3 . 01 68 6 v 1 [ he pph ] 3 M ar 2 02 0 Wigner function formalism and the evolution of thermodynamic quantities in an expanding magnetized plasma

By combining the Wigner function formalism of relativistic quantum kinetic theory with fundamental equations of relativistic magnetohydrodynamics (MHD), we present a novel approach to determine the proper time evolution of the temperature and other thermodynamic quantities in a uniformly expanding hot, magnetized, and weakly interacting plasma. The aim is to study the contribution of quantum corrections to this evolution. We first determine the corresponding Wigner function in terms of the solution of the Dirac equation in the presence of a constant magnetic field. Using this function, we then compute the energy-momentum tensor of the above-mentioned plasma, which eventually yields its energy density and pressure. Plugging these quantities in the energy equation of relativistic MHD, we arrive, after choosing an appropriate coordinate system, at a differential equation for the temperature as a function of the proper time. The numerical solution of this equation leads finally to the proper time evolution of the temperature. The latter is then used to determine the evolution of a large number of thermodynamic quantities in this expanding and magnetized plasma. We compare our results with other existing results from relativistic MHD. We also comment on the effect of point to point decaying magnetic fields on the thermodynamic properties of this plasma.


I. INTRODUCTION
The primary goal of modern heavy-ion collision (HIC) experiments at the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) is to produce a plasma of quarks and gluons, and to study its evolution from an early out of equilibrium stage to a late hadronization one. It is widely believed that at τ 0 ∼ 0.2 fm/c after the collision, the created hot plasma enters a state of local thermal equilibrium. There are strong pieces of evidence that at this stage, the dynamics of the created quark matter is well described by relativistic hydrodynamics (for a recent review see [1] and references therein), that, because of extremely large magnetic fields that are also created in noncentral HIC experiments [2][3][4][5], is to be extended to relativistic MHD [6]. Assuming firstly that the plasma expands uniformly in the longitudinal direction with respect to the beam direction, and secondly that the external magnetic field is aligned perpendicular to the plasma velocity, it was recently possible to extend the well-known 1 + 1-dimensional Bjorken solution of relativistic hydrodynamics [7,8] to the so-called "ideal transverse MHD" [9], and to determine the proper time evolution of the magnetic field after the onset of hydrodynamics. In [10], we introduced a more general framework to study nonideal transverse MHD, and determined apart from the evolution of electric and magnetic fields, the spacetime evolution of the temperature in a hot, electromagnetized, and uniformly expanding plasma. In order to explore, in particular, the effect of external electromagnetic fields on the spacetime evolution of the plasma's temperature, the latter is to be compared with * tabatabaeemehr sma@physics.sharif.ir † Corresponding author: sadooghi@physics.sharif.ir the Bjorken 1 + 1 dimensional solution in the absence of electromagnetic fields. In a parallel development, we generalized the method introduced in the framework of "anisotropic hydrodynamics" [11,12] to anisotropic MHD, and studied the effect of magnetization on the isotropization of a uniformly expanding plasma in and out of equilibrium [13]. Following the approach introduced in [11,12], we combined the equations of ideal transverse MHD with the Boltzmann equation of classical kinetic theory in a relaxation time approximation, and determined, in particular, the spacetime evolution of the temperature in this plasma. The latter is in principle affected by the evolution of the anisotropy parameter in this model. A comparison shows, however, no significant difference between this result and the Bjorken solution for the temperature [13].
In the present paper, we introduce a novel approach to determine the proper time evolution of the temperature and other thermodynamic quantities in a uniformly expanding, hot, magnetized, and weakly interacting plasma. To do this, we combine the Wigner function formalism of relativistic quantum kinetic theory [14] with the energy equation of relativistic MHD. The aim is to consider the quantum corrections arising from Landau levels, that arise in the presence of constant magnetic fields. In this way, we determine the proper time evolution of a number of thermodynamic quantities including the temperature T , energy density ǫ, transverse and longitudinal pressures p ⊥ , and p , transverse and longitudinal speeds of sound c s,⊥ , and c s, , and magnetic susceptibility χ m , in a uniformly expanding hot QED plasma in the presence of constant and varying magnetic fields B. 1 To this purpose, we go through the following steps: i) We first determine the Wigner function of a fermionic hot and magnetized plasma. To do this, we use the solution of the Dirac equation in a constant background magnetic field using the Ritus eigenfunction formalism [15]. Our results are comparable with those presented in [16].
ii) We then use this Wigner function to analytically determine the energy-momentum tensor of a hot plasma in a constant magnetic field. Using this tensor, we determine ǫ, p ⊥ , and p from its diagonal components as functions of T . We first show that the resulting expressions (yet in an integral form) are comparable with the corresponding expressions arising from an appropriate thermodynamic potential in the presence of a constant magnetic field. We then consider two different cases of massless (m = 0) and massive (m = 0) fermions. In the massless case, we determine these quantities in two different limits of eB ≪ T 2 (high-temperature or weak magnetic field limit) and eB ≫ T 2 (low-temperature or strong magnetic field limit). In the massive case, we determine these quantities only in the limit M 2 n ≪ T 2 , where M 2 n ≡ m 2 + 2neB is the magnetic mass. iii) We then consider the energy equation of relativistic MHD, that is given in terms of ǫ, p , and p ⊥ . In a certain Milne coordinate system, this turns out to be simply a first order differential equation in the proper time τ . By plugging the analytical expressions for the above thermodynamic quantities in the two aforementioned limits into this equation, we arrive at a differential equation for T as a function of τ . We solve this equation numerically, and arrive, in particular, at the proper time evolution of the temperature.
iv ) By plugging the resulting τ dependence of T into the analytical expressions for ǫ, p ⊥ , p , c s,⊥ , c s, , and χ m for eB ≪ T 2 and eB ≫ T 2 in the massless case, and M 2 n ≪ T 2 in the massive case, we arrive at the proper time evolution of these quantities in a constant background magnetic field B in these two limits. Finally, in order to consider the more realistic scenario of a decaying magnetic field, we assume that B is constant at each infinitesimal time interval δτ , but decays according to a certain profile B = B(τ ) for B. Using this τ dependent magnetic field, it is then possible to determine the τ dependence of the temperature in a decaying magnetic field. We consider a number of profiles for B(τ ), and determine the proper time evolution of T corresponding to these point to point varying magnetic fields.
