Modified plasma waves described by a logarithmic electrodynamics
Fernando Haas Patricio Gaete Leonardo P. R. Ospedal José Abdalla Helayël-Neto







 fernando.haas@ufrgs.br
Instituto de Física, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, 91501-970 Porto Alegre, RS, Brasil

 patricio.gaete@usm.cl 
Departamento de Física and Centro Científico-Tecnológico de Valparaíso-CCTVal,
Universidad Técnica Federico Santa María, Valparaíso, Chile

 leoopr@cbpf.br
Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, RJ, Brasil 

helayel@cbpf.br
Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, RJ, Brasil 





The propagation of plasma waves in a new non-linear, logarithmic electrodynamics model is performed. A cold, uniform, collisionless fluid plasma model is applied.  Electrostatic waves in magnetized plasma are shown to correspond to modified Trivelpiece-Gould modes, together with changes of the plasma and upper-hybrid frequencies, driven by the logarithmic electrodynamics effects. Electromagnetic waves are described by a generalized Appleton-Hartree dispersion relation. The cases of propagation parallel or perpendicular to the equilibrium magnetic field are analyzed in detail. In particular, generalized ordinary and extraordinary modes are obtained. We determine the changes, due to logarithmic electrodynamics, in the allowable and forbidden frequency bands of the new extraordinary mode. Estimates are provided about the strength of the ambient magnetic field, so that the non-linear electrodynamics effects become decisive. 




 
12.60.Cn, 26.60.Gy, 52.35.Fp, 52.35.Mw 

Keywords: logarithmic electrodynamics, electrostatic plasma waves, Appleton-Hartree equation, modified extraordinary mode.


Modified plasma waves described by a logarithmic electrodynamics
    J.A. Helayël-Neto
    December 30, 2023
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§ INTRODUCTION



In 1923, the experiments by Compton on the scattering of X-rays off electrons demonstrated that Einstein's light quanta carry not only energy, but also momentum, making it clear that they behave as true particles <cit.>. It was Lewis, in his 1926 article, The Conservation of Photons <cit.>, who coined the word photon to name the particles of light, though, to his sense, the photon was understood as a sort of atom of light. One year later, in the 1927 Solvay Meeting, entitled Electrons and Photons, Compton used the term photon as we understand it today <cit.>. 

Ever since, there started a broad activity aimed to study the light by light scattering, once the light quanta - the photons - were then understood as genuine elementary particles and, therefore, they could scatter each other. In 1930 and 1931, a series of papers was devoted to detect the collision between photons and to check whether or not the superposition principle was respected <cit.>. The perception was that the photon-photon scattering could unveil non-linear effects in the electromagnetic theory once deviations from the superposition principle were detected. However, only in 1933, a theoretical investigation, based on the 1928 Dirac's theory for the electron <cit.>, was proposed by Otto Halpern <cit.>, who claimed that virtual electron-positron pairs could be the actual origin of photon-photon collisions. This short - but deep and consequent - work provided a more qualitative framework to be applied in the description of the photon-photon scattering. Halpern's paper opened up a very intensive line of investigation in the immediate following years. Later on, in 1935, Euler and Kockel, both Heisenberg's students, based on the early developments of Quantum Electrodynamics (QED), derived the leading non-linear corrections to the Maxwell equations in vacuum <cit.>. The years 1933-1936 were a rich period for the inspection of non-linear electrodynamic models, when the Delbrück scattering <cit.>, the Breit-Wheeler effect <cit.>, the spontaneous decay of photons <cit.> and the Euler-Heisenberg <cit.> and Born-Infeld <cit.> models were investigated. Photon-photon scattering and the detection of physical phenomena in vacuum as a consequence of non-linear electrodynamics (NLED) remain a challenge up to present days for both theoretical and experimental physics <cit.>.

