Wigner function and pair production in parallel electric and magnetic fields
IICTS-USTC-18-20
Wigner function and pair production in parallel electric and magnetic fields
Xin-li Sheng,
1, 2
Ren-hong Fang, Qun Wang, and Dirk H. Rischke
2, 1 Interdisciplinary Center for Theoretical Study and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China Institute for Theoretical Physics, Goethe University,Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics,Central China Normal University, Wuhan, Hubei 430079, China
We derive analytical formulas for the equal-time Wigner function in an electromagnetic fieldof arbitrary strength. While the magnetic field is assumed to be constant, the electric field isassumed to be space-independent and oriented parallel to the magnetic field. The Wigner functionis first decomposed in terms of the so-called Dirac-Heisenberg-Wigner (DHW) functions and thenthe transverse-momentum dependence is separated using a new set of basis functions which dependon the quantum number n of the Landau levels. Equations for the coefficients are derived and thensolved for the case of a constant electric field. The pair-production rate for each Landau level iscalculated. In the case of finite temperature and chemical potential, the pair-production rate issuppressed by Pauli’s exclusion principle. I. INTRODUCTION
Quantum electrodynamics (QED) of strong electromagnetic (EM) fields has been studied for a very long time [1].In the initial stage of non-central heavy-ion collisions, the electromagnetic field can be as large as . × V / m atthe Relativistic Heavy Ion Collider (RHIC) [2] and even larger at the Large Hadron Collider (LHC). Such extremelystrong fields are generated by the fast-moving nuclei but will rapidly fall off with time [3]. The medium was estimatedto extend the lifetime of the fields and enhance the possibility of detecting the influence of strong EM fields [4, 5].A strong magnetic field leads to interesting effects related to the chiral anomaly of quantum chromodynamics(QCD). On the other hand, a strong electric field can lead to decay of the QED vacuum. When the field strength isnear or above the critical strength E c = m c /q (cid:126) [6–8], where m is the mass of the particle and q its electric charge,particle-antiparticle pairs can be created from vacuum. This process is commonly called Schwinger process in honorof Julian Schwinger, who derived the pair-production rate in a famous work [8]. The rate is exponentially suppressedbelow the critical field strength, which is about . × V / m for electron-positron production. Pair production isa nonlinear phenomenon and the corresponding experiment is important for studying QED beyond the perturbativeregime. The mechanism can occur in many systems such as in the early Universe, around neutron stars, and inheavy-ion collisions, while it is expected to appear in strong-laser experiments like the free-electron X-ray laser XFEL[9, 10] and the extreme-light infrastructure ELI [11].Although the Schwinger process has been studied for more than half a century, calculating the pair production inan arbitrary electromagnetic field is still a challenging problem. The case of a vanishing magnetic field B ( t, x ) = 0 and a space-independent electric field has been exhaustively discussed, where the problem can be translated intosolving the famous Vlasov equation of quantum kinetic theory [12–14]. It can be analytically solved for a constantelectric field E ( t ) = E and the Sauter-type field E ( t ) = E sech ( t/τ ) . Many theoretical methods are developed todeal with these two cases and go beyond these analytical benchmarks, such as directly through quantum field theory[8], WKB methods [15–17], instanton methods [18–20], the Wigner-function method [21–23], the numerical world-line loop method [24, 25], and holographic methods [26–28]. In principle, some methods such as the Wigner-functionmethod [21–23] can be applied to very general cases, but one faces a system of non-linear partial differential equations.However, the field configurations in cosmology or in heavy-ion collisions are much more complicated than the abovementioned cases. One might find an approximate solution by partitioning space-time into small cells and applying theanalytical results for a constant electric field in each cell. This, however, may generate uncontrollable uncertaintiesbecause an instanton study [29] showed that temporal inhomogeneities tend to enhance the pair production whilespatial ones tend to suppress it. Especially in heavy-ion collisions, where the EM fields vary rapidly in both spaceand time [5, 30, 31], a proper numerical treatment is necessary [32, 33].Nowadays many researchers are focusing on the Schwinger process in strong-laser experiments [34, 35]. The criticalfield strength E c for e + e − pair production corresponds to an average laser intensity I c = µ c E c (cid:39) . × W / cm .Unfortunately, such a large intensity is difficult to generate in an experiment. In the ELI project [11], the laser pulsecan only reach ∼ W / cm , which is three magnitudes lower than I c . The pair production in such a case is stronglysuppressed by a factor exp( − πE c /E ) (cid:39) − . Clearly, the critical intensity I c is not attainable for laser experimentsin the near future. Meanwhile, the electric field in heavy-ion collisions can reach eE ∼ m π c / (cid:126) (cid:29) eE c at RHIC [2], a r X i v : . [ h e p - ph ] F e b which provides realistic conditions to study pair production. In the recently discovered Dirac semimetals, masslessDirac fermions can be excited by an external electromagnetic field and may be experimentally observed through theirtransport properties. The production rate remains finite even if the Dirac fermions are massless [36], which is differentfrom the Schwinger process in vacuum.According to Maxwell’s equations, a varying electric field will generate a magnetic field. Analytical calculationsshow that a magnetic field which is parallel to the electric field can increase the pair-production rate [16, 37–40].Recently the enhancement of the pair-production rate due to parallel magnetic fields has been studied in string theory[41–43]. The pair production rate is modified by the thermal medium [44–46]. In this paper, we will reproduce theseresults via the Wigner-function method. On the other hand, the pair production in parallel electromagnetic fieldsis related to the chiral anomaly [47, 48] and to pseudoscalar condensation [49–51], which