It is worthwhile to notice that recently the Wigner function formalism is used to study the kinetic phenomena of chiral plasma [17,18] and different aspects of QCD phase transition within the Nambu-Jona-Lasinio model [19]. The method introduced in the present paper, however, is a novel application of this formalism, that, once combined with the main equations of relativistic (magneto)hydrodynamics yields the evolution of thermodynamic quantities in an expanding plasma.
The organization of this paper is as follows: In Sec. II, we present a number of review materials: In Sec. II A, we review the Ritus eigenfunction method, and present, in particular, an appropriate quantization for the fermionic fields which is used later to determine the Wigner function. In Sec. II B, we derive the thermodynamic quantities, p , p ⊥ , and ǫ, using an appropriate thermodynamic potential. The resulting expressions are later compared with the corresponding expressions arising from the energy-momentum tensor constructed in terms of the Wigner function. In Sec. II C, we present the main equations of ideal transverse MHD together with the Bjorken solution of 1 + 1 dimensional hydrodynamics. The Gubser temperature is also introduced in a slightly different form than the original one in [20]. Being interested in the evolution of T and other thermodynamic quantities in point to point decaying magnetic fields, we present in Sec. II D, various solutions for the proper time evolution of B.
Our main analytic results are presented in Sec. III: In Sec. III A, we derive the Wigner function, the energy-momentum tensor, and the resulting thermodynamic quantities, yet in an integral form. In Sec. III B, we perform high-and low-temperature approximations, and determine p , p ⊥ , and ǫ in these approximations. In Sec. IV, we present the numerical results for the τ dependence of T in high-and low-temperature approximations. These data are then used to determine the τ dependence of other thermodynamic quantities. Section V is devoted to concluding remarks. Comments on Gubser temperature as well as useful formulae in highand low-temperature approximations are presented in Appendices A and B.

II. REVIEW MATERIAL
To be self-content, we present in what follows a number of review materials. We start by introducing the solutions of the Dirac equation in the presence of a constant magnetic field in the Ritus eigenfunction formalism (Sec. II). In particular, we present the quantization of fermions with a positive charge. In Sec. III A, we use this quantization to determine the energy-momentum tensor of a magnetized plasma via the Wigner function formalism. In Sec. II B, we use the thermodynamic potential of a hot plasma in a constant background magnetic field, and present the corresponding analytical expressions for p , p ⊥ , and ǫ. In Sec. II C, we then briefly introduce the ideal relativistic transverse MHD. In particular, assuming the plasma to expand uniformly, we present the energy and Euler equation of ideal transverse MHD in the Milne coordinate system, as well as the Bjorken and Gubser solutions for the temperature. Finally, in Sec. II D, we present a number of phenomenological and theoretical solutions for the proper time dependence of the magnetic field. These expressions are then used in Sec. IV, to deter-mine the proper time dependence of the temperature for point to point decaying magnetic fields.

A. Fermions in a constant magnetic field
One of the mostly used methods to solve the Dirac equation of charged fermions in the presence of a constant magnetic field is the Ritus eigenfunction method [15]. We outline it in this section. A complete derivation of this method in a 3 + 1 dimensional spacetime in a multiflavor system is previously presented in [21]. In what follows, we present, in particular, an appropriate quantization for a positively charged fermion. We use this quantization in Sec. III A to determine the Wigner function and the energy-momentum tensor of a QED plasma.
Let us start with the Dirac equation for a charged fermion with mass m q , Here, Π (q) µ = i∂ µ + eqA ext µ , with e > 0 and q the charge of this fermion. By fixing the gauge field A ext µ as A ext µ = (0, 0, Bx, 0) with B > 0, we arrive at a magnetic field B aligned in the third spatial direction B = Be z . In order to solve (II.1), we use the ansatz ψ (q) q ), as the positive frequency solution and ψ as the negative frequency solution. Here, E (q) n,κ with κ = ±1 satisfies the Ritus eigenfunction relation with κ = +1 (κ = −1) for positive (negative) frequency solution. Using the above expressions for ψ (q) κ and the relation (II.2), it is possible to show that the spinors u as well as v are the free Dirac spinors satisfying as well as Moreover, as it is shown in [22][23][24], the Ritus momentum for a particle with charge q is given bỹ (II.3) with κ = ±1 for positive and negative frequency solutions, n labeling the Landau levels in the external magnetic field, and s q ≡ sgn(qeB). In this paper, we assume eB > 0. Following the method presented in [22][23][24], the Ritus eigenfunctions E (q) n,κ , κ = ±1 are given by with ξ sq κ ≡ (x − κs q p y ℓ 2 B )/ℓ b and ℓ b ≡ |qeB| −1/2 as well asp µ ≡ (p 0 , 0, p y , p z ) as well asx µ ≡ (t, 0, y, z). The physical, on the mass-shell fermion possesses the following energy dispersion relation: Here, E (q) n,κ is defined by with Π n ≡ 1 − δ n,0 , and the projectors P (q) ± given by As it turns out, P . The functions f Here, a n ≡ (2 n n! √ πℓ b ) −1/2 . As in [21], we introduce at this stage the following quantization for a positively charged fermion (q = +1) with mass m ≡ m + in the presence of a constant magnetic field: and the spinors u s,α ,ū s,α as well as v s,α ,v s,α satisfying s u s,α (p + )ū s,β (p + ) = (γ ·p + + m) αβ , In [21], we used the above quantization relations (II.10) to derive the propagator of a positively charged fermions. In the present paper, however, we use it to determine the Wigner function of a hot and magnetized fermionic plasma.