More recently, there has been a renewed interest in NLED in connection with new Physics beyond the Standard Model, specially the possibility of detecting non-linear electromagnetic vacuum effects induced by quantum gravity corrections to Maxwell electrodynamics <cit.>. In connection with black holes, one has focused on different NLED models to get a broad class of singularity-free black hole solutions <cit.>. In a recent paper <cit.>, a logarithmic electrodynamic action has been presented and inspected to analyze the thermodynamic implications of NLED on an AdS black hole solution. The present paper sets out to pursue an investigation of this particular (logarithmic) NLED in the context of a plasma, as it shall be described in the sequel.


The propagation of electromagnetic waves in a logarithmic electrodynamics context has already been derived in free space <cit.>. However, the analysis of wave propagation in a material medium governed by logarithmic electrodynamics has not yet been carried out. In a first approach to this subject, the present work considers a cold, non-relativistic, collisionless plasma, composed by electrons inn a fixed homogeneous ionic background. The choice of the simplest possible plasma allows to investigate the main new effects induced by logarithmic electrodynamics on some of the most salient plasma waves. It would be impossible to perform a similar analysis for all relevant plasma waves. For instance, in this first attempt, magnetohydrodynamic waves or waves taking into account the ions response (e.g. ion-acoustic waves) are not yet addressed. Similarly, high amplitude, non-linear waves are not considered here. Nevertheless, a rich variety of logarithmic- electrodynamics-driven essential new aspects of wave propagation in plasmas shall be identified. A basic result common to all classes of waves considered in our analysis is that logarithmic electrodynamic effects crucially depend on the parameter c  B_0/β, where c is the speed of light, B_0 is the equilibrium magnetic field and β is a fundamental parameter of the model, which we presently consider to be positive. The general result can be of help for the determination of β, which is here taken as a phenomenological parameter within logarithmic electrodynamics <cit.>. 

This work is organized according to the outline that follows. To go through a logical path, we start in Section II from the simplest plasma waves, viz. electrostatic waves, in both unmagnetized and magnetized plasma. In Section III,  electromagnetic perturbations are allowed, which are known to be described by the Appleton-Hartree equation <cit.>, for arbitrary propagation angle. Here, a modified Appleton-Hartree equation is obtained, with changes driven by the logarithmic electrodynamics. In Section IV, we treat the special case of waves propagating along the direction of the ambient magnetic field. In Section V, the perpendicular wave propagation is addressed to, along with the derivation of the corresponding modified ordinary and extraordinary modes. Section VI shows some physical estimates on the strength of the effects stemming from logarithmic electrodynamics. In Section , we cast our Concluding Comments.  



§ ELECTROSTATIC WAVES


According to <cit.>, logarithmic electrodynamics is described  by the following Lagrangian density


    ℒ = - 2 β^2 ε_0  ln[ 1 + 1/β √(s/2 ( c^2 | B|^2 - | E|^2 ) ) ]   
    + 2 β ε_0  √(s/2 ( c^2 | B|^2 - | E|^2 ) )  ,


where E and B denote the electric and magnetic fields, respectively. In addition, β is a fundamental physical parameter, c is the speed of light and s is a number such that s = -1 for | E| ≥ c | B| and s = 1 for c | B| > | E|.


The corresponding field equations, in the presence of charge and current densities ρ and j, read as below:


    ∇· D   =   ρ/ε_0  , 
    ∇× E   =    - ∂ B/∂ t  , 
    ∇· B   =    0   , 
    ∇× H   =   μ_0   j + 1/c^2∂ D/∂ t  ,


where ε_0 is the vacuum permittivity, μ_0 the vacuum permeability and the auxiliary fields D and H are given by


    D = √(2)β E/√(2)β + √(-s(| E|^2-c^2 | B|^2)) ,    H = √(2)β B/√(2)β + √(-s(| E|^2-c^2 | B|^2))  .




In a first approach, let us focus on the  electrostatic case, using fluid theory in the cold plasma limit, 


    ∂ n/∂ t + ∇·(n u)    =    0  , 
    ∂ u/∂ t +  u·∇ u   =    - e/m( E +  u× B_0)  , 
    ∇· D   =   e/ε_0 (n_0 - n)  .


