluid equations, except for quantum corrections to the pressure tensor.

This subsection is devoted to a brief exposition of the non-RQP theory. In section 3.2.1 we present the basic Wigner–Moyal kinetic formalism. We illustrate the application of this formalism for linear waves (both electrostatic and electromagnetic) in unmagnetized quantum plasmas in section 3.2.2. In section 3.2.3 we discuss the development of the quantum fluid theory and in section 3.2.4 we discuss spin effects in quantum plasmas. A very brief review of recent work on nonlinear quantum plasma processes is included in section 3.2.5.

3.2.1. The Wigner–Moyal formalism.

In order to build a kinetic quantum plasma theory, one starts by defining the Wigner distribution function [199]. The N-body Wigner function f_N can be defined in terms of the density matrix $\hat{\rho}$ in the coordinate representation, ρ(x, x'; t) ≡ 〈x|ψ(t)〉〈ψ(t)|x'〉, as

$$\begin{eqnarray} &&\fl f_{N}\left({\bit x}_{N},{\bit P}_{N};t\right) =\frac{1}{\left(2\pi\hbar\right)^{3N}}\nonumber\\ &&\times \int \rmd^{3N}{\bit y}_{N}\,\rho\left({\bit x}_{N}-\frac{{\bit y}_{N}}{2},{\bit x}_{N}+\frac{{\bit y}_{N}}{2};t\right) {\rme}^{-\rmi {{\bit P}_{N}\cdot {\bit y}_{N}/\hbar}}, \end{eqnarray} \tag{ 19 }$$

where x_N and P_N are 3N-dimensional vectors denoting the conjugate positions and canonical momenta coordinates of N particles. The momentum P used in this definition is the canonical momentum P = p + qA/c, where p = mv is the kinetic momentum of the particle, v is its velocity, and A is the electromagnetic vector potential. The expectation value of any observable, $\langle O\rangle = \Tr(\hat{\rho}\hat{O})$, is then written in the language of the Wigner function as

$$\begin{eqnarray} &&\fl \langle O\rangle = \int \rmd^{3N} {\bit x}_{N}\, \rmd^{3N}{\bit P}_{N} \, f_{N}\left({\bit x}_{N},{\bit P}_{N};t\right) O\left({\bit x}_{N},{\bit P}_{N};t\right) . \end{eqnarray} \tag{ 20 }$$

Of particular interest in kinetic plasma physics is the one-particle Wigner function f(x₁, P₁),

$$\begin{eqnarray} &&\fl f({{\bit x}_1,{\bit P}_1}) = \int f_{N}\left({\bit x}_{N},{\bit P}_{N};t\right)\, \rmd {\bit x}_2 \ldots \rmd {\bit x}_N\, \rmd {\bit P}_2 \ldots \rmd {\bit P}_N \nonumber\\ &&= {1\over{(2\pi\hbar)^3}}\, \int\limits_{-\infty}^{+\infty}\ \psi^*({\bit x}_1+{\bit y}/2)\, \psi({\bit x}_1-{\bit y}/2)\, \rme^{\rmi {{\bit P}_1\cdot {\bit y}}/\hbar}\, \rmd^3{\bit y} .\nonumber\\ \end{eqnarray} \tag{ 21 }$$

In the following we will consider only the one-particle WF and will omit the subscript '1' for brevity. In the spirit of Moyal's theory, the evolution of f(x, P) is determined from the Schrödinger equation, $\rmi\hbar \partial_t \psi = \hat{H}\, \psi$, or the corresponding von Neumann equation for the density matrix, $\rmi \hbar\, {\partial\hat{\rho}}/{\partial t}=[\hat{H},\hat{\rho}]$.

The special case of a plasma interacting with a purely electrostatic field, A = 0 (and hence P = p), allows one to investigate fairly easily important basic kinetic phenomena such as the Landau damping. The Hamiltonian in this case is

$$\begin{equation} \hat{H} = ({\hbar^{2}}/{2m})\nabla^{2} + V\left({\bit x},t\right) = {{\bit p}^{2}}/(2m)+V\left({\bit x},t\right), \end{equation} \tag{ 22 }$$

where V(x, t) = qφ(x, t), q is the particle charge, and φ(x, t) is the electrostatic potential. Then, the evolution equation becomes

$$\begin{eqnarray} &&\fl \frac{\partial f}{\partial t}+\frac{{\bit p}}{m} \cdot\nabla f = \left(2\pi\right)^{-3}\int \rmd^3{\bit y} \int \rmd^3{\bit s}\,\, \exp\left[\rmi{\bit y}\cdot \dot{\left({\bit s}-{\bit p}\right)}\right] \nonumber\\ &&\times \rmi q f\left({\bit x},{\bit s},t\right)\left[\frac{V\left({\bit x}+{\hbar{\bit y}}/{2},t\right) - V\left({\bit x} - {\hbar{\bit y}}/{2},t\right)}{\hbar}\right] , \nonumber\\ \end{eqnarray} \tag{ 23 }$$

which is equivalent to what is known as the Moyal kinetic equation [198],

$$\begin{eqnarray} \fl \frac{\partial f\left({\bit x,p,t},\right)}{\partial t} &=& \{\{f, H \}\} \nonumber\\ \fl &\equiv& -\, \frac{2}{\hbar}\,{\rm sin}\, \left[ \frac{\hbar}{2}\left(\frac{\partial^{\left(f\right)}} {\partial{\bit x}}\frac{\partial^{\left(H\right)}}{\partial{\bit p}}-\frac{\partial^{\left(f\right)}}{\partial{\bit p}}\frac{\partial^{\left(H\right)}} {\partial{\bit x}}\right) \right]\nonumber\\ \fl &&\times f\left({\bit x,p},t\right)H\left({\bit x,p},t\right) . \end{eqnarray} \tag{ 24 }$$

Here, {{f, g}} ≡ (iℏ)⁻¹ (f * g − g * f) = {f, g} + O(ℏ²) is the Moyal bracket, f * g is called the Moyal product, and {f, g} is the usual Poisson bracket. The superscripts (f) and (H) indicate which function the derivative operates on. The sine with a differential operator argument is defined in terms of its power series. To obtain a complete set of equations, equation (23) is supplemented with the Poisson equation

$$\begin{equation} \nabla^2 \phi = -\, 4 \pi \rho({\bit x},t) , \end{equation} \tag{ 25 }$$

where ρ(x, t) is the electric charge density.