B. Thermodynamic quantities arising from the thermodynamic potential of a magnetized QED plasma As aforementioned, in Sec. III A we use the above quantization (II.10), and determine the thermodynamic quantities ǫ, p , p ⊥ , · · · using the Wigner function formalism. We then compare the resulting expressions in the integral form with the expressions arising from the effective potential of a magnetized QED plasma at finite temperature T and zero chemical potential µ. It is given by [22,23] (II. 13) with β ≡ T −1 and the spin degeneracy factor α n ≡ 2 − δ n,0 . It arises from the free fermion determinant Ω eff = − 1 βV ln Z with V a three-dimensional volume and Z the fermionic partition function [25]. We use the standard definitions to determine the parallel pressure p , the entropy density s, and the magnetization M . We arrive at For later convenience, we have used an appropriate partial integration (PI) in (II.5) to reformulate the following integral: Combining the expressions for p , s, and BM , the energy density ǫ and the transverse pressure p ⊥ are given by and (II.20) In Sec. IV, we show that the matter (T -dependent) part of the above expressions for ǫ, p , and p ⊥ in the massless case coincides with the results arising from the energymomentum tensor determined in the Wigner function approach.

C. Bjorken flow and the ideal transverse MHD
The above expressions for the thermodynamic quantities are, in particular, functions of the temperature T and the magnetic field eB. Assuming thermodynamic equilibrium with a fixed temperature, the T dependence of the corresponding integrals to ǫ, p , p ⊥ , · · · can be numerically determined for a fixed eB and fermion mass m. The goal of the present paper is, however, to determine the proper time dependence of these thermodynamic quantities in a uniformly expanding ideal plasma, where, in particular, the temperature depends on the space and time. In this case, because of their dependence on T , all the other thermodynamic quantities become also spacetime dependent. Hence, to determine their dynamics, it is enough to determine the evolution of T for a fixed value of eB.
To do this, we use the ideal relativistic MHD. This turns out to be a useful tool to describe the dynamics of an ideal and locally equilibrated fluid in the presence of electromagnetic fields. Let us thus assume our magnetized plasma to be in local thermal equilibrium. Assuming furthermore its electric conductivity to be infinitely large, we can neglect the electric field, and concentrate only on the presence of a magnetic field. As it is described in [9,10,13], relativistic MHD is described by a set of equations consisting of the energy-momentum conservation, the homogeneous and inhomogeneous Maxwell equations, with the current The total energy-momentum T µν in (II.21) consists of two parts In an infinitely conducting fluid with vanishing electric field, the fluid part of T µν is given by [10,13] The latter arises from the definition of B µ in terms of the field strength tensor F µν , B µ ≡ 1 2 ǫ µναβ F να u β , leading to B µ = (0, B) in the local rest frame of the fluid u µ = (1, 0). As concerns the electromagnetic part of T µν , it is given by the standard Maxwell tensor [9,10], In (II.22), the dual field strength tensor is given bỹ Moreover, the magnetization tensor M µν in (II.23) is given by M µν ≡ −M b µν . Contracting (III.21) with u ν and ∆ αν = g αν − u α u ν , and assuming that the fluid velocity v is perpendicular to the direction of the magnetic field B, we arrive at the energy and Euler equations in transverse ideal MHD, respectively, For the plasma under consideration, we make a number of other assumptions. First, we assume that the magnetic field B is aligned in the z-direction, and that the fluid velocity v, being perpendicular to the direction of the magnetic field, is directed in the y direction. Then, assuming 2. a uniform expansion of the fluid in the longitudinal y direction, and 3. boost invariance along the y direction, 4. boost invariance of ǫ, p and B, we arrive at the Bjorken flow [7,8] Here η ≡ 1 2 ln t+y t−y is the boost variable in the Milne coordinates. In these coordinates, and for a plasma expanding in the y-direction, the proper time reads τ = t 2 − y 2 . Using this parametrization, we also obtain (II. 32) In these coordinates, the energy equation (II.29) is a first order differential equation in τ , In Sec. III B, we determine the thermodynamic quantities p , p ⊥ , and ǫ as functions of T in an integral form, and show that for massless fermions they coincide with those presented in (II.15). Plugging the corresponding expressions to these quantities into the energy equation (II.33) of the ideal relativistic MHD, it becomes a first order differential equation for T . In Sec. IV, we numerically solve this equation for a fixed value of B, and determine the proper time evolution of the temperature T . Plugging this numerical result back into p , p ⊥ , and ǫ, the proper time evolution of these and a number of other thermodynamic quantities are determined. The numerical results arisen in this approach is then compared with the 1 + 1 Bjorken solution [8] with T 0 the temperature at the initial time τ 0 , and κ = 3 appearing in the equation of state ǫ = κp of an ideal gas. The coefficient κ is related to the sound velocity c s by κ = c −2 s . Generalizing this result to a 3 + 1 dimensional flow, which describes a fluid expanding in both transverse and longitudinal directions, we get Let us notice that this result is only valid at r = 0, with r the distance to the center of the collision. In general, it is to be replaced by T = T 0 ρ0 ρ , where ρ ≡ √ τ 2 − r 2 and T 0 the temperature at ρ 0 . In Sec. IV, we refer to this solution as the Hubble solution at r = 0.
Another useful comparison is made with the temperature arisen from the 3 + 1 dimensional Gubser flow [20], 2 which describes a fluid expanding not only in the transverse y-direction but also in the x-z plane, . (II.37) The factor q that characterizes the Gubser temperature is typically given by q = 1/4.3 fm −1 . The parameter r is the radial distance from the origin in the x-z plane, and T 0 is the initial temperature at τ 0 and r 0 . Moreover, in (II.37), G(r 0 , τ 0 ; r 0 , τ 0 ) = 1.

D. Proper time evolution of the magnetic field in an expanding relativistic plasma
In this paper, for the sake of simplicity, we consider a plasma consisting of only one flavor with electric charge q = +1. It is not difficult to generalize our results to the more realistic case of an expanding plasma of quarks and gluons consisting of various quark flavors. This is the plasma that is created in the early stages of heavy-ion collisions. Very strong and short-living magnetic fields are also believed to be generated in the same stage of these collisions. As it turns out, after a few fm/c, they become too weak to be able to affect the dynamics of the QGP. The proper time dependence of the electromagnetic field created in HICs was indeed the subject of intensive theoretical and experimental studies in the last few years (see e.g. [2][3][4][5]10]). The question about the effect of spacetime dependent magnetic fields on the evolution of thermodynamic quantities is still open. One of the main purposes of this paper is to answer this question. The Ritus eigenfunction method, described in Sec. II A is however only valid for constant magnetic fields. Consequently, the proper time dependence of the temperature and other thermodynamic quantities are only valid for constant magnetic fields.
