Applicability of the Klein-Gordon equation for pair production in vacuum and plasma

In this paper, a phase-space description of electron-positron pair-creation will be applied, based on a Wigner transformation of the Klein-Gordon equation. The resulting theory is similar in many respects to the equations from the Dirac-Heisenberg-Wigner formalism. However, in the former case, all physics related to particle spin is neglected. In the present paper we compare the pair-production rate in vacuum and plasmas, with and without spin effects, in order to evaluate the accuracy and applicability of the spinless approximation. It is found that for modest frequencies of the electromagnetic field, the pair production rate of the Klein-Gordon theory is a good approximation to the Dirac theory, provided the matter density is small enough for Pauli blocking to be neglected, and a factor of two related to the difference in the vacuum energy density is compensated for.


I. INTRODUCTION
The interest in strong field physics has increased in the past decades due to the rapid development of the laser facilities [1,2].See, e.g., Ref. [3,4] for the recent developments in the research area.In the presence of a strong electromagnetic field, the vacuum can decay into electron-positron pairs.This process was first studied by Sauter in 1931 [5] and a rigorous mathematical description of this process was derived by Julian Schwinger in 1951 [6].Since then, the basic mechanism has been studied in a dynamical context and for more complicated field geometries.In particular, a study of the interplay between temporal and spatial variations of the fields has been done in Refs.[7][8][9][10].To maximize the number of produced particles in the case of sub-critical field strengths, a geometry of colliding laser pulses has been suggested in [11][12][13][14].
In spite of much progress in recent years [7][8][9][10][11][12][13][14], due to spin and chiral effects associated with Dirac fermions, various simplifying assumptions are typically needed when studying electron-positron pair production.A way to relax some of these assumptions concerning e.g. the electromagnetic field geometry, is to study scalar QED, where the spin of the particles is ignored.Scalar QED models, based on the Klein-Gordon equation, are considerably less complicated than those derived from the Dirac equation.While it is known that many qualitative features regarding pair-production are preserved when replacing fermionic QED with scalar QED [15][16][17], a better understanding of the differences/similarities are desirable, to know to what degree results from scalar QED can be trusted.The purpose of the present paper is to study pair production in vacuum and plasmas, with and without including the particle spin properties, to highlight differences and similarities between the bosonic and fermionic results.
Various models and methods can be used when studying electron-positron pair production from vacuum and in plasmas.This includes quantum kinetic theories (QKT), which can be applied for fermionic and bosonic pair production in electric fields [18].Treatments, where a bosonic model of pair production has been used, are given in Refs.[19,20], and works on both bosonic and fermionic pair production in electric and magnetic fields can be found in Ref. [21].Moreover, lattice Quantum Electrodynamics has been used to model scalar QED pair production [22].
In the present paper, we will use a phase-space approach, based on a gauge invariant Wigner transform of the evolution equations.In the fermionic case, this leads to the real-time Dirac-Heisenberg-Wigner (DHW) formalism, first derived in Ref. [23].For the bosonic case, this corresponds to making a (gauge-invariant) Wigner transform of the Klein-Gordon equation, as was first derived in Ref. [24] with applications presented e.g. in Refs.[24][25][26][27][28][29].We will refer to these equations as the Klein-Gordon-Wigner (KGW) equations.We note that since both KGW and DHW are based on a phase-space approach, they are conceptually similar, and this facilitates a direct comparison of the results.
An important conclusion from our study, which agrees with previous findings [6,30], is that KGW and DHW are in exact agreement for pair-production in the zero frequency limit, provided a factor of two is compensated for.The factor of two is closely related to the difference between the fermionic and bosonic vacuum expectation values.When the frequency of the applied field is increased, the spin polarization current in the DHWformalism grows, which is associated with a gradually larger deviation between the bosonic and fermionic results.For significant pair-production, the self-consistent field generated by the plasma currents becomes appreciable, and this tends to enhance the deviation between bosonic and fermionic dynamics.For even higher plasma densities, Pauli blocking can be important, which naturally cannot occur in the bosonic case.
The paper is organized as follows: In Section II we present the derivation of the KGW-formalism.In Section III we derive the linear response of a plasma to electromagnetic waves.We make a comparison both to the classical dispersion relation (which is recovered in the appropriate limit) and to the results based on the DHW formalism.Next, numerical results for large fields (Schwinger) pair production are presented in Section IV.Here we compared the particle production in KGW and DHW using different scaling parameters.Finally, we present our conclusions in Section V.

