dataset. The numbers indicate the directionality of the causal edges in the
graph.

FINISH

\---------------------------

Admin (to chat_manager):

export the matrix into csv file call "Auto_Coding_MAD_React.csv"

\---------------------------

Code_Writer (to chat_manager):

‘‘‘python

import pandas as pd

# Create the adjacency matrix as a Pandas DataFrame

adj_matrix_data = [[0, 1, 0, 0, 0, 1],

[0, 0, 0, 0, 0, 0],

[1, 0, 0, 0, 0, 0],

[0, 1, 0, 0, 1, 0],

[0, 0, 0, 0, 0, -1],

[0, 0, 0, 0, -1, 0]]

columns = [’Altitude’, ’Temperature’, ’Precipitation’, ’Longitude’,
’Sunshine’, ’Latitude’]

adj_matrix_df = pd.DataFrame(adj_matrix_data, columns=columns, index=columns)

# Export the adjacency matrix to a CSV file

adj_matrix_df.to_csv("Auto_Coding_MAD_React.csv")

‘‘‘

\---------------------------

>>>>>>>> EXECUTING CODE BLOCK (inferred language is python)...

Code_Executor (to chat_manager):

exitcode: 0 (execution succeeded)

Code output:

\---------------------------

Code_Writer (to chat_manager):

The adjacency matrix representing the causal relationships among the variables
Altitude, Temperature, Precipitation, Longitude, Sunshine, and Latitude has
been successfully exported to a CSV file named "Auto_Coding_MAD_React.csv".

\---------------------------

Code_Executor (to chat_manager):

\---------------------------

Code_Writer (to chat_manager):

I have successfully completed all the steps to find the causal relationship
among variables in the climate dataset:

1. Data Preparation: Loaded the dataset and handled missing values.

2. Graph Initialization: Created an initial fully connected, undirected graph.

3. Conditional Independence Testing: Implemented statistical tests to check for conditional independence between variable pairs.

4. Orientation Rules: Applied orientation rules to determine the directionality of edges based on conditional independence patterns and converted the graph into a directed acyclic graph (DAG).

5. Implemented the PC Algorithm for causal relationship discovery and obtained the adjacency matrix for the human-readable causal graph.

The adjacency matrix has been exported to a CSV file named
"Auto_Coding_MAD_React.csv".

I have provided a comprehensive solution to find the causal relationships in
the climate dataset. If you have any more tasks or questions, feel free to let
me know!

\---------------------------

Code_Executor (to chat_manager):

\---------------------------

Code_Writer (to chat_manager):

FINISH

\---------------------------


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

Haidar Al-Naseri<EMAIL_ADDRESS>Department of Physics, Umeå
University, SE–901 87 Umeå, Sweden  Gert Brodin<EMAIL_ADDRESS>Department
of Physics, Umeå University, SE–901 87 Umeå, Sweden

###### Abstract

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.

###### pacs:

52.25.Dg, 52.27.Ny, 52.25.Xz, 03.50.De, 03.65.Sq, 03.30.+p

##  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 [8, 7, 9, 10, 12, 11, 14, 13], 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 DHW-formalism 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

$\Big{[}(\partial_{\mu}-ieA_{\mu})(\partial_{\mu}+ieA_{\mu})+m^{2}\Big{]}\phi(\mathbf{r},t)=0$
(1)

where $e$ is the elementary charge, $m$ is the mass of the electron, and
$A^{\mu}=(A^{0},\mathbf{A})$, where $A^{0}$ $\mathbf{A}$ 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

$\Phi=\begin{pmatrix}\psi\\\ \chi\end{pmatrix}$ (2)

where

$\displaystyle\psi$
$\displaystyle=\frac{1}{2}\Big{(}\phi+\frac{i}{m}\frac{\partial\phi}{\partial
t}-\frac{eA^{0}}{m}\Big{)}$ (3) $\displaystyle\chi$
$\displaystyle=\frac{1}{2}\Big{(}\phi-\frac{i}{m}\frac{\partial\phi}{\partial
t}+\frac{eA^{0}}{m}\Big{)}$ (4)

