Kinetic Theory for Spin-1/2 Particles in Ultrastrong Magnetic Fields

PHYSICAL REVIEW E 102, 043203 (2020)

Kinetic theory for spin-1/2 particles in ultrastrong magnetic ﬁelds

Haidar Al-Naseri,1,* Jens Zamanian,1,† Robin Ekman,2,‡ and Gert Brodin 1,§ 1Department of Physics, Umeå University, SE–901 87 Umeå, Sweden 2Centre for Mathematical Sciences, University of Plymouth, Plymouth PL4 8AA, United Kingdom

(Received 5 June 2020; accepted 9 September 2020; published 6 October 2020)

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

DOI: 10.1103/PhysRevE.102.043203

I. INTRODUCTION In the atmospheres of pulsars and magnetars [18], the electron motion may become relativistic, and the magnetic field Quantum kinetic descriptions of plasmas are typically of strength can be ultrastrong, i.e., the Zeeman energy may be most interest for high densities and modest temperature [1]. comparable to or even larger than the electron rest mass en- For this purpose, typically the starting point is the Wigner- ergy [19]. Further information about pulsar properties can be Moyal equation [1–6], or various generalizations thereof also gained through the emission profiles, see, e.g., Refs. [20–22]. accounting for physics associated with the electron spin, Theoretical studies of wave propagation relevant for strongly such as the magnetic dipole force [7,8], spin magnetization magnetized objects have been made both with kinetic [23] [7,9], and spin-orbit interaction [9]. Both weakly [10,11] and and hydrodynamic [24,25] models. However, most previous strongly [12,13] relativistic treatments have been presented in theoretical studies starting from the Dirac equation (see also the recent literature. Refs. [11,12]) have been limited to cases where the magnetic Certain quantum phenomena depend strongly on the mag- field strength is well below the critical field B = m2c2/|q|h¯. nitude of the electromagnetic field, however, rather than on cr Our objective in this work is to derive a fully relativistic the density and temperature parameters. Phenomena such kinetic model of spin-1/2 particles, applicable for ultrastrong as radiation reaction (reviewed in, e.g., Ref. [14]) and pair magnetic fields, i.e., with creation fall into this category.1 Another field-dependent 2 2 9 phenomenon is Landau quantization [17], which becomes B ∼ Bcr = m c /|q|h¯ = 4.4 × 10 T, (1) prominent whenever the Zeeman energy due to the magnetic relaxing the conditions given in previous works. Assuming field is comparable to or larger than the thermal energy, or the that the electric field is low enough to avoid pair cre- Fermi energy, in the case of degenerate electrons. 2 3 ation (i.e., below the critical electric field Ecr = m c /|q|h¯ = 1.3 × 1018V/m), we can use the Foldy-Wouthuysen transformation [26,27] to separate particle states from antiparticle *[email protected] † states in the Dirac equation. Moreover, we limit ourselves to [email protected] the case where the characteristic spatial scale length of the ‡[email protected] fields is much longer than the Compton length, L = h¯/mc. §[email protected] c Making a Wigner transformation of the density matrix, our × Published by the American Physical Society under the terms of the approach results in an evolution equation for a 2 2 Wigner Creative Commons Attribution 4.0 International license. Further matrix, where the four components encode information re- distribution of this work must maintain attribution to the author(s) garding the spin states. However, the off-diagonal elements and the published article’s title, journal citation, and DOI. Funded of the matrix is associated with the spin-precession dynamics by Bibsam. which is too rapid to be resolved by the theory. Thus a further 1While the classical radiation reaction, where the response is a reduction is made, where only the diagonal components rep- smooth function of the orbit, is a useful approximation as long as the resenting the spin-up and the spin-down states relative to the emitted spectrum is soft (dominated by photons with energies well magnetic field remain. below mc2), the description based on QED (see, e.g., Refs. [15,16]), Together with Maxwell’s equations, we obtain a closed containing discrete probabilistic contributions due to the emission of system describing the plasma dynamics. A unique feature of high-energy quanta, is more generally applicable. the model is the energy expression, where the usual γ factor