can be verified using theresults of this paper. We will focus on these effects in future work.This paper is organized as follows: In Sec. II we will briefly introduce the equal-time Wigner function and theDirac-Heisenberg-Wigner (DHW) functions. General equations of motion for the DHW functions are also listed inthis section. In Sec. III we simplify the equations of motion for the DHW functions in a spatially homogeneouselectric field and give analytical solutions for a constant electric field. A constant magnetic field, which is parallelto the electric field, is taken into account in Sec. IV. The DHW functions reflect the behavior of the Landau levels.Analytical solutions are derived when both electric and magnetic fields are constant. In Sec. V we read off the pair-production rate from the DHW functions derived in Sec. IV. In Sec. VI we give a summary and provide an outlook tofuture work. Details about the auxiliary functions used in this paper and their properties are summarized in App. A.We take fermions to have positive unit charge q = + e and the electric and magnetic fields to point in the z -direction. We use the following notations for four-vectors: X = ( x µ ) = ( t, r ) = ( t, x T , z ) = ( t, x, y, z ) and P = ( p µ ) =( E, p ) = ( E, p T , p z ) = ( E, p x , p y , p z ) . We also use the following differential operators ∂ t = ∂∂t , ∇ x = ( ∂∂x , ∂∂z , ∂∂z ) and ∇ p = ( ∂ p x , ∂ p y , ∂ p z ) = ( ∂∂p x , ∂∂p y , ∂∂p z ) . Our units are natural Heaviside-Lorentz units, (cid:126) = c = k B = (cid:15) = µ = 1 .The metric tensor is g µν = diag(+ , − , − , − ) . II. DHW FUNCTIONS AND THEIR EQUATIONS OF MOTION
In this section we define the DHW functions as expansion coefficients of the equal-time Wigner function. The choiceof the gauge potential is to some degree arbitrary. Here we use the temporal gauge A = 0 , for which the EM fieldsare given by E = − ∂ t A and B = ∇ × A . In principle the EM fields include contributions from external fields andcontributions from all charged particles. But in this paper we will focus on the case of an external field only andneglect the interaction between particles, which corresponds to a free Fermi gas.The gauge-invariant Wigner operator is given by ˆ W ( X, P ) = (cid:90) d Y (2 π ) exp (cid:18) − iy µ p µ (cid:19) ¯ ψ (cid:18) X + Y (cid:19) ⊗ U (cid:18) X + Y , X − Y (cid:19) ψ (cid:18) X − Y (cid:19) , (1)where ψ is the Dirac field operator for spin-1/2 particles. This formula represents the Fourier transform with respectto the relative position Y of the direct product of two fermion field operators at space-time points X + Y and X − Y ,respectively. The gauge link between these two points renders the Wigner operator gauge-invariant and is defined as U (cid:18) X + Y , X − Y (cid:19) = exp (cid:20) − iey µ (cid:90) / − / dsA µ ( X + sY ) (cid:21) , (2)where A µ is the gauge potential, e.g. in this paper the electromagnetic potential. Taking the expectation value of theWigner operator in a state | Ω (cid:105) , we obtain the Wigner function W ( X, P ) ≡ (cid:104) Ω | ˆ W ( X, P ) | Ω (cid:105) . (3)The Wigner function, defined in eight-dimensional phase space ( x µ , p µ ) , is Lorentz covariant but does not have a clearphysical interpretation [52, 53]. By integrating over the energy p we obtain the corresponding equal-time Wignerfunction [54–56], which can be interpreted as a quasi-probability distribution in six-dimensional phase space ( x , p ) attime t . Such a procedure evidently breaks the Lorentz covariance, however, the equation of motion might simplify tothat of an initial-value problem. On the other hand, we adopt the Hartree approximation, i.e., we treat the quantumEM field as a semi-classical EM field, from which we derive the following formula for the equal-time Wigner function W ( t, x , p ) , W ( t, x , p ) = (cid:90) d y (2 π ) exp (cid:20) i y · p + ie (cid:90) / − / ds y · A ( t, x + s y ) (cid:21)(cid:28) Ω (cid:12)(cid:12)(cid:12)(cid:12) ¯ ψ (cid:18) t, x + y (cid:19) ⊗ ψ (cid:18) t, x − y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Ω (cid:29) . (4)The Hartree approximation ignores higher-loop radiative corrections and is a good approximation for strong EM fields.The equation of motion for the equal-time Wigner function can be derived from the Dirac equation. We consider anon-zero chemical potential µ associated with the conservation of fermion number, which, for the sake of simplicity,we assume to be constant in space-time, hence its derivatives vanish. An effective way to include the chemicalpotential is by adding a term + µ ˆ N to the Dirac–Hamilton operator ˆ H D , where ˆ N is the fermion-number operator.The corresponding Dirac equation reads [ iγ σ ( ∂ σ + ieA σ ) − m + µγ ] ψ ( X ) = 0 . (5)Taking the time derivative of Eq. (4) and simplifying the result using the Dirac equation, we obtain the followingequation of motion [21]: D t W = 12 D x · (cid:2) W, γ γ (cid:3) − i Π · (cid:8) W, γ γ (cid:9) + im (cid:2) W, γ (cid:3) , (6)where the operators D t , D x , and Π are generalized operators for time and spatial derivatives, as well as momentum,in the presence of an EM field, D t ≡ ∂ t + e (cid:90) / − / ds E ( t, x − is ∇ p ) · ∇ p , D x ≡ ∇ x + e (cid:90) / − / ds B ( t, x − is ∇ p ) × ∇ p , Π ≡ p + ie (cid:90) / − / ds s B ( t, x − is ∇ p ) × ∇ p . (7)For spatially homogeneous EM fields F µν ( t, x ) = F µν ( t ) , these operators become local, D t = ∂ t + e E ( t ) · ∇ p , D x = ∇ x + e B ( t ) × ∇ p , Π = p . (8)It can be easily checked that the equal-time Wigner function W ( t, x , p ) satisfies W † = γ W γ and can be decom-posed in terms of the 16 independent generators of the Clifford algebra Γ i = { , iγ , γ µ , γ γ µ , σ µν } , W ( t, x , p ) = 14 (cid:18) F + iγ P + γ µ V µ + γ γ µ A µ + 12 σ µν S µν (cid:19) , (9)where σ µν = i [ γ µ , γ ν ] is the anti-symmetric spin tensor. These 16 functions, commonly called Dirac-Heisenberg-Wigner (DHW) functions, are real functions of time t and six-dimensional phase space ( x , p ) . The tensor part can befurther decomposed into two vector functions T = S S S , S = S S S . (10)Some of these DHW functions have a clear physical meaning [54], e.g. F determines the mass density, V µ the vector-charge current density, A µ the chiral-charge current density, and S the magnetic-moment density. Substituting Eq.