# Schwinger mechanism of magnon-antimagnon pair production on magnetic field
inhomogeneities and the bosonic Klein effect

T. C. Adorno<EMAIL_ADDRESS>Department of Physics, School of
Mathematics and Physics, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road,
Suzhou Dushu Lake Science and Education Innovation District, Suzhou Industrial
Park, Suzhou 215123, People’s Republic of China.  S. P. Gavrilov
gavrilovsergeyp<EMAIL_ADDRESS>Department of Physics,
Tomsk State University, Lenin Ave. 36, 634050, Tomsk, Russia,  Department of
General and Experimental Physics, Herzen State Pedagogical University of
Russia, Moyka embankment 48, 191186, St. Petersburg, Russia;  D. M. Gitman
<EMAIL_ADDRESS>Department of Physics, Tomsk State University, Lenin Ave. 36,
634050, Tomsk, Russia,  P.N. Lebedev Physical Institute, 53 Leninskiy
prospect, 119991 Moscow, Russia,  Institute of Physics, University of São
Paulo, Rua do Matão, 1371, CEP 05508-090, São Paulo, SP, Brazil.

###### Abstract

It is known that similar to Dirac models of nanostructures, one can describe
low-energy exitations-magnons of lattice spin systems in terms of effective
field theories. The latter theory that describes antiferromagnets can be
mapped into scalar massless electrodynamics with an external electromagnetic
potential. Here we consider the case of a constant inhomogeneous magnetic
field applied to an antiferromagnet. Magnetic moment plays here the role of
the electric charge, and magnons and antimagnons differ from each other in the
sign of the magnetic moment. In the framework of such a consideration, it is
important to take into account the vacuum instability (the Schwinger effect)
under the magnon-antimagnon production on magnetic field inhomogeneities (an
analog of pair creation from the vacuum by electric-like fields). We
demonstrate how to use the strong field QED with $x$-steps developed by the
authors (SPG and DMG) to study the magnon-antimagnon pair production on
magnetic field inhomogeneities. Initial and final vacua and one-particle
states are constructed by the help of stationary plane waves satisfying the
Klein-Gordon equation. Mean numbers of magnons and antimagnons created from
the vacuum are expressed via overlap amplitudes of the stationary plane waves.
Observable physical quantities specifying the vacuum instability are
determined. We calculate and analyze the fluxes of energy and magnetic moments
of created magnons.Characteristics of the vacuum instability obtained for some
magnetic steps that allows exact solving the Klein-Gordon equation are
presented. In particular, we consider examples of magnetic steps with very
sharp field derivatives that correspond to a regularization of the Klein step.
In the case of smooth-gradient steps, we describe an universal behavior of the
flux density of created magnon pairs.The obtained results are quite general
and are not limited to the simple cubic-type lattice and the G-type
antiferromagnet. We also note that since the low-energy magnons are bosons
with small effective mass, then for the first time maybe the opportunity will
arise to observe the Schwinger effect in the case of the Bose statistics, in
particular, the bosonic Klein effect in laboratory conditions. Moreover, it
turns out that in the case of the Bose statistics appears a new mechanism for
amplifying the effect of the pair creation, which we call statistically-
assisted Schwinger effect.

##  I Introduction

Magnons, or quantized spin waves occur in various types of ordered magnets:
antiferromagnet, ferromagnet, and ferrimagnet. They present collective
magnetic excitations of the electron spin structure in a crystal lattice. The
emerging field of magnonics utilizes magnons for information processing; see
Ref. [1] for a review. Using magnons as information carriers has various
advantages, in particular, the low power-consumption. Although spin systems
are originally described as lattice models, similar to Dirac models of
nanostructures, one can describe their low-energy dynamics based on a
continuum field theory at energy scales much lower than the inverse lattice
spacing; see Ref. [2, 3] and references therein. Developed effective field
theory (EFT) descriptions of the dynamics of spin systems at low energies
suggests also a possibility of including in the model some external fields.
The magnon EFT can incorporate various symmetry-breaking terms. For example,
it can be a Zeeman term due to the coupling to an external magnetic field that
breaks the symmetry explicitly. It turn out that the magnon EFT that describes
antiferromagnets is relativistic-like. Our special interest is the case of an
inhomogeneous magnetic field applied to an antiferromagnet in the collinear
(homogeneous) ground state. It was recently shown [2] that the magnon EFT with
the easy-axis anisotropy can be mapped into electrodynamics of a charged
scalar field interacting with an external electromagnetic potential. The mass
of this field is determined by the sum of the easy-axis potential and the
ratio of magnetization and condensation parameters. Magnetic moment plays here
the role of the electric charge, and magnons and antimagnons differ from each
other in the sign of the magnetic moment. In the framework of such a
consideration, it is important to take into account the vacuum instability
(the Schwinger effect) under the magnon-antimagnon production on magnetic
field inhomogeneities (an analog of particle-antiparticle creation by constant
inhomogeneous electric-like fields). In this article, the leading imaginary
part of the one-loop effective action was calculated in the framework of the
semiclassical world-line formalism for the case of a linearly varying magnetic
field and agrees with the Schwinger’s formula for a constant electric field
[4]. One can see that in the presence of a constant inhomogeneous external
magnetic field the latter problem is technically reduced to the problem of
charged-particle creation from the vacuum by an electric potential step. It
should be noted that a number of articles have recently appeared that consider
magnon-antimagnon pair creation by a sufficiently high rectangular potential
step (the Klein step) and barrier, which are formed by magnetic field
inhomogeneities. [5, 6, 7, 8]. In these cases the world-line formalism does
not work and the problem is considered in the framework of the relativistic
quantum mechanics. In the relativistic quantum mechanics, problems of such
kind were considered in the relation to the Klein paradox in the pioneer works
[9, 10, 11] (a detailed historical review can be found in Refs. [12, 13]). We
note that to avoid a confusion, the Klein paradox should be distinguished from
the Klein tunneling through the square barrier. This tunneling without an
exponential suppression occurs when a particle is incident on a high barrier,
even when it is not high enough to create particles. It is known that attempts
to consider overlapping amplitudes as amplitudes of particle transmission and
reflection by the Klein step in the same manner as in the relativistic quantum
mechanics often leads to contradictions and paradoxes. As it is known
processes with changing the number of particles have to be considered in the
framework of quantum field theory (QFT). Recently, a consistent
nonperturbative treatment of the vacuum instability with respect to charged
particle creation was developed in the framework of strong-field quantum
electrodynamics (QED) with constant external electric potential steps (we call
them conditionally $x$-steps) in Refs. [14, 15, 16]. In the case of bosons,
the latter nonperturbative treatment is based on the existence of special
exact solutions of the Klein-Gordon equation with the corresponding $x$-step.
This makes it possible to uniformly consider effects of pair creation by the
$x$-steps of arbitrary form, including, in particular, the Klein step. We hope
that the present article will promote consistent application of strong field
QED methods in magnonics and will allow avoiding contradictions and non-
existent paradoxes in the interpretation of the obtained theoretical results.

In the present work we use the strong field QED to study the magnon-antimagnon
pair production on magnetic field inhomogeneities.111Here we are using the
natural system of units $\hslash=1$.. The article is organized as follows: In
Sect. II the EFT model describing the low-energy dynamics of magnons is mapped
into scalar electrodynamics with $x$-steps. In Sect. III, we construct a Fock
space realization of the $EFT$ model in the framework of strong-field QED with
$x$-steps. Initial and final one-particle states are constructed by the help
of stationary plane waves satisfying the Klein-Gordon equation. Initial and
final vacua are defined and initial and final states of the Fock space are
constructed. Mean numbers of magnons and antimagnons created from the vacuum
are expressed via overlap amplitudes of the stationary plane waves. Observable
physical quantities specifying the vacuum instability are determined. We
calculate and analyze the fluxes of energy and magnetic moments of created
magnons. In Sect. IV, we present characteristics of the vacuum instability
obtained for some magnetic steps that allows exact solving the Klein-Gordon
equation. In particular, we consider examples of magnetic steps with very
sharp field derivatives $\partial_{x}U$ that correspond to a regularization of
the Klein step. In the case of smooth-gradient steps, we describe an universal
behavior of the flux density of created pairs. In the last Sect. V, we
summarize the main results of the present work. Some details of the scalar
field quantization in the presence of critical potential steps are placed in
Appendix A. Examples of some exact solutions with $x$-steps are given in
Appendix B.

##  II EFT model describing low-energy dynamics of magnons

The system under consideration consists of localized spins which live on sites
of a cubic-type lattice. These sites are numbering by the index $n$. The
corresponding spin vector operators are denoted by $\mathbf{\hat{s}}^{n}$. It
is assumed that the spins are involved in the antiferromagnetic interaction.
Its original $SO\left(3\right)$ spin-rotation symmetry is explicitly (but
softly) broken due to an external magnetic field $\mathbf{B}$ and an
anisotropic interaction $C$ known as single-ion anisotropy. The Hamiltonian
describing such a system reads:

$\displaystyle\hat{H}_{\mathrm{spin}}$ $\displaystyle=$
$\displaystyle-\sum_{n}\sum_{i=1}^{d}J\delta^{ab}\hat{s}_{a}^{n}\hat{s}_{b}^{n+\hat{\imath}}$
(1) $\displaystyle-$ $\displaystyle\sum_{n}\left[\mu
B^{a}\left(\mathbf{r}_{n}\right)\hat{s}_{a}^{n}+C^{ab}\hat{s}_{a}^{n}\hat{s}_{b}^{n}\right]\
.$

Here $\hat{s}_{a}^{n}$ denote spin operator components on the site $n$
$\left(\left[\hat{s}_{a}^{n},\hat{s}_{b}^{n}\right]=i\epsilon_{ab}^{c}\
\hat{s}_{c}^{n}\right)$;$\ J>0$ is the antiferromagnetic interaction coupling
constant. To describe the nearest-neighbor pairs, the direction
$\hat{\imath}=1,2,\ldots,d$ with a spatial dimension $d$ is introduced. The
sum over $\hat{\imath}$ means summing over nearest neighboring spins and the
sum over $n$ means summing over sites $n$ of a cubic-type lattice.
$B^{a}\left(\mathbf{r}_{n}\right)$ are external magnetic field components on
the site $n$, they depend on the coordinates
$\mathbf{r}_{n}=\left(x_{n},y_{n},z_{n}\right)$ of the site; $\mu>0$ is the
modulus of spin magnetic moment; the single-ion anisotropic interaction is
presented by the term $C^{ab}\hat{s}_{a}^{n}\hat{s}_{b}^{n}$, where the
$C^{ab}$ is a constant symmetric rank-two tensor. Here $\hat{\imath}$ is a
vector of the length $l$ ($l$ is the lattice spacing) pointing in the
$i$-direction. The Kronecker delta $\delta^{ab}$ and the Levi-Civita symbol
$\epsilon_{abc}$ are used for the internal spin indices, $a,b=1,2,3$, and the
summation over the repeated indices is implied.

We only consider the simple cubic-type lattice and the G-type antiferromagnet,
in which the Néel order appears along all the spatial directions. In the
absence of explicit symmetry-breaking terms ($\mu B^{a}=0$ and $C^{ab}=0$)
Hamiltonian (1) enjoys $SO\left(3\right)$ symmetry. We can study effects of
symmetry-breaking terms using the background field (spurion) method if these
terms are small enough compared to the symmetric interaction ($\mu
B^{a},\;C^{ab}\ll J$).

A continuum field-theoretical description of magnons is given by a
$O\left(3\right)$ nonlinear sigma model, in which a three-component unit
vector $\mathbf{n}=\left(n^{1},n^{2},n^{3}\right)$ with $n^{a}n_{a}=1$ plays a
role as a dynamic degree of freedom. This unit vector expresses the Néel order
parameter. Taking the continuum limit of the background field,
$B^{a}\left(\mathbf{r}_{n}\right)\rightarrow B^{a}\left(\mathbf{r}\right)$,
and following the way described in Ref. [2], one can construct a $SO(3)$ gauge
invariant effective Lagrangian. The corresponding space-time is parametrized
by coordinates $X=\left(t,\mathbf{r}\right)$, $t=X^{0}$,
$\mathbf{r}=X^{j}=\left(x,y,z\right)$, $j=1,2,3$. The local $SO(3)$
transformation simply acts on the vector field $\mathbf{n}$ as
$\mathbf{n}\rightarrow g(X)\mathbf{n}$ with $g(X)\in SO(3)$, as in the lattice
case. One can identify the background field $B^{a}$ as the $SO(3)$ gauge field
on which the local $SO(3)$ transformation acts as follows:

$B^{a}\rightarrow g(X)B^{a}g^{-1}(X)+g(X)\partial_{0}g^{-1}(X)\ .$

As it will be seen further, the field $B^{a}$ play the same role as the zero
component of the electromagnetic potential, $A_{0}^{a}$, in the theory of a
charged scalar field. However, the constant piece of
$B^{a}\left(\mathbf{r}\right)$ is physical quantity and is used to tune the
collinear ground state.