2470-0045/2020/102(4)/043203(9) 043203-1 Published by the American Physical Society AL-NASERI, ZAMANIAN, EKMAN, AND BRODIN PHYSICAL REVIEW E 102, 043203 (2020)

from classical relativistic theory is replaced by an operator the spin degrees of freedom. In our derivation, we need to split in phase space, also depending on the magnetic field. Interest- the magnetic field B = B0 + δB into a strong but constant ingly, the Landau-quantized states in a constant magnetic field background B0 and a weak but freely varying perturbation turn out to be eigenfunctions of the energy operator of this δB. We will also not resolve dynamics on length scales com- theory, which is helpful when computing the thermodynamic parable to the de Broglie wavelength and not on timescales background state for the Wigner matrix. To demonstrate the comparable to the inverse Compton frequency. usefulness of the theory, we compute the dispersion relation for Langmuir waves propagating parallel to an external magnetic field. Finally, the consequences of the theory and III. DERIVATION OF THE THEORY applications to astrophysics are discussed AND CONSERVATION LAWS In this section we review the Foldy-Wouthuysen transfor- II. BACKGROUND AND PRELIMINARIES mation for strong fields (Sec. III A), and then we apply it to obtain a particle-only Hamiltonian that is applicable in In this section we will briefly summarize the fundamental ultrastrong magnetic fields (Ref. III B). Next, in Sec. III C, approaches we have used in order to derive a kinetic theory for we derive the corresponding kinetic equation in phase space. / spin-1 2 particles in ultrastrong magnetic field; see the next Finally, in Sec. III D, we present conservation laws for particle section for a technical derivation. number and energy. In the formulation of quantum mechanics in phase space [28] the system is described by a quasidistribution function W (r, p). This is in contrast to the Hilbert space formulation A. Foldy-Wouthuysen transformation where the system is described by a density matrix ρ(r, r ). To derive our theory for ultrastrong magnetic fields, we will The two formulations are related via the Wigner transforma- take the Dirac Hamiltonian tion. From the von Neumann equation for the density matrix, Hˆ = βm + Eˆ + Oˆ , (3) ∂ ρ = ˆ , ρ , ih¯ t ˆ [H ˆ ] (2) as our starting point. Here m is the mass, Eˆ = qφ(rˆ), Oˆ = one obtains an equivalent equation for W , involving ∂ W α · πˆ , and α and β are the Dirac matrices. Furthermore, pˆ is x π = − , and ∂pW . To lowest order inh ¯, this equation will be the the canonical momentum, ˆ pˆ qA(rˆ t ), q is the charge, Liouville equation for the Hamiltonian at hand. This gener- and φ and A are, respectively, the scalar and vector potentials. alizes to a many-particle system, and because the phase-space From now one we use units such that c = 1. formulation is analogous to classical statistical physics, the In Foldy and Wouthuysen’s seminal paper [26] the expan- Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierar- sion parameter ε was E/m, meaning that the expansion is chy [29] applies, and assuming exchange and correlation valid for sufficiently small energies, i.e., E represents relevant effects are small, one obtains an analog of the classical Vlasov energies (kinetic energy, magnetic dipole energy, etc.). In this equation. paper we will use the results of Refs. [27,32] where a modified While this approach is completely general (see Ref. [28] Foldy-Wouthuysen transformation was developed for the case for numerous applications in a wide range of fields), for the where the expansion parameter is instead the scale length of Dirac Hamiltonian in particular, the physical interpretation the fields. This transformation hence makes it possible to take is complicated due to that the Dirac Hamiltonian is mixing into account arbitrarily strong fields as long as we are only particle and antiparticle states. Through a Foldy-Wouthuysen concerned with variations on sufficiently long scale lengths. transformation [26,27], particle and antiparticle states can be An operator is called odd if it couples the upper and the decoupled to a given order in some small expansion param- lower pairs of components of the Dirac four-spinor and even eter ε. Using the resulting Hamiltonian HˆFW and applying a if it does not. The goal of a Foldy-Wouthuysen transformation Wigner transformation, one can get a kinetic theory for the is thus to obtain a Hamiltonian HFW that is even. The odd and quasidistribution W involving particles or antiparticles only. even terms of the Hamiltonian (3)areOˆ and Eˆ, respectively, For this to be physically valid, the pair production rate should satisfying [β,Eˆ] = 0 and {β,Oˆ }=0. be low when ε is small. In Ref. [32], Silenko found a unitary transformation of the The obtained kinetic equation is an evolution equation for Hamiltonian operator a2× 2 Wigner matrix, where the four components represent ˆ = ˆ ˆ − ∂ ˆ † + ∂ , the different spin states. There are several ways of treating the H U (H i t )U i t (4) spin degrees of freedom in a phase-space approach [11], e.g., where extending phase-space with spin dimensions [7], dividing the Wigner matrix into a scalar and a 3-vector [30], or using other ˆ + m ± βOˆ generalizations of the Wigner function for spin [31]. Uˆ (†) = √ , (5) 2ˆ (ˆ + m) The kinetic equation for the particles or antiparticles is closed with Maxwell’s equations with sources as moments of 3 where ˆ = m2 + Oˆ 2. Using the Dirac Hamiltonian Hˆ in W , e.g., the free charge density is ρ f (r) = qW (r, p) d p = q r| ρˆ |r. Eq. (4), one obtains In the next section, we present the derivation of a kinetic Hˆ = β ˆ + Eˆ + Oˆ , (6) theory for plasmas in ultrastrong magnetic fields, including