II. KGW-MODEL
In this section, we will make a brief derivation of the Klein-Gordon-Wigner (KGW) formalism that has previously been derived by Ref. [24], which the reader may consult for more details.The starting point is the Klein-Gordon equation where e is the elementary charge, m is the mass of the electron, and A µ = (A 0 , A), where A 0 A are the scalar vector potential, respectively.This equation contains a second-order derivative in time.To transform the Klein-Gordon equation to phase-space and obtain an explicit expression of the charge density in phase-space, we express the Klein-Gordon equation in the representation of Feshbach and Villars [31].In this representation, we have a first-order time-derivative.The Klein-Gordon field is expressed by the two-component wave function where In matrix representation we get where 1 is the identity matrix.This equation is first order in the time-derivative and has a Schrödinger-type of time evolution.The right-hand side of this equation can be interpreted as a Hamiltonian operator that will be used to find the time evolution in phase-space.Next, we use the gauge-invariant Wigner transformation Ŵ (r, p, t) where The Wigner function is defined as the expectation value of the Wigner operator where |Ω Ω| is the state of the system in the Hilbert space.To find an evolution equation in phase space, we take the time derivative of Eq. ( 6) and use the Hamiltonian operator in Eq. ( 5).We use the Hartree approximation where the electromagnetic field is treated as a non-quantized field.Finally, we have an equation of motion of the Wigner function where σ i , with i = 1, 2, 3, are the Pauli matrices and For longer macroscopic scales, the non-local operators given by the integrals can be expanded in powers of .Thus, for example in the long-scale limit, we may use D t = ∂ t + eE • ∇ p , and similarly for the other operators [24].The interpretation of the variables described by Eq. ( 9) is not simple.Thus we make an expansion of the Wigner function over the Pauli matrices σ i and the identity matrix 1 where the expansion coefficients f i , with i=0-3, will lead to four coupled partial differential equations.
This PDE-system is the final one that covers all physics of the Klein-Gordon equation.This system is closed by Maxwell's equations with the sources ρ = e d 3 pf 0 (15) Compared to the DHW-formalism [23] which is based on the Dirac equation, this system contains only four instead of 16 scalar equations.This makes the KGW-formalism simpler to solve numerically.The 4 scalars of the KGWformalism have a straightforward physical interpretation, as can be seen from the following observables where Q is the total charge, J is the current density, W is the particle energy and M is the momentum.Interpretations that can be done from the above expressions include, e.g., that ef 0 is the (phase space) charge density and ep/m(f 2 + f 3 ) the current density, as implied by the sources in Maxwell's equations.The f i -functions have the vacuum-contribution where = m 2 + p 2 .The expressions above are obtained by calculating the Wigner operator for the free particle Klein-Gordon equation and taking the vacuum expectation value.If we add a plasma to the background, we should modify the source terms for f 2 + f 3 and f 3 − f 2 .Moreover, the charge density f 0 should be non-zero.Adding electron and positron sources we first obtain f 2 +f 3 = (m/ )F and f 3 −f 2 = ( /m)F in Eq. ( 21) where Here f e,p can be viewed as classical electron/positron distribution functions.Note that we have a positive sign between the particle and vacuum sources, instead of a negative one as in the DHW-formalism [23,32].This is related to the physics of the Pauli-exclusion principle, which obviously is not included in the KGW-formalism, as the model is derived for spinless particles.By contrast, in the DHW-formalism, the Pauli-principle will be respected provided the correct expectation values for the vacuum states are implemented.Moreover, we can note that the particle contributions to F have an extra factor of 2 in Eq. ( 22), as compared to DHW-formalism.This is due to the fact that the magnitude of the vacuum contribution is only half as large for spinless particles.
The functions f e/p can be picked as any common background distribution function from standard kinetic theory, i.e., a Maxwell-Boltzmann, Synge-Juttner, or Fermi-Dirac distribution, depending on whether the characteristic kinetic energy is relativistic and whether the particles are degenerate.In conclusion, initial conditions for the KGWvariables involving a homogeneous medium with electron and positron distribution functions f e,p are given by with F given by Eq. ( 22).The above expression gives a time-independent solution to the KGW-equations in the absence of electromagnetic fields (assuming that the total charge density and current density are zero).The expressions in Eq. ( 23), describing the equilibrium states, are a necessary prerequisite for including dynamical scenarios, leading to particle pair creation as will be studied in the next sections.