(9) into the equation of motion (6) and projecting onto the 16 basis matrices, we find a system of partial differentialequations (PDEs) for the DHW functions D t G G G G = M − M − M − m − M m G G G G , (11)where the DHW functions have been divided into four groups and each group is composed of four functions [57], G = (cid:18) F S (cid:19) , G = (cid:18) V A (cid:19) , G = (cid:18) A V (cid:19) , G = (cid:18) P T (cid:19) . (12)In Eq. (11), we have introduced the two matrices M ≡ (cid:18) Π T Π D × x (cid:19) , M ≡ (cid:18) D T x D x − Π × (cid:19) , (13)where Π and D x were already defined in Eq. (7). For any three-dimensional column vector V , V T is the correspondingtransposed vector (line vector) and V × represents the anti-symmetric × matrix V × = − V z V y V z − V x − V y V x , (14)the elements of which are V × ij = − (cid:15) ijk V k . The differential equations 11 are equivalent to the ones in Refs. [21, 55] buthere we write them in a matrix form. When dealing with the Landau levels in a constant magnetic field, this matrixform allows for more compact formulas [57]. III. SPATIALLY HOMOGENEOUS ELECTRIC FIELD
In this section we will simplify the equations of motion (11) for the DHW functions in a spatially homogeneouselectric field and then give the solution for a constant electric field. The electric field is taken to point into the z -direction. In this case, the gauge potential is A ( t ) = A ( t ) e z with ∂ t A ( t ) = − E ( t ) . A similar procedure has beenadopted in Ref. [21], where the authors only discussed the pair production in vacuum. In a thermal environment,the low-energy states are occupied, which blocks the production of pairs into these states. In this section, a thermalequilibrium distribution is assumed at the initial time. Since collisions between particles are not included, all existingparticles are accelerated by the electric field and thus the distribution depends on the canonical momentum. We showthat, in the solution, the thermal distribution appears as an overall suppression factor, which does not influence thestructure of the PDE system. The basis used in this section is different from the one in Ref. [21], but both span thesame Hilbert space and thus are equivalent to each other. The system of PDEs and corresponding initial conditionsderived with the basis in this section provides a convenient framework to describe pair production in parallel electricand magnetic fields in Sec. IV.Let us first consider the DHW functions for a free gas of fermions. These can be derived by first quantizing thefield operators in terms of solutions for free particles, which can be found in any textbook of quantum field theory,and then inserting the field operators into the definition of the Wigner function. The result is (cid:18) F V (cid:19) free ( p ) = d s (2 π ) E p (cid:20) f F D ( E p − µ ) + f FD ( E p + µ ) − (cid:21) (cid:18) m p (cid:19) , V , free ( p ) = d s (2 π ) (cid:20) f F D ( E p − µ ) − f FD ( E p + µ ) + 1 (cid:21) . (15)here d s is the degeneracy of spin, which is d s = 2 for spin- particles, and f FD ( E p ∓ µ ) = 11 + exp[ β ( E p ∓ µ )] (16)is the Fermi-Dirac distribution for particles/anti-particles with energy E p and vector chemical potential µ , while β = T − is the inverse temperature. Note that the fermionic field operators in the definition (4) are not normal-ordered, therefore taking the expectation value in the state | Ω (cid:105) yields an additional ∓ , which appears in the squarebrackets in Eq. (15). Here V , V , and F are the charge, current, and mass densities, respectively. All other DHWfunctions vanish for a free gas of fermions, P = A = A = S = T = 0 , which can be proven using the completenessrelations for the Dirac spinors u ( k , s ) and v ( k , s ) .We now proceed to solve the equations of motion (11) for the DHW functions. Due to the absence of a magneticfield and translation invariance of the system, we can set the spatial derivative D x to zero and Π ≡ p . The matricesin Eq. (13) then simplify to M = (cid:18) p T p 0 × (cid:19) , M = (cid:18) × × − p × (cid:19) , (17)and D t = ∂ t + eE ( t ) ∂ p z . Then, the 16 equations of motion for the DHW functions can be divided into several groups.The equation for the charge density separates from the others and reads D t V ( t, p ) = 0 . (18)After integrating over the momentum p and neglecting the boundary terms (because there is no particle with infinite p z ), the above equation is nothing but the conservation of net charge. Furthermore, the ten equations of motion forthe DHW functions F , V , A , and T decouple from the other five for the functions P , A , and S . These latter oneswill no longer be considered, because their initial values are zero and thus they will remain zero for later times aswell. In matrix form we have D t w ( t, p ) = M ( p ) w ( t, p ) , (19)where w ( t, p ) = ( F , V , A , T ) T is a ten-dimensional vector consisting of ten DHW functions and M ( p ) is a × matrix M ( p ) = 2 p T p × − m p × − p m . (20)Inspired by the form (15) of the free DHW functions, we make the following ansatz for the solution of Eq. (19), w ( t, p ) = d s (2 π ) (cid:26) f F D (cid:0) E p + eδA ( t ) e z − µ (cid:1) + f FD (cid:0) E p + eδA ( t ) e z + µ (cid:1) − (cid:27) (cid:88) i =1 χ i ( t, p ) e i ( p T ) . (21)Here δA ( t ) ≡ A ( t ) − A ( t ) is the difference of the gauge potentials at time t and at initial time t . The distributionthus depends on the canonical momentum, which reflects the acceleration of fermions in an electric field. Since actingthe operator D t on p + eδA ( t ) e z gives zero, the term in the curly brackets in Eq. (21) behaves like a constant overallfactor and can be taken out of Eq. (19). The value of this term is in the range ( − , , which is the effect of Pauliblocking by particles already present in the thermal system. Note that, since the matrix w ( t, p ) has dimension ten, inprinciple we would need ten basis vectors e i in the ansatz (21). However, we actually only need three because theseform a closed sub-space under the operators D t and M ( p ) , while the initial conditions are also inside this sub-space.These basis vectors are e = e z , e ( p T ) = 1 m T m p T , e ( p T ) = 1 m T z × p T − m e z , (22)which are independent of t and p z , so that D t e i = 0 for all i = 1 , , . Here we have introduced the transverse mass m T ≡ (cid:112) m + p T , so that the three basis vectors are properly normalized, e i · e j = δ ij . We can also check that theyare closed under the operator M ( p ) , M ( p ) e e e = 2 − m T p z m T − p z e e e . (23)Inserting the ansatz (21) into Eq. (19) and using