043203-2 KINETIC THEORY FOR SPIN-1/2 PARTICLES IN … PHYSICAL REVIEW E 102, 043203 (2020) where Note that we kept all orders of the magnetic field. However, we only kept up to first order of the combination ofh ¯ with 1 Eˆ = Eˆ + ([Tˆ , [Tˆ ,β ˆ + Fˆ ]] − [Oˆ , [Oˆ , Fˆ ]] E and ∇r. In the expression for Hˆ there are still some odd 2Tˆ operators, but since Oˆ is linear in μBE and μB∂t B,they 1 should be smaller than ˆ. Thus it is fine to only include the − [ˆ ,[ˆ ,Fˆ ]]) , (7) Tˆ minor correction of the second transformation in (9). Thus the Hamiltonian Hˆ after the second transformation will have Oˆ = 1 β { + , Oˆ , Fˆ }−{Oˆ , ,Fˆ } 1 , FW ( ˆ m [ ] [ˆ ] ) (8) the same structure as in (10). However, the anticommutator in 2Tˆ Tˆ √ HˆFW in (10) is proportional to the square of Oˆ . Since we only = + Fˆ = Eˆ − ∂ where Tˆ 2ˆ(m ˆ) and ih¯ t . The Hamiltonian kept up to first order of E and ∂t B, we keep only the first two Oˆ H still has odd terms, the anticommutators in are linear in terms of HˆFW, Oˆ α Oˆ , i.e., linear in . However, the odd term in H should be μBm HˆFW = β ˆ + qφ(rˆ) + √ small compared to ˆ, after choosing an expansion parameter. 2ˆ (ˆ + m) Since we are interested in the effects of an ultrastrong mag- 2 netic field, our small expansion parameters will be the electric 2μBm m × ˆ · (πˆ × E − E × πˆ ) − 1 + 2 field strength E/Ecr (such that pair production is negligible) ˆ 2ˆ (m + ˆ) and the inverse scale length of the fields L /L, but we will c 1 make no assumption on the magnetic field strength. × (πˆ × B) · E √ , (14) 2ˆ (ˆ + m) Thus a second transformation can be preformed with the following operator: 2 2 where ˆ = m + πˆ − 2μBmˆ · B. Taking the limit of weak μ / i 1 B field, i.e., keeping up to first order in BB m, we recover the Uˆ = eiS , Sˆ =− Oˆ , . (9) 4 ˆ Hamiltonian in Ref. [27]. The Hamiltonian in (14) includes the spin orbit interaction, see Ref. [12] for more details, but Since Oˆ is small, only the major corrections are taken into since E B, it is negligible compared to the magnetic in- account. Finally, the transformed Hamiltonian is teraction. We will therefore neglect the spin-orbit interaction from now on, as the main idea of this paper is to study the 1 1 Hˆ = β + Eˆ + Oˆ 2, . effects of a strong magnetic field on the dynamics of a plasma. FW ˆ (10) 4 ˆ The transformed Hamiltonian is now This Hamiltonian has no odd terms, so we are done with ˆ = β 2 + π2 − μ ˆ · + φ . the transformation of the Dirac Hamiltonian. Until now, we HFW m ˆ 2 Bm B q (ˆx) (15) have presented the Foldy-Wouthuysen transformation derived This Hamiltonian includes all orders of μBB/m and is fully in Ref. [27]. relativistic and hence is suitable to be used in deriving a kinetic equation for plasma in an environment where the mag- B. Strong-field Hamiltonian netic field is of the order of the critical field Bcr. However, in order to obtain a kinetic equation in phase space (see next Our approach now is to calculate the commutators occur- subsection for more details), we will use the identities from ring in (6) to all orders in the magnetic field. In doing this Ref. [34], see (19) and (20) below. These identities can be we will expand some of them in a series using Ref. [33]; utilized for a function F that only depends on πˆ . However, see the Appendix for more details. After the calculation, the our Hamiltonian in Eq. (15) depends on both B(rˆ) and πˆ , and Hamiltonian H in (6) becomes thus we divide the magnetic field as Hˆ = β ˆ + Eˆ + Oˆ , (11) B(rˆ) = B0 + δB(rˆ), (16) where where B is a constant strong magnetic field (i.e., μ B /m ∼ 0 B 0 1) and δB(rˆ) is a varying magnetic field. Furthermore, a Tay- μ 2 im B 2 m = Eˆ = Eˆ + ihq¯ + (E × B) · πˆ lor series around B B0 can be done; see Ref. [35]formore Tˆ Tˆ ˆTˆ details. However, the Taylor series gets more complicated for the higher-order terms, and thus a restriction on δB(rˆ) needs to ihq¯ 1 μ δ / − (E × B) · πˆ + iˆ · (E × πˆ − πˆ × E) , (12) be done. Considering the case where B B m 1, similarly ˆ2 Tˆ to the previous assumption μ E/m 1, the Hamiltonian be- B comes imμ β πˆ Oˆ = B 2α · E( ˆ + m) − 2 · E α · πˆ 2 2 Tˆ ˆ HˆFW = β m + πˆ − 2μBmσ · B0 + qφ(rˆ). (17)