III. LINEAR WAVES
In this section, we consider the linear response of a plasma to an electromagnetic wave field, accounting also for the contribution from the vacuum expectation values.For our case with no background electromagnetic fields, as described in the previous section, we get the unperturbed vacuum contributions as the Wigner transform of the expectation value of the free Klein-Gordon field operators.Thus, Eq. ( 23) (including the plasma) applies for the background.For the wave perturbation, we consider electromagnetic waves with the following geometry The ions will not be treated dynamically but will constitute a constant neutralizing background.In the absence of ions, naturally, the electron and positron background charge densities must cancel for the background to be in equilibrium.However, in the presence of a constant (immobile) ion background, the electron and positron may differ, in particular, the positron background may vanish.
Next, in order to study linear theory, we divide the functions f i as where the upper indices denote unperturbed and perturbed quantities.Furthermore, we use Ampére's law to obtain an implicit dispersion relation that can be written as After some algebra, we can compute the current sources and deduce the explicit expression where Here the operators are given by ∆ 1 = 1 and ∆ 2 = k∇ pz /12 in the long-scale limit, where we have kept contributions to the lowest non-vanishing order.We can obtain the classical limit by letting −→ 0 in Eq. (27).Doing so, we get ∆ 2 = 0.The dispersion relation then reduces to which can be shown to agree with the standard result based on the relativistic Vlasov equation, after some straightforward algebra.Here, f e and f p behave as classical distribution functions with one important difference.Since, in the underlying model, electrons and positrons are described by the same evolution equations, we must model the case of two identical beams moving with the same beam momentum This accounts for the fact that positrons can be viewed as electrons moving backward in time.We note that the appearance of in the integration measure is just a matter of normalization, and not a sign of any remaining quantum features [33].Now, we want to compare the dispersion relation Eq. (27) with the DHW formalism.To be able to do that, we take the homogeneous limit of Eq. ( 27) (since only the electrostatic limit resulting from k −→ 0 has been computed in the DHW-case) and compare it with the corresponding result from the DHW-formalism [32,34].Letting k −→ 0 in Eq. ( 27), and including electrons only in the background plasma f = f e , we get For the DHW formalism, we apply the k −→ 0-result of the electrostatic case, see [32,34], and we obtain We can clearly see that the results agree in the classical limit, and that the denominators coincide, i.e., the physics related to wave-particle interaction is similar.Both for the KGW and DHW, the real part of the wave frequency ω r is given by the classical limit to a good approximation.The reason is that for ω ∼ m, the Fermi energy E F will be much larger than unity.Thus, even if ω ∼ m we will have ω E f , which, in turn, implies a minor quantum contribution to Eqs. (31) and (32).Nevertheless, there are also some differences between Eqs. (31) and (32).To focus on the most important one, we now concentrate on the damping given by the imaginary part of the wave frequency ω i , associated with the pole contribution of the momentum integral.Using the Plemelj formula where P is the principal value, we obtain from the KGWexpression Eq. ( 31) For the DHW-system, we instead obtain Here p res is the resonant momentum, making the denominators in the integrands zero, i.e. m 2 + p 2 res = 2 ω 2 /4.While some features of the two damping rates agree, it can be noted that there are also significant differences.These differences can be better understood in light of certain numerical results presented in the next section.Thus we will wait to make a more detailed comparison of the damping rates.