In the leading order of the derivative expansion at low-energies, namely,
preserving derivatives only up to the second order, the $SO(3)$ gauge
invariant effective Lagrangian can be written as:

$\mathcal{L}=\frac{f_{t}^{2}}{2}\left(D_{0}n^{a}\right)^{2}-\frac{f_{s}^{2}}{2}\left(\partial_{i}n^{a}\right)^{2}+rC^{ab}n_{a}n_{b}\
,$ (2)

where the covariant derivative $D_{0}$ with the $SO(3)$ background gauge field
is defined as:

$D_{0}n^{a}=\partial_{0}n^{a}-\epsilon_{bc}^{a}\ n^{b}\mu
B^{a},\;\partial_{0}=\frac{\partial}{\partial t}\,,$

and low-energy parameters $f_{t}$, $f_{s}$, and $r$ can be determined from the
underlying lattice model by the matching condition.

Suppose that our spin system possesses a potential with an easy-axis
anisotropy and develops the collinear ground state. We apply an inhomogeneous
magnetic field along the spin direction of the ground state. We assume that
the magnetic field points to the direction of axis $z$ and depends on the
coordinate $x$, $B^{a}\left(x\right)=B\left(x\right)\delta^{a3}$, and the sign
of the field is positive, $B\left(x\right)>0$. This field gives the collinear
ground state with the Néel vector pointing to the direction of axis $z$ as
$\left\langle\mathbf{n}\right\rangle=\left(0,0,1\right)$. Then one can
introduce magnon complex scalar fields $\Phi\left(X\right)$ and
$\Phi^{\ast}\left(X\right)$ as fluctuations on the top of the ground state,
which parametrize the vector $\mathbf{n}$ as

$\mathbf{n}=\left(\frac{\Phi+\Phi^{\ast}}{\sqrt{2}},\frac{\Phi-\Phi^{\ast}}{\sqrt{2}i},\sqrt{1-\Phi^{\ast}\Phi}\right),$

where the constraint $n^{a}n_{a}=1$ is explicitly solved. Substituting this
parametrization into Eq. (2), one obtains the effective Lagrangian of magnons
at the quadratic order of fluctuation fields around the ground state in the
following form:

$\displaystyle\mathcal{L}^{\left(2\right)}=f_{t}^{2}\left(D_{0}\Phi^{\ast}D_{0}\Phi-\Delta^{2}\Phi^{\ast}\Phi\right)-f_{s}^{2}\delta^{ij}\partial_{i}\Phi^{\ast}\partial_{j}\Phi,$
$\displaystyle
D_{0}\Phi=\left(\partial_{0}+iU\right)\Phi,\;D_{0}\Phi^{\ast}=\left(\partial_{0}-iU\right)\Phi^{\ast},$
(3)

where the notation $U=\mu B$ and
$rC^{ab}=\frac{1}{2}f_{t}^{2}\Delta^{2}\delta^{a3}\delta^{b3}$ are used. We
see that the field-theoretical description of magnons carrying magnetic moment
$\mu$ embedded into an external inhomogeneous magnetic field $B$ can be
realized in terms of scalar electrodynamics where a charged complex field
$\Phi\left(X\right)$ (with $\mu$ playing the role of an electric charge)
coupled to the zero component of the electromagnetic potential $A_{0}=B$. Here
the constant $v_{s}=f_{s}/f_{t}$ plays the role of the light speed and the
energy gap $\Delta$ plays the role of a mass term. We note that the constant
$v_{s}$ is not related to the speed of the light $c$ and is relatively small,
e.g. $\Delta\sim 1\mathrm{meV}$ and $v_{s}\sim 60\mathrm{m/s}$ for
antiferromagnetic $MnF_{2}$ [17].

It follows from the effective Lagrangian (3) that the corresponding wave
equation for the field $\Phi\left(X\right)$ is a modification of the Klein-
Gordon equation,

$\left(D_{0}^{2}-v_{s}^{2}\delta^{ij}\partial_{i}\partial_{j}+\Delta^{2}\right)\Phi\left(X\right)=0.$
(4)

Summarizing, we can say that in the example under consideration, the $EFT$
model describing low-energy dynamics of magnons can technically be identified
with the theory of a charged scalar field interacting with an external
constant electric field, which in our terminology is an $x$-step; see Refs.
[14, 15, 16]. In this theory, the wave equation describing the corresponding
charged particles has the form (4). A nonperturbative study (with respect to
the interaction with the external field) of various quantum effects in such a
system, in particular, the study of the vacuum instability, can use the
technique (with the necessary modifications due to the specifics of the wave
equation (4)) developed earlier by two coauthors (S.P. Gavrilov and D.M.
Gitman) for strong-field $QED$ with $x$-potential steps and presented in Refs.
[14, 15]. In the next section, we study the vacuum instability in the system
of low-energy magnons following the formulated idea.

##  III Consideration in the framework of strong-field QED

###  III.1 Solutions of the Klein-Gordon equation with critical $x$-steps

Solutions of the Klein-Gordon equation with critical $x$-steps are known in
the form of stationary plane waves. A complete set of such solutions reads:

$\displaystyle\phi_{m}\left(X\right)=\exp\left(-i\varepsilon
t+i\mathbf{p}_{\bot}\mathbf{r}_{\bot}\right)\varphi_{m}\left(x\right)\,,$
$\displaystyle\mathbf{r}_{\bot}=\left(0,y,z\right),\ \
m=(\varepsilon,\mathbf{p}_{\bot}).$ (5)

In fact these are stationary states with well defined the total energy of a
particle $\varepsilon$ and with definite momenta $\mathbf{p}_{\bot}$ in the
perpendicular to the axis $x$ directions. Substituting Eq. (5) into Eq. (4),
we obtain a second-order differential equation for the function
$\varphi_{m}(x)$,

$\displaystyle\left\\{v_{s}^{2}\partial_{x}^{2}+\left[\varepsilon-U\left(x\right)\right]^{2}-\pi_{\bot}^{2}\right\\}\varphi_{m}\left(x\right)=0\,,$
$\displaystyle\pi_{\bot}=\sqrt{v_{s}^{2}\mathbf{p}_{\bot}^{2}+\Delta^{2}},$
(6)

We chose the potential energy $U\left(x\right)$ of a scalar particle (here a
magnon carrying the positive magnetic moment $\mu$ that plays the role of an
effective electric charge) in the form of a monotonically decreasing function,
$\partial_{x}U\left(x\right)<0$, such that the kinetic energy of the particle,
$\varepsilon-U\left(x\right)>0$, would grow monotonously along the axis $x$.
The kinetic energy222The interpretation of these quantities as kinetic
energies of a particle and an antiparticle can be easily seen in the one-
particle sector of QED. In the general case their interpretation is discussed
in appendix A. of the antiparticle (an antimagnon carrying the negative
magnetic moment $-\mu$) is $U\left(x\right)-\varepsilon>0$, it grows
monotonously in inverse direction of the axis $x$.

If the field derivative $\partial_{x}U\left(x\right)$ (playing here the role
of the electric field in the scalar electrodynamics) acting on magnons
produces big enough work, the magnon-antimagnon pair production from the
corresponding vacuum may take place. We assume that the action of the field
derivative contributes significantly to mean numbers of created pairs in the
restricted region $S_{\mathrm{int}}$ between two planes $x=x_{\mathrm{L}}$ and
$x=x_{\mathrm{R}}$ during the sufficiently large (macroscopic) time period
$T$. It is either negligible or switches off in the macroscopic regions
$S_{\mathrm{L}}=\left(x_{\mathrm{FL}},x_{\mathrm{L}}\right]$ on the left of
the plane $x=x_{\mathrm{L}}$ and in
$S_{\mathrm{R}}=\left[x_{\mathrm{R}},x_{\mathrm{FR}}\right)$ on the right of
the plane $x=x_{\mathrm{R}}$. We also assume that the points $x_{\mathrm{FL}}$
and $x_{\mathrm{FR}}$ are separated from the origin by macroscopic but finite
distances. In this way, the magnetic field $B$ plays the role of an electric
potential step, which we call shortly an $x$-step. Its magnitude is:

$\delta
U=U_{\mathrm{L}}-U_{\mathrm{R}}>0\,,\;U_{\mathrm{L}}=U\left(-\infty\right)\,,\;U_{\mathrm{R}}=U\left(\infty\right)\,.$
(7)

We distinguish two types of $x$-steps, noncritical $\delta U<2\Delta,$ and
critical $\delta U>2\Delta$. The pair production from the vacuum occurs due to
the critical $x$-step.

Nonperturbative calculations of the vacuum instability effects in the
framework of strong-field QED with $x$-steps are possible if there are special
complete sets of exact solutions of the corresponding relativistic wave
equations (solutions of the Klein-Gordon equations in the case under
consideration) orthonormalized on a $t$ constant hyperplane; see Ref. [14]. In
the strong-field QED with $x$-step, which acts during the macroscopic time
$T$, one can construct such sets of solutions, see below. In our work these
solutions are chosen in the form of stationary plane waves with given real
longitudinal momenta $p^{\mathrm{L}}$ and $p^{\mathrm{R}}$ in the regions
$S_{\mathrm{L}}$ and $S_{\mathrm{R}}$. Moreover, it is necessary to construct
two types of complete sets of solution in the form (5). The first one
${}_{\;\zeta}\phi_{m}\left(X\right)$ is constructed with the help of the
functions $\varphi_{m}\left(x\right)$ denoted as
${}_{\;\zeta}\varphi_{m}\left(x\right)$ and the second
one,${}^{\;\zeta}\phi_{m}\left(X\right)$, with the help of functions
$\varphi_{m}\left(x\right)$ denoted as
${}^{\;\zeta}\varphi_{m}\left(x\right)$. Asymptotically, these functions have
the following forms:

${}_{\;\zeta}\varphi_{m}\left(x\right)=\
_{\zeta}\mathcal{N}\exp\left[ip^{\mathrm{L}}\left(x-x_{\mathrm{L}}\right)\right],\
x\in S_{\mathrm{L}},$ ${}^{\;\zeta}\varphi_{m}\left(x\right)=\
^{\zeta}\mathcal{N}\exp\left[ip^{\mathrm{R}}\left(x-x_{\mathrm{R}}\right)\right],\
x\in S_{\mathrm{R}},$ $\displaystyle
p^{\mathrm{L}}=\frac{\zeta}{v_{s}}\sqrt{\left[\pi_{0}\left(\mathrm{L}\right)\right]^{2}-\pi_{\bot}^{2}},\
\
p^{\mathrm{R}}=\frac{\zeta}{v_{s}}\sqrt{\left[\pi_{0}\left(\mathrm{R}\right)\right]^{2}-\pi_{\bot}^{2}},$
(8)

where $\zeta=\pm$, $\ {}_{\zeta}\mathcal{N}$ and ${}^{\zeta}\mathcal{N}$ are
normalization factors to be determined in what follows. We introduce the
quantities $\pi_{0}\left(\mathrm{L/R}\right)=\varepsilon-U_{\mathrm{L/R}}$.
Note that $\pi_{0}\left(\mathrm{R}\right)>\pi_{0}\left(\mathrm{L}\right)$.

By analogy with how this is done in the time-independent potential scattering
due to noncritical steps, it is assumed that there exist time-independent
observables in the presence of critical $x$-steps. For example, it seems quite
natural that the pair-production rate and the flux of created particles are
constant during the macroscopic time $T$. It means that a leading contribution
to the number density of created particle-antiparticle pairs is assumed to be
proportional to the large dimensionless parameter
$\sqrt{v_{s}\left|\partial_{x}U\right|}T$ and is independent from switching-on
and -off if this parameter satisfies inequality

$T\gg\left(v_{s}\left|\partial_{x}U\right|\right)^{-1/2}\max\left\\{1,\Delta^{2}/v_{s}\left|\partial_{x}U\right|\right\\}.$
(9)

It is clear that the process of pair creation is transient. Nevertheless, the
condition of the smallness of backreaction shows there is a window in the
parameter range of $\partial_{x}U$ and $T$ where the constant field
approximation is consistent [18].

For any two solutions $\Phi\left(X\right)$ and $\Phi^{\prime}\left(X\right)$
of the Klein-Gordon equation, the inner product on the hyperplane
$x=\mathrm{const}$ has the form of a longitudinal flux:

$\left(\Phi,\Phi^{\prime}\right)_{x}=i\int\left[\Phi^{\prime}\left(X\right)\partial_{x}\Phi^{\ast}\left(X\right)-\Phi^{\ast}\left(X\right)\partial_{x}\Phi^{\prime}\left(X\right)\right]dtd\mathbf{r}_{\bot}.$
(10)

Note, that this flux is proportional to the effective charge current. We
consider the system under consideration in a large space-time box that has a
spatial area $V_{\bot}=K_{y}K_{z}$ and the time dimension $T$, where all
$K_{y}$, $K_{z}$, and $T$ are macroscopically large. In general the wave
packets $\Phi(X)$ and $\Phi^{\prime}(X)$ can be decomposed into plane waves
$\phi_{m}(X)$ and $\phi_{m}^{\prime}(X)$. Along with the introduced plane
waves, it is assumed that all the solutions $\Phi(X)$ are periodic under
transitions from one box to another. Then the integration in integral (10)
over the transverse coordinates runs from $-K_{j}/2$ to $+K_{j}/2$, $j=y,z,$
and over the time $t$ from $-T/2$ to $+T/2$. We assume that the macroscopic
time $T$ is the system surveillance time. Under these suppositions, one can
verify, integrating by parts, that the inner product (10) does not depend on
$x$.