In Sec. IV, we show that it is possible to generalize the results arisen from our numerical approach for the case of a constant magnetic field to the case of a spacetime dependent magnetic field in an expanding plasma. To do this, we divide the time interval under study into a large number of infinitesimal time slices. We then assume the magnetic field to be constant in each infinitesimal time slice. Solving the aforementioned differential equation in this time slice, and then varying the magnetic field according to a given phenomenological or theoretical ansatz, we repeat the same computation for the next time slice. Repeating this procedure for all infinitesimal time interval, we arrive at the proper time dependence of the temperature for a point to point varying magnetic field (see Sec. IV for more details).
In what follows, we present a number of solutions for the spacetime dependence of the magnetic field.
As it is argued in [4], the early time dynamics of the magnetic field is described by where eB(0) is some constant initial value for the magnetic field at τ = 0 fm/c and t B ∼ 0.065 fm/c at RHIC top energies. In Sec. IV, we refer to this solution as the early time (E-time) solution. Another phenomenological expression, that describes the evolution of the magnetic field in near-central collisions is Here, r is the distance to the center of the collision, m π ∼ 140 MeV is the pion mass, a and b are two phenomenological parameters given by a = 78.2658 and b = 79.5457 fm −1 , and σ r = 3.5 fm for a zero impact parameter [26]. In Sec.IV, we refer to this solution as the phenomenological Gaussian (Pheno) solution. Apart from these phenomenological expressions, the evolution of the electromagnetic fields is derived from relativistic MHD. These kinds of derivation are based on a number of assumptions, including those listed in Sec. II C. In [9,10], it is also assumed that the magnetic field is perpendicular to its velocity, v. As it turns out, the expansion of the QGP plays a crucial role in the proper time evolution of the magnetic field. In the simplest case of ideal MHD described by 1 + 1 dimensional transverse MHD, and based on the Bjorken flow (II.31), the evolution of B is given by where B 0 is the magnetic field strength at the initial time τ 0 . In [27], we extended this ideal MHD result to a 3 + 1 dimensional Gubser flow [20]. To do this we , and CMHD for α0 = 0.1 (solid red) and α0 = 1 (dashed magenta) with the τ dependence of eB/m 2 π arising from IdMHD for the initial magnetic field a) eB0 = 5m 2 π and b) eB0 = 15m 2 π . As it is shown, the magnetic field arising from CMHD for α0 = 0.1 is very similar to the Bjorken expansion. Later, we will show that despite the difference between these curves, the temperature T ⋆ arising from these magnetic fields are similar (see Sec. IV).
solve the MHD equation in a conformally flat dS 3 × E 1 spacetime. Performing an appropriate Weyl transformation back into the Minkowski spacetime, we arrive at Here, similar to G(r, r 0 ; τ, τ 0 ) from (II.37), H(r 0 , τ 0 ; r 0 , τ 0 ) = 1. Moreover, according to [27], the factor α 0 is the ratio of the longitudinal and transverse components of the magnetic field α ≡ B y /B x 3 In Sec. IV, we refer to this solution as the conformal MHD (CMHD) solution. In Fig. 1, a comparison is made between the τ dependence of the most simple IdMHD solution (II.40) and the other solutions introduced in this section, including the E-time, Pheno and CMHD solutions for α 0 = 0.1 and α 0 = 1 from (II.38), (II.39), and (II.41) in the first few femtoseconds after the collision, ∆τ = 4 fm/c. Assuming the background magnetic field to be τ dependent, it is possible to determine the τ dependence of the temperature following the procedure described before. It turns out that despite obvious difference between the τ dependence of various solutions for B(τ ), the temperature arisen from these solutions are similar (see Sec. IV for more details).

III. THERMODYNAMIC QUANTITIES IN AN EXPANDING MAGNETIZED PLASMA: THE WIGNER FUNCTION APPROACH
In this section, we first derive in Sec. III A 1 the Wigner function of a fermionic system consisting of a positive charge (q = +1) in the presence of a constant magnetic field using the quantization given in (II.10). Using this result, we then derive the corresponding energy-momentum tensor to this system in Sec. III A 2. In Sec. III B, we present the corresponding expressions to a number of thermodynamic quantities including, among others, p , p ⊥ , and ǫ, and compare the resulting expressions, all in integral forms, with the results presented in (II.15), (II. 19) and (II.20). To evaluate these integrals in high and low-temperature approximations in Sec. III B, and present separately the analytical results corresponding to the massive and massless case.
A. The Wigner function, the energy-momentum tensor, and the resulting thermodynamic quantities

The Wigner function
In quantum statistical mechanics, the Wigner function W (r, k) is given by the thermal average of the normal ordered Wigner operatorŴ as [8], (III.2) Here, ψ andψ are fermionic field operators, and U (+, −), defined by is the Wilson line that guarantees the gauge invariance of the productψψ, once the point-splitting is performed. The latter is characterized by r ′ . In (III.2) and (III.3), r µ = (t, x, y, z) and r ′µ = (t ′ , x ′ , y ′ , z ′ ). In the presence of a constant magnetic field, aligned in the z-direction, i.e. for A µ = Bxe y , U (+, −) reads Plugging, apart from U (+, −) from (III.4), the quantization (II.10) of the field operators ψ andψ into (III.2), and using (II.12) as well as a †n ′ ,s ′ p ′ a n,s withp = (0, p y , p z ) and f F (E n , p z ) defined in (II.16), we arrive first at Here, = P + f + n + Π n P − f − n from (II.6). Plugging f ± n from (II.8) into (III.6), performing the integration over r ′µ = (t ′ x ′ , y ′ , z ′ ) by making, in particular, use of where L k m (u) is the associated Laguerre polynomial, and eventually performing the integral over p y and p z , we arrive after a lengthy but straightforward computation at the Wigner function for a hot and magnetized QED plasma, ± ±m)P ± or P ± (γ ·p (n) ± ± m)P ∓ , we used P ± γ µ = γ µ P ± , and P ± γ µ ⊥ = γ µ ⊥ P ∓ , and P 2 ± = P ± as well as P ± P ∓ = 0 [28]. Let us notice that the expression on the first line of (III.8) including L 0 (u) = 1 is the contribution from the lowest Landau level (LLL) while the remaining terms are those from higher Landau levels (HLL). In what follows, we use the above Wigner function to determine the energy-momentum tensor of this hot and magnetized QED plasma.