Eq. (23) we can derive the equations of motion for the coefficientfunctions χ i ( t ) , D t χ χ χ ( t, p ) = 2 m T − p z − m T p z χ χ χ ( t, p ) . (24)In order to solve this system of PDEs, we need to specify the initial condition. Here we choose the values of the DHWfunctions in the absence of an electric field. For an integrable electric field, which vanishes sufficiently rapidly for t → ±∞ , such as the Sauter-type field E ( t ) = E cosh − ( t/τ ) , we specify the initial condition for t → −∞ , wherewe take the DHW functions to assume the values given by Eq. (15).However, for a constant electric field E ( t ) = E , the momentum shift will be infinitely large if we take t →−∞ , because a constant field is not integrable. In reality, fermions will collide with each other, kinetic energy willbe converted to thermal energy, and the system will approach thermodynamical equilibrium. Here we make theassumption that the system is already in thermodynamical equilibrium at initial time t . We should find a solutionthat coincides with Eq. (15) when the field strength is sufficiently small, E → , i.e., χ χ χ ( t, p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E → = 1 E p p z m T . (25)The pair-production rate and the corresponding Wigner function have analytical solutions for both a constant fieldand a Sauter-type field, see Ref. [21] for details of the derivation from quantum kinetic theory, which we will notrepeat here. In a constant field E ( t ) = E the solutions do not depend on space-time coordinates, which is obviousbecause of translation invariance, χ χ χ ( p ) = d (cid:18) η, (cid:113) eE p z (cid:19) m T √ eE d (cid:18) η, (cid:113) eE p z (cid:19) m T √ eE d (cid:18) η, (cid:113) eE p z (cid:19) , (26)where η ≡ m T / ( eE ) is the dimensionless transverse mass square and the auxiliary functions are listed in Eq. (A1)of App. A. It is easy to check numerically that the solutions (26) satisfy the constraint (25) and the system (24) ofPDEs. The corresponding DHW functions can be readily derived by inserting Eq. (26) into Eq. (21). IV. PARALLEL AND SPATIALLY HOMOGENEOUS ELECTRIC AND MAGNETIC FIELDS
In this section, we will consider spatially homogeneous electric and magnetic fields which are parallel to each other.Without loss of generality, the fields are assumed to point into the z -direction. We also assume the magnetic field tobe constant in time. Then the solution can be simplified by separately considering the different Landau levels. Weprovide an analytical solution for the case when the electric field is also constant in time. A. Initial conditions
Analogously to the case without magnetic field, we choose the DHW functions in a pure magnetic field as initialcondition for the system (11) of PDEs. Since we consider this field to be constant in space and time, an analyticalsolution can be found. The corresponding covariant DHW functions in eight-dimensional phase space have beendetermined in Ref. [57]. In this paper, we set the axial chemical potential to zero, i.e., we do not consider the chiralmagnetic effect. Then, using the results of Ref. [57] the covariant DHW functions read (cid:18) G ( P ) G ( P ) (cid:19) = (cid:88) n =0 V ( n ) ( p , p z ) e ( n )1 ( p T ) (cid:18) mp + µ (cid:19) , G ( P ) = p z V (0) ( p , p z ) e (0)1 ( p T ) + (cid:88) n> V ( n ) ( p , p z ) (cid:20) p z e ( n )2 ( p T ) + √ neB e ( n )3 ( p T ) (cid:21) , G ( P ) = 0 , (27)where V ( n ) ( p , p z ) = 2(2 − δ n )(2 π ) δ (cid:26) ( p + µ ) − [ E ( n ) p z ] (cid:27)(cid:26) θ ( p + µ ) f F D ( p ) + θ ( − p − µ ) (cid:20) f F D ( − p ) − (cid:21)(cid:27) . (28)Here, E ( n ) p z = (cid:112) m + p z + 2 neB is the energy of the n th Landau level in a constant magnetic field and f F D is theFermi-Dirac distribution function. The basis vectors e ( n ) i are given in Eq. (A4) of the appendix. Since the pairproduction is a dynamical process, it is more convenient to use the equal-time formula. We emphasize that thecovariant DHW functions can be obtained from the equal-time ones by applying an additional Fourier transformationin t , i.e., t → p , and conversely, the equal-time DHW functions can be derived from the covariant ones by integratingover p . Here we give the equal-time DHW functions, G ( p ) = (cid:88) n =0 mE ( n ) p z C ( n )1 ( p z ) e ( n )1 ( p T ) , G ( p ) = (cid:88) n =0 C ( n )2 ( p z ) e ( n )1 ( p T ) , G ( p ) = p z E (0) p z C (0)1 ( p z ) e (0)1 ( p T ) + (cid:88) n> C ( n )1 ( p z ) 1 E ( n ) p z (cid:20) p z e ( n )2 ( p T ) + √ neB e ( n )3 ( p T ) (cid:21) , G ( p ) = 0 . (29)Here, C ( n )1 ( p z ) ≡ (cid:82) dp E ( n ) p z V ( n ) ( p , p z ) and C ( n )2 ( p z ) ≡ (cid:82) dp ( p + µ ) V ( n ) ( p , p z ) , respectively. The p -integrals canbe performed, yielding the result C ( n )1 ( p z ) = 2 − δ n (2 π ) (cid:20) f F D ( E ( n ) p z − µ ) + f F D ( E ( n ) p z + µ ) − (cid:21) ,C ( n )2 ( p z ) = 2 − δ n (2 π ) (cid:20) f F D ( E ( n ) p z − µ ) − f F D ( E ( n ) p z + µ ) + 1 (cid:21) . (30)The Fermi-Dirac distributions are the number densities in coordinate space for fermions/anti-fermions. The prefactor − δ n is the spin degeneracy of the various Landau levels. Comparing with Eq. (15) without the electromagneticfield, Eq. (29) has more non-vanishing components and depends on the Landau levels n . We will show later that in aconstant magnetic field, different Landau levels evolve independently. B. Equations of Motion
In the presence of a constant magnetic field, the operator for the generalized spatial differentiation, cf. second Eq.(8), becomes D x = e B × ∇ p , where the ordinary spatial gradient ∇ x has been dropped, since all considered fieldsare spatially homogeneous and the system is translation-invariant.The lowest Landau level is special since we only need the basis vector e (0)1 ( p T ) to describe the dynamics in thelowest Landau level. The reason is that, in a constant magnetic field, e (0)1 ( p T ) is an eigenvector for all operators D t , M , M appearing in the equation of motion (11), M e (0)1 ( p T ) = 2 p z e (0)1 ( p T ) , M e (0)1 ( p T ) = D t e (0)1 ( p T ) = 0 . (31)For the higher Landau levels, the situation is more complicated. In the last subsection we have shown that the basisvectors e ( n ) i , i = 1 , , , cf. Eq. (A4), are necessary to describe the equal-time DHW functions in a constant magneticfield. One can easily check that these basis vectors are not closed under the operator M defined in Eq. (13). In orderto construct a closed space under M , we need another basis vector, e ( n )4 , the definition of which is also given in Eq.