Here, n is the electrons' number density, u is the electrons' fluid velocity field, -e and m are respectively the electron charge and mass, B_0 is the equilibrium magnetic field and n_0 is the ions' background number density (with atomic number Z = 1). For simplicity, ions are supposed to be infinitely massive, which is appropriate for high frequency waves. Likewise, thermal or collisional effects are also disregarded. 

Linear electrostatic waves consider n = n_0 + δ n, u = δ u and E = δ E, where δ n, δ u and δ E are first order plane wave perturbations proportional to exp[i( k· r - ω t)], where k is the wave vector and ω is the (angular) wave frequency. Magnetic field perturbations shall start to be studied in Section III. Hence, at this stage, the H field is not required. In passing, we note that for a cold plasma and for linear waves with zero equilibrium fluid velocity, a non-relativistic treatment is sufficient.

Initially, in the unmagnetized case (B_0 = 0), we have, from Eq. (<ref>), that δ D = δ E. Therefore, in this situation, we detect no changes due to the logarithmic electrodynamics, at least for small amplitude, linear waves. The usual (non-propagating) electron plasma wave with ω^2 = ω_p^2 is recovered, where ω_p = [n_0 e^2/(m ε_0)]^1/2 is the plasma frequency. 

In the magnetized case, one has | B_0| = B_0 > | E|/c, since the magnetic field is finite and the electric field is a perturbation. Hence, from Eq. (<ref>), we have s = 1 and 


    D = √(2)β E/√(2)β + √(c^2 B_0^2 - | E|^2) .


Linearizing Eqs. (<ref>)-(<ref>) and assuming B = B_0 ẑ,  k = k x̂,  E = δ E x̂ (longitudinal wave) yield 


    ω^2 = ω̃_h^2 = ω̃_p^2 + ω_c^2  ,


where ω_c = e B_0/m is the electron's cyclotron frequency, 


    ω̃_̃p̃ = ω_p (1 + c B_0/√(2)β)^1/2


is a modified plasma frequency and ω̃_h is a modified upper-hybrid frequency.  In the limit B_0 ≪β, one re-obtains the usual upper hybrid wave ω^2 = ω_h^2 = ω_p^2 + ω_c^2. Otherwise, Eq. (<ref>) shows a modified upper hybrid wave due to log effects. 

With more generality, propagation in an arbitrary direction so that k∥ E but with k = k (sinθ, 0 , cosθ) gives a modified Trivelpiece-Gould dispersion relation, 


    ω^4 - (ω̃_p^2 + ω_c^2) ω^2 + ω̃_p^2ω_c^2cos^2θ = 0  ,


yielding


    ω^2 = 1/2[ω̃_p^2 + ω_c^2 ±((ω̃_p^2 - ω_c^2)^2 + 4 ω̃_p^2 ω_c^2 sin^2θ)^1/2]  .


For θ = π/2, we recover Eq. (<ref>). The modes following from Eq. (<ref>) are always stable (ω^2 ≥ 0). In the low field case c B_0 ≪β, we recover the usual Trivelpiece-Gould modes. Notice that the original work by Trivelpiece and Gould <cit.> has considered wave propagation along an arbitrary angle, see also Fig. 4.21 in <cit.>. 






§ GENERALIZED APPLETON-HARTREE EQUATION


We now turn to electromagnetic electron waves in cold, collisionless magnetized plasma in a fixed homogeneous ionic background. The relevant equations are the Faraday's law, Eq. (<ref>), the force equation, namely,


    ∂ u/∂ t +  u·∇ u = - e/m( E +  u× B)  ,


and the modified Ampère-Maxwell law, Eq. (<ref>), which can be recast as 


    ∇× H = - μ_0 n e  u + 1/c^2∂ D/∂ t  .



Following the usual procedure <cit.>, we are going to linearize around the homogeneous equilibrium n = n_0,  u = 0,  E = 0,  B =  B_0≠ 0. In passing, notice that from Eq. (<ref>) for B_0 = 0 gives δ D = δ E, δ H = δ B, whatever the sign of s, so that no logarithmic effects show up in this case, which is associated to the standard electromagnetic plasma wave ω^2 = ω_p^2 + c^2 k^2 for k· E = 0 and to the electron plasma mode ω^2 = ω_p^2 for k· E≠ 0. 