In a more general case of a plasma of spinless charged particles interacting with a general electromagnetic field according to the Hamiltonian

$$\begin{equation} \hat{H} = {1\over{2m}}\, \left[{\bit P} - {q {\bit A}\left({\bit x},t\right)/c} \right]^{2} + q\phi\left({\bit x},t\right) , \end{equation} \tag{ 26 }$$

a kinetic model based on the WF approach was developed recently by Tyshetskiy et al [200]. By treating the electromagnetic field as a self-consistent classical mean field as in the classical Vlasov theory (similar to the Hartree mean-field approximation in quantum mechanics), they obtained the following collisionless kinetic equation:

$$\begin{eqnarray} &&\fl \frac{\partial f\left({\bit x,p},t\right)}{\partial t}+\frac{{\bit p}}{m}\cdot\nabla f+q\left[{\bit E}+\frac{{{\bit p}\times {\bit B}}}{mc}\right]\cdot\nabla_{{\bit p}}f\nonumber\\ &&=I_{q}\left({\bit x},{\bit p},t\right) , \end{eqnarray} \tag{ 27 }$$

where the quantum interference integral I_q(x, p, t) represents the effect of interference of overlapping wave functions and where we have transitions from using the canonical momentum P to the kinetic momentum p = P − qA/c. Explicit expression for I_q(x, p, t) in terms of f(x, p, t) and (A, φ) reads [200]:

$$\begin{eqnarray*}&&\fl I_{q}\left({\bit x},{\bit p},t\right)= \frac{1}{\left(2\pi\right)^{3}}\int \rmd^3 {\bit y} \int \rmd^3{\bit s}\,\textrm{exp}\left[\rmi {\bit y} \cdot\dot{\left({\bit s}-{\bit p}\right)}\right] \\ &&\fl \times \bigg\{\rmi q f({\bit x},{\bit s},t) \left[\!{\bit y}\cdot\nabla\phi({\bit x},t)\!- \frac{\phi({\bit x}+\frac{\hbar{\bit y}}{2},t)\!- \phi({\bit x}\!-\frac{\hbar{\bit y}}{2},t)}{\hbar}\! \right]\\ &&\fl -\frac{\rmi q}{mc}f({\bit x},{\bit s},t) \Bigg[{\bit y}\cdot\nabla({\bit s}\cdot{\bit A} ({\bit x},t))\nonumber\\ &&\fl -\frac{{\bit s} \cdot{\bit A}({\bit x}+\frac{\hbar{\bit y}}{2},t) -{\bit s}\cdot{\bit A}({\bit x}-\frac{\hbar{\bit y}}{2}, t)}{\hbar}\Bigg] \\ &&\fl -\,\frac{q}{2mc}\Biggl[ (\nabla f({\bit x}, {\bit s},t)+f({\bit x},{\bit s},t) \nabla) + \frac{\rmi q}{c}f({\bit x},{\bit s},t)\\ &&\fl \times \Bigg(\nabla({\bit y}\cdot{\bit A}({\bit x}, t))-\frac{{\bit A}({\bit x} +\frac{\hbar{\bit y}}{2},t)-{\bit A} ({\bit x}-\frac{\hbar{\bit y}}{2},t)}{\hbar}\Bigg) \Biggr] \\ &&\fl \cdot\left[2{\bit A}\left({\bit x},t\right)-{\bit A} \left({\bit x}+\frac{\hbar{\bit y}}{2},t\right)- {\bit A}\left({\bit x}-\frac{\hbar{\bit y}}{2},t\right)\right] \bigg\} \\ \end{eqnarray*}$$

Equation (27) is a quantum analog of the classical Vlasov equation.

The limits of applicability of the one-particle WF formalism are set by the weak-coupling condition and by the non-relativistic condition. Weak coupling can be quantified by specifying that the interaction energy density be much smaller than the kinetic energy density: Γ = u_int/_kin ≃ 2e²/(l_int m〈v²〉) ≪ 1, which in the case of a fully degenerate plasma becomes v_F/c ≫ α ≡ e²/ℏc ≃ 1/137, where v_F ≡ (2E_F/m_e)^1/2 = (ℏ/m_e) (3π²n)^1/3 is the Fermi velocity. The condition that the non-relativistic approximation is valid for a degenerate plasma is simply v_F/c ≪ 1 as usual.

3.2.2. Kinetic theory of linear waves and instabilities in unmagnetized quantum plasmas.

An important part of classical plasma physics is the study of linear waves and instabilities. The general linear dispersion relation can be written in terms of the linear dielectric permittivity tensor _ij, which describes the response of a medium to small electromagnetic field perturbations, as Det[_ij(ω, k)] = 0. In an isotropic plasma the dielectric tensor can be divided into longitudinal and transverse parts,

$$\begin{eqnarray} &&\fl \epsilon_{ij}\left(\omega,{\bit k}\right)=\epsilon^{\rm long}\left(\omega,{\bit k}\right) \frac{k_{i}k_{j}}{k^{2}}+\epsilon^{\rm trans}\left(\omega,{\bit k}\right)\left(\delta_{ij}-\frac{k_{i}k_{j}}{k^{2}}\right) ,\nonumber\\ \end{eqnarray} \tag{ 28 }$$

For a single-species unmagnetized non-RQP, the Wigner–Moyal formalism leads to (e.g., [200, 201])

$$\begin{eqnarray} &&\fl \epsilon^{\rm long}\left(\omega,{\bit k}\right) =1+\frac{4\pi e^{2}}{k^{2}}\nonumber\\ &&\times\int\frac{\rmd^3{\bit p}}{\hbar}\frac{f_{0}\left({\bit p} + {\hbar{\bit k}}/2 \right) - f_{0}\left({\bit p} - {\hbar{\bit k}}/{2}\right)}{\omega-{\bit k}\cdot{\bit p}/m}, \end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray} &&\fl \epsilon^{\rm trans}\left(\omega,{\bit k}\right)=1-\frac{\omega_{\rm p}^{2}}{k^{2}}+\frac{2\pi e^{2}}{m^{2}\omega^{2}}\nonumber\\ &&\times \int\frac{\rmd^3{\bit p}}{\hbar}\, p_{\perp}^{2}\, \frac{f_{0}\left({\bit p} + {\hbar{\bit k}}/{2}\right) - f_{0}\left({\bit p} - {\hbar{\bit k}}/{2}\right)}{\omega-{\bit k}\cdot{\bit p}/m}, \end{eqnarray} \tag{ 30 }$$

where f₀ is the equilibrium Wigner distribution function, p_⊥ is the component of p perpendicular to k, and

$$\begin{equation} \omega_{\rm p} \equiv \sqrt{{4\pi n e^2}\over{m}} \end{equation} \tag{ 31 }$$

is the classical plasma frequency.

The generalization of this tensor to magnetized quantum plasmas with a background magnetic field B₀ has not yet been carried out.

Electrostatic waves and instabilities. Electrostatic (e.g., Langmuir) waves leading to a fully developed plasma microturbulence in ultra-magnetized RQPs have been invoked by Beloborodov and Thompson [58] as a mechanism for effective resistive dissipation of the intense electric currents at the interface between a twisted magnetosphere of an active magnetar and its surface (see section 2.2.1 and section 4.4). The full theory of how this turbulence develops in the presence of an ultra-strong magnetic field has not yet been developed, so, as a first step, it is important to understand the behavior of these waves and instabilities in an unmagnetized quantum plasma. This understanding will hopefully pave the way toward a future more comprehensive theory.