(A4). Acting with M , M onto these basis vectors and using the relations (A9) gives for all higher Landau levels n > M e ( n ) i ( p T ) = (cid:88) j =1 ( c ( n )1 ) Tij e ( n ) j ( p T ) ,M e ( n ) i ( p T ) = (cid:88) j =1 ( c ( n )2 ) Tij e ( n ) j ( p T ) , (32)where the coefficient matrices are c ( n )1 ≡ p z √ neB p z √ neB , c ( n )2 ≡ − √ neB − p z −√ neB p z . (33)Note that the transpose of these matrices appears in Eq. (32).We have already seen in Eq. (29) that, when the electric field vanishes, the DHW functions can be expressed interms of the basis vectors e ( n ) i . Taking Eq. (29) as initial condition one can straightforwardly conclude that the DHWfunctions will stay in the space spanned by e ( n ) i when they evolve according to the equation of motion (11). This isbecause D t acting on e ( n ) i gives zero, while we have already seen that these basis vectors form a closed subset whenacting with M , onto them, see Eq. (32). We thus make the ansatz G i ( t, p ) = f (0) i ( t, p z ) e (0)1 ( p T ) + (cid:88) n> (cid:88) j =1 f ( n ) ij ( t, p z ) e ( n ) j ( p T ) , (34)where i, j = 1 , , , . Since the magnetic field is assumed to be constant in time, the basis vectors e ( n ) i are alsoindependent of time. Inserting Eq. (34) into the equation of motion (11) for the DHW functions, and using theorthogonality relations (A7) and (A8) for the basis vectors, we can derive the equations of motion for the functions f (0) i and f ( n ) ij . For the lowest Landau level we obtain D t f (0)1 f (0)2 f (0)3 f (0)4 ( t, p z ) = 2 p z − m − p z m f (0)1 f (0)2 f (0)3 f (0)4 ( t, p z ) . (35)The equations for the higher levels are D t f ( n )1 f ( n )2 f ( n )3 f ( n )4 ( t, p z ) = c ( n )1 − c ( n )2 − c ( n )2 − m − c ( n )1 m f ( n )1 f ( n )2 f ( n )3 f ( n )4 ( t, p z ) , (36)where f ( n ) i ≡ ( f ( n ) i , f ( n ) i , f ( n ) i , f ( n ) i ) T is a four-dimensional column vector. We observe that, on account of the orthog-onality relations (A7), (A8), the equations for the different Landau levels separate from each other, which greatlyfacilitates the solution of the equations of motion. C. Lowest Landau level
The spin of the fermion in the lowest Landau level with positive/negative charge is parallel/anti-parallel to themagnetic field. The equation for f (0)2 , cf. the second line in Eq. (35), decouples from the other equations and givesrise to the conservation of net fermion number in the lowest Landau level. In order to see this, we note that the netfermion number density V ( t, p ) is the first component of G in Eq. (12). The lowest Landau level contributes just f (0)2 ( t, p z )Λ (0)+ ( p T ) , cf. Eqs. (34) and (A4). Acting with D t = ∂ t + eE ∂ p z on that and integrating over p yields withthe definition n (0) ≡ (cid:82) d p f (0)2 ( t, p z )Λ (0)+ ( p T ) the conservation law ∂ t n (0) = (cid:90) d p (cid:20) D t f (0)2 ( t, p z ) (cid:21) Λ (0)+ ( p T ) = 0 , (37)where we have integrated by parts and neglected the boundary term. The equation D t f (0)2 ( t, p z ) = 0 , together with f (0)2 ( t, p z ) (cid:12)(cid:12)(cid:12) E → = C ( n )2 ( p z ) has the special solution f (0)2 ( t, p z ) = C (0)2 ( p z − eE t ) . (38)This solution describes an overall acceleration of all charged particles. We note that in this paper we focus on a freefermion gas, so there are no collisions to prevent the acceleration.The equations of motion for the other three functions f (0) i =1 , , are coupled with each other. In order to simplify theproblem, we make an ansatz which splits off the thermal distribution functions, (cid:110) f (0)1 , f (0)3 , f (0)4 (cid:111) = (cid:110) χ (0)1 , χ (0)2 , χ (0)3 (cid:111) C (0)1 ( p z − eE t ) . (39)where C (0)1 is defined in Eq. (30). Here, p z + eA ( t ) = p z − eE t is the canonical momentum. When acting with D t on f (0) i , we only need to consider its effect on χ i , because D t C (0)1 ( p z − eE t ) = 0 . Thus, the equations of motion for the χ i are the same as the one for the corresponding f (0) i , cf. Eq. (35), D t χ (0)1 χ (0)2 χ (0)3 ( t, p z ) = 2 p z − m − p z m χ (0)1 χ (0)2 χ (0)3 ( t, p z ) . (40)Comparing the ansatz (39) with the initial condition (29), i.e., for E → , we find χ (0)1 χ (0)2 χ (0)3 ( t, p z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E → = 1 E (0) p z mp z , (41)The system (40) of PDEs with the initial condition (41) coincides with the PDE system (24) in a pure electric field(substituting χ → χ (0)2 , χ → χ (0)1 , χ → − χ (0)3 and setting p T = 0 ). One can therefore immediately give the solutionfor a constant electric field E ( t ) = E , χ (0)1 χ (0)2 χ (0)3 ( p z ) = m √ eE d (cid:18) η (0) , (cid:113) eE p z (cid:19) d (cid:18) η (0) , (cid:113) eE p z (cid:19) − m √ eE d (cid:18) η (0) , (cid:113) eE p z (cid:19) , (42)with d i defined in Eq. (A1) and η (0) = m /eE . Multiplying Eq. (42) with C (0)1 ( p z − eE t )) gives the functions f (0)1 , f (0)3 , and f (0)4 in a constant electric field, f (0)1 f (0)3 f (0)4 ( p z ) = m √ eE d (cid:18) η (0) , (cid:113) eE p z (cid:19) C (0)1 ( p z − eE t ) d (cid:18) η (0) , (cid:113) eE p z (cid:19) C (0)1 ( p z − eE t ) − m √ eE d (cid:18) η (0) , (cid:113) eE p z (cid:19) C (0)1 ( p z − eE t ) . (43)Inserting these functions into Eq. (34), one obtains the contribution from the lowest Landau level to the DHWfunctions. D. Higher Landau levels
For the higher Landau levels ( n > ) we can read off from Eqs. (29), (34) that, when switching off the electric field,the only functions which do not vanish are f ( n )11 , f ( n )21 , f ( n )32 , and f ( n )33 . Writing down the equations of motion (36) forthe f ( n ) ij functions for the higher Landau levels using Eq. (33) we observe that f ( n )24 , f ( n )42 , and f ( n )43 couple with f ( n )11 , f ( n )32 , and f ( n )33 in the presence of an electric field. The corresponding six basis functions form a closed sub-space. Theother nine functions are decoupled to three independent groups, (cid:110) f ( n )12 , f ( n )13 , f ( n )31 , f ( n )41 (cid:111) , (cid:110) f ( n )22 , f ( n )23 , f ( n )34 , f ( n )44 (cid:111) and (cid:110) f ( n )14 (cid:111) , each forms a closed set of homogeneous PDEs. However, since all of them have vanishing values when theelectric field is zero, all of them will stay zero during the further evolution, even after switching on the electric field.In the following, we therefore focus on the seven non-trivial