A. Preliminaries
Before presenting the numerical results, we discuss the issue of renormalization.For large momentum where we can use the approximation ≈ p, it is straightforward to see that the integrand in Eq. ( 31) scales as 1/p, and thus the integral has a logarithmic divergence.In principle, this needs to be dealt with, but in practice, for a numerical solution, this is almost automatically solved by a numerical cut-off in the momentum integration that acts as an effective regularization.Due to the slow growth of the logarithmic divergence, in the presence of a numerical momentum cut-off, simply ignoring the issue of renormalization does not result in a significant error.The technical aspects are the same as for the DHW-case, which has been described in more detail in Ref. [34,35].
To able to compare the results of the numerical solution of Eq. ( 14) with the ones from the DHW-formalism in [35], we consider the homogeneous-limit of Eq. ( 14) Apparently, in the homogeneous limit, f 0 decouples, and we only need to solve three equations.To simplify the numerical solution, we define new variables as deviations from the vacuum state, i.e. where With this choice, we make sure that our basic variables become small for large momenta, such that a momentum cut-off is possible.By introducing fi , we remove the vacuum contribution from the variables that we solve numerically.Obviously, this does not mean that vacuum physics is not included in our solution, we only do this to handle certain numerical technicalities.The PDE-system is now For notational convenience, we omit the tilde for variables in what follows.In order to solve the system numerically, we need to make some further adaptions.The next step is to make the additional change of variable introduce the canonical momentum q given by q = p + eA (41) and use normalized variables: , where ω ce is the Compton frequency.The final system resulting from these changes, which will be solved numerically is together with Ampere's law where η = α/π, here α is the fine-stucture constant.Note that, for notational convenience, we omit the subscript "n" for variables in what follows.The system Eq.(42)-Eq.( 43) is solved numerically using a phase-corrected staggered leapfrog method [36].Typical parameters of the simulations are a time-step of the order of ∆t ∼ 0.001, a parallel momentum step ∆q ∼ 0.01, and a perpendicular momentum step ∆p ⊥ ∼ 0.1.In spite of the rather good resolution in parallel momentum, the q-dependence of the produced data tends to look noisy in momentum space.This is due to the zitterbewegung effect which produces increasingly short scales.Nevertheless, the dynamics of the larger scales are not sensitive to the small-scale momentum details, i.e. changing ∆q does not affect the results presented in this paper.To confirm that the numerical scheme produces sound results, we study the energy conservation law.An energy conservation law of the system Eqs.( 42) and ( 43) can be written in the form For the numerical resolutions used in the runs presented below, the total energy of the system is conserved within a relative error typically less than 10 −4 .