As an example, the dispersion relation for one-dimensional longitudinal electrostatic waves in a plasma of quantum electrons with a fixed ion background can be written as (e.g., [202])

$$\begin{eqnarray} &&\fl 1 + \frac{\omega_{\rm p}^{2}}{k^{2}}\int_{-\infty}^{\infty} \rmd v_{z}\frac{f_{0}\left(v_{z}\right)}{\left(v_{z}- \frac{\omega}{k}\right)^{2}-\left(\frac{\hbar k}{2m_{\rm e}} \right)^{2}} \nonumber\\ &&\fl + \rmi \pi\frac{\omega_{\rm p}^{2}}{k^{2}} \frac{4m}{\hbar k}\left[f_{0}\left(v_{z}+ \frac{\hbar k}{2m_{\rm e}}\right)-f_{0} \left(v_{z}-\frac{\hbar k}{2m_{\rm e}}\right)\right] = 0 . \end{eqnarray} \tag{ 32 }$$

For a fully degenerate (T = 0) Fermi–Dirac plasma, Langmuir (and also ion-acoustic, IAWs) waves were investigated by Eliasson and Shukla [203]. They obtained an explicit expansion of the real part of the frequency in powers of k in the long wavelength limit k → 0,

$$\begin{equation} \omega^{2}=\omega_{\rm p}^{2}+\frac{3}{5}k^{2}v_{\rm F}^{2}+\left(1+\frac{48}{175}\frac{m^{2}v_{\rm F}^{2}}{\hbar^{2}\omega_{\rm p}^{2}}\right)\frac{\hbar^{2}k^{4}}{4m^{2}}. \end{equation} \tag{ 33 }$$

and also mentioned the possibility of Landau damping even in the completely degenerate case, although did not present a specific calculation or expression for it.

Melrose and Mushtaq [197] applied for the first time the longitudinal dielectric tensor to electrostatic waves in plasmas of arbitrary degeneracy. They examined the fully degenerate and non-degenerate limits for Langmuir waves and IAW's in more detail, utilizing a power series expansion in the degeneracy parameter exp(μ/k_BT), and derived both the real and imaginary parts of the frequency, as well as the screening length.

Recently, Rightley and Uzdensky (2014, in preparation) developed a general semi-analytical procedure for studying linear electrostatic waves and instabilities in unmagnetized quantum plasmas with arbitrary equilibrium distribution, including arbitrary degree of degeneracy, drifting sub-populations, etc. This provides a single coherent framework for exploring a wide range of kinetic phenomena, including Buneman, bump-on-tail, ion-acoustic instabilities, over broad parameter ranges, and for investigating the role of quantum effects. Electrostatic streaming instabilities, such as two-stream and bump-on-tail, have been also investigated for nonrelativistic quantum Schrödinger–Poisson plasmas using a different approach, namely, a multi-stream model cast in terms of Cauchy distributions, which yielded analytically tractable results [204, 205]. It was established that no analog of the Penrose functional exists for electrostatic waves in quantum plasmas, though the stability of single-peaked distribution functions remains valid. Haas and Bret [206] investigated the Buneman instability using a cold fluid model with collisions introduced phenomenologically. They found that the presence of the quantum Bohm pressure term tended to stabilize the system in the linear regime and led to oscillatory behavior in the nonlinear regime not present in the purely classical case. General, realistic studies with Fermi–Dirac electrons of arbitrary degeneracy are yet to appear.

Electromagnetic waves and instabilities. The dispersion relation for electromagnetic waves propagating through a non-RQP of fully degenerate electrons and stationary ions,

$$\begin{equation} \omega^{2}=\omega_{\rm p}^{2}+\left(kc\right)^{2} + ({\hbar k^2}/{2m})^{2}, \end{equation} \tag{ 34 }$$

was obtained by Zhu and Ji [207] as a non-relativistic limit of a more general expression. Dispersion relations for electromagnetic waves in plasmas of arbitrary degeneracy have yet to appear in the literature.

Bret and Haas [208] derived the full electromagnetic response tensor for a non-RQP and applied it to the onset and growth of the electromagnetic filamentation instability. Linear Weibel instability was studied by Haas [209], who found that quantum effects tend to weaken or suppress the instability, and also by Tsintsadze [210], who found a novel oscillatory Weibel instability. Haas et al [211] also studied the nonlinear saturation of the quantum Weibel instability numerically.

3.2.3. Non-relativistic quantum fluid theory.

As in the classical case, the kinetic description of quantum plasmas often carries more information than is actually needed to adequately describe certain phenomena. The relative simplicity of fluid theory is often worth the sacrifice in the domain of validity, especially when one deals with nonlinear physics. This justifies developing a fluid theory, using fluid quantities defined via the moments of the WF

$$\begin{eqnarray} &&\fl n\left({\bit x},t\right)=\int \rmd^{3} {\bit P}\, f\left({\bit x},{\bit P};t\right) , \end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray} &&\fl {\bit u}\left({\bit x},t\right)=\frac{1}{mn} \int \rmd^{3}{\bit P} \, f\left({\bit x},{\bit P};t\right) \left({\bit P} - q {\bit A}/c\right) , \end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray} &&\fl \boldsymbol{ \Pi} \left({\bit x},t\right) = \frac{1}{m^{2}}\int \rmd^{3} {\bit P}\, f\left({\bit x},{\bit P};t\right)\left({\bit P} - q{\bit A}/c\right)\nonumber\\ &&\otimes \left({\bit P}-q{\bit A}/c\right) - nm\, {\bit u}\otimes{\bit u} , \end{eqnarray} \tag{ 37 }$$

that obey evolution equations systematically derived by taking the moments of the quantum kinetic (e.g., Moyal) equation. Thus, Manfredi and Haas [212] derived a set of QHD equations and then Haas [213] obtained QMHD equations. It was found that the continuity equation,

$$\begin{equation} \partial_{t} n + \nabla\cdot\left(n{\bit u}\right) = 0 , \end{equation} \tag{ 38 }$$

is the same as in the classical case. The momentum equation, assuming for simplicity a scalar pressure, Π_ij = Pδ_ij, becomes

$$\begin{eqnarray} &&\fl \partial_{t}{\bit u}+\left({\bit u}\cdot\nabla\right){\bit u} = \frac{q}{m}\left({\bit E}+{{{\bit u}\times{\bit B}}\over{c}}\right)-\frac{1}{mn}\nabla P\nonumber\\ &&+\frac{\hbar^{2}}{2m^{2}}\nabla\left(\frac{\nabla^{2}\sqrt{n}}{\sqrt{n}}\right) \end{eqnarray} \tag{ 39 }$$

and is identical to its classical counterpart except for the additional pressure term proportional to ℏ². This quantum term can be important when there are structures on the scale of the thermal de Broglie wavelength and when the parameter ℏω_p/k_BT approaches or exceeds unity. Haas [213] goes on to discuss two-fluid QMHD with phenomenological collisional terms, ideal QMHD, and a class of magnetostatic QMHD equilibria.

Fluid theory for quantum plasmas is still an actively developing and controversial topic; a criticism of quantum plasma fluid theory was recently provided by Vranjes et al [214].