functions f ( n )11 , f ( n )21 , f ( n )24 , f ( n )32 , f ( n )33 , f ( n )42 , and f ( n )43 .The equation for f ( n )21 , D t f ( n )21 = 0 , decouples from the others. As discussed in the previous subsection, this equationis nothing but the conservation of net charge in each Landau level. The solution is f ( n )21 ( t, p z ) = C ( n )2 ( p z − eE t ) , (44)where p z − eE t describes the overall acceleration of all existing particles by the electric field in the z -direction.0As already mentioned above, the other six functions, f ( n )11 , f ( n )24 , f ( n )32 , f ( n )33 , f ( n )42 , f ( n )43 , satisfy a six-dimensionalsystem of PDEs. They can be further decoupled by introducing the following linear combinations, (cid:32) g ( n )1 g ( n )3 g ( n )4 g ( n )2 (cid:33) = 1 m ( n ) (cid:18) m √ neB √ neB − m (cid:19) (cid:32) f ( n )11 f ( n )24 f ( n )33 f ( n )42 (cid:33) , (45)where the effective mass at level n is m ( n ) ≡ √ m + 2 neB . Then we get the following two groups of equations D t g ( n )1 g ( n )2 f ( n )32 ( t, p z ) = 2 − p z p z − m ( n ) m ( n ) g ( n )1 g ( n )2 f ( n )32 ( t, p z ) , (46)and D t g ( n )3 g ( n )4 f ( n )43 ( t, p z ) = 2 − p z p z m ( n ) − m ( n ) g ( n )3 g ( n )4 f ( n )43 ( t, p z ) , (47)In this way, g ( n )1 , g ( n )2 , and f ( n )32 decouple from g ( n )3 , g ( n )4 , and f ( n )43 . When the electric field vanishes, we find from Eqs.(29), (34), and (45) that g ( n )3 , g ( n )4 , and f ( n )43 vanish. Under the time evolution determined by Eq. (47) this will remainthe case after switching on E . Therefore, we only need to focus on the equations for g ( n )1 , g ( n )2 , and f ( n )32 . Analogousto the treatment of the lowest Landau level, we assume that the solutions have the following form, (cid:110) g ( n )1 , g ( n )2 , f ( n )32 (cid:111) = (cid:110) χ ( n )1 , χ ( n )2 , χ ( n )3 (cid:111) C ( n )1 ( p z − eE t ) . (48)Since D t C ( n )1 ( p z − eE t ) = 0 , the system of PDEs for { χ ( n )1 , χ ( n )2 , χ ( n )3 } reads D t χ ( n )1 χ ( n )2 χ ( n )3 ( t, p z ) = 2 − p z p z − m ( n ) m ( n ) χ ( n )1 χ ( n )2 χ ( n )3 ( t, p z ) . (49)The initial values can be deduced by first reading off the functions f ( n ) ij via a comparison of Eq. (29) with Eq. (34)and then using Eq. (45), χ ( n )1 χ ( n )2 χ ( n )3 ( t, p z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E → = 1 E ( n ) p z m ( n ) p z . (50)The system (49) of PDEs and the initial condition (50) coincide with the PDE system (24) in a pure electric field(replacing χ → χ ( n )3 , χ → χ ( n )1 , χ → χ ( n )2 , and setting p T = 2 neB ). Then the solutions for a constant electric field E ( t ) = E are straightforward to write down, χ ( n )1 χ ( n )2 χ ( n )3 ( p ) = m ( n ) √ eE d (cid:18) η ( n ) , (cid:113) eE p z (cid:19) m ( n ) √ eE d (cid:18) η ( n ) , (cid:113) eE p z (cid:19) d (cid:18) η ( n ) , (cid:113) eE p z (cid:19) , (51)with d i defined in Eq. (A1) and η ( n ) = ( m + 2 neB ) / ( eE ) .Now that we have found the solution for the χ ( n ) i , we can insert it into the ansatz (48) and obtain g ( n )1 , g ( n )2 , and1 f ( n )32 . Then using the inverse of the transformation (45), one can compute all non-vanishing functions, (cid:32) f ( n )11 f ( n )33 (cid:33) = (cid:18) m √ neB (cid:19) √ eE d (cid:18) η ( n ) , (cid:114) eE p z (cid:19) C ( n )1 ( p z − eE t ) , (cid:32) f ( n )24 f ( n )42 (cid:33) = (cid:18) √ neB − m (cid:19) √ eE d (cid:18) η ( n ) , (cid:114) eE p z (cid:19) C ( n )1 ( p z − eE t ) ,f ( n )32 = d (cid:18) η ( n ) , (cid:114) eE p z (cid:19) C ( n )1 ( p z − eE t ) , (52)together with f ( n )21 from Eq. (44). The remaining ten functions are zero. V. PAIR-PRODUCTION RATE
In the last section we have derived the DHW functions in constant electric and magnetic fields. In this section wewill relate the DHW functions to pair production. Note that, in the presence of an electric field, the system cannotremain in thermodynamical equilibrium.Let us first consider a multi-pair system, where the particles are described by the plane-wave solutions of the freeDirac equation. Inserting these wave functions into the definition of the Wigner function and then projecting ontothe unit matrix and the gamma matrices γ , we obtain the contribution of fermion/anti-fermion pairs to the DHWfunctions [54], F = 2 mE p [ n pair ( p ) − , V = 2 p E p [ n pair ( p ) − , (53)where n pair ( p ) is the number density of pairs in phase space. The Pauli principle implies that ≤ n pair ( p ) ≤ . Thedensity of pairs will change due to the pair-production process caused by the electric field. The corresponding rate isgiven by ddt n pair = 12 ddt (cid:90) d p m F + p · V E p , (54)where n pair = (cid:82) d p n pair ( p ) is the number of pairs. Equation (54) can be proven by inserting Eq. (53) into theright-hand side.Analogously, for a multi-pair system in a constant background magnetic field, the on-shell energy is E ( n ) p z = (cid:112) m + p z + 2 neB . If there is pair production by an electric field in the system, its rate in the n th Landau levelcan then be calculated via ddt n ( n )pair = 12 ddt (cid:90) d p m F ( n ) + p · V ( n ) E ( n ) p z . (55)Here, F ( n ) and V ( n ) represent the DHW functions corresponding to the n th Landau level. Employing Eq. (12) andthe ansatz (34), we get ddt n ( n )pair = 12 ddt (cid:90) d p (cid:20) η ( n ) E ( n ) p z (cid:114) eE d (cid:18) η ( n ) , (cid:114) eE p z (cid:19) + p z E ( n ) p z d (cid:18) η ( n ) , (cid:114) eE p z (cid:19)(cid:21) C ( n )1 ( p z − eE t )Λ ( n )+ ( p T ) , (56)where C ( n )1 is given by Eq. (30). The integration over p T can be performed using Eq. (A6). Replacing the kineticmomentum p z by the canonical momentum q z = p z − eE t we obtain the pair-production rate in the n th Landau levelin parallel electric and magnetic fields and a thermal background, ddt n ( n )pair = (cid:90) dq z (cid:20) − f FD ( E ( n ) q z − µ ) − f FD ( E ( n ) q z + µ ) (cid:21) ddt n ( n )vac ( t, q z ) , (57)2where ddt n ( n )vac ( p z ) is the pair-production rate in vacuum for given quantum numbers p z and n , ddt n ( n )vac ( t, q z ) = − (cid:18) − δ n (cid:19) e BE (2 π ) ddq z (cid:40) η ( n ) E ( n ) q z + eE t (cid:114) eE d (cid:20) η ( n ) , (cid:114) eE ( q z + eE t ) (cid:21) + q z + eE tE ( n ) q z + eE t d (cid:20) η ( n ) , (cid:114) eE ( q z + eE t ) (cid:21)(cid:41) . (58)The Fermi-Dirac distributions in the square bracket in Eq. (57) describe the suppression of pair production due tothe Pauli