B. Effects due to the vacuum background
Before solving the Eq.(42)-Eq.( 43) for more complicated scenarios, we can check the validity of the numerical solution by considering constant electric fields E = E 0 .The rate for creating a fermionic pair from vacuum dn f /dt is well-known [6,30] By comparison, the rate for creating a bosonic pair as described by the KGW-theory, is dn b /dt = 0.5dn f /dt.This is a direct consequence of the factors given in Eq. ( 22), where the relative contribution from the vacuum sources is half of the contribution seen in the DHW-formalism.
To validate the numerics, we will solve Eq. ( 42) and the corresponding equations in the DHW-formalism [32] for the case of constant electric fields E 0 and only vacuum initially.Since our goal is to check the validity of the numerics by comparing with theory without back-reaction effects, we also neglect back-reaction, i.e., we don't solve Eq. ( 43) self-consistently.Then, we calculate the number of produced particles n k in the KGW-formalism and n D in the DHW-formalism, with n k given by In Fig. 1, we compare the numerical values of n k/D with dn b/f /dt for different values of E 0 .As can be seen, the numerical values follow the analytical ones with good agreement.Interestingly, nothing prevents us from making a toy model where we use spinless dynamics for the evolution equations, but increase the vacuum expectation values by a factor of two in the KGW-model.Doing this, for constant fields n k will get the same behavior as n D and thus the slightly modified version of the KGW-formalism gives the exact same results as the DHW-formalism C. High frequency effects Now, we turn to the problem of time-dependent electric fields.We solve Eq. (42)-Eq.(43) for the case with no plasma and only vacuum initially using the following representation of the time-dependent pulse where τ 0 is the phase-shift, b is the pulse length and ω = N ω ce /100.See Fig. 2 for a temporal plot of the electric pulse.We use n k Eq. ( 46) to find the ratio n k /n D of the produced pair in KGW-formalism and n D in the DHW-formalism [32].For constant fields E = E 0 , given by the rate presented for fermions in Eq. ( 45) and for bosons, we have For the case with a time-dependent electric field, in Fig. 3, we plot the ratio n k /n D versus N .We can note that for small N , i.e., in the constant field limit, the ratio is very close to 0.5 as expected.This is a good agreement between numerical and analytical solutions.For N = 200, we have ω = 2m and it is thus possible to create an electron-positron pair from a single quanta.For this case, the relative rate of the KGW pair production has a minimum.Generally, the ratio n k /n D tends to be smaller than 0.5 when processes involving few quanta (as opposed to the pure Schwinger mechanism) are possible to occur.This is because the probability of producing pairs from few quanta is smaller in the KGW-description than in the DHW-description.Looking specifically at one-quanta processes, this fact is confirmed in the linear damping results (where the linear damping is due to single quanta pair creation) as given by Eq. ( 34)-Eq.( 35).For frequencies ω ≈ 2m e , we have p res close to zero, and thus the created pairs have very small momentum.From Eq. ( 34), the damping of the wave in the KGW-formalism scales as p 3 res , compared to the DHW-formalism, where the wave damping is linear in p res .For higher frequencies, we have larger p res and this lead to a somewhat larger value of n k /n D , but it is still well below 0.5.The reason for the saturation of pair creation for high frequencies in the KGW-model is the scaling ω i ∼ ω −4 r in Eq. ( 34), more than compensating for the scaling ω i ∼ p 3 res .Since the KGW-formalism is spinless, only free currents are created.By contrast, the total current in the DHWformalism is the sum of free current, the magnetization current [37], and the polarization current J p [38].Note, however, that in the absence of spatial variations, the   magnetization current vanishes.For pulses with ω ω ce , the polarization current is small due to J p = ∂P/∂t, and the slow (compared to the Compton frequency) temporal scale.Here P is the polarization due to the particle spin properties.The smallness of the polarization current is confirmed in Fig. 4, where the polarization and total currents are plotted over time by solving the equations of the DHW-formalism using the pulse in Eq. (47).For the upper panel of Fig. 4 where we used N = 2, i.e., ω = 0.02ω ce , the polarization current is roughly only around 1% of the total current.When the frequency is increased to N = 20, the polarization current increases to about 10% of the total current.As the ratio n k /n D = 1/2 comes from the difference in the vacuum contribution only, the deviation from this value depends on the different dynamics of the KGW and DHW descriptions.For the present case, as long as the polarization current density J p associated with the spin is small compared to the total current density, the dynamical influence of spin effects is comparatively small.Thus it is not surprising that the ratio n k /n D = 1/2 is relatively unaffected by the spin dynamics.However, a careful comparison of Fig. 3 and Fig. 4 shows that the deviation from n k /n D = 1/2 coincides with an increase in the relative importance of the spin dynamics.
Next, we would like to make contact between the an-alytical and numerical calculations.Specifically, we will study where in momentum space the pairs are produced.In Fig. 5, contour curves over the pair momentum density, i.e. the integrand in Eq. ( 46) is shown.The two figures in the upper panel show the contour for DHWexpression n D (p ⊥ , q) (the figure to the left) and the KGWexpression n k (p ⊥ , q) (the figure to the right) using the pulse in Eq. ( 47) with ω = 0.02ω cr , τ 0 = π, b = 20 and E 0 = 1.In the upper panel, we can see that created pairs have the same momentum distribution in both of the models.It should be emphasized that we here have used the toy model with the vacuum contribution twice as large as for a scalar field in the KGW calculations, to better mimic the behavior of fermions.
In the lower panel of Fig. 5, we have used the pulse in Eq. ( 47) with ω = 5ω ce , τ 0 = π, b = 2 and E 0 = 2.For such a high frequency, single quanta pair-creation processes dominate.Clearly, the polarization current J p is important in this regime.As can be seen, in spite that pairs are created close to the region of resonant momenta, satisfying the resonance condition m 2 + p 2 res = 2 ω 2 , the different models have different momentum distributions nevertheless.For the KGW-model (the figure to the right), the pairs created tend to have a higher value of |q|, and a smaller value of p ⊥ .For the DHW-model, the distribution in momentum space is the opposite.
To clarify the reason for this difference, we will rewrite the previous expressions Eqs.(31) and (32) in cylindrical momentum coordinates.In this case, the dispersion relation Eq. ( 31) for the KGW-case becomes and the dispersion relation for the DHW-case is written where ⊥ = m 2 + p 2 ⊥ .Note that we have regained p z rather than q here [39] and also keep the un-normalized variables of Section III.The key to understanding the lower panels of Fig. 5 are the scalings with p z and p ⊥ of the integrands in Eqs.(49) and (50).What contributes to the damping rates is the pair creation that results from the pole contribution.That is, pair creation comes from an integration over resonant momenta p res .The resulting pair creation will have a distribution in momentum space reflecting the magnitude of the integrand as a function of p z and p ⊥ along the curve given by a constant p res .As we are now considering pair-creation in vacuum rather than a plasma, we can inspect the integrand in Eq. ( 49) with only the vacuum contribution present, in which case the integrand reads Noting that the momenta which contributes to singlequanta pair creation will be close to the resonant momenta, we see that for resonant values with large p z (q) and small p ⊥ , we will get the dominant pair creation contribution.This is in agreement with the lower right panel of Fig. 5 as p ⊥ ∼ 0 gives a negligible pair creation rate.
For the left figure of the lower panel in Fig. 5, we have the contour curves of the momentum distribution for the created pairs in the DHW-formalism.As noted above, we see that the created pairs tend to have a high perpendicular momentum.This effect has been seen in previous works [32,40].Moreover, it is further confirmed by considering only the vacuum contribution to the integrand in Eq. ( 50) We note that for resonant momenta, the integrand is largest for high perpendicular momentum and has a smaller value when |p z | is large, explaining why the momentum distribution of the single-quanta pair creation in the DHW-formalism has the momentum distribution that we see in Fig. 5.
Next, we will include the effects due to back-reaction, i.e., we will solve Eq. ( 43) together with Eq. (42) using the pulse in Eq. (47).We start from vacuum and divide the electric field into the self-consistent electric field, given by the current sources generated by the created pairs, and the external electric field which is prescribed and given by the same type of temporal profile as before.
In Fig. 6, we have a plot of the self-consistent electric field (as the prescribed part is the same) over time comparing the results from DHW and KGW.In the first panel, we have used N = 2, i.e., ω = 0.02ω ce .For this low frequency, we see a very good agreement between DHW and KGW.The good agreement is dependent on the choice to double the background vacuum contribution in the KGW-model, i.e., we have imitated initial conditions for the fermions, even though the dynamical evolution follows that of a scalar field, neglecting the spin.
In the second panel, we use a slightly higher frequency N = 4.While the agreement is still good, we see a graudally growing phase-shift between the DHW and KGW models.While the fraction n K /n D was almost unchanged when the frequency was increased to this value, when the self-consistent dynamic is involved, we see a higher sensitivity to the frequency.Increasing the frequency even higher to 0.1ω ce , the difference between DHW and KGW is even more pronounced.Here, we see a more dramatic phase-shift at an early stage and a qualitatively different evolution of the wave amplitude.Thus, modeling electronpositron pair creation with the KGW-model requires lower frequencies in the case of self-consistent dynamics.