h¯ 1 This Hamiltonian is the ﬁnal one that will be used in next + ˆ · ∂ B α · πˆ , (13) section to derive the kinetic equation. Note that all operators ˆ t ˆ T in (17) are even, and thus we will let β → 1 and ˆ → σ from μ = / now on. where B qh¯ 2m is the Bohr magneton and δ Note that even though B does not appear explicitly in Eq. σ ˆ = 0 . (17), the perturbed magnetic ﬁeld will still be contained in the 0 σ Lorentz force, as will be seen below, because πˆ contains the

043203-3 AL-NASERI, ZAMANIAN, EKMAN, AND BRODIN PHYSICAL REVIEW E 102, 043203 (2020) full vector potential A from which the full magnetic field is scalar equations for W+ and W− as follows: obtained. This is an example of how the phase-space formulation can make the physics of a system more explicit. ∂ + 1 · ∇ + + 1 × · ∇ = . tW± p rW± q E p B pW± 0 ± ± C. Gauge-invariant Wigner function and kinetic equation (25) Now we want to derive a kinetic equation on phase space The kinetic equation in Eq. (25) is our main result in this work. using the Hamiltonian (17). We will use the gauge-invariant The new effects of this kinetic equation are hiding in ±. First, Wigner function defined by Stratonovich [34], we have all orders of the spin magnetic moment, compared to 3 previous models [11,12] where only the first-order correction 1 3 Wαβ (r, p, t ) = d λ exp is included. Note also that the equations for W+ and W− are 2πh¯ decoupled and can be solved separately. Moreover, we have λ 1/2 momentum derivatives in , which turn to be energy opera- × i · + τ + τλ p q d A(r ) tors, see Sec. IV for more details. h¯ 1/2 Equation (25) describes the dynamics of an ensemble of λ λ / × ραβ r + , r − , t , (18) spin-1 2 particles in an ultrastrong magnetic field in the mean- 2 2 field approximation. In this approximation, the electric and for each spinor-space component of the density matrix, in- magnetic fields are generated by the sources via diced by α,β. Thus, we write out Eq. (2) and use the identities ∇ · E = ρ f , and ∇ × B = j f + ∂t E, (26) [34] where ρ and j are the free charge and current density, f f F[−ih¯∇r − qA(r)]ραβ (r, r ) respectively, → − / ∇ + μ × ∇ αβ , , F[p ih¯ 2 r im BB p]W (r p) (19) 3 ρ f = q d pW±, (27) ± ∇ − ρ , F[ih¯ r qA(r)] αβ (r r ) 3 1 → + / ∇ − μ × ∇ , . j = q d p pW±. (28) F[p ih¯ 2 r im BB p]Wαβ (r p) (20) f ± ± Using the Hamiltonian in (17) and keeping up to first order in Under the assumptions we have made, the bound sources ∇ as we did in the derivation of the Hamiltonian, the kinetic r arising from the spin are negligible, but these and have been equation is included in other models, e.g., Refs. [7,11]. A more thorough 1 1 discussion of our model, including comparison with previous ∂ Wαβ + p · ∇ Wαβ + q E + p × B · ∇ Wαβ = , t r p 0 works [8–13,36–38] is found in Sec. VI. (21) where B is the full magnetic field and D. Conservation laws = m2 + p2 − 2mμ σ · B − m2μ2 (B ×∇ )2. (22) To check the validity of the derived model, we derive the B 0 B 0 p conservation law of energy and the mass continuity. Starting Also note that is a function of σ and that the momentum with the mass continuity, the number density of spin-up (spin- 3 derivatives act on everything to the right of the operator. Thus, down) particles n± can be given by n± = d pW±.Toshow / that this quantity is conserved we take the time derivative of in the second and fourth terms of (21), 1 acts also on p.In order to get a scalar theory, we can Taylor-expand around σ it and use Eq. (25). Since is independent of r,itistrivialto show that = 1 + I + 1 − σ , 2 ( + −) 2 ( + −) z (23) 1 ∂ + ∇ · 3 = . t n± r d p pW± 0 (29) where I is the identity matrix and ± The number densities are separately conserved because transi- = 2 + 2 ∓ μ − 2μ2 ×∇ 2. ± m p 2m BB0 m B(B0 p) (24) tions between spin-up and spin-down states require absorption or emission of quanta with energies on the order of m.Thisis Next, we note that if initially Wαβ has no off-diagonal ele- based on the assumption that μBB0/m ∼ 1 holds, since this ments, let us say that W = W+, W = W−, then the evolution 11 22 is the regime where the strong-field quantum effects of our (21)forW+ and W− will decouple into separate equations for model are most significant. Note that this is closely related to the spin-up and spin-down populations, as defined relative to the approximation made when deriving Eq. (25). B . While limiting ourselves to such initial conditions may 0 Moving to the conservation of energy, the total energy seem unwarranted, the only thing left out by this restriction is density is the spin precession dynamics. However, since the timescale 1 for spin precession is the inverse Compton frequency, we = 2 + 2 + 3 . note that the present theory, based on the assumption ∂/∂t Etot (E B ) d p ±W± (30) 2 ± c/Lc, is not designed to resolve the spin precession dynamics anyway. Hence, from now on, we will be using the above We want now to show that the energy is conserved, taking representation for Wαβ, in which case (21) decouples into the the time derivative of Etot, and using Maxwell’s equations