3.2.4. Non-relativistic quantum spin-1/2 plasma (Pauli plasma).

Particle spin introduces a new level of complexity into the dynamics of plasmas, especially in magnetized plasmas. In non-relativistic quantum mechanics for spin-1/2 particles, spin is incorporated by generalizing the wave-function to a two-component spinor (Ψ_↑, Ψ_↓) corresponding to 'up' and 'down' spin components, respectively. For electrons (with charge q = −e), the evolution equation for this spinor in a magnetic field B is the Pauli equation:

$$\begin{eqnarray} \fl \rmi \hbar\, \partial_{t}\Psi &=& \hat{H}\, \Psi \nonumber\\ \fl &=& \biggl[ - \frac{\hbar^{2}}{2m}\left(\nabla+\frac{\rmi e}{\hbar c}{\bit A}\right)^{2}{\bit I} - e \phi + \mu_{\rm B}{\bit B}\cdot{\boldsymbol \sigma} \biggr]\, \Psi \end{eqnarray} \tag{ 40 }$$

where I is the 2 × 2 unit matrix, σ is the vector of Pauli matrices, σ = (σ_x, σ_y, σ_z) and μ_B is the electron magnetic moment eℏ/2m_e. It should be noted that this equation neglects direct spin–spin interactions, which can influence wave dispersion and may generate macroscopic-scale magnetic fields. These interactions can in principle be included with the more general Hamiltonian

$$\begin{equation} \hat{H}= -\, \frac{\hbar^2}{2m}\left[{\boldsymbol \sigma\cdot}\left(\nabla+\frac{\rmi e}{\hbar c}{\bit A}\right)\right]^{2}-e\phi. \end{equation} \tag{ 41 }$$

In an ultra-strong magnetic field, however, spin–spin interactions can be expected to be dominated by the external field.

Marklund and Brodin [215] used the Pauli equation along with a Madelung decomposition of the wavefunction to derive a set of quantum spin magnetized fluid equations, referred to as the spin quantum hydrodynamics (SQHD). In addition to the usual fluid quantities (the density, velocity, and the pressure tensor), the quantum spin fluid theory includes one more vector, the spin density S_s for species s, and hence one more equation—the one governing the evolution of S_s:

$$\begin{equation} \left(\partial_{t}+{\bit u}_{s}\cdot\nabla\right){\bit S}_{s}= -\, \frac{2\mu_{s}}{\hbar}\, {\bit B}\times{\bit S}_{s} . \end{equation} \tag{ 42 }$$

In addition, the interaction of the spin density vector with the magnetic field results in an extra force in the momentum equation, $(2n_s \mu_s /\hbar)\, S_s^j \nabla B_j$, where n_s and μ_s are the number density and the magnetic moment of the particles of species s. Otherwise, the set of fluid equations is identical to the one for spinless quantum plasmas (see equations (38) and (39)), including the quantum pressure-like term $\hbar^2 /(2m^2)\, \nabla (n^{-1/2}\,{\nabla^{2}\sqrt{n}})$ in the equation of motion.

Marklund and Brodin [215] then applied their spin fluid theory to derive the linear susceptibility tensor in the low frequency (ω/Ω_c ≪ 1, where Ω_c is the cyclotron frequency) and highly magnetized (μB₀/k_BT ≫ 1) limits. They found that spin can significantly alter the dispersion of the fast and slow magnetosonic waves, which may have important implications for cold laboratory plasmas and also for ultra-magnetized astrophysical plasmas. This spin fluid model was subsequently also applied to study the influence of spin on Alfvén waves [216]. It was found that, for wave phenomena occurring on time-scales shorter than the spin-flip timescale but longer than the electron cyclotron time, spin-up and spin-down electrons essentially behave as two separate fluids, which influences the nonlinear aspects of the wave propagation.

A recent critique of some of the SQHD theories of spin-1/2 quantum plasmas was offered by Krishnaswami et al [217].

3.2.5. Nonlinear processes in quantum plasmas.

Basic quantum mechanics is not capable of accounting for all the physics occurring in the presence of super-critical fields, and one needs to turn to a relativistic quantum field theory (QFT). In QFT one deals with second quantized field operators instead of complex-number wavefunctions. Correspondingly, RQP physics extends the Wigner function formalism to relativistic systems in terms of quantized fields. The WF is then replaced by a Wigner operator. For physics excluding nuclear matter and neutrinos, the applicable QFT is quantum electrodynamics (QED). For spin-1/2 fermions of QED the Wigner operator is a 4 × 4 matrix, built from a direct product of two Dirac bispinors corresponding to spin-up and spin-down particles and antiparticles.

The main applications of RQP physics include laboratory laser-produced pair plasmas; quark–gluon plasmas, e.g., in relativistic heavy-ion collision experiments (although this area lies beyond QED); and various astrophysical applications described in this review.

A complete description of a plasma in terms of QED is difficult to incorporate in the already complex problems of plasma physics. However, some key phenomena predicted by QED have been isolated and included in a number of studies in the context of ultra-magnetized NSs. Such phenomena include polarization of virtual pairs leading to non-trivial optical behavior of the vacuum (vacuum polarization, see section 3.1.3), anisotropy in Compton and Coulomb scattering, and modifications to the emission of synchrotron radiation.

In this subsection we outline the recent progress in building the fundamental theoretical framework of RQP physics, first for spinless (KG) particles (section 3.3.1) and then for spin-1/2 (Dirac) fermions (section 3.3.1).

3.3.1. Relativistic quantum kinetic theory without spin (KG plasma).

Mendonca [222] used a relativistic scalar Wigner function and the KG equation $-\,c^{-2}\, \partial_t^2 \psi + \nabla^2\psi = (mc/\hbar)^2\, \psi$, to derive a relativistic quantum Vlasov kinetic equation for unmagnetized plasmas composed of spinless (scalar) charged particles (KG plasma), treating electromagnetic field as a classical mean field. He then used it to derive the dispersion relations for longitudinal and transverse linear waves. In the high frequency limit, he arrived at specific forms for the dispersion relation for longitudinal electrostatic and transverse electromagnetic waves for degenerate electrons in a stationary neutralizing ion background:

$$\begin{equation} \omega_{\rm L}^{2}=\frac{\omega_{\rm p}^{2}}{\gamma_{\rm F}} \left[1+\frac{k^{2}}{\omega^{2}}\left(1-\frac{\omega^{2}}{k^{2}c^{2}}\right)v_{\rm F}^{2} + \left(\frac{\hbar k}{2m}\right)^{2}\gamma_{\rm F}^{-2}\right] , \end{equation} \tag{ 43 }$$

$$\begin{equation} \omega_{\rm T}^{2}=\left(kc\right)^{2}+\frac{\omega_{\rm p}^{2}}{\gamma_{a}^{2}}, \end{equation} \tag{ 44 }$$

where

$$\begin{equation} \gamma_{\rm F}\equiv\langle\gamma\rangle=\int\gamma f_{0}\left({\bit v}\right) \rmd {\bit v}=\sqrt{1+v_{\rm F}^2/c^2}, \end{equation} \tag{ 45 }$$

and

$$\begin{equation} \gamma_{a}\equiv\sqrt{1+\frac{eA_{0}}{mc}} , \end{equation} \tag{ 46 }$$

with A₀ being the amplitude of the (circularly polarized) wave.