Nontrivial solutions ${}_{\zeta}\phi_{m}\left(X\right)$ and
${}^{\;\zeta}\phi_{m}\left(X\right)$ exist only for the quantum numbers $m$
that obey the following relations:

$\left[\pi_{0}\left(\mathrm{L/R}\right)\right]^{2}>\pi_{\bot}^{2}\Longleftrightarrow\left\\{\begin{array}[]{l}\pi_{0}\left(\mathrm{L/R}\right)>\pi_{\bot}\\\
\pi_{0}\left(\mathrm{L/R}\right)<-\pi_{\bot}\end{array}\right..$ (11)

In the case of critical $x$-steps and for $2\pi_{\bot}\leq\delta U\ $there
exist five ranges $\Omega_{k}$, $k=1,...,5$, of quantum numbers $m$,

$\displaystyle\pi_{0}\left(\mathrm{L}\right)\geq\pi_{\bot}\
\mathrm{\;if\;}m\in\Omega_{1},$ $\displaystyle\
\left|\pi_{0}\left(\mathrm{L}\right)\right|<\pi_{\bot},\;\pi_{0}\left(\mathrm{R}\right)>\pi_{\bot}\mathrm{\;if\;}m\in\Omega_{2},$
$\displaystyle\pi_{0}\left(\mathrm{L}\right)\leq-\pi_{\bot},\;\pi_{0}\left(\mathrm{R}\right)\geq\pi_{\bot}\mathrm{\;if\;}m\in\Omega_{3},$
$\displaystyle\pi_{0}\left(\mathrm{L}\right)<-\pi_{\bot},\;\left|\pi_{0}\left(\mathrm{R}\right)\right|<\pi_{\bot}\
\mathrm{\;if\;}m\in\Omega_{4},$
$\displaystyle\pi_{0}\left(\mathrm{R}\right)\leq-\pi_{\bot}\
\mathrm{\;if\;}m\in\Omega_{5}\ ,$ (12)

where the solutions$\ {}_{\zeta}\phi_{m}\left(X\right)$ and $\
{}^{\zeta}\phi_{m}\left(X\right)$ have similar forms and properties for a
given $\pi_{\bot}$. The manifold of all the quantum numbers $m$ is denoted by
$\Omega,$ so that $\Omega=\Omega_{1}\cup\cdots\cup\ \Omega_{5}$. In the ranges
$\Omega_{2}$ and $\Omega_{4}$ we deal with standing waves completed by linear
superpositions of solutions $\ {}_{\pm}\phi_{m}\left(X\right)$ and $\
{}^{\pm}\phi_{m}\left(X\right)$ with corresponding longitudinal fluxes that
are equal in magnitude for a given $m$. In fact, $\phi_{m}\left(X\right)$ for
$m\in\Omega_{2}$ are wave functions that describe an unbounded motion in
$x\rightarrow\infty$ direction while $\phi_{m}\left(X\right)$ for
$m\in\Omega_{4}$ are wave functions that describe an unbounded motion toward
$x=-\infty$. It was demonstrated in the framework of strong-field QED with
$x$-steps; see sections V and VII and Appendices C1 and C2 in Ref. [14], by
using one-particle mean currents and the energy fluxes that, depending on the
asymptotic behavior in the regions $S_{\mathrm{L}}$ and $S_{\mathrm{R}}$, the
plane waves $\ {}_{\pm}\phi_{m}\left(X\right)$ and $\
{}^{\pm}\phi_{m}\left(X\right)$ are identified unambiguously as components of
the initial and final wave packets of particles and antiparticles.

The plane waves are subjected to the following orthonormality conditions:

$\displaystyle\left(\ {}_{\zeta}\phi_{m},\
_{\zeta^{\prime}}\phi_{m^{\prime}}\right)_{x}=\zeta\delta_{\zeta,\zeta^{\prime}}\delta_{m,m^{\prime}}\,,$
$\displaystyle\left(\ {}^{\zeta}\phi_{m},\
^{\zeta^{\prime}}\phi_{m^{\prime}}\right)_{x}=\zeta\delta_{\zeta,\zeta^{\prime}}\delta_{m,m^{\prime}}.$
(13)

In fact, integrals (13) represent the flux densities of particles with given
$m$. The normalization factors with respect to the inner product (10) are:

$\ {}_{\zeta}\mathcal{N}=\ _{\zeta}CY,\;\ ^{\zeta}\mathcal{N}=\
^{\zeta}CY,\;Y=\left(V_{\bot}T\right)^{-1/2},$ $\
{}_{\zeta}C=\left|2p^{\mathrm{L}}\right|^{-1/2},\;\
^{\zeta}C=\left|2p^{\mathrm{R}}\right|^{-1/2}.$ (14)

In the $K_{j}\rightarrow\infty$ and $T\rightarrow\infty$ limits, one has to
replace $\delta_{m,m^{\prime}}$ by the factor
$\delta\left(\varepsilon-\varepsilon^{\prime}\right)\delta\left(\mathbf{p}_{\bot}-\mathbf{p}_{\bot}^{\prime}\right)$
in the normalization conditions (13) and then to set
$Y=\left(2\pi\right)^{-\left(d-1\right)/2}$.

Stationary plane waves of type (8) are usually used in potential scattering
theory, where they represent one-particle states with corresponding conserved
longitudinal currents. It is clear that in the ranges $\Omega_{1}$ and
$\Omega_{2}$ we have deal with states of a particle whereas in the ranges
$\Omega_{4}$ and $\Omega_{5}$ the plane waves describe states of an
antiparticle. In these ranges particles and antiparticles are subjected to the
scattering and the reflection only. Such one-particle consideration does not
work in the range $\Omega_{3}$, where the many-particle quantum field theory
consideration is essential. Note that the range $\Omega_{3}$ is often referred
to as the Klein zone. In contrast to the case $\Omega_{1}$ (and $\Omega_{5}$),
where signs of $\pi_{0}\left(\mathrm{L}\right)$ and
$\pi_{0}\left(\mathrm{R}\right)$ coincide, they are opposite in the Klein
zone. This reflects the fact that the interpretation of overlapping between
amplitudes ${}_{\zeta}\phi_{m}\left(X\right)$ and $\
{}^{\zeta}\phi_{m}\left(X\right)$ using the quantities
$\pi_{0}\left(\mathrm{L/R}\right)$ by analogy with potential scattering theory
can be erroneous.

Indeed, it is known that attempts to consider the overlapping amplitudes in
the range $\Omega_{3}$ as amplitudes of particle transmission and reflection
as its works in the relativistic quantum mechanics led (and often still lead
researchers) to contradictions and paradoxes, before the advent of consistent
consideration in the framework of QFT. Therefore, we will discuss this issue
again below.

It is assumed that each pair of solutions $\ {}_{\zeta}\phi_{m}\left(X\right)$
and $\ {}^{\zeta}\phi_{m}\left(X\right)$ with given quantum numbers
$m\in\Omega_{1}\cup\Omega_{3}\cup\Omega_{5}$ is complete in the space of
solutions with each given $m$. Due to Eq. (13) the corresponding mutual
decompositions of such solutions have the form:

$\ {}^{\zeta}\phi_{m}\left(X\right)$ $\displaystyle=$ $\
{}_{+}\phi_{m}\left(X\right)g\left({}_{+}\left|{}^{\zeta}\right.\right)-\
_{-}\phi_{m}\left(X\right)g\left({}_{-}\left|{}^{\zeta}\right.\right),$
$\;{}_{\zeta}\phi_{m}\left(X\right)$ $\displaystyle=$ $\
{}^{+}\phi_{m}\left(X\right)g\left({}^{+}\left|{}_{\zeta}\right.\right)-\
^{-}\phi_{m}\left(X\right)g\left({}^{-}\left|{}_{\zeta}\right.\right),$ (15)

where decomposition coefficients $g$ are given by the relations:

$\displaystyle\left(\ {}_{\zeta}\phi_{m},\
^{\zeta^{\prime}}\phi_{m^{\prime}}\right)_{x}=\delta_{m,m^{\prime}}g\left(\
{}_{\zeta}\left|{}^{\zeta^{\prime}}\right.\right),$ $\displaystyle g\left(\
{}^{\zeta^{\prime}}\left|{}_{\zeta}\right.\right)=g\left(\
{}_{\zeta}\left|{}^{\zeta^{\prime}}\right.\right)^{\ast},\ \
m\in\Omega_{1}\cup\Omega_{3}\cup\Omega_{5}\ .$ (16)

Substituting Eq. (15) into the orthonormality conditions (13), we derive the
following unitary relations for the decomposition coefficients:

$\displaystyle g\left(\ {}^{\zeta^{\prime}}\left|{}_{+}\right.\right)g\left(\
{}_{+}\left|{}^{\zeta}\right.\right)-g\left(\
{}^{\zeta^{\prime}}\left|{}_{-}\right.\right)g\left(\
{}_{-}\left|{}^{\zeta}\right.\right)=\zeta\delta_{\zeta,\zeta^{\prime}}\ ,$
$\displaystyle g\left(\ {}_{\zeta^{\prime}}\left|{}^{+}\right.\right)g\left(\
{}^{+}\left|{}_{\zeta}\right.\right)-g\left(\
{}_{\zeta^{\prime}}\left|{}^{-}\right.\right)g\left(\
{}^{-}\left|{}_{\zeta}\right.\ \right)=\zeta\delta_{\zeta,\zeta^{\prime}}\ .$
(17)

In particular, these relations imply that

$\displaystyle\left|g\left({}_{-}\left|{}^{+}\right.\right)\right|^{2}=\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{2},\;\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{2}=\left|g\left({}_{-}\left|{}^{-}\right.\right)\right|^{2},$
$\displaystyle\frac{g\left({}_{+}\left|{}^{-}\right.\right)}{g\left({}_{-}\left|{}^{-}\right.\right)}=\frac{g\left({}^{+}\left|{}_{-}\right.\right)}{g\left({}^{+}\left|{}_{+}\right.\right)}.$
(18)

One can see that all the coefficients $g$ can be expressed via only two of
them, e.g. via $g\left({}_{+}\left|{}^{+}\right.\right)$ and
$g\left({}_{+}\left|{}^{-}\right.\right)$. However, even the latter
coefficients are not completely independent, they are related as follows:

$\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{2}-\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{2}=1.$
(19)

Nevertheless, in what follows, we will use both coefficients
$g\left({}_{+}\left|{}^{-}\right.\right)$ and
$g\left({}_{+}\left|{}^{+}\right.\right)$ in our consideration. This maintains
a certain symmetry in important relations.

In canonical quantization of field theory, state vectors in the Fock space are
global objects, that is, the definition of vacuum and particle(antiparticle)
states has to be realized in the whole space in a given time instant. To this
end one has to use a time-independent inner product of solutions of the Klein-
Gordon equation. In the case under consideration, this inner product must be
adopted for the stationary plane waves (5). We recall that the inner product
between two solutions of the Klein-Gordon equation can be defined on
$t$-const. hyperplane as a charge (in the case under consideration, the role
of an effective charge plays the magnetic moment). Note that physical states
are wave packets that vanish on the remote boundaries, that is why the
effective charge of the scalar field is finite. It allows one to integrate by
parts in the inner product neglecting boundary terms. The latter property
provides the inner product to be time independent. However, considering the
stationary plane waves (5) that are generalized states, which do not vanish at
the spatial infinity, one should take some additional steps. In the case under
consideration, the motion of particles in the $x$ direction is unlimited,
therefore the corresponding wave functions cannot be subjected to any periodic
boundary conditions in the $x$-direction without changing their physical
meaning. For this reason one has to use a special kind of the volume
regularization; see section C2 and appendix B in Ref. [14] and section 2.1 in
Ref. [15] for details. Following Refs. [14, 15], we mean that the strong field
$\partial_{x}U$ under consideration is located inside the region
$S_{\mathrm{int}}$ during the time $T$. Consequently, causally related to the
area $S_{\mathrm{int}}$ can there be only such parts of the areas
$S_{\mathrm{L}}=\left(x_{\mathrm{FL}},x_{\mathrm{L}}\right]$ and
$S_{\mathrm{R}}=\left[x_{\mathrm{R}},x_{\mathrm{FR}}\right)$, which are
located from it at distances not exceeding $v_{s}T$. The field derivative
$\partial_{x}U$ is either negligible or switches off in these macroscopic
regions. We assume that there exist some macroscopic but finite parameters
$K^{\left(\mathrm{L/R}\right)}$ of the volume regularization which are in
spatial areas where the contribution of the field $\partial_{x}U$ is
negligible,
$\left|x_{\mathrm{FL}}\right|>K^{\left(\mathrm{L}\right)}\gg\left|x_{\mathrm{L}}\right|>0$
and $x_{\mathrm{FR}}>K^{\left(\mathrm{R}\right)}\gg x_{\mathrm{R}}>0$.