As in the previous Section, in the magnetized case for linear waves, we must set s = 1. We have 


    D = √(2)β E/√(2)β + √(c^2 | B|^2 - | E|^2) ,    H = √(2)β B/√(2)β + √(c^2 | B|^2 - | E|^2) .


Assuming n = n_0 + δ n,  u = δ u,  E = δ E,  B =  B_0 + δ B, for small amplitude perturbation, yields, in particular,


    δ D = H_0/B_0δ E ,    H =  H_0 + δ H ,


where H_0 = | H_0|, 


    H_0 = √(2)β B_0/√(2)β + c B_0 ,   δ H = H_0/B_0(δ B - c  B_0   H_0·δ B/√(2)β B_0)


and 


    H_0/B_0 = √(2)β/√(2)β + c B_0 .


The parameter H_0/B_0 ≤ 1 plays a significant rôle. For instance, from Eq. (<ref>), we have ω̃_p = (B_0/H_0)^1/2ω_p. 

Assume plane wave perturbations proportional to exp[i( k· r - ω t)], for B_0 = B_0ẑ, k = k (sinθ, 0, cosθ), so that θ is the angle between k and the equilibrium magnetic field. The linear wave analysis in this context is well-known <cit.>. We shall isolate the linearized velocity field δ u = (δ u_x, δ u_y, δ u_z) from Eq. (<ref>) in terms of δ E = (δ E_x, δ E_y, δ E_z), yielding as usual 


    δ u_x = e/m(ω_c δ E_y + i ωδ E_x)/(ω_c^2 - ω^2) ,   δ u_y = e/m(-ω_c δ E_x + i ωδ E_y)/(ω_c^2 - ω^2) ,   δ u_z = - i e δ E_z/mω .


Using k×δ E = ωδ B and inserting the results from Eq. (<ref>) into the modified Ampère-Maxwell law (<ref>) yields a linear homogeneous system for the electric field components, 


    [ S - η^2cos^2θ         -i  D   η^2cosθsinθ;          i  D   S - α(θ)η^2             0;   η^2cosθsinθ             0 P - η^2sin^2θ ][ δ E_x; δ E_y; δ E_z ] = 0  ,


where Eqs. (<ref>) and (<ref>) were also needed. Here, η = ck/ω is the refraction index, α(θ) = cos^2θ + (H_0/B_0)sin^2θ  and 


    D = ω_c ω̃_p^2/ω(ω_c^2-ω^2) ,    S = 1 + ω̃_p^2/ω_c^2-ω^2 ,    P = 1 - ω̃_p^2/ω^2 .


are modified D (Difference), S (Sum) and P (Plasma) coefficients. 
In the limit H_0/B_0 → 1, one recovers the usual result <cit.>. 
Setting P = 0 we get ω^2 = ω̃_p^2, while setting S = 0 yields the modified upper-hybrid wave in Eq. (<ref>). 

The determinant of the matrix in Eq. (<ref>) must vanish. Following, as closely as possible the traditional notation <cit.>, we find 


    A η^4 - B η^2 + C = 0  ,


where 


    A    =   α(θ) ( Ssin^2θ +  Pcos^2θ)  , 
    
    B    =    R Lsin^2θ +  P S  [α(θ) +  cos^2θ]  ,
    
    C    =    P R L ,    R =  S +  D ,    L =  S -  D ,


also introducing modified R (Right) and L (Left) coefficients. 