In the weakly relativistic limit β_F ≡ v_F/c ≪ 1, analogous results were also obtained by Zhu and Ji [207] who started from the general kinetic equation for the Dirac plasma and then neglected the spin terms. They obtained:

$$\begin{eqnarray} &&\fl \omega_{\rm L}^{2}=\omega_{\rm p}^{2}\left(1-\frac{1}{2}\beta_{\rm F}^{2}\right)+\frac{3}{5}\left(kv_{\rm F}\right)^{2}\left(1-\frac{3}{2}\beta_{\rm F}^{2}\right)\nonumber\\&&+\left(\frac{\hbar k}{2m}\right)^{2} \left(1-\frac{3}{2}\beta_{\rm F}^{2}\right), \end{eqnarray} \tag{ 47 }$$

$$\begin{eqnarray} &&\fl \omega_{\rm T}^{2}=\left(kc\right)^{2}+\omega_{\rm p}^{2}\left(1-\frac{1}{2}\beta_{\rm F}^{2}\right)+\left(\frac{\hbar k}{2m}\right)^{2}\left(1-\frac{3}{2}\beta_{\rm F}^{2}\right). \end{eqnarray} \tag{ 48 }$$

However, the case of a strongly magnetized KG plasma has not yet been properly worked out. It is still not clear how this formalism can be used for strongly magnetized case, where the perpendicular motion of particles is quantized in discrete Landau states.

3.3.2. Relativistic quantum kinetic theory with spin 1/2 (Dirac plasma).

3.3.3. Relativistic hydrodynamics.

The topic of light propagation through an ultra-magnetized plasma in magnetar magnetospheres is important for two reasons. First, it affects radiative processes, such as radiative cooling and particle creation, that control the physical conditions in, e.g., a reconnection layer in SGR flares, see section 4.5. Second, since direct observation of radiation is often our only diagnostic tool (which is true in astrophysics in general), our interpretation of various phenomena in magnetars is subject to modification due to the propagation of the photons to us across the magnetosphere.

In sharp contrast to the case of weak magnetic fields, an ultra-strong field in a magnetar magnetosphere (and, to a lesser extent, slightly subcritical, ∼10¹² G, fields in magnetospheres of normal NSs) strongly affects the propagation of high-energy photons even in a vacuum, in the absence of plasma. This is due to the so-called QED vacuum polarization effect or vacuum birefringence reviewed in section 3.1. The effect arises due to the interaction of light with virtual pairs in the presence of the strong magnetic field and corresponds to one-loop corrections in the QED (Euler–Heisenberg) Lagrangian. There has been a lot of work on understanding the effect of vacuum polarization on electromagnetic waves, motivated by both fundamental physics and astrophysics (e.g., magnetar) applications (see, e.g., [187, 188]).

Even though, formally speaking, the propagation of photons through a QED vacuum is not within the scope of plasma physics, it is still important to review it because it changes the energy-exchange processes between radiation and pairs. Overall, the picture can be summarized as follows (see, e.g., [84] for review). First, it is important to distinguish between the two photon polarizations (for non-parallel propagation): the ordinary mode (the O-mode), whose electric field vector has a component parallel to the background magnetic field, and the extraordinary mode (the X-mode), whose electric field is perpendicular to the background magnetic field. The propagation of X-mode photons across the field is strongly suppressed by the QED process of photon splitting. The attenuation coefficient for this process becomes insensitive to B for B > B_*, but is a strong function of the photon energy $({\sim}\epsilon_{\rm ph}^5)$. This means that, basically, only photons with energies _ph ≲ 0.1 m_ec² ≃ 50 keV are able to escape the strong-field region of size L ∼ R_NS ∼ 10⁶ cm around a magnetar. In addition, X-mode photons with energies exceeding the threshold for pair creation with one particle created at the zeroth and the other at the first excited Landau level (this threshold is $\epsilon_{\rm ph} = (1+\sqrt{1+2b})\,m_{\rm e} c^2 \simeq 3 \, {\rm MeV}$ for b = 10) are subject to additional attenuation due to one-photon pair production. For these pair-producing photons this process is even more important than photon splitting. The process of one-photon pair production also leads to heavy absorption of pair-producing (MeV) O-mode photons. This process and its inverse essentially ensure that these MeV photons are in equilibrium with pairs in environments with high plasma energy density, e.g., inside a reconnection layer during an SGR flare. In contrast, the photon splitting process does not affect O-photons. Sub-MeV O-photons thus can propagate essentially freely across a vacuum magnetic field.

An ultra-strong magnetic field also affects the nonlinear behavior of vacuum electromagnetic waves. In particular, Heyl and Hernquist [192, 193] found that, nonlinearly, electromagnetic waves steepen and tend to form shocks as they propagate across an ultra-strong magnetic field.

QED effects associated with an ultra-strong magnetic field are also important to take into account when considering electromagnetic radiation propagating through a plasma. In particular, an ultra-strong field affects various photon–particle interaction processes, such as Thomson/Compton scattering, photon–photon pair creation and annihilation, and cyclotron/synchrotron emission and absorption (e.g., [63, 84, 230–232]). For example, it was shown that Compton scattering of X-photons is strongly suppressed by the magnetic field: $\sigma_{\rm es}^{(X)}/\sigma_{\rm T} \simeq (\epsilon_{\rm ph}/b\, m_{\rm e} c^2)^2$ for a photon with energy _ph ≪ b m_ec² propagating perpendicular to the magnetic field [230]. In contrast, the Compton scattering of O-mode photons is not strongly suppressed and so this process often provides the main source of opacity for sub-MeV O-mode photons. The dispersion relation for normal electromagnetic modes propagating in a relativistic pair plasma in an ultra-strong field was derived by Arons and Barnard [231] in the context of NS magnetospheres. Resonant cyclotron scattering and Comptonization in a non-relativistic warm plasma immersed in an inhomogeneous magnetic field, as well as their observational implications for AXPs, were analyzed by Lyutikov and Gavriil [63].

All these processes are important to take into account when modeling observable radiative signatures of high-energy processes in magnetospheres of magnetars and, to some degree, even of normal NSs.

As we have seen in section 2, many of the plasma-physical aspects of magnetar flares and SN/GRB central engines involve large-scale MHD dynamics believed to be similar to MHD processes taking place in various other astrophysical systems across the Universe. For example, MHD dynamo in a rotating convective star is likely to be the main mechanism for the generation of magnetic fields both in young NSs and in main-sequence stars, such as the Sun. In addition, γ-ray flares in active SGR systems may happen in a manner dynamically similar to solar flares. In particular, they may involve magnetospheric field-line twisting by footpoint motions on the NS crust, subsequent inflation of magnetic flux ropes leading to the loss of equilibrium and/or the development of the kink instability, which creates current sheets and thus sets up the macroscopic conditions for reconnection. Likewise, MHD processes in long-GRB collapsar central engines probably share a lot of similarities with other BH accretion-disk/jet systems such as XRBs and AGNs. For example, MRI turbulence may provide a mechanism for angular momentum transport in a collapsar accretion disk and, at the same time, through its MHD dynamo action, generate a large-scale magnetic field. The latter then facilitates the launching, collimation and acceleration of a relativistic magnetic tower jet. In the millisecond magnetar scenario for GRBs, a similar magnetic-tower-like jet may be collimated by the interaction of the rapidly rotating magnetar magnetosphere with the surrounding infalling stellar envelope, as has been illustrated, e.g., with the 'pulsar-in-a-cavity' model [119, 120], or with the relativistic pulsar-wind model [160–162] (see section 2.3 and also [122]).