D. Non-degeneracy effects
In this subsection, we would like to illustrate the mechanism of Pauli blocking.This mechanism is always present in the DHW formalism, but not in the KGW-formalism as the model is spinless.However, whether or not Pauli blocking is an important feature dynamically, depends on the number of free states where pairs can be created.In particular, a degenerate initial distribution will block the low-energy states in momentum space, which is the region where particles are most easily created.Thus, in this subsection, we will start with a plasma initially and solve Eqs.(42) and (43) and the corresponding equations in the DHW-formalism.As initial conditions in the runs, we let E(t = 0) = E 0 and A(t = 0) = 0.The initial plasma is represented by a Fermi-Dirac distribution, where µ is the chemical potential and T is the temperature (we normalize both µ and T with the electron mass).For µ T , the chemical potential fulfills µ ≈ E F , where E F The number density n D/K (q) = dp ⊥ p ⊥ n D/K (q, p ⊥ ) as a function of the canonical momentum is plotted for E0 = 4, µ = 4 and T = 0.02 for DHW and KGW.Here, the solid curve is for n(t=0) and the .... is for n(t=T).
is the Fermi energy, in which case the distribution will be degenerate (i.e., we will have f ≈ 1, i.e. all electron states filled, for energies smaller than the Fermi energy, and f ≈ 0 otherwise).Naturally, non-degenerate initial distributions are still possible when using Eq.(53), by letting µ < T .Also, by picking a large but negative chemical potential, we get a Maxwell-Boltzmann type of distribution.
In Fig. 7, we demonstrate the Pauli blocking mechanism.In the upper panel, we have considered the DHWformalism using a degenerate distribution with µ = 4 and T = 0.02, where all the low-energy electron states are filled, preventing further pair-creation in the filled region.As shown, due to a strong initial field E 0 = 4, pairs are still created at a high rate, but the low-energy region is perfectly blocked.This is seen in Fig. 7 by comparing the momentum distribution n D (q) after an oscillation period with the initial momentum distribution n D (t = 0).Specifically, it should be noted that the two curves coincide (no further pair-creation) for a low or modest parallel momentum.In the lower panel of Fig. 7, we have used the KGW-formalism using the same input data as in the upper panel.We note that pairs are still created for low parallel momentum, in contrast to the DHW case.The effect is expected as the KGW model does not include the effect of the Pauli blocking dynamically, even if a degenerate distribution is chosen as an initial condition.