In matrix representation we get

$i\frac{\partial\Phi}{\partial
t}=\Bigg{[}\frac{1}{2m}\Big{(}-i\frac{\partial}{\partial\mathbf{r}}-e\mathbf{A}\Big{)}^{2}\begin{pmatrix}1&1\\\
-1&-1\end{pmatrix}\\\ +m\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}+eA^{0}\mathbf{1}\Bigg{]}\Phi$ (5)

where $\mathbf{1}$ 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

$\hat{W}(\mathbf{r},\mathbf{p},t)\\\ =\int
d^{3}z\exp\bigg{(}i\mathbf{p}\cdot\mathbf{z}+ie\int^{1/2}_{-1/2}d\lambda\mathbf{z}\cdot\mathbf{A}(\mathbf{r}+\lambda\mathbf{z},t)\bigg{)}\hat{C}(\mathbf{r},t)$
(6)

where

$\hat{C}(\mathbf{r},t)=\left\\{\Phi(\mathbf{r}+z/2,t),\Phi(\mathbf{r}-z/2,t)\right\\}$
(7)

The Wigner function is defined as the expectation value of the Wigner operator

$W(\mathbf{r},\mathbf{p},t)=\bra{\Omega}\hat{W}(\mathbf{r},\mathbf{p},t)\ket{\Omega}$
(8)

where $\ket{\Omega}\bra{\Omega}$ 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

$iD_{t}W(\mathbf{r},\mathbf{p},t)=-\frac{i}{2}\hat{O}_{1}\left\\{\sigma_{3}+i\sigma_{2},W\right\\}\\\
+\hat{O}_{2}\left[\sigma_{3}+i\sigma_{2},W\right]+m\left[\sigma_{1},W\right]$
(9)

where $\sigma_{i}$, with $i=1,2,3$, are the Pauli matrices and

$\displaystyle\hat{O}_{1}$
$\displaystyle=\frac{\mathbf{p}\cdot\nabla}{m}+\frac{\mathbf{p}}{m}\cdot
e\int^{1/2}_{1/2}d\lambda\mathbf{B}(x+i\hbar\lambda\nabla_{p})\times\nabla$
(10) $\displaystyle\hat{O}_{2}$
$\displaystyle=\frac{\nabla^{2}}{4m}-\frac{p^{2}}{m}-\frac{e\hbar^{2}}{12m}\nabla\cdot(\mathbf{B}\times\nabla)$
$\displaystyle+2\frac{\mathbf{p}}{m}\cdot
ie\hbar\int^{1/2}_{1/2}d\lambda\lambda\mathbf{B}(x+i\hbar\lambda\nabla_{p})\times\nabla_{p}$
(11) $\displaystyle D_{t}$
$\displaystyle=\partial_{t}+e\int_{1/2}^{1/2}d\lambda\mathbf{E}(x+i\hbar\lambda\nabla_{p})\cdot\nabla_{p}$
(12)

For longer macroscopic scales, the non-local operators given by the integrals
can be expanded in powers of $\hbar$. Thus, for example in the long-scale
limit, we may use $D_{t}=\partial_{t}+e\mathbf{E}\cdot\nabla_{p}$, 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 $\sigma_{i}$ and the identity matrix
$\mathbf{1}$

$W(\mathbf{r},\mathbf{p},t)=f_{3}(\mathbf{r},\mathbf{p},t)\mathbf{1}+\sum_{i=1}^{3}f_{3-i}(\mathbf{r},\mathbf{p},t)\sigma_{i}$
(13)

where the expansion coefficients $f_{i}$, with i=0-3, will lead to four
coupled partial differential equations.

$\displaystyle D_{t}f_{0}$ $\displaystyle=-\hat{O}_{1}(f_{2}+f_{3})$
$\displaystyle D_{t}f_{1}$ $\displaystyle=-\hat{O}_{2}(f_{2}+f_{3})+2mf_{2}$
$\displaystyle D_{t}f_{2}$
$\displaystyle=\hat{O}_{2}f_{1}+\hat{O}_{1}f_{0}-2mf_{1}$ $\displaystyle
D_{t}f_{3}$ $\displaystyle=-\hat{O}_{2}f_{1}-\hat{O}_{1}f_{0}$ (14)