Following Refs. [14, 15] we propose the time-independent inner product between
two solutions $\Phi(X)$ and $\Phi^{\prime}(X)$ of the Klein-Gordon equation on
$t$-constant hyperplane as

$\displaystyle\left(\Phi,\Phi^{\prime}\right)=\frac{1}{v_{s}^{2}}\int_{V_{\bot}}d\mathbf{r}_{\bot}\int\limits_{-K^{\left(\mathrm{L}\right)}}^{K^{\left(\mathrm{R}\right)}}\Psi^{{\dagger}}\left(X\right)\sigma_{1}\Psi\left(X\right)dx\mathbf{\
,}$
$\displaystyle\Psi\left(X\right)=\left(\begin{array}[]{c}i\partial_{0}-U\left(x\right)\\\
1\end{array}\right)\Phi\left(X\right),$ (22)

where the integral over the spatial volume $V_{\bot}$ is completed by an
integral over the interval
$\left[K^{\left(\mathrm{L}\right)},K^{\left(\mathrm{R}\right)}\right]$ in the
$x$ direction and $\sigma_{1}$ is a Pauli matrix. The parameters
$K^{\left(\mathrm{L}/\mathrm{R}\right)}$ are assumed sufficiently large in
final expressions. First, we note that states with different quantum numbers
$m$ are independent, therefore decompositions of wave packets $\Phi$ into the
plane waves (5) in Eq. (22) do not contain interference terms. That is why it
is enough to consider Eq. (22) only for a particular case of plane waves
${}^{\zeta}\phi_{m}$ and ${}_{\zeta}\phi_{m}$ with equal $m$. Assuming that
the areas $S_{\mathrm{L}}$ and $S_{\mathrm{R}}$ are much wider than the area
$S_{\mathrm{int}}$,

$K^{\left(\mathrm{L}\right)}-\left|x_{\mathrm{L}}\right|,K^{\left(\mathrm{R}\right)}-x_{\mathrm{R}}\gg
x_{\mathrm{R}}-x_{\mathrm{L}},$ (23)

and the potential energy $U\left(x\right)$ is a continuous function, the
principal value of integral (22) is determined by integrals over the areas
$x\in\left[-K^{\left(\mathrm{L}\right)},x_{\mathrm{L}}\right]$ and
$x\in\left[x_{\mathrm{R}},K^{\left(\mathrm{R}\right)}\right]$, where the field
derivative $\partial_{x}U$ is negligible small. Thus, it is possible to
evaluate integrals of the form (22) for any form of the external field, using
only the asymptotic behavior (8) of functions in the regions
$S_{\mathrm{L}}$and $S_{\mathrm{R}}$ where particles are free. The form of the
field $\partial_{x}U$ in the area $S_{\mathrm{int}}$ affects only coefficients
$g$ entering into the mutual decompositions of the solutions given by Eq.
(15). One can see that the norms of the plane waves  ${}_{\zeta}\phi_{m}$ and
${}_{\zeta}\phi_{m}$ with respect to the inner product (22) are proportional
to the macroscopically large parameters
$\tau^{\left(\mathrm{L}\right)}=K^{\left(\mathrm{L}\right)}/v^{\mathrm{L}}$
and
$\tau^{\left(\mathrm{R}\right)}=K^{\left(\mathrm{R}\right)}/v^{\mathrm{R}}$,
where
$v^{\mathrm{L}}=v_{s}^{2}\left|p^{\mathrm{L}}/\pi_{0}\left(\mathrm{L}\right)\right|>0$
and
$v^{\mathrm{R}}=v_{s}^{2}\left|p^{\mathrm{R}}/\pi_{0}\left(\mathrm{R}\right)\right|>0$
are absolute values of the longitudinal velocities of particles in the regions
$S_{\mathrm{L}}$ and $S_{\mathrm{R}}$, respectively; see Sect. IIIC.2 and
Appendix B in Ref. [14] for details.

It was shown (see Appendix B in Ref. [14]) that the following couples of plane
waves are orthogonal with respect to the inner product (22)

$\left({}_{\zeta}\phi_{m},^{-\zeta}\phi_{m}\right)=0,\ \
m\in\Omega_{1}\cup\Omega_{5}\ ;\ \
\left({}_{\zeta}\phi_{m},^{\zeta}\phi_{m}\right)=0,\ \ m\in\Omega_{3}\ ,$ (24)

if the parameters of the volume regularization
$\tau^{\left(\mathrm{L/R}\right)}$ satisfy the condition

$\tau^{\left(\mathrm{L}\right)}-\tau^{\left(\mathrm{R}\right)}=O\left(1\right),$
(25)

where $O\left(1\right)$ denotes terms that are negligibly small in comparison
with the macroscopic quantities $\tau^{\left(\mathrm{L/R}\right)}$. One can
see that $\tau^{\left(\mathrm{R}\right)}$ and $\tau^{\left(\mathrm{L}\right)}$
are macroscopic times of motion of particles and antiparticles in the areas
$S_{\mathrm{R}}$ and $S_{\mathrm{L}}$, respectively and they are equal,

$\tau^{\left(\mathrm{L}\right)}=\tau^{\left(\mathrm{R}\right)}=\tau.$ (26)

It allows one to introduce an unique time of motion $\tau$ for all the
particles in the system under consideration. This time can be interpreted as a
system monitoring time during its evolution. Under condition (25) the norms of
the plane waves on the $t$-constant hyperplane are

$\displaystyle\left(\ {}_{\zeta}\phi_{m},\ _{\zeta}\phi_{m}\right)=\left(\
{}^{\zeta}\phi_{m},\ ^{\zeta}\phi_{m}\right)=\kappa\mathcal{M}_{m}\,,$
$\displaystyle\kappa=\left\\{\begin{array}[]{ll}1&\mathrm{if\ \
}m\in\Omega_{1}\\\ -1&\mathrm{if\ \ }m\in\Omega_{5}\end{array}\right.\,,$ (29)
$\displaystyle\left(\ \phi_{m},\ \phi_{m}\right)=\mathcal{M}_{m}\;\mathrm{if}\
\ m\in\Omega_{2}\,,$ $\displaystyle\left(\ \phi_{m},\
\phi_{m}\right)=-\mathcal{M}_{m}\;\mathrm{if}\ \ m\in\Omega_{4},$
$\displaystyle\left(\ {}_{\zeta}\phi_{m},\ _{\zeta}\phi_{m}\right)=-\left(\
{}^{\zeta}\phi_{m},\ ^{\zeta}\phi_{m}\right)=\mathcal{M}_{m},\;\mathrm{if}\ \
m\in\Omega_{3},$
$\displaystyle\mathcal{M}_{m}=2\frac{\tau}{T}\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{2}\
\ \mathrm{if}\ \ m\in\Omega_{1}\cup\Omega_{5},$
$\displaystyle\mathcal{M}_{m}=2\frac{\tau}{T}\;\mathrm{if}\ \
m\in\Omega_{2}\cup\Omega_{4}$
$\displaystyle\mathcal{M}_{m}=2\frac{\tau}{T}\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{2}\
\ \mathrm{if}\ \ m\in\Omega_{3}\ ;$ (30)

see Appendix B in Ref. [14] for details. The $\pm$ signs of integrals in Eq.
(30) correspond to the signs of the effective charge (magnetic moment) of a
particle; see appendix A) for details.

Thus, there are constructed two linearly independent couples of complete on
the $t$-constant hyperplane states with a given $m$ that are either initial
(”in”) or final (”out”) states,

$\displaystyle\mathrm{in:}\mathrm{\
}\phi_{m_{1}}^{\left(\mathrm{in},+\right)}=\ _{+}\phi_{m_{1}},\
\phi_{m_{1}}^{\left(\mathrm{in},-\right)}=\ ^{-}\phi_{m_{1}};$
$\displaystyle\phi_{m_{5}}^{\left(\mathrm{in},+\right)}=\
^{+}\phi_{m_{5}},\mathrm{\ }\phi_{m_{5}}^{\left(\mathrm{in},-\right)}=\
_{-}\phi_{m_{5}};$ $\displaystyle\phi_{m_{3}}^{\left(\mathrm{in},+\right)}=\
_{-}\phi_{m_{3}},\;\phi_{m_{3}}^{\left(\mathrm{in},-\right)}=\
^{-}\phi_{m_{3}};\ \ $ $\displaystyle\mathrm{out:}\mathrm{\
}\phi_{m_{1}}^{\left(\mathrm{out},-\right)}=\ _{-}\phi_{m_{1}},\
\phi_{m_{1}}^{\left(\mathrm{out},+\right)}=\ ^{+}\phi_{m_{1}};$
$\displaystyle\phi_{m_{5}}^{\left(\mathrm{out},-\right)}=\
^{-}\phi_{m_{5}},\mathrm{\ }\phi_{m_{5}}^{\left(\mathrm{out},+\right)}=\
_{+}\phi_{m_{5}};$ $\displaystyle\phi_{m_{3}}^{\left(\mathrm{out},+\right)}=\
_{+}\phi_{m_{3}},\ \ \phi_{m_{3}}^{\left(\mathrm{out},-\right)}=\
^{+}\phi_{m_{3}},\ \;m_{k}\in\Omega_{k}.$ (31)

Note that standing waves $\phi_{m}\left(X\right)$ for
$m\in\Omega_{2}\cup\Omega_{4}$ are the same for initial and final sets. In the
ranges $\Omega_{1}$ and $\Omega_{2}$ we have deal with states of a particle
whereas in the ranges $\Omega_{4}$ and $\Omega_{5}$ the plane waves describe
states of an antiparticle. In these cases, description of the one-particle
scattering and reflection in the framework of strong-field QED is the same as
in the framework of the potential scattering. In particular, the sign of the
flux density, given by Eq. (13), allows one to determine initial and final
states, this fact was already used above. In doing so we take into account the
charge conjugation, which implies that the physical longitudinal momentum of
an antiparticle differs in sign from the quantum numbers $p^{\mathrm{L}}$ and
$p^{\mathrm{R}}$; see appendix A for more details. We also note that in the
range $\Omega_{3}$ the choice of initial and final states is not so obvious
and will be justified below.

###  III.2 Quantization in terms of particles

In the absence of an explicit time evolution of physical quantities presented
by the plane waves (5), we can nevertheless give an interpretation of states
in the ranges  $\Omega_{1}$ , $\Omega_{3}$, and $\Omega_{5}$ considering one-
particle mean values of the operators of the effective charge, the kinetic
energy, the number of particles, and fluxes of the energy and the effective
charge (that is, effective current) through the surfaces $x=x_{\mathrm{L}}$
and $x=x_{\mathrm{R}}$. The operators of these physical quantities of the
quantum scalar field are presented in appendix A. Next, we mainly consider the
range $\Omega_{3}$.

Physically it make sense to believe that the field of the $x$-step,
$\partial_{x}U\left(x\right)$, should be considered as a part of a time-
dependent inhomogeneous field $E_{\mathrm{pristine}}$ directed along the
$x$-direction, which was switched on sufficiently fast before the time instant
$t_{\mathrm{in}}$, by this time it had time to spread to the whole area
$S_{\mathrm{int}}$ and disappear in the macroscopic regions $S_{\mathrm{L}}$
and $S_{\mathrm{R}}$. Then it was switched off sufficiently fast just after
the time instant $t_{\mathrm{out}}=t_{\mathrm{in}}+T$. The field
$E_{\mathrm{pristine}}=\partial_{x}U\left(x\right)$ in the regions
$S_{\mathrm{L}}$, $S_{\mathrm{int}}$ and $S_{\mathrm{R}}$ from
$t_{\mathrm{in}}$ till $t_{\mathrm{out}}$, that is, it acts as a constant
field concentrated in a restricted space area $S_{\mathrm{int}}$ during the
sufficiently large period of time $T$. In general the vacuum instability is
manifest in the fact that the initial (true) vacuum,
$\left|0,\mathrm{true\;in}\right\rangle$, in which the system was before
turning on the field $E_{\mathrm{pristine}}$, differs significantly from the
final (true) vacuum, $\left|0,\mathrm{true\;out}\right\rangle$, in which the
system appears after turning off this field. The difference of the true final
vacuum from the initial one is specified by the total vacuum-to-vacuum
transition probability,

$P_{v}^{\mathrm{true}}=\left|c_{v}^{\mathrm{true}}\right|^{2},\;c_{v}^{\mathrm{true}}=\left\langle
0,\mathrm{true\;out}\right.\left|0,\mathrm{true\;in}\right\rangle,$ (32)

that can be represented as a trace in the phase space of a complete set of
quantum numbers,

$\ln P_{v}^{\mathrm{true}}=\sum_{m}\ln
p_{m}^{\mathrm{true}},\;p_{m}^{\mathrm{true}}=\left(1+N_{m}^{\mathrm{true}}\right)^{-1},$
(33)

where $N_{m}^{\mathrm{true}}$ is the differential mean number of pairs created
from the vacuum; see, e.g., [19, 20]. Note that $p_{m}^{\mathrm{true}}$ is the
probability that the partial vacuum state with a given $m$ remains a vacuum.
In the case of the field $E_{\mathrm{pristine}}$ that is constant during the
macroscopic time $T$, the quantity $\ln P_{v}^{\mathrm{true}}$ appears to be
proportional to a total number of created pairs,
$N^{\mathrm{true}}=\sum_{m}N_{m}^{\mathrm{true}}$ and then is proportional to
the large dimensionless parameter $\sqrt{v_{s}\left|\partial_{x}U\right|}T$ .
In particular, we have: $\ln P_{v}^{\mathrm{true}}\approx-N^{\mathrm{true}}$
if $N_{m}^{\mathrm{true}}<1$.