exclusion principle. Summing Eq. (57) over all Landau levels yields the total pair-production rate.In a medium where the chemical potential is zero but the temperature is nonzero the suppression factor is − f FD ( E ( n ) q z ) = tanh βE ( n ) qz , which suppresses the production of pairs with small energies. This factor agrees with theresult of Ref. [45]. However, the integral in Eq. (57) is hard to calculate numerically, because the auxiliary functions d and d are highly oscillatory at large q z , which makes the integration converge too slowly. This problem can besolved by separating the vacuum contribution from the thermal contribution. The pair-production rate in the n thLandau level in vacuum can be analytically calculated, using the asymptotic behavior of the d , functions, cf. App.A, ddt n ( n )vac = (cid:90) dq z ddt n ( n )vac ( t, q z ) = − (cid:18) − δ n (cid:19) e BE (2 π ) (cid:20) d (cid:18) η ( n ) , −∞ (cid:19) + d (cid:18) η ( n ) , + ∞ (cid:19)(cid:21) = (cid:18) − δ n (cid:19) e BE π exp (cid:18) − π m + 2 neBeE (cid:19) . (59)The total rate from all Landau levels is ddt ∞ (cid:88) n =0 n ( n )vac = e E B π exp (cid:18) − πm eE (cid:19) coth (cid:18) πBE (cid:19) , (60)which was previously derived in Refs. [37, 38, 40]. We see that the rate will be enhanced for B (cid:29) E compared tothat without the magnetic field [8]. Similarly we can also derive the production rate of chiral charge dn /dt in astrong magnetic field, which gives the anomaly with the pair production [58, 59].The thermal contribution in Eq. (57) for the n th Landau level is ddt n ( n )thermal = − (cid:90) dq z (cid:20) f FD ( E ( n ) q z − µ ) + f FD ( E ( n ) q z + µ ) (cid:21) ddt n ( n )vac ( t, q z ) . (61)The Fermi–Dirac distributions provide an exponential suppression ∼ e − E ( n ) qz for large q z , thus the q z -integral convergesquickly. In order to show the thermal suppression in a physically intuitive way, we introduce the ratio r of the thermalto the vacuum contribution. This ratio is a function of time t and the three dimensionless parameters e ˜ E ≡ eE [ m ( n ) ] , ˜ T ≡ Tm ( n ) , and ˜ µ ≡ µm ( n ) , where m ( n ) = √ m + 2 neB is the effective mass in the n th Landau level. The totalpair-production rate in the n th Landau level is given by ddt n ( n )pair = (cid:20) r ( t, ˜ E , ˜ T , ˜ µ ) (cid:21) ddt n ( n )vac . (62)In order to show the thermal influence on pair production, we choose the time t = 0 , which is when the canonicalmomentum equals the kinetic one. Figure 1 shows the function r (0 , ˜ E , ˜ T , ˜ µ ) at finite dimensionless temperature andchemical potential. The values stay between − and for all parameters considered, which describes the thermalsuppression of pair production as demanded by the Pauli exclusion principle: a quantum state has a higher probabilityto be occupied at higher temperature or higher chemical potential; this occupation will block the production of newpairs with the same quantum numbers. When the electric field is strong enough, pairs with higher energies, which havesmaller thermal occupation numbers, are more likely to be excited. Thus the suppression is inversely proportional tothe electric field strength. VI. SUMMARY
In this paper we have analytically calculated the Wigner function as well as the Schwinger pair production inconstant and parallel electric and magnetic fields. We have derived the equation of motion for the equal-time Wigner3
Figure 1: The ratio of the thermal contribution to the total pair-production rate to the vacuum contribution for a constantelectric field. Left panel: dependence on electric field strength for temperature ˜ T = 1 and chemical potential ˜ µ = 0 , , .Right panel: ˜ µ = 0 and ˜ T = 0 . , , . function, whose sixteen components, the so-called DHW functions, have definite physical meanings. One can relatethe Schwinger pair-production rate to some of these functions. For the case of a pure constant electric field, we tookthe vacuum values for the sixteen DHW functions as initial condition. Then, we obtained an analytic solution for thesystem of PDEs for the DHW functions. For parallel electric and magnetic fields, we adopted a similar method tocalculate the DHW functions. We showed that the contributions of different Landau levels separate from each other.Under the replacement p T → neB the system of PDEs and the condition when the electric field vanishes coincidewith those in a pure electric field for each Landau level. This provides us with a new method for calculating thepair production in parallel electric and magnetic fields. Analytical solutions for the DHW functions for the case ofconstant electric and magnetic fields, together with the pair-production rate in each Landau level are derived. Ourresults can be directly generalised to the case of finite temperature and chemical potential. The calculation shows thatthe pair-production rate is thermally suppressed and the suppression is proportional to the thermodynamic variables T and µ . More energetic pairs can be created in a stronger electric field, which are less likely to be Pauli-blocked bythe thermal distribution, and this leads to a decrease of the suppression factor.The equation of motion for the Wigner function is equivalent to the Dirac equation if we adopt the classical-fieldapproximation. However, the Wigner function contains sixteen independent components, which leads to a sixteen-dimensional system of PDEs. Due to advances in computer technology in the past few decades, it becomes possibleto numerically solve this PDE system in some simplified cases. In this paper, we have found a set of basis functionsin the presence of a constant magnetic field. These basis functions provide us with a way to replace the continuoustransverse momenta p x , p y by the discrete Landau level index n . The parameter space of the Wigner function isthen simplified from six-dimensional phase space ( x , p ) to the four-dimensional space spanned by ( x , p z ) plus onediscrete parameter n , which makes the system of PDEs more amenable for a numerical solution. However, the caseconsidered in this paper, i.e., homogeneous and parallel electromagnetic fields, is effectively only a (1 + 1) -dimensionalproblem, whereas the fields in real experiments are more likely to be space-time dependent. Nevertheless, the way ofdecomposing the Wigner function presented here may inspire future works and may be a convenient starting pointfor the Wigner-function approach. Acknowledgments.