043203-4 KINETIC THEORY FOR SPIN-1/2 PARTICLES IN … PHYSICAL REVIEW E 102, 043203 (2020) together with the kinetic equation, we get Dirac equations for individual particle states have the same spatial dependence for the wave function, we can adopt the ∂ E + ∇ · K = 0, (31) t tot r expression for the Wigner function from Ref. [7] (based on where K is the energy flux the Pauli equation) with some relatively minor adjustments. (1) Contrary to Ref. [7], we have made no Q transform 3 K = E × B + d p pW±. (32) to introduce an independent spin variable, and thus the spin- ± dependence of Ref. [7] reduces to W±. This is precisely what one would expect: the Poynting vector (2) The Wigner function of Ref. [7] must be expressed in 2 + 2 / → 2 + 2 / for the fields and the kinetic energy flux for the particles as terms of the momentum, i.e., m(vx vy ) 2 (px py ) 2m. required in a relativistic theory. Since we have neglected po- (3) The nonrelativistic energy of Ref. [7] is replaced by larization and magnetization, there is no Abraham-Minkowski the relativistic expression Eq. (33) of the Dirac theory. dilemma in this model; see Ref. [13] and references therein for (4) The normalization of the Wigner function must be a related discussion. adopted to fit the present case. With these changes, the Wigner function for an energy eigenstate Wn± can be written IV. BACKGROUND WIGNER FUNCTION IN 0 = − nφ , A CONSTANT MAGNETIC FIELD Wn± gn,±(pz )( 1) n(p⊥) (34) In principle we can compute the time-independent solu- where = tions for W in a constant magnetic field B B0zˆ by solving 2 2 p⊥ 2p⊥ the Dirac equation for this geometry, making a sum over dif- φ (p⊥) = exp − L , (35) n ω2 n ω2 ferent particle states and then perform the Foldy-Wouthuysen mh¯ ce mh¯ ce and Wigner transformations of Sec. III C. Except for the where Ln denotes the Laguerre polynomial of order n and Foldy-Wouthuysen transformation, this was done in a covari- gn±(pz ) is a function that is normalizable but otherwise ar- ant approach in Ref. [36]. However, here we will take a shorter bitrary. The number of particles in each Landau quantized route to arrive at the same results. Noting that for a constant eigenstate n0n,± obeys the condition magnetic field, both the Dirac equation and the Pauli equation results in electrons obeying a quantum harmonic oscillator n 3 n ± = g ±(p )(−1) φ (p⊥) d p equation, we can make a trivial generalization of the Pauli 0n n z n case [7]. First we need to find the Wigner function correspond- ⇒ ing to the Landau quantized energy eigenstates. Both for the Pauli and the Dirac equations, the spatial dependence of the (2πh¯)3 n ± = g ±(p )dp . (36) wave function in Cartesian coordinates can be expressed as 0n 2 n z z a Hermite polynomial times a Gaussian function [39] only 0 the energy eigenvalues are different. Specifically, applying Naturally, a general time-independent solution W± to the Dirac theory, the energy of the Landau quantized states Eqs. (25) can be written as a sum over the energy eigenstates become (34), according to ω 2 0 = 0 = − nφ . h¯ ce pz W± Wn± gn,±(pz )( 1) n(p⊥) (37) E ± = m 1 + (2n + 1 ± 1) + , (33) n m m2 n n where n = 0, 1, 2,..., corresponds to the different Landau That the factor φn(p⊥) gives us the proper Wigner function levels for the perpendicular contribution to the kinetic energy, for the Landau quantized eigenstates can be confirmed by the index ± represents the contribution from the two spin an independent check. Since the expression (40) contains no 2 states, and the term proportional to pz gives the continuous dependence on the azimuthal angle in momentum space, we dependence on the parallel kinetic energy. Since the Pauli and can write