V. CONCLUSION
In this paper, we have compared pair-creation in the KGW-formalism, based on a Wigner transform of the Klein-Gordon equation, and pair-creation in the DHWformalism, based on a Wigner transform of the Dirac equation.Firstly, studying high-frequency pair creation in low amplitude linear theory, we have noted that while the DHW-formalism and the KGW-formalism share qualitative features (e.g.pair-creation when the same resonance condition is fulfilled), the expression for the wave damping rates due to pair creation are fundamentally different.
Generally, the agreement between KGW and DHW is better for lower frequencies of the electromagnetic field.
In particular, if one compensates for the larger magnitude of the vacuum background in the DHW-case (by picking a vacuum background a factor two larger than for a genuine scalar field), the KGW and DHW equations agree perfectly for the pair-production rate in the zero-frequency limit.
Increasing the frequency of the electric field, the agreement gradually deteriorates.The lack of agreement is correlated with a simultaneous growth of the relative contribution of the polarization current density to the total current density.As the polarization current is a direct consequence of the electron spin, this result is to be expected.
Studying pair-production including the back-reaction from the self-consistent current density created by the pairs, the sensitivity to a finite frequency is increased.That is, we need a lower frequency of the electromagnetic field to still get a good agreement between the DHWtheory and the KGW-theory.Practically speaking ω ∼ 0.04ω c or smaller, Moreover, for a self-consistent theory it is crucial to magnify the vacuum background by a factor two in the KGW-case (as compared to a genuine scalar field), as the nonlinear dynamics otherwise will be heavily modified, in case the self-consistent response has the wrong magnitude.
Furthermore, we have studied the dynamics for large plasma densities, where Pauli blocking is prominent.As expected, in the DHW-formalism pairs can not be created in a region of momentum space where the states are occupied.The same does not apply for the KGW-formalism, naturally.
The present study is motivated by a desire to use the KGW-formalism whenever it is a valid approximation, as the 16 scalar equations of the DHW-formalism (as compared to the 4 scalar equations of the KGW-formalism) are considerably more difficult to solve, both numerically and analytically.While the present study has certain restrictions, e.g. with only minor exceptions we have considered electrostatic fields and the homogeneous limit, we expect that many of the present findings apply in more general scenarios.In particular, we expect that the relatively good agreement between the DHW-formalism and the KGW-formalism for low frequencies generalizes to cases of spatial variations and electromagnetic fields, at least qualitatively.However, more studies are needed to give a definitive answer.and P. Johansson, Semi-relativistic effects in spin-1/2 quantum plasmas, New Journal of Physics 14, 073042 (2012).
[39] We note that the difference between q and pz is not important for high frequency pair creation as this process will take place also for small/modest electric field, which leads to a small A.

Figure 1 .
Figure 1.The number of produced pairs from KGW (n k ) and DHW (nD) using the numerical solutions together with the corresponding analytical results dn b /dt and dn f /dt.

Figure 3 .
Figure 3.The ratio n k /nD is plotted as a function of N , displaying the validity of the KGW-formalism to model fermions at different regimes.Here we have used E0 = 1.

Figure 4 .
Figure 4. Polarization current Jp and total current Jtot plotted over time for N = (2, 20) in the upper and lower panels, respectively.Note the different scales for Jp and Jtot

Figure 6 .
Figure 6.The time profile of the self-consistent electric field from the KGW and DHW-formalism for three values of the frequency of the initial electric pulse N = (0.02, 0.04, 0.1).

Figure 7 .
Figure 7.The number density n D/K (q) = dp ⊥ p ⊥ n D/K (q, p ⊥ ) as a function of the canonical momentum is plotted for E0 = 4, µ = 4 and T = 0.02 for DHW and KGW.Here, the solid curve is for n(t=0) and the .... is for n(t=T).