After standard rearrangements,  
the solution to Eq. (<ref>) can be expressed as 


    η^2 = 1 -  2(A-B+C)/2A-B ±√(B^2-4 A C) ,

or 


    η^2 = 1 - ω̃_p^2/ω^2/Q ,    Q = Q_0 ± F/Q_1


where 


    Q_0    =    1 + (1-2α(θ))ω_c^2ω̃_p^2sin^2θ+(α(θ)-1)ω_c^2(ω^2+ω̃_p^2)-(α(θ)-1)(ω^2-ω̃_p^2)^2/2((ω^2-ω̃_p^2)(ω̃_p^2+(α(θ)-1)ω^2)-(α(θ)-1)ω_c^2ω^2sin^2θ) , 
    
    Q_1    =   [(2-α(θ))ω^2-ω̃_p^2]ω̃_p^2 + (α(θ)-1)ω^4 - (α(θ)-1)ω_c^2ω^2sin^2θ/(ω^2-ω_c^2)ω^2/2 ,
    
    F^2    =    [ R L- (H_0/B_0) P S]^2 sin^4θ + 4 α(θ)  P^2  D^2 cos^2θ .



In the low field limit H_0/B_0 → 1, one has 


    Q = 1 - ω_c^2 sin^2θ/2(ω^2-ω_p^2)±(ω_c^4 sin^4θ/4(ω^2-ω̃_p^2)^2 + ω_c^2 cos^2θ/ω^2)^1/2 ,


which is the standard result <cit.>. 


Equation (<ref>) is a generalized Hartree-Appleton equation, describing electromagnetic wave propagation in cold, uniform  plasma governed by a logarithmic electrodynamics. In the low field limit, the usual Appleton-Hartree equations <cit.> are recovered, in the collisionless case. Since ion motion was neglected, to apply Eq. (<ref>) the frequency must be large in comparison with the ion cyclotron frequency.

Since the analysis of Eq. (<ref>) is quite involved in general, we consider the special cases of propagation parallel or perpendicular to the ambient magnetic field. 




§ PROPAGATION PARALLEL TO B_0


For θ = 0, we have α(θ) = 1, irrespective of the strength of B_0. We also find, from Eq. (<ref>), that Q = 1 ±ω_c/ω, leading to a modified right-hand circularly polarized (RCP) wave,


    c^2 k^2/ω^2 = 1 - ω̃_p^2/ω(ω-ω_c)


and a modified left-hand circularly polarized wave (LCP) wave, 


    c^2 k^2/ω^2 = 1 - ω̃_p^2/ω(ω+ω_c)


where the only change in comparison with the usual RCP and LCP waves is the replacement ω_p →ω̃_p. 
The properties of RCP and LCP waves are well-known <cit.> and are reproduced here provided the larger effective plasma frequency is used. 

However, the parallel propagation case admits another possibility, since for θ = 0 all three coefficients A, B, C in Eq. (<ref>) become proportional to P. Setting P = 0, one regains the modified electron plasma oscillations ω^2 = ω̃_p^2. 



§ PROPAGATION PERPENDICULAR TO B_0


For θ = 90^∘, one has α(θ) = H_0/B_0 and two wave modes, described below. 



 §.§ Modified ordinary mode


Taking the minus sign in Eq. (<ref>), we have Q = 1 and ω^2 = ω̃_p^2 + c^2 k^2, which is the ordinary (O) mode, modified by the presence of the new plasma frequency ω̃_p, instead of the usual one. With this proviso, the standard analysis of the O-mode applies <cit.>. 



 §.§ Modified extraordinary mode


Taking the plus sign in Eq. (<ref>), we are left with an involved expression of Q and the dispersion relation


    H_0/B_0c^2 k^2/ω^2 = (ω^2-ω̃_R^2)(ω^2-ω̃_L^2)/ω^2(ω^2- ω̃_h^2) ,


where


    ω̃_R = 1/2(ω_c+(ω_c^2+4ω̃_p^2)^1/2)  ,   ω̃_L = 1/2(-ω_c+(ω_c^2+4ω̃_p^2)^1/2)  .


and where ω̃_h is the modified upper-hybrid frequency defined in Eq. (<ref>). 
Apart from the overall factor H_0/B_0 on the left-hand side of Eq. (<ref>), we have the same result as the dispersion relation of the well-known extraordinary (X) mode, with the replacements ω_p →ω̃_p, ω_h →ω̃_h. 