2. The energy-momentum tensor and thermodynamic quantities p , p ⊥ , and ǫ The energy-momentum tensor of a fermionic system is given by [29] with the Lagrangian density (III.10) Here, − → D µ = − → ∂ µ − ieA µ and ← − D µ = ← − ∂ µ + ieA µ denote the right (→) and left (←) covariant derivatives. As it turns out, the thermal average of this operator leads to T µν = : T µν : Plugging the Wigner function (III.8) into (III.11), and using to perform the trace over Dirac matrices, we arrive after some works first at To perform the integration over k ⊥ , we use as well as (III.15) Moreover, we use symmetry arguments 4 to perform the following integral for i = x, y: (III. 16) We finally arrive at (III.17) Here, α n = 2 − δ n,0 , as introduced previously in Sec. II B. The energy-momentum tensor presented in (III.17) is symmetric and diagonal. Its diagonal elements are given by (III.18) Here, as expected, the trace of the energy-momentum tensor is proportional to the fermion mass, and reads It vanishes for m = 0, as expected from conformal symmetry for massless fermions. Identifying, at this stage, T 00 with the energy density ǫ, T 11 = T 22 with the transverse pressure p ⊥ , and T 33 with the longitudinal pressure p , it turns out that the results for these quantities in the massless case coincide, as aforementioned, with the matter (T dependent) part of ǫ, p , and p ⊥ from (II.15), (II. 19) and (II.20), respectively. The quantity BM , including the magnetization M , is 4 Here, k 2 i , i = x, y is first replaced by 1 2 k 2 ⊥ , and then (II.16) is used.
given by BM = p − p ⊥ , or To present the analytic results for p , p ⊥ , and ǫ in the high-and low-temperature approximations, let us introduce first the following integral [25]: with x ≡ k z /T , y ≡ M 2 , and M a generic mass. According to our results from previous section, p , p ⊥ , and ǫ are then given by where y n ≡ βM n with β = T −1 is defined in terms of the magnetic mass M n ≡ m 2 + 2neB 1/2 . In what follows, we present the results for p , p ⊥ , and ǫ in the high-and low-temperature approximations in the massive and massive cases separately.

High temperature approximation for the massive and massless fermions
In the high-temperature expansion, h ℓ -integrals in (III.23) are evaluated according to [30], (see also Appendix B for more details), This leads in particular to (see also [25]) (III. 26) In (III.25), Γ(z) = (z − 1)!, are the gamma-, Riemann zeta-, and digamma functions, respectively. Moreover, γ E is the Euler-Mascheroni constant, Assuming the summation over Landau levels n in (III.24) is limited to N , the corresponding results to p , p ⊥ , and ǫ for nonvanishing fermion mass read n ln y n π + γ E , (III. 27) with y n = βM n , as before. Here, N n=0 α n = 1 + 2N is used. In the massless case, for eB ≪ T 2 , we arrive, in particular, at n ln z n π + γ E , (III. 28) with z n ≡ β √ 2neB. Setting, N = 0 in (III.28), we obtain p ⊥ = 0, and ǫ = p = eBT 2 12 , as expected from (III.21). 5 The results presented in (III.28) can be in particular regarded as the high-temperature approximation of the T dependent part of p , p ⊥ , and ǫ arising from the effective action (II.13) [see (II.15), (II. 19) and (II.20) for details].
In the next section, we use these quantities to determine the proper time evolution of the temperature. Plugging the corresponding numerical results for T back into (III.27), (III.28) and (III.32), we arrive at the proper time evolution of p , p ⊥ , and ǫ in the highand low-temperature approximations in the massless and massive cases for a constant magnetic field.

IV. NUMERICAL RESULTS
In this section, we present a number of numerical results arising from the analytical ones in the previous sections. In Sec. IV A, we determine the τ dependence of T by numerically solving the energy equation arising from the relativistic MHD. The necessary inputs are the pressures p and p ⊥ as well as the energy density ǫ, which were determined analytically in the Wigner quantum kinetic theory approach. We also present a number of comparisons between our results with the τ dependence of T arising from relativistic hydrodynamics (Bjorken, Hubble, and Gubser flow) for zero magnetic fields. The corresponding results are also compared with the resulting temperatures for a point to point varying magnetic field.
In Sec. IV B, we use our results for the τ dependence of T to determine the τ dependence of p , p ⊥ , and ǫ.
To answer the question of whether a constant magnetic field has a significant impact on the τ dependence of these quantities is answered by comparing our results with the corresponding results to a 1 + 1 dimensional Bjorken flow. The τ dependence of other thermodynamic quantities that are particularly defined in terms of p , p ⊥ , and ǫ is presented in Sec. IV C. These quantities include χ m , c s, and c s,⊥ . In Sec. IV D, we combine the τ dependence of these quantities and that of T , and determine the T dependence of these quantities in the high-and low-temperature approximations.