Importantly, in order to be successful, the large-scale magnetic field in a magnetic-tower (or any other magnetically powered) GRB jet should be able to survive the propagation through the progenitor star. Determining under what conditions this is at all possible and in what form (i.e., with what kind of structure) the magnetic energy emerges at large distances, is an important and nontrivial problem of extreme plasma astrophysics. The reason why this is a difficult issue is that there are large-scale MHD processes that present strong obstacles to the jet propagation. Generically, the magnetic field in a GRB jet may develop spatial and temporal intermittency and have a complex substructure [119, 120, 233]. The evolution of this substructure may lead to the formation of thin current-sheet configurations that can be classified according to the type of the underlying magnetic geometry, as was discussed by McKinney and Uzdensky [233]. For example, field-line opening in an inflating magnetosphere driven by differential rotation (e.g., [79, 80]) or by relativistic rotation beyond the light cylinder (e.g., [234–236]) may naturally lead to the formation of thin current sheets, across which magnetic field changes sharply. In addition, it is generally quite unlikely that the large-scale magnetic field driving the jet at the base would have a simple, e.g., dipole-like, geometry maintained for the duration of the event. For example, if the field is produced by the MRI dynamo in a collapsar accretion disk, then, as numerical simulations of MRI turbulence show [237], the toroidal field emerging above the disk into the corona changes its polarity in a quasi-periodic fashion on a time scale of about 10 orbits. This implies that the toroidal field in the jet pumped over the entire (several seconds) duration of the event may experience hundreds of direction reversals and, correspondingly, may contain hundreds of magnetic 'stripes' separated by current sheets (similar to the striped pulsar wind [233]). Finally, even if the magnetic field near the central engine is somehow well-ordered, it may develop irregularities over time as it propagates out. In particular, the highly-twisted magnetic field in the jet can develop ideal MHD instabilities, such as the kink [119, 172, 238], leading to current sheet formation and reconnection onset, similar to sawtooth crashes in tokamaks.

Importantly, current sheets thus produced become potential sites for the onset of magnetic reconnection, which therefore becomes an important issue for magnetically driven GRB models. The effectiveness of reconnection governs where and whether the Poynting flux in the jet is dissipated and thus determines the fate of the jet, i.e., whether it crumbles or survives, with important observational consequences [171, 173, 176, 233, 238, 239]. In particular, if reconnection proceeds successfully early on, e.g., while the jet is still making its way through the progenitor star, then it may lead to the break-up of a single large-scale magnetic structure (like a magnetic tower) into a train of multiple smaller toroidal plasmoids [119]. This picture is rather similar to the cyclic behavior involving reconnection suggested for magnetospheres of accreting young stars [80, 240]. The typical sizes of such plasmoids and their production rate may then influence the variability of dissipation at later times and larger distances in the GRB jet, leading to observable timing signatures. In turn, they are controlled by detailed reconnection dynamics and determining them requires a thorough understanding of reconnection process in this extreme environment, which we will discuss in section 4.5.

Despite this general similarity in the macro-scale MHD dynamics in active magnetar magnetospheres and central engines of SNe and GRBs on the one hand, and the more familiar and well-studied solar corona and accretion disk/jet systems on the other, the thermodynamics of these two types of environments is very different. Many of the differences are due to the presence of radiation (both photon and neutrino).

In particular, in SN/GRB central engines, and also in putative reconnection layers in magnetar flares (see section 4.5), we are dealing with extreme environments that are very dense and relativistically hot (k_BT > m_ec²). This means that, even though the plasma inertia may be dominated by the baryons (in the SN/GRB case), the particle number density, the pressure, and the entropy are all dominated by a hot closely coupled plasma of pairs and photons in local thermodynamic equilibrium (LTE). The electron, positron and photon numbers are not conserved but instead are constantly maintained by pair and photon creation and annihilation processes near the levels dictated by the LTE condition, which can be expressed as an analog of the Saha equilibrium. The densities and pressures then essentially become unique functions of the temperature and the lepton fraction (which in the SN/GRB environments is affected by the deleptonization processes associated with neutrino losses). As a result, all the thermodynamic properties of such a plasma, including the effective equation of state, are significantly altered relative to the more familiar standard plasmas where no particles are created or destroyed.

Furthermore, radiation controls most of the transport processes in these environments—thermal conduction, resistivity, viscosity. Importantly, the radiative processes are strongly affected by ultra-strong magnetic fields as we discussed in section 4.1. At relatively modest Thomson optical depths (perhaps across a current sheet in a magnetar magnetosphere) radaitive cooling by photon diffusion may be significant and may control the energy balance. Also, radiative Silk damping due to photon thermal conduction may affect sound waves, etc.. However, in the case of SN/GRB central engines, the environments of interest are so dense and optically thick that the photon mean free path is negligibly short. Then, photon-mediated transport processes (thermal conduction, viscosity, etc.) can be completely ignored on the length- and time-scales dictated by the global MHD dynamics. The same is also true for transport due to electrons, positrons, ions, and, under some circumstances, neutrons.

However, this does not necessarily mean that the plasma in SN and GRB central engines can be described by simple ideal MHD on the time scales of interest. The reason for this is that the role of photon radiative energy transport is taken up by neutrinos. Neutrino cooling and heating can play a dominant role in governing the thermodynamics of these systems. A truthful description of neutrino transport is complicated by the large number of different neutrino processes that may need to be taken into account, depending on the conditions. These processes include neutrino–neutrino scattering; neutrino–anti-neutrino annihilation; neutrino–electron (and positron) scattering; neutral–current neutrino scattering off of free nucleons, α particles, and nuclei; charged-current neutrino and anti-neutrino absorption and emission by free nucleons and nuclei, and so on (see, e.g., [241–243] for through reviews.). Comprehensive modeling of neutrino transport processes in SN/GRB central engines is further complicated by the substantial range of neutrino optical depths that need to be covered. Whereas inside NSs and in the densest parts of collapsar accretion disks neutrinos are optically thick and one can employ the radiation diffusion approximation, the surrounding areas (e.g., the post-bounce shock region in SN or at the base of the jet in GRBs) are optically thin to neutrinos which hence leak through them easily. These lower-density regions are actually very important: it is here that one would like to see the neutrinos that come out from the hotter denser central regions to deposit enough of their energy to revive a SN shock or to create a high-entropy photo-leptonic fireball to produce a GRB explosion.

Furthermore, since the cross-sections of various reactions between neutrinos and normal matter are often rather strong functions of the neutrino energy, one has to treat neutrinos of different energies differently. In particular, the boundary between optically thick and optically thin regions—the neutrino-sphere—may strongly depend on the neutrino energy. Perhaps the only feature of neutrino transport that may make its detailed modeling simpler than, say, photon transport in a stellar atmosphere is the fact that neutrino cross-sections are smooth functions of energy, with no small-scale energy structure (i.e., the absence of spectral lines), for which exact or approximate analytical expressions are available [241–243].

Finally, nuclear composition of extreme astrophysical environments of core-collapse SN/GRB central engines is not fixed and can be far from that dictated by nuclear statistical equilibrium. It often has to be evolved self-consistently via a nontrivial network of nuclear reactions, which also affects the thermodynamics of the plasma.

These remarks point to the desirability of developing more realistic theoretical and numerical models of basic astrophysical MHD processes that pay a more careful attention to the questions of thermodynamics. One needs to go beyond the basic purely dynamical models that treat heating and cooling in a greatly simplified way, e.g., with a rudimentary equation of state.