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

$\displaystyle\rho$ $\displaystyle=e\int d^{3}pf_{0}$ (15) $\displaystyle{\bf
J}$ $\displaystyle=\frac{e}{m}\int d^{3}p\,\mathbf{p}(f_{2}+f_{3})$ (16)

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 KGW-formalism
have a straightforward physical interpretation, as can be seen from the
following observables

$\displaystyle Q$ $\displaystyle=e\int d^{3}pd^{3}xf_{0}$ (17)
$\displaystyle\mathbf{J}$ $\displaystyle=\frac{e}{m}\int
d^{3}p\,\mathbf{p}(f_{2}+f_{3})$ (18) $\displaystyle W$ $\displaystyle=\int
d^{3}pd^{3}x\Big{[}\frac{p^{2}}{2m}(f_{2}+f_{3})+mf_{3}\Big{]}$ (19)
$\displaystyle\mathbf{M}$ $\displaystyle=\int
d^{3}pd^{3}x\,\mathbf{p}(f_{0}-f_{1})$ (20)

where $Q$ is the total charge, $\mathbf{J}$ is the current density, $W$ is the
particle energy and $\mathbf{M}$ 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 $e\mathbf{p}/m(f_{2}+f_{3})$ the current density, as
implied by the sources in Maxwell’s equations.

The $f_{i}$-functions have the vacuum-contribution

$\displaystyle f_{0}=f_{1}=0$ $\displaystyle f_{2}+f_{3}=\frac{m}{\epsilon}$
$\displaystyle f_{3}-f_{2}=\frac{\epsilon}{m}$ (21)

where $\epsilon=\sqrt{m^{2}+p^{2}}$. 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 $f_{2}+f_{3}=(m/\epsilon)F$ and
$f_{3}-f_{2}=(\epsilon/m)F$ in Section II where

$F=1+2f_{e}(\mathbf{p})+2f_{p}(\mathbf{p})$ (22)

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 KGW-
variables involving a homogeneous medium with electron and positron
distribution functions $f_{e,p}$ are given by

$\displaystyle f_{0}$ $\displaystyle=2f_{p}-2f_{e}$ $\displaystyle f_{1}$
$\displaystyle=0$ $\displaystyle f_{2}+f_{3}$
$\displaystyle=\frac{m}{\epsilon}F$ $\displaystyle f_{3}-f_{2}$
$\displaystyle=\frac{\epsilon}{m}F$ (23)

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 Section II, 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, Section II (including the plasma) applies
for the background. For the wave perturbation, we consider electromagnetic
waves with the following geometry

$\displaystyle\mathbf{E}$ $\displaystyle=E\mathbf{e}_{x}$
$\displaystyle\mathbf{k}$ $\displaystyle=k\mathbf{e}_{z}$
$\displaystyle\mathbf{B}$ $\displaystyle=B\mathbf{e}_{y}$ (24)

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

$f_{i}(z,\mathbf{p},t)=f_{i}^{0}(\mathbf{p})+f_{i}^{1}(\mathbf{p})e^{i(kz-\omega
t)}$ (25)

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

$D(k,\omega)=\omega^{2}-k^{2}+i\omega J_{x}\\\
=\omega^{2}-k^{2}+\frac{ie\omega}{(2\pi\hbar)^{3}E}\int
d^{3}p\frac{p_{x}}{m}(f_{3}^{1}+f_{2}^{1})=0$ (26)

After some algebra, we can compute the current sources and deduce the explicit
expression

$D(k,\omega)=\omega^{2}-k^{2}+\frac{e^{2}}{(2\pi\hbar)^{3}}\int
d^{3}p\frac{p_{x}}{\omega^{2}-k^{2}-4\epsilon^{2}+4k^{2}p_{z}^{2}/\omega^{2}}\Bigg{[}(\omega^{2}-2(p+\hbar
k/2)^{2}-4m^{2})\times\\\ \Big{[}\Delta_{1}\nabla_{p_{x}}-\frac{2\hbar
k}{\omega^{2}}\Big{(}\frac{\hbar\mathbf{k}}{12}-2\mathbf{p}\Delta_{2}\Big{)}\cdot(\mathbf{e}_{y}\times\nabla_{p})\Big{]}\Big{(}\frac{2m^{2}+p^{2}}{2m\epsilon}F-\frac{p^{2}}{2m\epsilon}F\Big{)}-2\Delta_{1}\nabla_{p_{x}}\frac{p^{2}}{\epsilon}F+8\frac{k}{\omega}p_{x}\Delta_{1}\nabla_{p_{z}}(f_{e}-f_{p})\\\
-4\frac{k^{2}}{\omega^{2}}p_{z}(p_{x}\nabla_{p_{z}}-p_{z}\nabla_{p_{x}})\Delta_{1}\frac{F}{\epsilon}-4\Big{(}(p+\hbar
k/2)^{2}+2m^{2}\Big{)}\frac{\hbar
k}{\omega^{2}}\Big{(}\frac{\hbar\mathbf{k}}{12}-2\mathbf{p}\Delta_{2}\Big{)}\cdot(\mathbf{e}_{y}\times\nabla_{p})\frac{F}{\epsilon}\Bigg{]}=0$
(27)