A. The τ dependence of T in constant and decaying magnetic fields In Sec. III, we determined the analytical T dependence of p , p ⊥ , and ǫ in high-and low-temperature approximations in the massless [(III.28) and (III.32)] and massive cases [(III.27)]. Plugging the corresponding expressions into the energy equation (II.33), and solving the resulting first order differential equation numerically, we arrive at the τ dependence of the temperature T in these approximations. The fermion mass m appearing in (III.27), is chosen to be the thermal mass of the electron [31], Here, m 0 = 0.5 MeV is the electron's rest mass, and e its electric charge. It is given by e ≡ √ 4πα with the fine structure constant α = 1/137. As concerns the initial value of the proper time, τ 0 and T (τ 0 ), we choose τ 0 = 0.2 fm/c and T 0 ≡ T (τ 0 ) = 600 MeV. The summation over Landau levels are truncated at N = ⌊    3. (color online). The τ dependence of T arising from a constant magnetic field eB = 5m 2 π (red diamond) is compared with the τ dependence of T arising from the energy equation with point to point varying magnetic fields. The initial magnetic fields for different solutions are eB0 = 5m 2 π and eB0 = 15m 2 π for eB ≪ T 2 and eB ≫ T 2 , respectively. According to Fig. 1, the B field arising from CMHD for α0 = 0.1 almost coincides with the B field arising from IdMHD. Although in eB ≪ T 2 the τ dependence of T arising from CMHD with α0 = 0.1, 1 almost coincides with the τ dependence of T arising from IdMHD, this is not the case in eB ≫ T 2 . Small oscillations appearing for the CMHD solution in the low-temperature approximation sustain once smaller intervals are chosen during which the magnetic field is assumed to be constant. aforementioned differential equation in the high-and low-temperature approximations. In Fig. 2(a), corresponding to eB ≪ T 2 , the initial value of the magnetic field is chosen to be eB 0 = 0.5, 5m 2 π and in Fig. 2(b), corresponding to eB ≫ T 2 , the initial value of the magnetic field is chosen to be eB 0 = 5, 15m 2 π . The results demonstrated in Fig. 2 are for massless fermions. We repeated the same computation for massive fermions by plugging (III.27) for eB ≪ T 2 into (II. 33), and arrived at the same result as presented in Fig. 2(a). We thus conclude that once the magnetic field is chosen to be constant, the fermion thermal mass has not a crucial effect on the τ dependence of T .
At this stage, following the procedure described in Sec. II D, we determined the proper time dependence of T for a point to point decaying magnetic field. For simplicity, we choose B(τ ) from (II.40) arising in the ideal transverse MHD (IdMHD). The corresponding results are denoted by δB HI for the high-temperature expansion with the initial magnetic field eB 0 = 2.5m 2 π [see black crosses in Fig. 2(a)] and δB LI for low-temperature expansion with the initial magnetic field eB 0 = 15m 2 π FIG. 4. The τ dependence of the temperature for the highand low-temperature approximations with constant magnetic fields eBH = 5m 2 π and eBL = 15m 2 π , as well as δBHI and δBLI results for point to point varying magnetic fields according to IdMHD solution for the B field are compared with the τ dependence of the 1 + 1 dimensional Bjorken, 3 + 1 dimensional Hubble, and Gubser flows at r = 0. According to this result, large magnetic fields slow up the fast decay of the temperature in the early time.
Let us notice that according to the above results for the high-temperature approximation, after an abrupt decay within 1 fm/c, the temperature remains almost constant around 300 MeV. Changing the initial magnetic field up to one order of magnitude does not affect this result. As concerns the low-temperature expansion, there is almost no difference between the τ dependence of T for constant (red curves) and varying (black crosses) magnetic fields. In the low-temperature approximation, however, the temperature decays fast, the result for eB = 5m 2 π and eB = 15m 2 π are different, while the results for constant eB = 15m 2 π and a point to point varying magnetic field with the initial magnetic field eB 0 = 15m π 2 almost coincide [see Fig. 3(b)].
In Fig. 4, the τ dependence of the temperature T is plotted for different solutions: eB H and eB L denote the results arising from the solution of the energy equation for constant magnetic fields eB = 5m 2 π and eB = 15m 2 π in the high-(H) and low-(L) temperature approximations, respectively. The solutions presented in (II.34), (II.35), and (II.36) for vanishing magnetic fields are denoted by "Bjorken", "Hubble at r = 0" and "Gubser at r = 0", respectively. The results corresponding to the solution of the energy equation with point to point varying magnetic fields are denoted by δB HI and δB LI in the high-and low-temperature approximations. In this case, the subscript "I" indicates the decay of the magnetic field according to (II.40) in the 1 + 1 di-mensional ideal MHD. As demonstrated in Fig. 4, the temperature decays very fast for the 3 + 1 dimensional Hubble solution. Weak magnetic fields slow up this decay significantly, and keep the temperature almost constant in a relatively large proper time interval. As concerns the Gubser solution, it interpolates between the B = 0 Bjorken solution in the early times and the low-temperature (large magnetic field) approximation in the late time, as is also denoted in [27].
B. The τ dependence of p , p ⊥ , and ǫ in constant and decaying magnetic fields Once the τ dependence of the temperature is determined by solving the energy equation either with constant or with a point to point varying magnetic field, it is possible to plug the corresponding results back into the analytical expressions for p , p ⊥ , and ǫ from (III.28) and (III.32) in the high-and low-temperature approximations, eB ≪ T 2 and eB ≫ T 2 , for massless fermions, and (III.27) in the eB ≪ T 2 approximation for massive fermions. In Fig. 5, we plotted the ratios p (τ )/p ,0 , p ⊥ (τ )/p ⊥,0 and ǫ(τ )/ǫ 0 arising from this computation. Here, p ,0 , p ⊥,0 and ǫ 0 are the corresponding quantities at the initial proper time τ 0 = 0.2 fm/c. In Figs. 5(a), 5(b), 5(e), and 5(f), we have compared the results arising from the numerical computation of p (τ )/p ,0 and ǫ(τ )/ǫ 0 with the corresponding quantities arising in 1 + 1 dimensional relativistic hydrodynamics (Bjorken solutions) in the absence of magnetic fields [8]. They are given by For an ideal gas with ǫ = 3p, κ = 3. According to these results, the data arising in the low-temperature (large magnetic field) approximation, eB ≫ T 2 , almost coincide with the corresponding results from (IV.2) from Bjorken flow [see Figs. 5(b) and 5(f)]. In the hightemperature (weak magnetic field) limit eB ≪ T 2 , however, magnetic field slows up the decay of p , and ǫ [see Figs. 5(a) and 5(e)]. As concerns the results for the point to point varying magnetic field, once the decay is given by (II.40) for 1 + 1 dimensional ideal MHD, the results coincide with the numerical data from high-and low-temperature approximations. In Fig. 5, these results are denoted by black crosses. In the eB ≪ T 2 and eB ≫ T 2 cases, the initial magnetic field are chosen to be eB 0 = 2.5m 2 π and eB 0 = 15m 2 π , respectively.