As discussed in section 2.2, the magnetic field outside an active magnetar is dynamic; both persistent x-ray emission and SGR gamma-ray flares are believed to be powered by the dissipative decay of the electrical currents induced in the force-free magnetosphere by the footpoint displacements on the NS surface. In particular, the question of how the electron–positron plasma carrying these currents through the magnetosphere interacts with the surface becomes an important and nontrivial issue (e.g., [58]). As the intense high-energy relativistic particle beam impinges on the neutron star's crust, it is stopped over a relatively short distance. The resulting dissipation of the beam's energy heats up the star's surface locally, perhaps melting the crust, and powers an intense hard x-ray emission in AXPs. Morphologically, this process is similar to two-ribbon solar flares, in which the hard x-ray and white-light emission from the solar surface is powered by high-energy electrons accelerated by reconnection in the corona high above the surface of the Sun and streaming down along the reconnected magnetic field lines. The underlying plasma physics of the beam–surface interaction, however, may be quite different in the two cases. In the case of magnetars, this interaction may involve collective collisionless plasma processes, such as the excitation and subsequent dissipation of small-scale microscopic Langmuir and ion-acoustic turbulence by kinetic plasma instabilities [58]. As discussed in section 3.1, under the conditions of interest, these processes take place on Compton (QED) scales and may be affected by the presence of an ultra-strong magnetic field. This means that their proper treatment and analysis requires developing a good understanding of RQP physics, the foundations of which are outlined in section 3.

Magnetic reconnection is a fundamental plasma process involving a rapid change of magnetic topology and often leading to a violent release of magnetic energy. How reconnection occurs, how fast it is, what its onset conditions are, and what are its outcomes, are all fundamental plasma-physical questions that have direct astrophysical implications. Magnetic reconnection of magnetar-strength fields (B > B_* ≃ 4 × 10¹³ G) is one of the most important extreme plasma astrophysics problems with applications to magnetar magnetospheres and GRB central engines and jets (see sections 2.2, 2.3 and 4.2). It is the most dramatic example of HED astrophysical reconnection. The physical conditions in the hot, dense lepto-photonic plasma in an ultra-magnetizied reconnection layer are very different from those in the conventional, low-density heliospheric plasmas. A rich variety of exotic physical effects (e.g., special relativity, radiative effects, pair creation, and QED effects due to the ultra-strong field) come into play in a most powerful way in this RQP, making this problem intellectually exciting. Understanding the basic physics relevant to this problem represents a new open frontier in magnetic reconnection research. Perhaps the main goals of ultra-strong magnetic field reconnection research are to establish how rapidly and in what form the dissipated energy emerges from the reconnection region. This includes calculating potentially observable radiative characteristics of the reconnection layer, such as its photospheric temperature and flare duration. Comparing the results of such calculations with observations would enable one to assess whether, for example, reconnection is a plausible mechanism for SGR giant flares and for GRBs.

Developing a detailed quantitative picture of this process, with all the bells and whistles of QED in an ultra-strong field, has not yet been accomplished. Generally speaking, this is a multi-scale phenomenon, with scales ranging from the Compton scale l_C = ℏ/m_ec, to the scales relevant to the radiative transfer in, and the growth of, the optically thick pair coat described below (from the photon mean free path l_mfp ∼ α⁻³ l_C to the photosphere thickness δ_ph), to the astrophysical scales of relevance (NS radius R_NS ∼ 10⁶ cm).

As a first step toward understanding reconnection of ultra-strong fields, Uzdensky [73] made simple preliminary estimates that enable one to identify the key physical ingredients of the process. This work should serve as the basis for any rigorous theoretical model of this process. The key findings can be summarized as follows. When the energy of an ultra-strong magnetic field is dissipated, the resulting internal plasma energy density is so high that the plasma inside the reconnection region becomes relativistically hot and one expects copious production of photons and electron–positron pairs in LTE with each other (see section 3.1). The temperature T₀ and the equilibrium pair density n_pairs at the center of the reconnection layer can be estimated in terms of the reconnecting magnetic field B₀ from the cross-layer pressure balance: $P_{\rm rad, 0} + P_{\rm pairs, 0} = B_0^2/8\pi$, where $P_{\rm rad, 0}= a T_0^4 /3 = (\pi^2/45)\, [m_{\rm e} c^2/(l_{\rm C}^3)] \, \theta_{\rm e}^4$ and P_pairs,0 ≃ (7/4) P_rad are the layer's radiation pressure and the relativistic pair plasma pressure, respectively. One then obtains θ_e ≡ k_BT₀/m_ec² ≃ (45/22 π³α)^1/4 b^1/2 ≃ 1.73 b^1/2, where, once again, b ≡ B₀/B_* and α is the fine structure constant. Correspondingly, the pair density inside the layer, determined by the LTE equilibrium condition in terms of T₀, becomes (in the ultrarelativistic limit θ ≫ 1 corresponding to b ≫ 1)

$$\begin{eqnarray} \fl n_{\rm pairs} & \simeq 0.183\, (k_{\rm B} T/{\hbar c})^3 \simeq 0.183\ l_{\rm C}^{-3}\, \theta_{\rm e}^3\nonumber\\ & \simeq 3.2\times 10^{30}\ {\rm cm}^{-3}\, \theta_{\rm e}^3 . \end{eqnarray} \tag{ 49 }$$

Thus, the characteristic density scale in an extreme reconnection layer is one particle per $l_{\rm C}^3$. This density is much higher than, and hence essentially independent from, the ambient plasma density in a magnetar magnetosphere. This justifies our assumption that the baryonic contribution to the pressure inside the layer should be negligible in the context of SGR flares.

The above very high pair plasma density implies that the global reconnection layer of length L ∼ R_NS ∼ 10⁶ cm is highly collisional in the traditional sense that the corresponding Sweet–Parker layer thickness $\delta_{\rm SP}(L)= \sqrt{L\eta/V_{\rm A}}$, where η is the magnetic diffusivity and V_A is the Alfvén velocity, is much larger than the characteristic collisionless length scales, such as the electron skin depth d_e, or the electron Larmor radius ρ_e [73, 148, 233]. This means that the classical collisional and Compton-drag resistivities should dominate over other non-ideal terms in the generalized Ohm law and hence that collisional radiative resistive MHD description should be valid on this scale.

However, since the corresponding global Lundquist number S = L V_A/η for this environment is huge, S ∼ 10¹⁸ [73], much higher than the critical number S_c ∼ 10⁴ for the onset of the secondary tearing (aka plasmoid) instability [244, 245], the global Sweet–Parker layer is violently tearing-unstable [246]. Then, reconnection proceeds in the highly dynamic plasmoid-dominated regime, via a hierarchical chain of secondary plasmoids and current sheets [246–252]. This provides a path toward forming very small dissipative structures—the elementary current sheets at the bottom of the hierarchy—that develop on microscopic plasma-physical scales, i.e., the Larmor radius or the collisionless skin depth (which, as we saw in section 3.1, are on the Compton length-scale) or the critical Sweet–Parker scale δ_c ≃ 100 η/V_A [251]. This suggests that even in the very dense and highly collisional extreme astrophysical environments considered in this review, detailed plasma physics beyond the standard resistive MHD may still be important in controlling energy dissipation processes. If the classical conventional reconnection theory can serve us as a guide, the effective reconnection rate should be bracketed between the purely resistive-MHD plasmoid-regime rate of 0.01 B₀V_A and the often-quoted canonical collisionless reconnection rate of 0.1 B₀V_A [73]. Which one of these limits applies should depend on the ratio δ_c/d_e ∼ 100 ηV_A/d_e, and preliminary estimates suggest that, for the physical conditions of interest, these two scales, δ_c and d_e, are roughly comparable. Whether this general picture holds, and how it is modified by the QED effects, intense radiation, and ultra-strong magnetic fields, is not at all clear. This calls for the development of a more precise physical description of microscopic current layers on the Compton scale, e.g., calculating plasma resistivity and other non-ideal-MHD effects, based on RQP theory.