where

$\displaystyle\Delta_{1}$ $\displaystyle=\int^{1/2}_{1/2}d\lambda\cos{(\hbar
k\lambda\nabla_{pz})}$ (28) $\displaystyle\Delta_{2}$
$\displaystyle=\int^{1/2}_{1/2}d\lambda\lambda\sin{(\hbar
k\lambda\nabla_{pz})}$ (29)

Here the operators are given by $\Delta_{1}=1$ and $\Delta_{2}=\hbar
k\nabla_{pz}/12$ in the long-scale limit, where we have kept contributions to
the lowest non-vanishing order. We can obtain the classical limit by letting
$\hbar\longrightarrow 0$ in Eq. 27. Doing so, we get $\Delta_{2}=0$. The
dispersion relation then reduces to

$D(k,\omega)=\omega^{2}-k^{2}-\frac{e^{2}}{\hbar^{3}\pi^{2}}\int
dp\,p^{2}\bigg{[}\Big{(}1-\frac{p^{2}}{3\epsilon^{2}}\Big{)}\frac{f_{e}+f_{p}}{\epsilon}\\\
-\frac{kp^{2}}{3\epsilon^{2}}\Big{(}\frac{1}{\omega+kp_{z}/\epsilon}-\frac{1}{\omega-
kp_{z}/\epsilon}\Big{)}\frac{\partial(f_{e}+f_{p})}{\partial p_{z}}\bigg{]}=0$
(30)

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 $p_{d}$ as $f_{e}(p-p_{d})=f_{p}(p+p_{d})$. This
accounts for the fact that positrons can be viewed as electrons moving
backward in time. We note that the appearance of $\hbar$ 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 $k\longrightarrow 0$ has been computed in
the DHW-case) and compare it with the corresponding result from the DHW-
formalism [32, 34]. Letting $k\longrightarrow 0$ in Eq. 27, and including
electrons only in the background plasma $f=f_{e}$, we get

$\omega^{2}=-\frac{4e^{2}}{\pi^{2}\hbar^{3}}\int
dp\frac{p^{2}\epsilon}{\hbar^{2}\omega^{2}-4\epsilon^{2}}\times\\\
\bigg{[}\Big{(}1-\frac{p^{2}}{3\epsilon^{2}}-\frac{\hbar^{2}\omega^{2}}{4\epsilon^{2}}\Big{)}f-\frac{\hbar^{2}\omega^{2}p^{2}}{24\epsilon^{4}}\bigg{]}$
(31)

For the DHW formalism, we apply the $k\longrightarrow 0$\- result of the
electrostatic case, see [32, 34], and we obtain

$\omega^{2}=-\frac{4e^{2}}{\pi^{2}\hbar^{3}}\int
dp\frac{p^{2}\epsilon}{\hbar^{2}\omega^{2}-4\epsilon^{2}}\times\\\
\Big{(}1-\frac{p^{2}}{3\epsilon^{2}}\bigg{)}\bigg{(}f-\frac{\hbar^{2}\omega^{2}}{4\epsilon^{2}}\bigg{)}$
(32)

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
$\omega_{r}$ is given by the classical limit to a good approximation. The
reason is that for $\hbar\omega\sim m$, the Fermi energy $E_{F}$ will be much
larger than unity. Thus, even if $\hbar\omega\sim m$ we will have
$\hbar\omega\ll E_{f}$, 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 $\omega_{i}$,
associated with the pole contribution of the momentum integral. Using the
Plemelj formula