The existence of the large parameter $\sqrt{v_{s}\left|\partial_{x}U\right|}T$
, satisfying inequality (9), allows one to approximate these true vacua in a
special way such that details of the system time evolution are not needed for
calculating leading contributions to total characteristics of the vacuum
instability. This possibility is due to the fact that both fast switching-on
and -off produce particle-antiparticle pairs with quantum numbers in a tiny
range of the kinetic energy, such that one can neglect the corresponding
contributions to total characteristics of vacuum instability that are
determined by sums over all the kinetic energies; see Ref. [15] for more
details. Neglecting effects of fast switching-on and -off in calculating the
total characteristics, one can approximate these quantities using instead of
the true vacua $\left|0,\mathrm{true\;in}\right\rangle$ and
$\left|0,\mathrm{true\;out}\right\rangle$ some time-independent states
$\left|0,\mathrm{in}\right\rangle$ and $\left|0,\mathrm{out}\right\rangle$
respectively, which approximate the formers in a certain sense. Here we choose
these states as ones with minimum kinetic energies (kinetic energies of the
states $\left|0,\mathrm{in}\right\rangle$ and
$\left|0,\mathrm{out}\right\rangle$ are the same) and such that the leading
contribution to the quantity $\ln P_{v}^{\mathrm{true}}$ is determined by the
probability $P_{v}$ of the transition from the vacuum
$\left|0,\mathrm{in}\right\rangle$ to the vacuum
$\left|0,\mathrm{out}\right\rangle$,

$P_{v}=\left|c_{v}\right|^{2},\;c_{v}=\left\langle
0,\mathrm{out}\right.\left|0,\mathrm{in}\right\rangle,$ (34)

such that

$\ln P_{v}^{\mathrm{true}}=\ln
P_{v}\left\\{1+O\left(\left[\sqrt{v_{s}\left|\partial_{x}U\right|}T\right]^{-1}\right)\right\\}.$
(35)

This approximation describes the situation when the leading contribution to
the total number of created pairs, $N^{\mathrm{true}}$, is formed as a sum
over the terms $N_{m}^{\mathrm{cr}}$, which represent slowly varying parts of
$N_{m}^{\mathrm{true}}$, and do not depend on oscillations related to fast
switching-on and -off of the field $E_{\mathrm{pristine}}$. In the same
approximation, we obtain:

$\displaystyle
N^{\mathrm{true}}=N^{\mathrm{cr}}\left\\{1+O\left(\left[\sqrt{v_{s}\left|\partial_{x}U\right|}T\right]^{-1}\right)\right\\},$
$\displaystyle N^{\mathrm{cr}}=\sum_{m\in\Omega_{3}}N_{m}^{\mathrm{cr}}\ ,$
(36)

and

$\ln P_{v}=\sum_{m\in\Omega_{3}}\ln
p_{m},\;p_{m}=\left(1+N_{m}^{\mathrm{cr}}\right)^{-1}\ ;$ (37)

see appendix A in Ref. [14] for details. The numbers $N_{m}^{\mathrm{cr}}$
depend only on the time-independent part of the field $E_{\mathrm{pristine}}$
and can be found by using the decomposition coefficients $g$ , given by Eq.
(16), as will be demonstrated further in this section.

In the following, namely the states $\left|0,\mathrm{in}\right\rangle$ and
$\left|0,\mathrm{out}\right\rangle$ are called the initial and the final
vacua, respectively. We consider magnon and antimagnon excitations of these
vacua as initial and final particles, respectively. Mean fluxes of the
effective charge and the energies of the vacuum states __
$\left|0,\mathrm{in}\right\rangle$__ and __
$\left|0,\mathrm{out}\right\rangle$ are quite distinct. This is what allows
the unambiguously defined the vacua $\left|0,\mathrm{in}\right\rangle$ and
$\left|0,\mathrm{out}\right\rangle$ and to construct initial and final states
of the Fock space, using the plane waves (5); see Appendix A for details.

One can decompose the Heisenberg operators of the scalar fields
$\hat{\Phi}\left(X\right)$ and $\hat{\Phi}\left(X\right)^{\dagger}$ into
solutions of either the initial or final complete sets (31). It is useful to
represent $\hat{\Phi}\left(X\right)$ (and $\hat{\Phi}\left(X\right)^{\dagger}$
respectively) as sums of five operators,
$\hat{\Phi}\left(X\right)=\sum_{k=1}^{5}\hat{\Phi}_{k}\left(X\right)$, where
the operators $\hat{\Phi}_{k}\left(X\right)$ are defined in the ranges
$\Omega_{k}$; see Eqs. (135) and (136) in appendix A for details. The
operators $\hat{\Phi}_{k}\left(X\right)$, $k=1,2$, are decomposed via the
creation and annihilation operators $a$ and $a^{{\dagger}}$ of the magnons,
while the operators $\hat{\Phi}_{k}\left(X\right)$, $k=4,5$, are decomposed
via the creation and annihilation operators $b\ $and $b^{{\dagger}}$ of
antimagnons. In the range $\Omega_{3}$ the operators
$\hat{\Phi}_{3}\left(X\right)$ and $\hat{\Phi}_{3}^{\dagger}\left(X\right)$
are decomposed via creation and annihilation operators of both magnons and
antimagnons.,

$\displaystyle\hat{\Phi}_{3}\left(X\right)$ $\displaystyle=$
$\displaystyle\sum_{m\in\Omega_{3}}\mathcal{M}_{m}^{-1/2}\left[\
{}_{-}a_{m}(\mathrm{in})\ _{-}\phi_{m}\left(X\right)\right.$ (38)
$\displaystyle+$ $\displaystyle\left.\ {}^{-}b_{m}^{\dagger}(\mathrm{in})\
^{-}\phi_{m}\left(X\right)\right]$ $\displaystyle=$
$\displaystyle\sum_{m\in\Omega_{3}}\mathcal{M}_{m}^{-1/2}\left[\
{}_{+}a_{m}(\mathrm{out})\ _{+}\phi_{m}\left(X\right)\right.$ $\displaystyle+$
$\displaystyle\left.\ {}^{+}b_{m}^{\dagger}(\mathrm{out})\
^{+}\phi_{m}\left(X\right)\right]\,.$

Here $\mathcal{M}_{m}$ are normalization factors given by Eq. (30). All the
operators labeled by the argument ”in” are interpreted as the operators of the
initial particles, whereas all the operators labeled by the argument ”out” are
interpreted as the operators of the final particles. This identification can
be confirmed as follows. The equal-time commutation relations given by Eq.
(128) in appendix A yield the standard commutation rules for the creation and
annihilation in- or out-operators introduced. The vacuum vectors
$\left|0,\mathrm{in}\right\rangle$ and $\left|0,\mathrm{out}\right\rangle$ are
null vectors for all annihilation operators $a$ and $b\ $given by Eq. (136) in
appendix A. In particular, in the range $\Omega_{3}$ nonzero commutators are:

$\displaystyle\left[\ {}_{\mp}a_{m}(\mathrm{in/out}),\
_{\mp}a_{m^{\prime}}^{\dagger}(\mathrm{in/out})\right]=\delta_{m,m^{\prime}},$
$\displaystyle\left[\ {}^{\mp}b_{m}(\mathrm{in/out}),\
^{\mp}b_{m}^{\dagger}(\mathrm{in/out})\right]=\delta_{m,m^{\prime}},$ (39)

and one-particle states of initial(final) magnon and antimagnon have the form:

$\
{}_{\mp}a_{m^{\prime}}^{\dagger}(\mathrm{in/out})\left|0,\mathrm{in/out}\right\rangle,\;\
^{\mp}b_{m}^{\dagger}(\mathrm{in/out})\left|0,\mathrm{in/out}\right\rangle.$

Using the both alternative decompositions (38) for
$\hat{\Phi}_{3}\left(X\right)$ and the orthonormality conditions (13), one can
find the following linear canonical transformation between the introduced in-
and out- creation and annihilation operators

${}_{+}a_{m}(\mathrm{out})$ $\displaystyle=$ $\displaystyle
g\left({}^{-}\left|{}_{+}\right.\right)^{-1}g\left({}^{-}\left|{}_{-}\right.\right)\;_{-}a_{m}(\mathrm{in})$
$\displaystyle-$ $\displaystyle
g\left({}^{-}\left|{}_{+}\right.\right)^{-1}\;{}^{-}b_{m}^{\dagger}(\mathrm{in})\,,$
${}^{+}b_{m}^{\dagger}(\mathrm{out})$ $\displaystyle=$
$\displaystyle-g\left({}_{-}\left|{}^{+}\right.\right)^{-1}\;_{-}a_{m}(\mathrm{in})$
(40) $\displaystyle+$ $\displaystyle
g\left({}_{-}\left|{}^{+}\right.\right)^{-1}g\left({}_{-}\left|{}^{-}\right.\right)\;^{-}b_{m}^{\dagger}(\mathrm{in})\,.$

One can verify that the transformation is unitary; see section VIID in Ref.
[14]. The inverse transformation reads

${}_{-}a_{m}(\mathrm{in})$ $\displaystyle=$ $\displaystyle
g\left({}^{+}\left|{}_{-}\right.\right)^{-1}g\left({}^{+}\left|{}_{+}\right.\right)\
_{+}a_{m}(\mathrm{out})$ $\displaystyle+$ $\displaystyle
g\left({}^{+}\left|{}_{-}\right.\right)^{-1}\
{}^{+}b_{m}^{\dagger}(\mathrm{out})\,,$ ${}^{-}b_{m}^{\dagger}(\mathrm{in})$
$\displaystyle=$ $\displaystyle g\left({}_{+}\left|{}^{-}\right.\right)^{-1}\
_{+}a_{m}(\mathrm{out})$ (41) $\displaystyle+$ $\displaystyle
g\left({}_{+}\left|{}^{-}\right.\right)^{-1}g\left({}_{+}\left|{}^{+}\right.\right)\
^{+}b_{m}^{\dagger}(\mathrm{out})\,.$

The $\pm$ sign of integrals in Eq. (30) corresponds to the sign of the
effective charge (magnetic moment) of a particle, which justifies the above
interpretation of states as magnons and antimagnons. in the ranges
$\Omega_{k},$ $k=1,2,4,5$. In the ranges $\Omega_{1}$ and $\Omega_{5}$ the
flux densities of particles through the surfaces $x=x_{\mathrm{L}}$ and
$x=x_{\mathrm{R}}$ given by Eq. (13) allow one to define initial and final
states of particles. In particular, it can be seen that the sign of the flux
densities of magnons with a given $m$ is equal to $\zeta$ in the range
$\Omega_{1},$ while the sign of the flux densities of antimagnons. with a
given $m$ is equal to $-\zeta$ in the range $\Omega_{5}$. The corresponding
reasons in the framework of QFT are presented in appendix A. One can see that
the partial vacua in the Fock subspaces with a given $m$ are stable in
$\Omega_{k},$ $k=1,2,4,5$. In the ranges $\Omega_{1}$ and $\Omega_{5}$ we meet
a realization of rules of the potential scattering theory in the framework of
QFT and see that relative probabilities of reflection and transmission (under
the condition that the vacuum remain the vacuum),

$\left|R_{m}\right|^{2}=\left|g\left({}_{+}\left|{}^{+}\right.\right)^{-1}g\left({}_{-}\left|{}^{+}\right.\right)\right|^{2},\;\left|T_{m}\right|^{2}=\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{-2},$
(42)

coincide with mean currents of reflected particles
$J_{R}=\left|R_{m}\right|^{2}$ and transmitted particles
$J_{T}=\left|T_{m}\right|^{2}$. The correct result $J_{R}+J_{T}=1$ follows
from the unitary relation (19).

In contrast to the ranges $\Omega_{1}$ and $\Omega_{5},$ one can see from Eq.
(30) that in the range $\Omega_{3}$ the magnetic moment of the particle states
${}_{\zeta}\phi_{m}$ is positive while the magnetic moment of the particle
states $\ {}^{\zeta}\phi_{m}$ is negative. This allows us to use the
magnon/antimagnon classification for the creation and annihilation operators
in Eq. (38). Considering diagonalized forms of the kinetic energy and magnetic
moment operators given by Eq. (141) in appendix A, we see that such
identification is also confirmed in the framework of QFT. In particular, we
see that the spectrum of the kinetic energy operator is positively defined and
one-particle states ${}_{\zeta}\phi_{m}$ represent magnons with the kinetic
energy ${}_{\zeta}E_{m}>0$ and the magnetic moment $\mu,$ whereas one-particle
states ${}^{\zeta}\phi_{m}$ are antimagnons with the kinetic energy
$-\;^{\zeta}E_{m}>0$.