∂ 1 ∂ 2μ B p2 = + 2 / 2 − μ2 2 + ∓ B 0 + z , ± m 1 p⊥ m BB0 (38) ∂ p⊥ p⊥ ∂ p⊥ m m2

|qB | when ± acts on φn(p⊥). Computing ±φn(p⊥) by Taylor- ω = 0 where ce m is the electron cyclotron frequency, con- expanding the square root to inﬁnite order, using the ﬁrming that φn(p⊥) generates the proper energy eigenvalues properties of the Laguerre polynomials, and then converting for the perpendicular kinetic energy and the spin degrees of the sum back to a square root, it is straightforward to verify freedom. the relation Next we turn our attention to the case of a background state in thermodynamic equilibrium. Given the ω 2 1/2 h¯ ce pz energy eigenstates, we only need to apply Fermi-Dirac φ (p⊥) = m 1 + (2n + 1 ± 1) + φ (p⊥), ± n m m2 n statistics, in which case the total thermodynamic Wigner TB TB (39) function W = ± W± , including both spin states, is

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given by n n 2(−1) φ (p⊥) W TB = 0 n , (40) π 3 exp [(E ,± − μ )/k T ] + 1 (2 h¯) n,± n c B

where n0 is the total electron number density of the plasma, μc is the chemical potential, and T is the temperature. Naturally, the expressions presented above are of most significance for relativistically strong magnetic fields, when Landau quantization is pronounced. As a consequence, the formula (40) will reduce to more well-known expressions when the limith ¯ωce/m 1 is taken. Specifically, (40) will become a relativistically degenerate Fermi-Dirac distribution in case we let T = 0 and μc = EF h¯ωce, where EF is the Fermi energy. Alternatively, for kBT EF and kBT h¯ωce, (40) reduces to a Synge-Juttner distribution. To give a concrete illustration, in Fig. 1 we have made a bar chart for the normalized number density n0n±/n0 in the different energy states for a few values of the temperature and magnetic field, under the assumption that the density is low enough for the system to be nondegenerate, i.e., assuming T TF . As will be demonstrated in the next section, to a large degree the electrons behave as a multispecies system, where each particle species has its own rest mass, as given by Eq. (33) but with pz = 0. This is because the separation between Landau levels is on the order of the rest mass, and all excitation quanta with energies of that order have been neglected. Noting that the effective number density of each “species” (discrete energy state) is given by − nφ n0 3 2( 1) n(p⊥) ± ≡ , n0n 3 d p (41) (2πh¯) exp [(En± − μc )/kBT ] + 1

we see that n0n± essentially will be determined by the Boltz- mann factors of (40). However, for the cases where our model Eq. (25) is of most interest, the magnetic ﬁeld B0 is strong enough to make relativistic Landau quantization prominent. Thus, in the next section and for the remainder of this paper, we will consider background distributions W0± where simpli- ﬁcations based onh ¯ωce/m 1 do not apply.

V. LINEAR WAVES FIG. 1. The normalized number density for different energy β = μ / The operator in the square root in Eq. (25)givestheim- states Em at different values of the parameters BB0 m and τ = / pression that it is very technical and complex to apply the kBT m. model in studying, e.g., waves in plasma. In this section we consider electrostatic waves in a homogeneous plasma by 0 = been shown to be energy eigenvalues when acting on W±,we using Eq. (25). We consider the wave vector k kez and −1 act on both sides by (ω − kp / ±) , such that the perturbed express the momentum p in cylindrical coordinates p⊥, ϕp, z distribution function becomes and pz. To linearize Eq. (25), we separate variables according = 0 , + 1 , , , = = to W± W±(p⊥ pz ) W±(z p⊥ pz t ), E E1ez, and B −iqE ∂W 0 −iqE ∂W 0 1 = ± = n± , B0ez, where the subscripts 0 and 1 denote unperturbed and W± (43) ω − kp / ∂ p ω − kp /E ± ∂ p perturbed quantities, respectively. Moreover, the perturbed z ± z n z n z 1 = quantities follow the wave plane ansatz according to W± 0 ˜ 1 ikz−iωt where in the second equality we used that W± is a sum of W±e . Equation (25)isnow eigenfunctions of ±, and the summation is over the Landau 0 kp ∂W± levels indexed by n. Using Poisson’s equation, the dielectric ω − z W 1 =−iqE . ± 1 ∂ (42) tensor for the electrostatic case is ± pz 2 ∂ 0 0 q 1 Wn± The unperturbed Wigner function W± is in general given by a D(k,ω) = 1 + d3 p . (44) k ω − kp /E ± ∂ p static solution of the form (37). Since the operators in ± have n,± z n z