$\frac{1}{u-a}=P\frac{1}{u-1}+i\pi\delta(u-a)$ (33)

where $P$ is the principal value, we obtain from the KGW-expression Eq. 31

$\omega_{i}=-\frac{e^{2}p_{res}^{3}}{6\hbar^{4}\omega_{r}^{4}\pi}\Big{(}1+2f(p_{res}\Big{)}$
(34)

For the DHW-system, we instead obtain

$\omega_{i}=-\frac{e^{2}p_{res}}{12\pi\hbar^{2}\omega}\Big{(}1+\frac{2m^{2}}{\hbar^{2}\omega_{r}^{2}}\Big{)}\Big{(}1-f(p_{res})\Big{)}$
(35)

Here $p_{res}$ is the resonant momentum, making the denominators in the
integrands zero, i.e. $m^{2}+p_{res}^{2}=\hbar^{2}\omega^{2}/4$. 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.

##  IV Numerical results

###  IV.1 Preliminaries

Before presenting the numerical results, we discuss the issue of
renormalization. For large momentum where we can use the approximation
$\epsilon\approx p$, 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 Section II with
the ones from the DHW-formalism in [35], we consider the homogeneous-limit of
Section II

$\displaystyle D_{t}f_{0}$ $\displaystyle=0$ $\displaystyle D_{t}f_{1}$
$\displaystyle=\frac{p^{2}}{m}(f_{2}+f_{3})+2mf_{2}$ $\displaystyle
D_{t}f_{2}$ $\displaystyle=-\Big{(}\frac{p^{2}}{m}+2m\Big{)}f_{1}$
$\displaystyle D_{t}f_{3}$ $\displaystyle=\frac{p^{2}}{m}f_{1}$ (36)

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.

$\tilde{f}_{i}=f_{i}-f_{i,vac}$ (37)

where

$\displaystyle f_{1,vac}$ $\displaystyle=0$ $\displaystyle f_{2,vac}$
$\displaystyle=\frac{1}{2m\epsilon}(m^{2}-\epsilon^{2})$ $\displaystyle
f_{3,vac}$ $\displaystyle=\frac{1}{2m\epsilon}(m^{2}+\epsilon^{2})$ (38)

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
$\tilde{f}_{i}$, 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

$\displaystyle D_{t}\tilde{f}_{1}$
$\displaystyle=\frac{p^{2}}{m}(\tilde{f}_{2}+\tilde{f}_{3})+2m\tilde{f}_{2}$
$\displaystyle D_{t}\tilde{f}_{2}$
$\displaystyle=\frac{eEp_{z}}{2m\epsilon}\Big{(}1+\frac{m^{2}}{\epsilon^{2}}\Big{)}-\Big{(}\frac{p^{2}}{m}+2m\Big{)}\tilde{f}_{1}$
$\displaystyle D_{t}\tilde{f}_{3}$
$\displaystyle=-\frac{eEp_{z}}{2m\epsilon}p^{2}+\frac{p^{2}}{m}\tilde{f}_{1}$
(39)

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

$\displaystyle f_{1}$ $\displaystyle=f_{1}$ $\displaystyle f_{+}$
$\displaystyle=f_{2}+f_{3}$ (40) $\displaystyle f_{-}$
$\displaystyle=f_{2}-f_{3}$

introduce the canonical momentum $q$ given by

$q=p+eA$ (41)

and use normalized variables: $t_{n}=\omega_{ce}t$, $q_{n}=q/m$,
$p_{n\perp}=p_{\perp}/m$, $E_{n}=E/E_{cr}$, $A_{n}=eA/m$, where $\omega_{ce}$
is the Compton frequency. The final system resulting from these changes, which
will be solved numerically is