C. The τ dependence of χm, c s, , and c s,⊥ in constant and decaying magnetic fields Using the numerical results for T, p , p ⊥ , and ǫ from the previous section, it is possible to compute a number of other thermodynamic quantities. Let us start with the magnetic susceptibility χ m , that is defined  5. (color online). The τ dependence of p (τ )/p ,0 (panels a and b), p ⊥ (τ )/p ⊥,0 (panels c and d), and ǫ(τ )/ǫ0 (panels e and f) from (III.28) and (III.32) with T arising from the numerical solution of the energy equation (II.33) is compared with the corresponding data from point to point decaying magnetic fields δBHI and δBLI, and the expressions from (IV.2) in a 1+1 dimensional Bjorken solution of relativistic hydrodynamics. In the low-temperature (large magnetic field) approximation, these results almost coincide. At high-temperature (weak magnetic field), the magnetic field slows up the decay of ǫ and p ⊥ , once compared with the Bjorken solutions (IV.2) (blue solid lines for eB = 0 in panels a,b,e, and f).
by the ratio χ m = M/B. Here, M is the magnetization of the medium, and is determined by the difference between the longitudinal and transverse pressure, BM = p − p ⊥ , as defined also in Sec. II B. Using the results for p , and p ⊥ , that are plotted in Figs. 5(a)-5(d), we plotted the τ dependence of Fig. 6. The red curves in 6(a) and 6(b) correspond to χ m in a background constant magnetic field eB = 0.5m 2 π in the high-T and eB = 15m 2 π in the low-T approximations. The blue crosses correspond to the results arising from a point to point decaying magnetic field according to for the high-and low-temperature approximations with a constant background magnetic field as well as for a point to point decaying magnetic field. As it turns out, whereas for constant magnetic fields χm decreases with τ (red curves), it increases with increasing τ for a decaying magnetic field according to (II.40). The latter result is expected from [13], arising in a classical kinetic theory approach.
(II.40). Whereas in the constant magnetic field χ m decreases with increasing τ , it increases abruptly once the magnetic field decreases from point to point. These results coincide with the results demonstrated in [13], where the τ dependence of χ m in a uniformly expanding plasma is studied using the classical kinetic theory methods of anisotropic MHD. Assuming that the magnetic field decreases with τ according to (II.40), it is shown that within the first few fm/c χ m increases. Later on, after reaching a maximum, it decreases in the late time τ ≃ 10 fm/c. Apart from this, χ m turns out to be positive. This indicates that the plasma under investigation is a paramagnet. Comparing the values of χ m in the high-and low-T cases, it turns out that χ m for eB = 15m 2 π is two order of magnitude smaller than χ m for eB = 5m 2 π .
Let us now turn to the numerical results arising for longitudinal and transverse speed of sound, c s, and c s,⊥ . They are defined by [32], 6 Here, the primes denote the derivative with respect to the temperature T . In Fig. 7, the τ dependence of c 2 s, and c 2 s,⊥ are plotted in the high-and low-temperature approximations [see Figs. 7(a) and 7(c) for the high-T and 7(a) and 7(c) for the low-T approximations]. They are determined by making use of the corresponding numerical data to p , p ⊥ , and ǫ from Fig. 5. As it is shown in these plots, whereas in the high-temperature (weak magnetic field) approximation, the results corresponding to a point to point decaying magnetic field (almost) coincides with those corresponding to constant magnetic fields, in the low-temperature (large magnetic field) approximation the corresponding results to a point to point decaying magnetic field is quite different from those corresponding to constant magnetic fields. The qualitative difference between the τ dependence of c s, , and c s,⊥ is also observed in the results presented in [13], arising from a classical kinetic theory approach.
D. The T dependence of χm, c s, , and c s,⊥ in constant magnetic fields The T dependence of thermodynamic quantities is of particular interest in the thermal field theory. Among others, lattice QCD is one of the most prominent numerical methods to compute this dependence. In order to have a comparison with lattice data, we present, in this section, the T dependence of χ m , c s, , and c s,⊥ arisen from our previous numerical results. Using the τ dependence of T demonstrated in Sec. IV A, and combining the resulting data with the data arisen from the τ dependence of χ m , c s, , and c s,⊥ , the T dependence of these quantities are determined. In Fig. 8, the T dependence of χ m (T )/χ m,0 is presented in the high-(orange circles) and low-(blue squares) temperature approximations. Here, χ m,0 is the value of χ m at τ = τ 0 . The proper time interval between two successive points in each curve is δτ = 0.05 fm/c. As it turns out, for a given temperature, the value of χ m arising from the high-T approximation is 10−30% larger than the corresponding value arising from the low-T approximation. Moreover, according to these results, χ m decreases with decreasing temperature. Larger magnetic fields have a s, and c 2 s,⊥ is similar to that presented in [13], arising in a classical kinetic theory approach.
larger impact on the decay of χ m within a fixed proper time interval. Here, this interval is chosen to be ∆τ = 4 fm/c. As concerns the T dependence of the longitudinal and transverse speed of sound, c 2 s, and c 2 s,⊥ , we combined the data corresponding to the τ dependence of T from Fig. 2 and those from c 2 s, and c 2 s,⊥ from Fig. 7, and arrived at the plots demonstrated in Fig.  9. According to these results, in the high-temperature approximation c 2 s, (c 2 s,⊥ ) decreases (increases) monotonically with decreasing temperature, and increasing the magnetic field up to one order of magnitude (from 0.5m 2 pi to 5m 2 π ) does not significantly affect them. In contrast, the T dependence of c 2 s, (c 2 s,⊥ ) exhibits a nonmonotonic decrease (increase) with decreasing T in the low-temperature (large magnetic field) limit. Increasing the magnetic field up to one order of magnitude affects this behavior significantly, in particular in the late time (low-T ).