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Note that if we seth ¯ to zero in En± in the dielectric tensor, then expressions. Here, however, the eigenvalue equation ε W0 = the denominator in the second term of the dielectric tensor is En,±W0 determines the background state for a given Landau the same as for the relativistic Vlasov equation [40]. level (bar a constant for the number density), and there is no 0 The background distribution Wn± will be divided into its need to go outside the kinetic theory itself to find proper initial eigenstates, which for the case of thermodynamic equilibrium, conditions. (40), will depend on the temperature and the magnetic field, To avoid some technical difficulties related to the operator see Fig. 1. As a result, the dispersion relation (44) is that of orderings, we have here divided the magnetic field into an a relativistic multispecies plasma where each species has its ultrastrong but constant part (B0zˆ) and a fluctuating part δB. own rest mass, En±. This approach allows for the treatment of large classes of problems in magnetar atmospheres, for example linear and nonlinear wave propagating in homogeneous backgrounds, VI. SUMMARY AND DISCUSSION even up to relativistic wave amplitudes, as long as the flutuat- In the present paper we have derived a kinetic model for ing part fulfills μB|δB|m. plasmas immersed in a relativistically strong magnetic field, Of course, a full modeling of the magnetar surroundings, i.e., with a field strength of the order of the critical field. covering the dipole nature of the background source field, is Based on a Foldy-Woythausen transformation and a Wigner beyond the scope of such a theory. Moreover, in order to focus transformation, an evolution equation for the spin-up and on the physics due to ultrastrong magnetic fields, the present spin-down components W± has been found. Besides having theory excludes effects such as the magnetic dipole force, the two components, the main difference to a classical relativistic spin magnetization, and the spin-orbit interaction included in model is that the γ factor in that theory is replaced by an some previous models [10,11], which can be justified for the operator containing momentum derivatives. Since our theory long scale lengths and moderate frequencies that we focus is formulated in phase space, we stress that such operators are on here. In this context, however, it should be noted that fundamentally different from the operators of Hilbert space, omission of the spin-orbit interaction is closely related to a and to the best of our knowledge, this effect has not been seen correction term of the free current density (see, e.g., Eq. (21) in any previous quantum kinetic models. of Ref. [10]). While the additional term kept by Ref. [10]isa Comparing the Wigner function of our model with pre- small correction for the conditions studied in this paper, it can vious quantum kinetic models [8–13] and [36–38], we note contribute with currents perpendicular to B0 thatmaybeof that many of them focus on other physical effects, including importance for certain problems, specifically for geometries short scale effects (of the order the de Broglie length) and/or where the (otherwise larger) parallel currents vanish. This and spin dynamics. The latter type of processes is filtered out other possible extensions of the current theory is a project for in our case, due to the short timescale for spin precession future research. associated with strong fields. Most other works studying the To illustrate the usefulness of the present theory, we have Wigner function for electrons make the assumption that the computed the dispersion relation for Langmuir waves in a electromagnetic field strength is well below the critical field strong magnetic field for a relativistic temperature. To a large strength, and hence the strong-field effects of our paper are not extent, we find that the electrons behave as if they are divided included. A notable exception consists of papers based on the into different species. More concretely, each Landau level of Dirac-Heisenberg-Wigner (DHW) approach [36–38], where a the background plasma contributes to the susceptibility with kinetic theory is formulated based on a direct Wigner trans- a term similar to the classical relativistic expression, but with 1/2 formation of the Dirac equation. This is contrasted with our its own effective mass mn± = m[1 + (2n + 1 ± 1)¯hωce/m] . approach, where a Foldy-Wouthuysen transformation is made We expect this result to generalize to some other problems, but before the Wigner-transformation, in order to separate particle not be completely general, as for certain problems, the differ- states from antiparticle states. An obvious disadvantage with ence between the standard γ factor and the energy expression our approach, compared to the DHW approach, is that some of of the current theory will be apparent. A more complete study the physics of the Dirac equation, notably pair creation, is lost. of the effects due to ultrastrong magnetic fields is a project for For cases where the separation of particles and antiparticles future research. is applicable, however, the Foldy-Wouthuysen transformation leads to considerable simplifications. This is illustrated by the 16 coupled components of the Wigner function in the DHW APPENDIX: COMMUTATORS approach [37], as compared to the relative simplicity of the The Hamiltonian in (6) contains some commutators of present evolution Eq. (25), where the two spin components functions of operators. To calculate these commutators, we are decoupled. need to expand them in a series. We present here some of the As pointed out above, a noted feature of the present calculations that were done in order to obtain the result in (11). theory is that the energy becomes an operator, which has First, both Tˆ and ˆ are functions of Oˆ , since these functions consequences when studying the background Wigner func- can be expanded in a Taylor series of Oˆ , the commutator tion. Classically, or in less advanced quantum mechanical models, the evolution equation does not predict the detailed [Oˆ , Oˆ n] = 0, (A1) momentum dependence for a given Landau level. Thus the background expression has been put in by hand, and one thus, we have that has to return to the starting point of the theory (e.g., the Pauli or Dirac equations for single particles) to find proper [Tˆ , Oˆ ] = [ˆ ,Oˆ ] = 0. (A2)