$\displaystyle\partial_{t_{n}}f_{1}$
$\displaystyle=\epsilon_{n}^{2}f_{+}+f_{-}$
$\displaystyle\partial_{t_{n}}f_{+}$
$\displaystyle=\frac{(q_{n}-A_{n})}{\epsilon_{n}^{3}}E_{n}-2f_{1}$ (42)
$\displaystyle\partial_{t_{n}}f_{-}$
$\displaystyle=\frac{q_{n}-A_{n}}{\epsilon_{n}}E_{n}-2\epsilon_{n}^{2}f_{1}$

together with Ampere’s law

$\frac{\partial E_{n}}{\partial t_{n}}=-\eta\int d^{2}p_{n}(q_{n}-A_{n})f_{+}$
(43)

where $\eta=\alpha/\pi$, here $\alpha$ is the fine-stucture constant. Note
that, for notational convenience, we omit the subscript ”n” for variables in
what follows. The system Section IV.1-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 $\Delta t\sim 0.001$, a parallel
momentum step $\Delta q\sim 0.01$, and a perpendicular momentum step $\Delta
p_{\bot}\sim 0.1$. 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 $\Delta q$
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 Sections
IV.1 and 43 can be written in the form

$\frac{d}{dt}\left[\frac{E^{2}}{2}+\frac{\eta}{2}\int
d^{2}p\Big{(}(\epsilon^{2}+1)f_{+}-f_{-}\Big{)}\right]=0$ (44)

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
$10^{-4}$.

###  IV.2 Effects due to the vacuum background

Figure 1: The number of produced pairs from KGW ($n_{k}$) and DHW ($n_{D}$)
using the numerical solutions together with the corresponding analytical
results $dn_{b}/dt$ and $dn_{f}/dt$.  Figure 2: The electric pulse is plotted
over time for three different frequencies $N=(10,100,500)$.

Before solving the Section IV.1-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]

$\frac{dn_{f}}{dt}=\frac{1}{4\pi^{3}}\Big{(}\frac{eE}{\hbar}\Big{)}^{2}\exp{\Big{(}-\frac{\pi
E_{cr}}{E}\Big{)}}$ (45)

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 Section IV.1 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

$n_{k}=\frac{1}{(2\hbar\pi)^{3}}\int
d^{3}p\frac{1}{2\epsilon}\Big{[}(\epsilon+1)f_{+}-f_{-}\Big{]}$ (46)

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

###  IV.3 High frequency effects

Now, we turn to the problem of time-dependent electric fields. We solve
Section IV.1-Eq. 43 for the case with no plasma and only vacuum initially
using the following representation of the time-dependent pulse

$E(t)=E_{0}{\rm sech}\Big{(}\frac{t}{b}-\tau_{0}\Big{)}\sin{\omega t}$ (47)

where $\tau_{0}$ is the phase-shift, $b$ is the pulse length and
$\omega=N\omega_{ce}/100$. 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

$\frac{n_{k}}{n_{D}}=\frac{1}{2}$ (48)

Figure 3: The ratio $n_{k}/n_{D}$ is plotted as a function of $N$, displaying
the validity of the KGW-formalism to model fermions at different regimes. Here
we have used $E_{0}=1$.

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 $\hbar\omega=2m$ 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
$\hbar\omega\approx 2m_{e}$, 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 $p_{res}^{3}$, 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
$\omega_{i}\sim\omega_{r}^{-4}$ in Eq. 34, more than compensating for the
scaling $\omega_{i}\sim p_{res}^{3}$.

Figure 4: Polarization current $J_{p}$ and total current $J_{tot}$ plotted
over time for $N=(2,20)$ in the upper and lower panels, respectively. Note the
different scales for $J_{p}$ and $J_{tot}$

Since the KGW-formalism is spinless, only free currents are created. By
contrast, the total current in the DHW-formalism 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 $\omega\ll\omega_{ce}$, the polarization
current is small due to $J_{p}=\partial P/\partial t$, 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.,
$\omega=0.02\omega_{ce}$, 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.

Figure 5: Color mappings of the momentum distribution of the created pairs
from DHW $n_{D}(q,p_{\bot})$ (the first column) and KGW $n_{k}(q,p_{\bot})$
(the second column). In the first panel, we have used $\omega=0.02\omega_{ce}$
, $\tau_{0}=\pi$, $b=20$ and $E_{0}=1$ and in the second panel we have
$\omega=5\omega_{ce}$, $\tau_{0}=\pi$, $b=2$ and $E_{0}=2$.