V. CONCLUDING REMARKS
In this paper, we combined the Wigner function formalism of relativistic quantum kinetic theory with the energy equations of relativistic MHD to determine the proper time evolution of the temperature in an expanding hot and magnetized QED plasma. This novel approach takes, in particular, important quantum corrections, including the contribution of Landau levels in the distribution function of this fermionic system, into account. Using the corresponding numerical results for the proper time dependence of the temperature, we also determined the evolution of a number of other thermodynamic quantities in this plasma.
We used the solution of the Dirac equation to explicitly determine the Wigner function of this plasma. This leads to the corresponding energy-momentum, which, for its part, gives rise to the longitudinal and transverse pressures with respect to the external magnetic fields, p and p ⊥ , as well as the energy density of this plasma, ǫ. For further use, we determined these quantities in a high and low-temperature approximation. For  FIG. 8. (color online). The T dependence of χm(T )/χm,0 is plotted in the high (orange circles) and low (blue squares) temperature approximations. Here, we used the τ dependence of T and χm. The proper time interval between two successive points in each curve is δτ = 0.05 fm/c. According to these results, larger magnetic fields have a larger impact on the decay (increase) of χm within a fixed τ (T ) interval. massless fermions, these approximations are characterized by eB ≪ T 2 and eB ≫ T 2 . They can also be interpreted as weak and strong magnetic field approximations, respectively. Plugging these expressions in the energy equation of relativistic MHD, and solving the resulting first order differential equation numerically, we arrived at the proper time evolution of the temperature in these high-and low-temperature approximations. Plugging these numerical results for T in the analytical expressions for p , p ⊥ and ǫ, the proper time evolution of these quantities were determined. We compared our results with the proper time evolution of T, p , p ⊥ , and ǫ arising from the well-known Bjorken solution of relativistic hydrodynamics. As concerns the τ dependence of the temperature, we showed that weak (strong) magnetic field decreases (increases) the slope of the decay of T [see Figs. 2(a) and (b)]. A point to point decaying magnetic field does not change this picture significantly. In the case of pressures and energy density, weak magnetic fields (large T ) decreases the slope of the decay of p and ǫ, while strong magnetic field does not affect the decay of these quantities too much.
We notice that the Bjorken solution is in particular based on the equation of state of a noninteracting ideal gas, ǫ = c 2 s p, where c s = 1/ √ 3 is the speed of sound. To analogously determine the relation between the pressure and energy density in this model, we plotted the proper time evolution of the longitudinal and transverse speed of sound with our own data for p , p ⊥ , and ǫ. The results demonstrated in Fig. 7 show that the proper time evolution of c 2 s,i , i = , ⊥ is strongly affected by strong magnetic fields eB ≫ T 2 . The effect of point to point decaying magnetic fields on the evo-lution of these quantities is also noticeable only in the case of strong magnetic fields.
We finally eliminated the parameter τ in the data corresponding to T (τ ) and χ m (τ ) as well as c 2 s,i (τ ), i = , ⊥, and arrive at the T dependence of these quantities. The results are plotted in Figs. 8 and 9. They indicate that χ m decreases with decreasing T , and, as concerns the T dependence of c 2 s,i , i = , ⊥, that strong magnetic fields significantly affect the T dependence of these quantities. We emphasize that the T dependence of c 2 s, and c 2 s,⊥ are totally different: Whereas c 2 s, increases, c 2 s,⊥ decreases with increasing T . It would be interesting to confirm these results using lattice QCD at finite T and in the presence of constant magnetic fields. As we have noticed before, in [13] we used the classical kinetic theory in combination with anisotropic MHD, and studied the proper time as well as the temperature dependence of the same thermodynamic quantities that are studied in the present paper. A comparison of the τ and T dependence of these quantities here with those presented in [13] shows the effect of the quantum corrections, included in the Wigner function, on the evolution of these quantities. It would be interesting to combine these two approaches to find a method where the Fermi-Dirac distribution function appearing in the nonvanishing components of T µν in (III.18) is replaced with the anisotropic distribution function appearing in [13], and to take, in this way, the effects of plasma viscosities and quantum corrections on the thermodynamic properties of the plasma into account. To do this, it is necessary to appropriately revise (III.5), in order to incorporate the anisotropy parameter appearing in [13] in the partition function on the right-hand side of this relation. It is worthwhile to emphasize that the method presented in [13] is a phenomenological method. It is, in particular, engineered according to the recipe introduced in [11,12] to consider, apart from the effect of plasma viscosities [11,12], the effect of its magnetization (net spin) on 1) its isotropization 2) its thermodynamic properties. In contrast, the approach introduced in the present paper considers the interplay between the spin and the magnetic field from first principles, as the Wigner function is built from quantized fermions that satisfy the Dirac equation in which the spin of the fermions is inherently included.
Recently, the Wigner function approach is used in many areas of the QGP physics, including, among others in the spin hydrodynamics [34]. It would be interesting to study the coupling of the magnetic field and spin in this context. An attempt in this direction is made in [18]. The analytical result for the Wigner function in the presence of a constant magnetic field, which is presented in this paper may be used in this framework. In particular, it would be interesting to study the effect of the spin-magnetic field coupling on the proper time evolution of the temperature and other thermodynamic quantities, using the method presented in this paper.
Another way to extend the results of this paper is FIG. 9. (color online). The T dependence of the longitudinal and transverse speed of sound c 2 s, and c 2 s,perp is plotted in the high (panels a and c) and low (panels b and d) temperature approximations. Increasing the magnetic field up to one order of magnitude does not affect the results in the high-temperature (weak magnetic field) limit. In contrast, in the low-temperature (strong magnetic field) limit the qualitative behavior of c 2 s, and c 2 s,perp is strongly affected by increasing the magnetic field up to a factor of three, especially in the low-T regime, T ≤ 300 MeV.
to consider a rotating plasma in a constant magnetic field. As it is shown in [35], the Dirac equation, in this case, has an analytical solution, that can be used to determine the corresponding Wigner function. The latter can then be used to determine the proper time dependence of the temperature and other thermodynamic quantities by carrying out the method presented in this paper.