The analysis of the modified X-mode is more involved than for the other waves considered so far, but it may be performed with the aid of the definitions of cutoff and resonance. We recall <cit.> that a cutoff happens whenever the refraction index, η, goes to zero, while a resonance occurs if η becomes infinity. In general, a wave is absorbed at a resonance and reflected at a cutoff. From Eq. (<ref>), we have that the location of cutoffs and resonances for the modified X-mode are the same as for the standard case, but in terms of the new plasma and upper-hybrid frequencies. Therefore, the standard analysis apply, with the net result shown in Fig. <ref>. Notice that the X-wave only propagates for ω̃_L < ω < ω̃_H or for ω > ω̃_R. 



It is apparent, from Eq. (<ref>), that ω̃_L increases due to the logarithmic electrodynamic effects, with the corresponding increase of the left forbidden band in Fig. <ref>. In addition, for the allowed band ω̃_L < ω < ω̃_h, one has


    ω̃_h - ω̃_L = ω_h - ω_L - Δω_p^2 (1/√(ω_c^2+4ω_p^2)-1/2√(ω_c^2+ω_p^2)) +  O(Δ^2)  ,   Δ = c B_0/√(2)β ,


while for the forbidden band  ω̃_h < ω < ω̃_R  one has


    ω̃_R - ω̃_h = ω_R - ω_h + Δω_p^2 (1/√(ω_c^2+4ω_p^2)-1/2√(ω_c^2+ω_p^2)) +  O(Δ^2)  ,


where ω_R,L is the limit of ω̃_R,L in standard electrodynamics. The correction terms in Eq. (<ref>) and (<ref>) are always negative (positive), showing a smaller allowed band and bigger forbidden band due to the log effects. The conclusion can be shown to be true for an arbitrary order too. 

In particular, for strongly magnetized plasmas, where ω_c ≫ω_p, one has, up to O(Δ), 


    ω̃_h - ω̃_L = ω_c (1-ω_p^2/2ω_c^2) - Δω_p^2/2ω_c ,   ω̃_R - ω̃_h = ω_p^2/2ω_c(1 + Δ)   .


For convenience, we recall that ω_p^2/ω_c^2 = μ_0 m n_0 c^2/B_0^2 = 1.02  n_0/B_0^2, so that Eq. (<ref>) requires B_0 ≫√(n_0), always in terms of S. I. units. 

Similarly, for high density plasmas, such that ω_p ≫ω_c, one has


    ω̃_h - ω̃_L = ω_c (1+3ω_c/4ω_p) - 3Δω_c^2/16ω_p ,   ω̃_R - ω̃_h = ω_c/2(1 -  3ω_c/4ω_p) + 3Δω_c^2/16ω_p ,


up to O(Δ).



§ PHYSICAL ESTIMATES


We may present at least an estimate of the strength of the new effects, shown in the previous Sections, concerning wave propagation in plasma. We notice that the main changes come from the parameter 


    ω̃_p/ω_p = (1 + c B_0/√(2)β)^1/2 .


If we estimate β/c ∼ 10^11  T, we find β∼ 10^19 V/m, which is one order of magnitude larger than the Schwinger limit m^2 c^3/(eħ) ∼ 10^18 V/m, where ħ is the reduced Planck's constant. Therefore, significant changes in plasma wave propagation may occur, for strongly magnetized plasma with B_0 ∼ 10^11  T. Although the result has been obtained in the cold, uniform, collisionless and non-relativistic approximation, it is expected that the main conclusion remains true for more general plasmas, together with additional features. 

A possible candidate for such strongly magnetized plasmas would be the surface of a magnetar, with magnetic fields B ∼ 10^10-10^12  T, where standard atomic nuclei compose a solid lattice together along with a sea of electrons <cit.>. However, in this case, more involved and detailed models would be necessary, with a more appropriate equation of state in connection with quantum and general- relativistic effects <cit.>. Nevertheless, the relevance of the parameter ω̃_p/ω_p will certainly show up also in a more detailed treatment. Ultra-strong magnetic fields beyond the QED limit are also found in other extreme astrophysical environments,  such as the interior of neutron stars, Central Engines of Supernovae and Gamma-Ray Bursts (GRB) and inner parts of GRB jets <cit.>. Finally, the attainability of the Schwinger limit in laboratory with extreme power lasers should be mentioned <cit.>.