Next, we would like to make contact between the analytical 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 DHW-expression $n_{D}(p_{\bot},q)$ (the figure to the left) and
the KGW-expression $n_{k}(p_{\bot},q)$ (the figure to the right) using the
pulse in Eq. 47 with $\omega=0.02\omega_{cr}$ , $\tau_{0}=\pi$, $b=20$ 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
$\omega=5\omega_{ce}$, $\tau_{0}=\pi$, $b=2$ 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 $m^{2}+p_{res}^{2}=\hbar^{2}\omega^{2}$,
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 $\left|q\right|$, and a smaller value of $p_{\bot}$. 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

$\omega^{2}=\frac{e^{2}}{4\pi^{2}\hbar^{3}}\int
dp_{\bot}p_{\bot}dp_{z}\frac{1/\epsilon}{\hbar^{2}\omega^{2}-4\epsilon^{2}}\times\\\
\bigg{[}2f(\hbar^{2}\omega^{2}-4\epsilon_{\perp}^{2})+\hbar^{2}\omega^{2}\frac{p_{z}^{2}}{\epsilon^{2}}\bigg{]}$
(49)

and the dispersion relation for the DHW-case is written

$\omega^{2}=-\frac{e^{2}}{2\pi^{2}\hbar^{3}}\int dp_{\bot}p_{\bot}dp_{z}\
\frac{\epsilon_{\bot}^{2}/\epsilon}{\hbar^{2}\omega^{2}-4\epsilon^{2}}\times\\\
\bigg{(}f-\frac{\hbar^{2}\omega^{2}}{4\epsilon^{2}}\bigg{)}$ (50)

where $\epsilon_{\bot}=\sqrt{m^{2}+p_{\bot}^{2}}$. 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 $p_{\bot}$ 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 $p_{\bot}$ 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

$\int
dp_{\bot}p_{\bot}dp_{z}\frac{p_{z}^{2}/\epsilon^{3}}{\hbar^{2}\omega^{2}-4\epsilon^{2}}$
(51)

Noting that the momenta which contributes to single-quanta pair creation will
be close to the resonant momenta, we see that for resonant values with large
$p_{z}$ ($q$) and small $p_{\perp}$, we will get the dominant pair creation
contribution. This is in agreement with the lower right panel of Fig. 5 as
$p_{\bot}\sim 0$ 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

$\int dp_{\bot}p_{\bot}dp_{z}\
\frac{\epsilon_{\bot}^{2}/\epsilon^{3}}{\hbar^{2}\omega^{2}-4\epsilon^{2}}$
(52)

We note that for resonant momenta, the integrand is largest for high
perpendicular momentum and has a smaller value when $\left|p_{z}\right|$ 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.

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 $N=(0.02,0.04,0.1)$.

Next, we will include the effects due to back-reaction, i.e., we will solve
Eq. 43 together with Section IV.1 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., $\omega=0.02\omega_{ce}$. 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 $0.1\omega_{ce}$, 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 electron-positron pair creation with the KGW-model requires lower
frequencies in the case of self-consistent dynamics.

###  IV.4 Non-degeneracy effects

Figure 7: The number density $n_{D/K}(q)=\int
dp_{\bot}p_{\bot}n_{D/K}(q,p_{\bot})$ as a function of the canonical momentum
is plotted for $E_{0}=4$, $\mu=4$ and $T=0.02$ for DHW and KGW. Here, the
solid curve is for n(t=0) and the …. is for n(t=T).

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 Sections IV.1 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,

$f(q,p_{\bot})=\frac{1}{1+\exp{(\epsilon-\mu)/T}}$ (53)

where $\mu$ is the chemical potential and $T$ is the temperature (we normalize
both $\mu$ and $T$ with the electron mass). For $\mu\gg T$, the chemical
potential fulfills $\mu\approx E_{F}$, where $E_{F}$ is the Fermi energy, in
which case the distribution will be degenerate (i.e., we will have $f\approx
1$, i.e. all electron states filled, for energies smaller than the Fermi
energy, and $f\approx 0$ otherwise). Naturally, non-degenerate initial
distributions are still possible when using Eq. 53, by letting $\mu<T$. 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 DHW-formalism using a degenerate distribution with $\mu=4$
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 DHW-
formalism, 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 DHW-theory and the KGW-theory.
Practically speaking $\omega\sim 0.04\omega_{c}$ 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.

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