QW is supported in part by the 973 program under Grant No. 2015CB856902 and by NSFCunder Grant No. 11535012. XLS is supported in part by China Scholarship Council and the Deutsche Forschungs-gemeinschaft (DFG) through the grant CRC-TR 211 "Strong-interaction matter under extreme conditions". DHRacknowledges support by the High-End Visiting Expert project of the State Administration of Foreign Experts Affairs(SAFEA) of China and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through theCollaborative Research Center CRC-TR 211 “Strong-interaction matter under extreme conditions” – project number315477589 - TRR 211.4
Figure 2: u -dependence of the auxiliary functions d i ( η, u ) , i = 1 , , for η = 2 (left panel) and η = 0 . (right panel). Appendix A: Auxiliary functions
We have introduced three auxiliary functions in the solutions for the DHW functions, d ( η, u ) = − e − πη η (cid:12)(cid:12)(cid:12)(cid:12) D − − iη/ ( − ue i π ) (cid:12)(cid:12)(cid:12)(cid:12) ,d ( η, u ) = e − πη e i π D − − iη/ ( − ue i π ) D iη/ ( − ue − i π ) + c.c.,d ( η, u ) = e − πη e − i π D − − iη/ ( − ue i π ) D iη/ ( − ue − i π ) + c.c., (A1)where D ν is the parabolic cylinder function. These functions satisfy the following differential equations, ddu d ( η, u ) = ηd ( η, u ) ,ddu d ( η, u ) = − ud ( η, u ) ,ddu d ( η, u ) = − d ( η, u ) + ud ( η, u ) . (A2)We plot the d i as function of u in Fig. 2. We observe that that all these functions are convergent for u → −∞ , butonly d is obviously convergent for u → + ∞ . The functions d and d are highly oscillatory in a finite region for large u , thus d /u and d /u converge to zero when u → + ∞ . Moreover, we have lim u →−∞ d ( η, u ) = − , lim u → + ∞ d ( η, u ) = 1 − e − πη . (A3)Four groups of basis vectors are used in the expansion of the DHW functions in a constant magnetic field. They arefunctions of the Landau-level index n and the transverse momentum p T , or its modulus p T = √ p T · p T , respectively, e ( n )1 ( p T ) = Λ ( n )+ ( p T ) T Λ ( n ) − ( p T ) , e ( n )2 ( p T ) = Λ ( n ) − ( p T ) T Λ ( n )+ ( p T ) , e ( n )3 ( p T ) = √ neBp T Λ ( n )+ ( p T ) p T , e ( n )4 ( p T ) = √ neBp T Λ ( n )+ ( p T ) − p y p x . (A4)Here, the Λ ( n ) ± functions are defined as Λ ( n ) ± ( p T ) ≡ ( − n (cid:20) L n (cid:18) p T eB (cid:19) ∓ L n − (cid:18) p T eB (cid:19)(cid:21) exp (cid:18) − p T eB (cid:19) , n > , (cid:18) − p T eB (cid:19) , n = 0 , (A5)5where L n ( x ) is the n th Laguerre polynomial. For the lowest Landau level, n = 0 , we have e (0)3 = e (0)4 = 0 and e (0)1 = e (0)2 . These basis vectors allow us to separate the p T dependence. When integrating over transverse momentum p T , Λ ( n )+ ( p T ) gives the density of states for Landau level n , while Λ ( n ) − ( p T ) gives zero for all n > , π ) (cid:90) d p T Λ ( n )+ ( p T ) = eB π π ) (cid:90) d p T Λ ( n ) − ( p T ) = 0 , ( n (cid:54) = 0) . (A6)The basis vectors e ( n ) i for i = 1 , , , and n = 0 , , , · · · are orthogonal with respect to an inner product, (cid:90) d p T e ( m ) Ti ( p T ) e ( n ) j ( p T ) = 2 πeBδ mn δ ij , (A7)for n > , together with (cid:90) d p T e (0) T ( p T ) e (0)1 ( p T ) = 4 πeB, (cid:90) d p T e ( n ) Ti ( p T ) e (0)1 ( p T ) = 0 . (A8)We can also check that the functions Λ ( n ) ± in Eq. (A5) satisfy the following relations, eB∂ p x Λ ( n )+ ( p T ) = − p x Λ ( n ) − ( p T ) ,eB∂ p x Λ ( n ) − ( p T ) = − p x (cid:18) − neBp T (cid:19) Λ ( n )+ ( p T ) , (A9)which are used to derive Eqs. (31) and (32). [1] W. Greiner, B. Muller, and J. Rafelski, QUANTUM ELECTRODYNAMICS OF STRONG FIELDS (1985).[2] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nucl. Phys.
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