Since transformations (40) and (41) entangle annihilation and creation
operators, the vacua $\left|0,\mathrm{in}\right\rangle$ and
$\left|0,\mathrm{out}\right\rangle$ are essentially different. The total
vacuum-to-vacuum transition amplitude $c_{v}$ is formed due to the vacuum
instability in the range $\Omega_{3}$. Differential mean numbers
$N_{m}^{a}\left(\mathrm{out}\right)$ and $N_{m}^{b}\left(\mathrm{out}\right)$,
$m\in\Omega_{3}$, of the magnons and antimagnons respectively, created from
the vacuum are equal,
$N_{m}^{b}\left(\mathrm{out}\right)=N_{m}^{a}\left(\mathrm{out}\right)=N_{m}^{\mathrm{cr}}$,
and have the forms:

$\displaystyle N_{m}^{a}\left(\mathrm{out}\right)$ $\displaystyle=$
$\displaystyle\left\langle 0,\mathrm{in}\left|\
{}_{+}a_{m}^{\dagger}(\mathrm{out})\
_{+}a_{m}(\mathrm{out})\right|0,\mathrm{in}\right\rangle$ $\displaystyle=$
$\displaystyle\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2},$
$\displaystyle N_{m}^{b}\left(\mathrm{out}\right)$ $\displaystyle=$
$\displaystyle\left\langle 0,\mathrm{in}\left|\
{}^{+}b_{m}^{\dagger}(\mathrm{out})\
^{+}b_{m}(\mathrm{out})\right|0,\mathrm{in}\right\rangle$ (43)
$\displaystyle=$
$\displaystyle\left|g\left({}_{-}\left|{}^{+}\right.\right)\right|^{-2},$

where the coefficients $g$ are given by Eq. (16).

To distinguish initial and final states in the range $\Omega_{3}$ , one needs
to consider one-particle mean values of the operators of the fluxes, of the
energy and the effective charge (that is, the magnetic moment current) through
the surfaces $x=x_{\mathrm{L}}$ and $x=x_{\mathrm{R}}$, given by Eqs. (133)
and (134) in appendix A. In the beginning we note that in the range
$\Omega_{3}$ the spatial distribution of physical states, presented by wave
packets of plane waves, is the same as in the ranges $\Omega_{2}$ and
$\Omega_{4}$. Therefore, it can be shown that particles (magnons) can be
situated only in the region $S_{\mathrm{R}}$, whereas antiparticles
(antimagnons) can be situated only in the region $S_{\mathrm{L}}$. The field
$\partial_{x}U\left(x\right)$ does not allow particles to penetrate through
the region $S_{\mathrm{int}}$, and turns them in the opposite direction. For
the plane waves such a behavior can be easily seen in the case of weak
external fields (but still strong enough, $\delta U>2\pi_{\bot}$, to provide
the existence of the $\Omega_{3}$-range) using a semiclassical approximation.
If $N_{m}^{\mathrm{cr}}$ tends to zero, then
$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{2}\rightarrow\infty$
and, at the same time,
$\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{2}\rightarrow\infty$ in
accordance to relation (19). Relations (15) imply that for an arbitrary
$m\in\Omega_{3}$ the magnon densities $\left|\
{}_{\zeta}\phi_{m}\left(X\right)\right|^{2}$ are concentrated in the region
$S_{\mathrm{R}},$ whereas the antimagnon densities $\left|\
{}^{\zeta}\phi_{m}\right|^{2}$ are concentrated in the region
$S_{\mathrm{L}}$.

In the general case when the quantities $N_{m}^{\mathrm{cr}}$ are not small,
it is natural to expect a similar behavior, namely: the region
$S_{\mathrm{L}}$ is not available for magnons, and the region $S_{\mathrm{R}}$
is not available for antimagnons. However, when the quantities
$N_{m}^{\mathrm{cr}}$ are not small, the latter property may hold only for the
corresponding wave packets, but not for the separate plane waves. That means
that these plane waves may be different from zero in the whole space. Namely
this fact leads often to a misinterpretation, since the behavior of these
plane waves looks like the one in the ranges $\Omega_{1}$ and $\Omega_{5}$,
where they represent one-particle densities both in the region
$S_{\mathrm{L}}$ and $S_{\mathrm{R}}$. However, this similarity is misleading.
Indeed, within our context it is assumed that the magnons and antimagnons in
one of corresponding asymptotic regions may occupy quasistationary states,
i.e., they should be described by wave packets that pertain their form a
sufficiently long time in these regions. Note, that in the ranges $\Omega_{1}$
and $\Omega_{5}$, the sign of the longitudinal momentum $p^{\mathrm{L/R}}$ is
related to the sign of the mean energy flux in the region $S_{\mathrm{L/R}}$.
In the range $\Omega_{3}$ the magnon states ${}_{\zeta}\phi_{m}$ are states
with a definite quantum number $p^{\mathrm{L}}$, whereas the antimagnon states
${}^{\zeta}\phi_{m}$ are states with a definite quantum number
$p^{\mathrm{R}}$. This fact together with relation (15) implies, for example,
that a partial wave ${}_{+}\phi_{m}$ of a magnon, in the region where this
particle can really be observed, i.e., in the region $S_{\mathrm{R}}$, is
always a superposition of two waves ${}^{+}\phi_{m}$ and ${}^{-}\phi_{m}$ with
opposite signs of the quantum number $p^{\mathrm{R}}$. Thus, the sign of the
mean energy flux in the region $S_{\mathrm{R}}$ cannot be related to the sign
of an asymptotic momentum in this region. Similarly, one can see, for example,
that the partial wave ${}^{+}\phi_{m}$ of an antimagnon, in the region
$S_{\mathrm{L}}$, is always a superposition of two waves with quantum number
$p^{\mathrm{L}}$ of opposite signs and, therefore, the sign of the mean energy
flux cannot be related to the sign of an asymptotic momentum in the region
where this particle can really be observed. However, as it will be
demonstrated, these are states with well-defined asymptotic energy flux and,
therefore, with a corresponding well-defined asymptotic field momentum.
Namely, these properties of the constituent plane waves are responsible for
the fact that stable magnon wave packets can exist only in the region
$S_{\mathrm{R}}$, whereas stable antimagnon wave packets can exist only in the
region $S_{\mathrm{L}}$; see appendix D in Ref. [14] for details.

Taking into account such a space separation of the magnons and antimagnons one
can use the one-particle mean values of fluxes, of the kinetic energy, and the
magnetic moment, given by Eq. (143) in appendix A, to differ initial and final
states in the range $\Omega_{3}$. So if the flux of the magnetic moment and
the kinetic energy in the region $S_{\mathrm{R}}$ coincides with the
acceleration direction of a magnon in the region $S_{\mathrm{int}}$, then the
state under consideration is a final state of the magnon, since such a
particle can only move away from the region $S_{\mathrm{int}}$
($x\rightarrow\infty$). And vice versa, if these fluxes are opposite to the
acceleration direction of a magnon in the region $S_{\mathrm{int}}$, then the
state under consideration is an initial state of the magnon, since such a
particle can only move to the region $S_{\mathrm{int}}$. In the case of
antimagnons, the direction of the flux of the kinetic energy coincides with
the direction of the flux density, but is opposite to the direction of the
flux of the magnetic moment. The antimagnons do exist in the region
$S_{\mathrm{L}}$ only. Therefore, if the direction of the flux of the kinetic
energy in the region $S_{\mathrm{L}}$ coincides with the acceleration
direction of the antimagnon in the region $S_{\mathrm{int}}$, then the state
under consideration is the final state of an antimagnon. And vice versa, if
the direction of the flux of the kinetic energy in the region $S_{\mathrm{L}}$
is opposite to the acceleration direction of an antimagnon in the region
$S_{\mathrm{int}}$, then the state under consideration is an initial state of
an antimagnon. Namely in such a manner initial and final states in Eq. (31)
are defined.

###  III.3 Observable physical quantities specifying the vacuum instability

Above, using representations (43) only differential mean number of magnon-
antimagnon pairs created from the vacuum were calculated. However, one can
obtain additional characteristics of the vacuum instability. This section is
devoted to their study.

The vacuum-to-vacuum transition amplitude $P_{v}$ is related to the mean
number $N_{m}^{\mathrm{cr}}$ by Eq. (37). On the other hand, this probability
can be represented via the imaginary part of a one-loop effective action $S$
by the seminal Schwinger formula [4],

$P_{\mathrm{v}}=\exp\left(-2\mathrm{Im}S\right).$ (44)

A relation of this representation with the one that follows from the locally
constant field approximation for the Schwinger’s effective action was found in
Ref. [21]. The probabilities of the magnon reflection and the magnon-
antimagnon pair creation can be expressed via the mean numbers
$N_{m}^{\mathrm{cr}}$ as follows:

$\displaystyle P(+|+)_{m,m^{\prime}}$ $\displaystyle=$ $\displaystyle|\langle
0,\mathrm{out}|\
_{+}a_{m}(\mathrm{out})\;_{-}a_{m}^{\dagger}(\mathrm{in})|0,\mathrm{in}\rangle|^{2}$
$\displaystyle=$
$\displaystyle\delta_{m,m^{\prime}}\left(1+N_{m}^{\mathrm{cr}}\right)^{-1}P_{v}\,,$
$\displaystyle P(+-|0)_{m,m^{\prime}}$ $\displaystyle=$ $\displaystyle|\langle
0,\mathrm{out}|\ _{+}a_{m}(\mathrm{out})\
^{+}b_{m}(\mathrm{out})|0,\mathrm{in}\rangle|^{2}$ (45) $\displaystyle=$
$\displaystyle\delta_{m,m^{\prime}}N_{m}^{\mathrm{cr}}\left(1+N_{m}^{\mathrm{cr}}\right)^{-1}P_{v}\,,$

The probabilities of the antimagnon reflection and the magnon-antimagnon pair
annihilation coincide with the quantities $P(+|+)$ and $P(+-|0)$,
respectively. In the case of bosons in a given state $m$ any number of pairs
can be created from the vacuum and from the one particle state. By this reason
probabilities (45) are not representative if the mean numbers
$N_{m}^{\mathrm{cr}}$ are large. In the partial state with a given $m$ the
probability of the creation of any pairs with given $m$ is $1-p_{m}$ where
$p_{m}$ is the probability that the partial vacuum state remains a vacuum,
given by Eq. (37). If all the mean numbers $N_{m}^{\mathrm{cr}}$ are
sufficiently small, $N_{m}^{\mathrm{cr}}\ll 1$, then the simple relations
$p_{m}\approx 1-N_{m}^{\mathrm{cr}}$ and $1-P_{v}\approx N^{\mathrm{cr}}\ll 1$
hold true in the leading approximation. In this case $P(+|+)_{m,m}\approx 1$
and $P(+-|0)_{m,m}\approx N_{m}^{\mathrm{cr}}$. Therefore, information about
the quantity $P_{v}$ allows one to estimate the total number
$N^{\mathrm{cr}}$. It is in this case that the Schwinger’s effective action
approach [4] to calculating $P_{v}$ turns out to be useful. We note that this
approach is a base of a number of approximation methods; see, e.g., Ref. [25]
for a review. In this relation, it should be noted that the probability
$P_{v}$ by itself is not very useful in the case of strong fields when
$P_{v}\ll 1$.

Taking into account Eq. (43), the total number of pairs created from the
vacuum reads:

$N^{\mathrm{cr}}=\sum_{m\in\Omega_{3}}N_{m}^{\mathrm{cr}}=\sum_{m\in\Omega_{3}}\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}.$
(46)

Magnons and antimagnons created with quantum numbers $m$ leaving the area
$S_{\mathrm{int}}$ enter the areas $S_{\mathrm{L}}$ and $S_{\mathrm{R}}$,
respectively. At the same time, the magnons continue to move in the $x$
direction with a constant velocity $v^{\mathrm{R}}$. The motion of the magnons
forms the flux density

$\left\langle j_{x}\right\rangle_{m}=N_{m}^{\mathrm{cr}}(TV_{\perp})^{-1}$
(47)

in the area $S_{\mathrm{R}}$, while the antimagnon motion in the opposite
direction with the constant velocities $-v^{\mathrm{L}}$ forms the flux
density $-\left(j_{x}\right)_{m}$ in the area $S_{\mathrm{L}}$. Here it is
taken into account that differential mean numbers of created magnons and
antimagnons with a given $m$ are equal. The total flux densities of the
magnons and the antimagnons are

$\left\langle j_{x}\right\rangle=\sum_{m\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{m}=N^{\mathrm{cr}}(TV_{\perp})^{-1}$ (48)

and $-\left\langle j_{x}\right\rangle$, respectively. The effective charge
(the magnetic moment) current density of both created  magnons and antimagnons
is $J_{x}^{\mathrm{cr}}=\mu\left\langle j_{x}\right\rangle$. It is conserved
in the $x$-direction.