The above reconnection rate range (0.01 − 0.1 B₀V_A) has strong implications for magnetar flares [73]. It translates into a very short reconnection time of about 1 msec, much shorter than the typical observed duration of the main spike (0.25 s). This implies that the reconnection process itself is not the main energy-release bottleneck and that it does not govern the flare duration. Instead, the duration may be governed by the escape of radiation from the hot fireball ejected out of the reconnection region, and/or by the slow driving by the NS crustal motions [53, 56, 70].

The second important role of the pairs produced in the reconnection process is their effect on the propagation of photons out of the layer, which affects both the radiative cooling of the layer and its observational appearance. The former is important because the equilibrium pair density, resistivity, and radiation pressure are all strong functions of temperature, and hence the issues of thermodynamics and thermal transport are critical.

In the case of reconnection of magnetar-strength fields, the compactness of the reconnection region is very large. Therefore, the high pair density is not limited to the thin current layer proper in which ohmic dissipation of the magnetic energy takes place. Instead, copious pair production by high-energy γ-ray photons emitted by the layer quickly dresses it in a continuously growing coat of electron–positron pairs [73]. As it grows, this pair coat may become much thicker than the current layer proper, and thus may appear to an outside observer as a somewhat flattened fireball. The coat becomes optically thick to Compton scattering and to one-photon pair creation and photon-splitting, even in the direction across the layer. This ensures good local thermal coupling between the photons and the plasma—in stark contrast with conventional low-energy-density reconnection found in, e.g., solar flares or the Earth magnetosphere.

The reconnection problem then becomes, fundamentally, a time-dependent radiative transfer problem; this aspect is essential to understanding the energetics of reconnection in this environment. On length scales much larger than the photon mean free path λ_ph, but smaller than the thickness of the growing pair coat, the problem can be treated in the radiation diffusion approximation. The radiative transfer equations are supplemented by the expressions for the local pair density and the photon mean-free-path λ_ph. In most of the pair coat, deep below the photosphere, the plasma-radiation thermal coupling is good and the LTE density is determined by an analog of the Saha formula; it is a strong function of the local temperature. The photon mean free path, then, also becomes a strong function of the temperature but, in addition, is greatly affected by the ultra-strong magnetic field through the QED effects described in section 4.1. Near the pair coat's photosphere, the situation becomes even more complicated: the LTE and radiative diffusion approximations both break down and so one needs to build a numerical kinetic model of radiative and pair-production processes taking place here.

Because of the large optical depth τ, photons diffuse gradually across the dressed layer of thickness H. However, in the case of a moderately subcritical reconnecting field (B₀ ≪ B_*) and hence a sub-relativistic layer temperature and moderate coat optical depth, the photon diffusion time t_diff ∼ τH/c may be shorter than the global advection time t_adv ∼ L/c along layer of length L. Then, photons can escape the layer before being advected out by the plasma flow if τ < L/H. In this case, a significant fraction of the dissipated magnetic energy promptly leaves the layer by rapid radiative diffusion across it, instead of being advected out along the layer as in traditional reconnection. In this strong cooling regime [253], radiative cooling becomes the main mechanism of removing the dissipated energy; it governs the thermodynamics of the layer and drastically affects its appearance to an external observer. In particular, in the case of steady state reconnection without guide field, one can show that the optical depth across the dressed layer τ, and the reconnection rate cE = v_recB₀ are related simply by $\tau \simeq T_0^4/T_{\rm ph}^4 \sim B_0/E \sim c/v_{\rm rec}$ [73]. Thus, the thicker the layer, the slower the reconnection process.

It turns out, however, that the strong-cooling assumption (i.e., a stationary balance between reconnection ohmic heating and radiative cooling) is not valid in the case of super-critical, magnetar-strength magnetic fields [73]. The reason for this is that the central temperature in the layer is expected to be ∼1.7 b^1/2 m_ec² (about 3 MeV for B ∼ 10B_*), whereas the photosphere of the pair coat forms at k_BT_ph ≃ 20–30 keV (because λ_ph,mfp due to Thomson scattering in an equilibrium pair plasma becomes comparable to the global system size L ∼ R_NS ∼ 10⁶ cm at this temperature). In a steady state, this would imply a huge optical depth of order $\tau\sim T_0^4/ T_{\rm ph}^4> 10^5\, b^2$, and then the radiative diffusion time would be much longer than both the observed flare time and the global advection time along the layer, t_adv ∼ L/c. This means that the reconnection-powered growth of the pair coat is not in the stationary strong-cooling regime and has to be considered as a time-dependent process. In the context of SGR flares, most of the magnetic energy dissipated in a reconnection event directly powers the continuous production of pairs and photons that make up the growing pair coat. The reconnection process can thus be viewed as an efficient conversion of magnetic energy into the energy of a relativistically hot lepto-photonic fireball surrounding the reconnection layer. On time-scales of order the global Alfvén (or light-) advection time t_adv ∼ L/c, this newly created pair plasma gets torn apart into two halves by the large-scale reconnection outflow, which directly leads to the formation of two optically thick, dense and hot plasma blobs, as discussed in section 2.2.2. One of them can be associated with the ejected fireball that powers the main SGR flare spike emission and the other with the fireball trapped by the reconnected post-flare magnetic loops close to the star and responsible for the long-lasting tail emission.

Some of the above considerations for reconnection in extreme ultra-magnetized environments also have implications for the workings of central engines of SNe and GRB jets. In particular, the inferred reconnection rate may be important for the survival of the inner parts of GRB jets [148, 233]. Deep inside the star, where the plasma in a reconnection layer is highly collisional, reconnection might not be fast enough to destroy the magnetic jet, which then may be able to survive the propagation to larger distances [148]. Eventually, well outside the star, the magnetic field in the jet drops, reconnection layers cool, pairs annihilate, and the plasma becomes optically thin and collisionless. Then, an ongoing reconnection process can switch to the fast collisionless reconnection regime, leading to a renewed magnetic energy release [233]. For typical collapsar parameters, McKinney and Uzdensky [233] found that this collisionality-induced reconnection switch is expected to trigger the transition to the fast reconnection at a relatively large distance (r ∼ 10¹³–10¹⁴ cm), giving the flow enough range to accelerate to a high Lorentz factor (γ ∼ 100–1000). It was concluded that this reconnection switch mechanism allows for efficient conversion of electromagnetic energy into prompt emission with radiation signatures consistent with observations.