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We can now rewrite (6)as where in the last equality, we neglected the commutator of ˆ and E j in accordance with the long-scale approximation. For 1 Hˆ = β ˆ + Eˆ + ([Tˆ , [Tˆ , Fˆ ]] − 2β[ˆ ,Fˆ ]Oˆ the commutator of ˆ andπ ˆ j, we need to expand it in a series. 2Tˆ However this time, we have a commutator of functions that + 2β[Oˆ , Fˆ ]ˆ + 2mβ[Oˆ , Fˆ ] − [ˆ ,[ˆ ,Fˆ ]] depend on bothp ˆ andr ˆ. Using the result from Ref. [33], we 1 expand the commutator in a series − [Oˆ , [Oˆ , Fˆ ]]) . (A3) ∞ Tˆ (ih¯)k ∂kπˆ ∂k ˆ ∂k ˆ ∂kπˆ ,π E = j − j E [ˆ ˆ j] j k k k k j k! ∂ ri ∂ pi ∂ ri ∂ pi Looking ﬁrst at the commutator of ˆ and Fˆ (the same result k=1 ˆ Fˆ can be used to the commutator of T and ), we have πˆ ≈ ihq · E × B , ¯ ( ) (A8) [ˆ ,Fˆ ] = [ˆ ,qφ(rˆ)] + ih¯∂t .ˆ (A4) ˆ where in the last equality we only kept up to k = 1inthe ,φ To calculate the commutator [ˆ (rˆ)], we expand the func- summation since higher-order terms vanish in the long-scale tions in the commutators in a series [33], approximation. Finally, we have ∞ (ih¯)k πˆ [ˆ ,φ(rˆ)] =− ˆ (k)φk (rˆ) [ˆ ,[ˆ ,Fˆ ]] = (ihq¯ )2 · (E × B). (A9) k! ˆ2 k=1 ˆ , ˆ , Fˆ ∂ ˆ Note that [T [T ]] is calculated in the same way, ≈− ∇ φ , ih¯ i (rˆ) (A5) 2 ∂πˆ i 1 m [Tˆ , [Tˆ , Fˆ ]] = (ihq¯ )2 2 + πˆ · (E × B). (A10) where in the last equality, higher derivative terms were ne- Tˆ ˆ glected in accordance with the long-scale approximation. Now we will calculate the commutator of Oˆ and Fˆ Thus, we have [Oˆ , Fˆ ] = ihq¯ α · E. (A11) πˆ [ˆ ,Fˆ ] = ihq¯ · E. (A6) We did not need to do any approximation in calculating this ˆ commutator since it is linear in Oˆ . Finally, we calculate Next, we calculate the commutator [Oˆ , [Oˆ , Fˆ ]] = ihq¯ (αiπˆ iα jE j − α jE jαiπˆ i ) 1 =− ˆ · π × − × π , [ˆ ,[ˆ ,Fˆ ]] = ihq¯ [ˆ ,πˆ · E] hq¯ ( ˆ E E ˆ ) (A12) ˆ where in the last equality we have used = 1 ,π + π , ihq¯ ([ˆ ˆ j]E j ˆ j[ˆ E j]) α α = δ + ε . ˆ i j ij i ijk k 1 = ihq¯ [ˆ ,πˆ ]E , (A7) Using (A6) and (A9)to(A12)in(A3), we get the Hamiltonian ˆ j j Hˆ in (11).

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