During the time $T,$ the created magnons carry the magnetic moment
$\mu\left\langle j_{x}\right\rangle_{m}T$ over the unit area $V_{\bot}$ of the
surface $x=x_{\mathrm{R}}$. This magnetic moment is evenly distributed over
the cylindrical volume of the length $v^{\mathrm{R}}T$. Thus, the magnetic
moment density of the magnons created with a given $m$ is $\mu
j_{m}^{0}\left(\mathrm{R}\right)$, where
$j_{m}^{0}\left(\mathrm{R}\right)=\left\langle
j_{x}\right\rangle_{m}/v^{\mathrm{R}}$ is the number density of the magnons.
During the time $T,$ the created antimagnons carry the magnetic moment
$\mu\left\langle j_{x}\right\rangle_{m}T$ over the unit area $V_{\bot}$ of the
surface $x=x_{\mathrm{L}}$. Taking into account that this magnetic moment is
evenly distributed over the cylindrical volume of the length
$v^{\mathrm{L}}T$, we can see that the magnetic moment density of the
antimagnons created with a given $m$ is $-\mu
j_{m}^{0}\left(\mathrm{L}\right)$, where
$j_{m}^{0}\left(\mathrm{L}\right)=\left\langle
j_{x}\right\rangle_{m}/v^{\mathrm{L}}$ is the number density of the magnons.
The total magnetic moment density of the created particles reads:

$\rho^{\mathrm{cr}}\left(x\right)=\mu\left\\{\begin{array}[]{c}-\sum_{m\in\Omega_{3}}j_{m}^{0}\left(\mathrm{L}\right),\
\ x\in S_{\mathrm{L}}\\\
\sum_{m\in\Omega_{3}}j_{m}^{0}\left(\mathrm{R}\right),\ x\in
S_{\mathrm{R}}\end{array}\right..\ $ (49)

Due to a relation between the velocities $v^{\mathrm{L}}$ and
$v^{\mathrm{R}}$, the total number densities of the created magnons and
antimagnons are the same,

$\sum_{m\in\Omega_{3}}j_{m}^{0}\left(\mathrm{L}\right)=\sum_{m\in\Omega_{3}}j_{m}^{0}\left(\mathrm{R}\right)\
.$

We also note that the created magnons and antimagnons are spatially separated
and carry magnetic moments that tend to smooth out the inhomogeneity of the
external magnetic field.

In the same manner, one can derive some representation for the nonzero
components of EMT of the created particles:

$\displaystyle
T_{\mathrm{cr}}^{00}(x)=\left\\{\begin{array}[]{c}\sum_{m\in\Omega_{3}}j_{m}^{0}\left(\mathrm{L}\right)\left|\pi_{0}\left(\mathrm{L}\right)\right|,\
x\in S_{\mathrm{L}}\\\
\sum_{m\in\Omega_{3}}j_{m}^{0}\left(\mathrm{R}\right)\pi_{0}\left(\mathrm{R}\right),\mathrm{\
}x\in S_{\mathrm{R}}\end{array}\right.,$ (52) $\displaystyle
T_{\mathrm{cr}}^{11}(x)=\left\\{\begin{array}[]{c}\sum_{m\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{m}\left|p^{\mathrm{L}}\right|,\mathrm{\ }x\in
S_{\mathrm{L}}\\\ \sum_{m\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{m}\left|p^{\mathrm{R}}\right|,\mathrm{\ }x\in
S_{\mathrm{R}}\end{array}\right.,$ (55) $\displaystyle
T_{\mathrm{cr}}^{kk}(x)=\left\\{\begin{array}[]{c}\sum_{m\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{m}\left(p_{k}\right)^{2}/\left|p^{\mathrm{L}}\right|,\mathrm{\
}x\in S_{\mathrm{L}}\\\ \sum_{m\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{m}\left(p_{k}\right)^{2}/\left|p^{\mathrm{R}}\right|,\mathrm{\
}x\in S_{\mathrm{R}}\end{array}\right.,\ k=2,3,$ (58) $\displaystyle
T_{\mathrm{cr}}^{10}(x)=\left\\{\begin{array}[]{c}-\frac{1}{v_{s}}\sum_{m\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{m}\left|\pi_{0}\left(\mathrm{L}\right)\right|,\mathrm{\
}x\in S_{\mathrm{L}}\\\ \frac{1}{v_{s}}\sum_{m\in\Omega_{3}}\left\langle
j_{x}\right\rangle_{m}\pi_{0}\left(\mathrm{R}\right),\mathrm{\ }x\in
S_{\mathrm{R}}\end{array}\right..$ (61)

Here $T_{\mathrm{cr}}^{00}(x)$ and $T_{\mathrm{cr}}^{kk}(x)$, $k=1,2,3$, are
energy density and components of the pressure of the particles created in the
areas $S_{\mathrm{L}}$ and $S_{\mathrm{R}}$ respectively, whereas
$T_{\mathrm{cr}}^{10}(x)v_{s}$, for $x\in S_{\mathrm{L}}$ or $x\in
S_{\mathrm{R}}$, is the energy flux density of the created particles through
the surfaces $x=x_{\mathrm{L}}$ or $x=x_{\mathrm{R}}$ respectively. In a
strong field, or in a field with the sufficiently large potential step $\delta
U$ , the energy density and the pressure along the direction of the axis $x$
are near equal.

Let us consider effects of the backreaction on the external field due to the
vacuum instability, to establish the so-called consistency conditions. We
assume that the volume $V=V_{\bot}\left(x_{\mathrm{R}}-x_{\mathrm{L}}\right)$
contains the area
$S_{\mathrm{int}}=\left(x_{\mathrm{L}},x_{\mathrm{R}}\right).$ The total
energy of the created particles in the volume $V$ is given by the
corresponding volume integral of the energy density
$T_{\mathrm{cr}}^{00}\left(t,x\right).$ The corresponding energy conservation
low reads:

$\frac{\partial}{\partial
t}\int_{V_{\bot}}d\mathbf{r}_{\bot}\int_{x_{\mathrm{L}}}^{x_{\mathrm{R}}}T_{\mathrm{cr}}^{00}\left(t,x\right)dx=-\oint_{\Sigma}v_{s}T_{\mathrm{cr}}^{k0}(x)df_{k}\
,$ (62)

where $\Sigma$ is a surface surrounding the volume $V$ and $df_{k}$,
$k=1,2,3$, are the components of the surface element $d\mathbf{f}$. Taking
into account that $T_{\mathrm{cr}}^{00}\left(t,x\right)$ does not depend on
the transversal coordinates, and $T_{\mathrm{cr}}^{k0}(x)=0$ for $k\neq 1,$ we
find using Eq. (61) that the rate of the energy density change of the created
particles in the region $S_{\mathrm{int}}$ per unit of the spatial area
$V_{\bot}$ is:

$\displaystyle\frac{\partial}{\partial
t}\int_{x_{\mathrm{L}}}^{x_{\mathrm{R}}}T_{\mathrm{cr}}^{00}\left(t,x\right)dx$
$\displaystyle=$ $\displaystyle
v_{s}\left[\left.T_{\mathrm{cr}}^{10}(x)\right|_{x\in
S_{\mathrm{L}}}-\left.T_{\mathrm{cr}}^{10}(x)\right|_{x\in
S_{\mathrm{R}}}\right]$ (63) $\displaystyle=$ $\displaystyle-\delta Uj_{x}\ .$

It characterizes the loss of the energy that the created particles carry away
from the region $S_{\mathrm{int}}$. At the same time, the constant rate (63)
determines the power of the constant effective field
$E_{\mathrm{pristine}}=\partial_{x}U\left(x\right)$ spent on the pair
creation. Integrating this rate over the time duration of the field from
$t_{\mathrm{in}}$ to $t_{\mathrm{out}}$, and using the notation

$\Delta
T_{\mathrm{cr}}^{00}\left(x\right)=-\int_{t_{\mathrm{in}}}^{t_{\mathrm{out}}}\frac{\partial}{\partial
t}T_{\mathrm{cr}}^{00}\left(t,x\right)dt\ ,$

we find the total energy density of created pairs per unit of the area
$V_{\bot}$ as

$\int_{x_{\mathrm{L}}}^{x_{\mathrm{R}}}\Delta
T_{\mathrm{cr}}^{00}\left(x\right)dx=\delta
U\frac{N^{\mathrm{cr}}}{V_{\perp}}\ .$ (64)

In strong-field QED it is usually assumed that just from the beginning there
exists a classical effective field having a given energy. The system of
particles interacting with this field is closed, that is, the total energy of
the system is conserved333One can, however, imagine an alternative situation
when these effective charges are getting out of the regions $S_{\mathrm{L}}$
and $S_{\mathrm{R}}$ with the help of the work done by an external storage
battery. For example, dealing with graphene devices, it is natural to assume
that the constant electric strength on the graphene plane is due to the
applied fixed voltage, i.e., we are dealing with an open system of fermions
interacting with a classical electromagnetic field. In that case there would
be no backreaction problem. Note that the evolution of the mean
electromagnetic field in the graphene, taking into account the backreaction of
the matter field to the applied time-dependent external field, was considered
in Ref. [26].. It is clear that due to pair creation from the vacuum, the
constant effective field $E_{\mathrm{pristine}}=\partial_{x}U\left(x\right)$
is losing its energy and should depleted with time. Thus, the applicability of
the constant field approximation, which is used in the formulation of strong
field QED with $x$-step, is limited by the smallness of the backreaction. The
relation (64) allows one to find conditions that provide this smallness, we
call these relations the consistency conditions. These conditions can be
obtained from the requirement that the energy density given by Eq. (64) is
essentially smaller than the energy density of the constant effective field
per unit of the area $V_{\bot}$.

Note that the presence of the matter in the initial state increases the mean
number of created bosons. It is an obvious consequence of the Bose-Einstein
statistics. In the case of fermions, the presence of the matter at the initial
state prevents the pair creation. Assuming that
$N_{m}^{(+)}\left(\mathrm{in}\right)$ and
$N_{m}^{(-)}\left(\mathrm{in}\right)$ are the mean numbers of particles and
antiparticles with quantum numbers $m$ at the initial time instant, one
obtains that the differential mean numbers of final particles and
antiparticles are

$N_{m}^{(\zeta)}=\left(1+N_{m}^{\mathrm{cr}}\right)N_{m}^{(\zeta)}(\mathrm{in})+N_{m}^{\mathrm{cr}}\left[1+N_{m}^{(-\zeta)}(\mathrm{in})\right]\,,$
(65)

respectively. The differential mean numbers of particles and antiparticles
created by the external field are given by an increment $\Delta
N_{m}^{(\zeta)}=N_{m}^{(\zeta)}-N_{m}^{(\zeta)}(\mathrm{in})$. One can see
that the increments of the numbers of particles and antiparticles are equal,

$\displaystyle\Delta N_{m}^{(+)}=\Delta N_{m}^{(-)}=\Delta N_{m}\,,$
$\displaystyle\Delta
N_{m}=N_{m}^{\mathrm{cr}}\left[1+N_{m}^{(+)}(\mathrm{in})+N_{m}^{(-)}(\mathrm{in})\right]\,.$
(66)

In contrast to the previously used methods for studying the production of
bosonic pairs by external fields, our approach allows us to consider the case
of special inhomogeneous external fields supporting the spatial separation of
particles and antiparticles (in the case under consideration, these are
magnons and antimagnons) in the Klein zone. In such a way, one can see that
the equal increments of mean numbers of particles in the area $S_{\mathrm{R}}$
and antiparticles in the area $S_{\mathrm{L}}$ do not depend on the symmetry
between the mean numbers of particles and antiparticles in the initial state.
For example, assuming the absence of the initial antiparticles,
$N_{m}^{(-)}(\mathrm{in})=0$, with the number of initial particles being not
zero in the Klein zone, $N_{m}^{(+)}(\mathrm{in})\neq 0$, one can see that the
number of created antiparticles is growing in comparison with the one created
from the vacuum, $\Delta
N_{m}=N_{m}^{\mathrm{cr}}\left[1+N_{m}^{(+)}(\mathrm{in})\right]$. Therefore,
the flux of created antiparticles in the area $S_{\mathrm{L}}$ is growing
proportionally to the flux of coming particles from the area $S_{\mathrm{R}}$.
Such a behavior can be called statistically-assisted Schwinger effect.