§ CONCLUDING COMMENTS AND NEW PROSPECTS


In a first attempt to analyze plasma waves in a logarithmic electrodynamics, we have chosen a cold uniform plasma immersed in a magnetic field, not only because of the mathematical tractability, but also because a large number of wave modes in more detailed treatments may be associated with the modes obtained within the simplest approach. In the words by Stix <cit.>, “the 
cold-plasma model gives, in fact, a remarkably accurate description of the common small-amplitude perturbations that are possible for a hot plasma". 

Within the cold-plasma model, we have found a modified Trivelpiece-Gould dispersion relation for electrostatic waves in magnetized plasma, along with adequate changes of the plasma and upper-hybrid frequencies, due to the logarithmic electrodynamics effects. Allowing for electromagnetic waves yields a generalized Appleton-Hartree dispersion relation. The cases of propagation parallel or perpendicular to the equilibrium magnetic field are analyzed in detail, so that, in particular, generalized ordinary and extraordinary modes are found.  We determined the changes, due to logarithmic electrodynamics, in the allowable and forbidden frequency bands of the new extraordinary mode. 
According to physical estimates, non-linear electrodynamics effects become unavoidable for ultra-strong magnetic fields, which exist in extreme astrophysical plasma environments. Such developments are relevant for the determination of the β parameter entering the basic new NLED Lagrangian density. 



NLED models appear as effective photonic descriptions that take into account the sum over the quantum effects of virtual (charged) particle-antiparticle pairs. So, from the very onset, they are supposed to correctly describe electromagnetic effects associated to waves whose (wave)lengths are much bigger than the Compton wavelength of the charged particle whose quantum effects have been integrated over. In the case of electronic matter, we are talking about typical frequencies higher than 10^20 Hz. So, our results for the logarithmic electrodynamics give a good description in this region of frequencies.

There is a number (around 15) of NLED models in the literature. Among those, Born-Infeld electrodynamics is widely studied in many scenarios and, besides eliminating the singularity in the electric sector, it exhibits a more involved structure of electric and magnetic fields than the logarithmic model we have contemplated here, for it also involves the Lorentz-invariant quantity E· B. So, as a further step, we wish to consider Born-Infeld model in a plasma medium, and this investigation would allow a new estimation of the Born-Infeld parameter, based on Plasma Physics. In the case of Born-Infeld and those models for which E· B is present, the D- and H-fields are respectively modified by the additions of a term in B and E. This brings about a new feature that is not contemplated in our present contribution. We shall be soon reporting on that elsewhere,  providing new insights about the existing connections <cit.> between Born-Infeld theory and wave propagation at arbitrary angles, now in the context of a warm magnetized plasma.


Acknowledgments: 
F. H. acknowledges the support by Con­se­lho Na­cio­nal de De­sen­vol­vi­men­to Cien­tí­fi­co e Tec­no­ló­gi­co (CNPq). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. LPRO is supported by the PCI-DB funds (CNPq/MCTIC). He expresses his gratitude to COSMO - CBPF for the friendly hospitality. P. G. was partially supported by Fondecyt (Chile) grant 1180178 and by Proyecto Basal FB0821. 




99



a A. H. Compton, Phys. Rev. 22, 409 (1923).
b G. N. Lewis, Nature, No. 2981 118, 874 (1926).
c A. H. Compton, Discordance between the Experiment and the Electromagnetic Theory of Radiation, talk published in the Proceedings Electrons and Photons of the 1927 Solvay Meeting, Eds. Gauthier-Villars et cie., Paris, 1928.
d G. E. M. Jauncey and A. L. Hughes, Phys. Rev.  35 1439, (1930);
A. L. Hughes and G. E. M. Jauncey, Phys. Rev. 36, 773 (1930);
S. Vavilov, Phys. Rev. 36, 1590 (1930);
A. K. Das, Phys. Rev. 37, 94 (1931).
e P. A. M. Dirac, Proc. Roy. Soc. London A 117, 610 (1928);  118, 351 (1928);  126, 360 (1930).
f O. Halpern, Phys. Rev. 44, 855 (1933).
g H. Euler and B. Kockel, Die Naturwissenschaften 23, 246 (1935);
H. Euler, Annalen der Physik 5,  Series 26:5, 398 (1936).
h M. Delbrück, Zeit. für Phys. 84, 144 (1933), addendum to the article by L. Meitner and H. Kösters published in Zeit. für Phys. 84, 137 (1933).
i G. Breit and J. A. Wheeler, Phys. Rev. 46, 1087 (1934).
j M. Bronstein, Physikalische Zeitschrift der Sowjetunion 10, 686 (1936);
M. Bronstein, JTEP 7, 335 (1937). 
k W. Heisenberg and H. Euler, Zeit. für Phys. 98, 714 (1936).
This work may also be found in arXiv: physics/0605038 [physics.hist-phys].
l M. Born and L. Infeld, Proc. Roy. Soc. London A 144, 425 (1934).
MM M. Marklund, Nat. Photonics 4, 72 (2010).
m J. Bardeen, presented at GR5, Tiflis, U.S.S.R., and published in the Conference Proceedings in the U.S.S.R. (1968).
n E. Ayon-Beato and A. Garcia, Phys. Rev. Lett. 80, 5056 (1998);
K.A. Bronnikov, Phys. Rev. D 63, 044005 (2001);
K.A. Bronnikov, Int. J. Mod. Phys. D 27 1841005 (2018);
I. Dymnikova and E. Galaktionov, Class. Quantum Gravity 32, 165015 (2015);
S.H. Mehdipour and M.H. Ahmadi, Nucl. Phys. B 926, 49 (2018);
S.I. Kruglov, Ann. Phys. 378, 59 (2017);
S. Nojiri and S.D. Odintsov, Phys. Rev. D 96, 104008 (2017);
B. Toshmatov, Z. Stuchlik and B. Ahmedov, Phys. Rev. D 95, 084037 (2017);
Z.Y. Fan and X. Wang, Phys. Rev. D 94, 124027 (2016).
o X.M. Kuang, B. Liu and A. Övgün, Eur. Phys. J. C 78, 840 (2018).
arXiv P. Gaete, J. A. Helayël-Neto and L. P. R. Ospedal,  Coulomb's law modification driven by a logarithmic electrodynamics, arXiv:1811.05018v1 (2018).
AH E. V. Appleton, J. Inst. Elect. Engr. 71, 642 (1932).
TG A. W. Trivelpiece and R. G. Gould, J. Appl. Phys. 30, 1784 (1959). 
Chen F. F. Chen, Introduction to Plasma Physics and Controlled Fusion vol. I, 2nd. ed. (Plenum, New York, 1984).

Stix T. H. Stix, Waves in Plasmas (Springer, New York, 1992). 
Swanson D. G. Swanson, Plasma Waves (Academic Press, London, 1989). 
Bittencourt J. A. Bittencourt, Fundamentals of Plasma Physics 3rd. Ed. (Springer, New York, 2004). 
magnetar V. M. Kaspi and A. M. Beloborodov, Annual Rev. Astron. Astrophys. 55, 261 (2017).  
Shapiro S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars (Wiley, Weinheim, 2004). 
Uz D. A. Uzdensky and S. Rightley, Rep. Prog. Phys. 77, 036902 (2014). 
Bu S. S. Bulanov, T. Z. Esirkepov, A. G. R. Thomas, J. K. Koga and S. V. Bulanov, Phys. Rev. Lett. 105, 220407 (2010).
Trines  D. A. Burton, R. M. G. M. Trines, T. J. Walton and H. Wen, J. Phys. A: Math. Gen. 44, 095501 (2010).