That is why operating with the concept of probability turns out to be
unfruitful in the case when the mean number $N_{m}^{\mathrm{cr}}$ is not
relatively small. In our considerations the presence of particles in the
initial state implies that these are ingoing particles and the mean numbers
$N_{m}^{(\zeta)}(\mathrm{in})$ are proportional to densities of ingoing
fluxes,

$\left\langle
j_{x}^{\left(\zeta\right)}(\mathrm{in})\right\rangle_{m}=N_{m}^{(\zeta)}(\mathrm{in})(TV_{\perp})^{-1}.$

Densities of outgoing fluxes are:

$\left\langle
j_{x}^{\left(\zeta\right)}\right\rangle_{m}=N_{m}^{(\zeta)}(TV_{\perp})^{-1}.$

Both ingoing and outgoing magnons are situated in the area $S_{\mathrm{R}}$
while both ingoing and outgoing antimagnons are situated in the area
$S_{\mathrm{L}}$. For example, assuming the absence of initial antiparticles,
$N_{m}^{(-)}(\mathrm{in})=0$, the presence of particles in the initial state,
$N_{m}^{(+)}(\mathrm{in})\neq 0$, leads to the fact that the density of the
outgoing particle flux turns out to be more than the density of incoming
particle flux,

$\left\langle j_{x}^{\left(+\right)}\right\rangle_{m}\left/\left\langle
j_{x}^{\left(+\right)}(\mathrm{in})\right\rangle_{m}\right.=1+N_{m}^{\mathrm{cr}}\left(1+1\left/N_{m}^{(+)}(\mathrm{in})\right.\right).$

Thus, the flux proportional to $N_{m}^{\mathrm{cr}}$ of particles born from
the vacuum is added to the total flux of reflected particles. A similar
picture is observed for antiparticle fluxes in the case when
$N_{m}^{(+)}(\mathrm{in})=0$ while $N_{m}^{(-)}(\mathrm{in})\neq 0$. In the
areas of $\Omega_{3}$ adjoining the borders of the ranges $\Omega_{2}$ and
$\Omega_{4}$, the pair creation is absent, $N_{m}^{\mathrm{cr}}\rightarrow 0$,
and the only the total reflection takes place. However, in general, in the
Klein zone, fluxes due to the total reflection cannot be separated from the
fluxes due to the pair creation, that is one more reason not to use
probabilities of the reflection.

##  IV Examples of exact results with x-steps

In this section, we present a collection of external magnetic fields that
allow calculating magnon pair production characteristics based on exact
solutions of Eq. (6). For the sake of convenience, we discuss examples
separately and list pertinent results only. Further details are placed in
Appendix B.

###  IV.1 Differential quantities

####  IV.1.1 $L$-constant step

The $L$-constant magnetic step is a model of magnetic field inhomogeneity that
grows linearly with $x$ within $S_{\mathrm{int}}$ and is constant outside of
it, $\left.B\left(x\right)\right|_{x\leq
x_{\mathrm{L}}}\neq\left.B\left(x\right)\right|_{x\geq x_{\mathrm{R}}}$. We
call this field “$L$-constant” magnetic step due to its analogy with the
“$L$-constant electric field”, which is a type of electric field that creates
electron-positron pairs from the vacuum if it is strong enough; see Ref. [31]
for a discussion. The field has the following form:

$B\left(x\right)=\left\\{\begin{array}[]{ll}B^{\prime}L/2\,,&x\in
S_{\mathrm{L}}=\left(-\infty,-L/2\right]\,,\\\ -B^{\prime}x\,,&x\in
S_{\mathrm{int}}=\left(-L/2,L/2\right)\,,\\\ -B^{\prime}L/2&x\in
S_{\mathrm{R}}=\left[L/2,+\infty\right)\,,\end{array}\right.$ (67)

where $B^{\prime}>0$, $L>0$, and we set $x_{\mathrm{L}}=-L/2=-x_{\mathrm{R}}$
for simplicity.

Beyond the intermediate interval potential energies are constants,
$U_{\mathrm{L}}=+\mu B^{\prime}L/2$ and $U_{\mathrm{R}}=-\mu B^{\prime}L/2$,
and exact solutions to Eq. (6) are plane waves, classified according to Eqs.
(8). As for the intermediate interval, $S_{\mathrm{int}}$, we perform a change
of variable

$\xi\left(x\right)=\frac{\varepsilon+\mu B^{\prime}x}{\sqrt{v_{s}\mu
B^{\prime}}}\,,$ (68)

to rewrite Eq. (6) as

$\left(\frac{d^{2}}{d\xi^{2}}+\xi^{2}-\lambda\right)\varphi_{m}\left(\xi\right)=0\,,\
\ \lambda=\frac{\pi_{\bot}^{2}}{v_{s}\mu B^{\prime}}\,.$

This is Weber’s Parabolic Cylinder differential equation [32], whose
independent sets of solutions are $D_{\nu}\left[\left(1-i\right)\xi\right]$,
$D_{-\nu-1}\left[\left(1+i\right)\xi\right]$ or
$D_{\nu}\left[-\left(1-i\right)\xi\right]$, and
$D_{-\nu-1}\left[-\left(1+i\right)\xi\right]$, where $\nu=-i\lambda/2$.

With the aid of the exact solutions (151) and the coefficient (156) discussed
in Appendix B, the differential mean numbers of magnon-antimagnon pairs
created from the vacuum by the external field (43) has the form:

$\displaystyle N_{m}^{\mathrm{cr}}$ $\displaystyle=$
$\displaystyle\frac{8e^{-\pi\lambda/2}}{\sqrt{\xi_{1}^{2}-\lambda}\sqrt{\xi_{2}^{2}-\lambda}}$
(69) $\displaystyle\times$
$\displaystyle\left|f_{1}^{\left(-\right)}\left(\xi_{2}\right)f_{2}^{\left(-\right)}\left(\xi_{1}\right)-f_{2}^{\left(-\right)}\left(\xi_{2}\right)f_{1}^{\left(-\right)}\left(\xi_{1}\right)\right|^{-2}\,.$

Here, $\xi_{1}=\xi\left(x_{\mathrm{L}}\right)$,
$\xi_{2}=\xi\left(x_{\mathrm{R}}\right)$, and

$\displaystyle f_{1}^{\left(\pm\right)}\left(\xi\right)$ $\displaystyle=$
$\displaystyle\left(1\pm\frac{i}{\sqrt{\xi^{2}-\lambda}}\frac{d}{d\xi}\right)D_{-\nu-1}\left[\pm\left(1+i\right)\xi\right]\,,$
$\displaystyle f_{2}^{\left(\pm\right)}\left(\xi\right)$ $\displaystyle=$
$\displaystyle\left(1\pm\frac{i}{\sqrt{\xi^{2}-\lambda}}\frac{d}{d\xi}\right)D_{\nu}\left[\pm\left(1-i\right)\xi\right]\,.$
(70)

Optimal conditions for the magnon pair production occur when step (67) is high
enough and stretches over a wide region of the space, characterized by the
inequalities

$\sqrt{\frac{\left|\mu
B^{\prime}\right|}{v_{s}}}L\gg\max\left\\{1,\frac{\Delta^{2}}{v_{s}\mu
B^{\prime}}\right\\}\,.$ (71)

If these conditions are met and $\sqrt{\lambda}$ is fixed, in the sense that
$\sqrt{\lambda}<K_{\perp}$, where $K_{\perp}$ is a reasonably large number
obeying the conditions $\sqrt{\left|\mu B^{\prime}\right|/v_{s}}L/2\gg
K_{\perp}^{2}\gg\max\left\\{1,\Delta^{2}/v_{s}\mu B^{\prime}\right\\}$, then
$\left|\xi_{1}\right|$ and $\xi_{2}$ are large

$\displaystyle\xi_{2}\geq\sqrt{\frac{\left|\mu
B^{\prime}\right|}{v_{s}}}\frac{L}{2}\,,\ \ -\sqrt{\frac{\left|\mu
B^{\prime}\right|}{v_{s}}}\frac{L}{2}\leq\xi_{1}\leq-K\,,$ $\displaystyle
K_{\perp}^{2}<K\ll\sqrt{\frac{\left|\mu
B^{\prime}\right|}{v_{s}}}\frac{L}{2}\,,$ (72)

which means that we can use asymptotic representations of Weber parabolic
cylinder functions (WPCFs) given by Eqs. (1)-(3) in Sec. 8.4 of Ref. [32],
vol. 2, to show that the mean numbers acquire the form

$N_{m}^{\mathrm{cr}}=\exp\left(-\pi\lambda\right)\left[1+O\left(\left|\xi_{1}\right|^{-3}\right)+O\left(\xi_{2}^{-3}\right)\right]\,.$
(73)

In the limit where the inhomogeneity of the field spreads over the entire
$x$-axis, i.e., when $L\rightarrow\infty$ (thus
$\left|\xi_{1}\right|\rightarrow\infty\,,\ \xi_{2}\rightarrow\infty$), we
obtain:

$N_{m}^{\mathrm{cr}}\rightarrow
N_{m}^{\mathrm{uni}}=\exp\left(-\pi\lambda\right).$ (74)

This is a well-known expression that was originally obtained in the context of
electron-positron pair creation from the vacuum by a constant uniform electric
field [34]. Its maximum value $\max
N_{m}^{\mathrm{uni}}=\left.N_{m}^{\mathrm{uni}}\right|_{\mathbf{p}_{\perp}=\mathbf{0}}$
becomes pronounced if the derivative $B^{\prime}$ is of the order of the
critical value $B_{\mathrm{c}}^{\prime}=\Delta^{2}/v_{s}\mu$, which plays the
role of the Schwinger’s critical field [4] in the case under consideration.

Another configuration of the external field worth of discussion is when its
spacial inhomogeneity varies “abruptly” along the $x$-direction. We call this
configuration “sharply-varying” or “steep” field configuration. A steep
$L$-constant magnetic step is characterized by the set of inequalities,

$\delta U=\left|\mu B^{\prime}\right|L=\mathrm{const}>2\Delta,\ \ \delta
UL/v_{s}\ll 1\,,$ (75)

which, in turn, implies in the conditions

$\max\left(\left|\pi_{0}\left(\mathrm{L}\right)\right|\frac{L}{v_{s}},\pi_{0}\left(\mathrm{R}\right)\frac{L}{v_{s}}\right)\ll
1\,,$ (76)

as quantum numbers are bounded in the Klein zone $\Omega_{3}$. As a result,
coefficients involving asymptotic momenta are also small since
$\left|p^{\mathrm{L/R}}\right|<\left|\pi_{0}\left(\mathrm{L/R}\right)\right|$.
In this case, the argument of the WPCFs are sufficiently small in $\Omega_{3}$
and we may use the power-series expansion to show that

$\displaystyle
N_{m}^{\mathrm{cr}}\approx\frac{4\left|p^{\mathrm{L}}\right|\left|p^{\mathrm{R}}\right|}{\left|\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|+i\sigma\right|^{2}}\,,$
$\displaystyle\sigma=\left[\left|p^{\mathrm{L}}\right|\left|p^{\mathrm{R}}\right|+\left(i+\lambda\right)\frac{\left|\mu
B^{\prime}\right|}{v_{s}}\right]L\,.$ (77)

Notice that the limit $L\rightarrow 0$ is admissible provided the difference
$\left|\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|\right|$ is
larger compared $\left|\sigma\right|$. In particular, if
$\left|\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|\right|\gg\left|\sigma\right|$,
then the mean number (77) admits form

$N_{m}^{\mathrm{cr}}\approx\frac{4k}{\left(1-k\right)^{2}}\,,$ (78)

where $k=\left|p^{\mathrm{R}}\right|/\left|p^{\mathrm{L}}\right|$. It is in
agreement with results obtained at $p_{\bot}=0$ in Refs. [9, 10, 12, 13] for
the case of the Klein step formed by an electric field. Additionally, Eq. (78)
can also be reproducible by other magnetic steps, as shall be seen below.

####  IV.1.2 Sauter-like step

The Sauter-like magnetic step444We name the field (79) “Sauter-like”, in
reference to F. Sauter [11], who first solved relativistic wave equations for
a charged particle with a potential step of the form $-\alpha
E\tanh\left(x/\alpha\right)$. – a more realistic, smoothed version of the
$L$-constant magnetic step – is another example of magnetic field
inhomogeneity for which exact solutions of Eq. (6) are known. The Sauter (or
Sauter-like) electric field is a popular example of an electric field that may
violate the vacuum stability; see Ref. [14] for an extensive discussion of the
phenomenon in QED with $x$-steps.

In the present case, the Sauter-like magnetic step has the form:

$B\left(x\right)=-B^{\prime}L_{\mathrm{S}}\tanh\left(x/L_{\mathrm{S}}\right)\,,\
\ B^{\prime}>0\,,\ \ L_{\mathrm{S}}>0\,.$ (79)

At remote regions $x\rightarrow\mp\infty$, the magnetic field is constant
$B\left(\mp\infty\right)=\pm B^{\prime}L_{\mathrm{S}}$, which means that
$U_{\mathrm{L}}=\mu B^{\prime}L_{\mathrm{S}}=-U_{\mathrm{R}}$. Therefore, the
magnitude of the potential step (7) in this case is $\delta
U=U_{\mathrm{L}}-U_{\mathrm{R}}=2\mu B^{\prime}L_{\mathrm{S}}$.

Performing the change of variable

$\chi\left(x\right)=\frac{1}{2}\left[1+\tanh\left(x/L_{\mathrm{S}}\right)\right]\,,$
(80)

and seeking for solutions in the form

$\varphi_{m}\left(x\right)=\chi^{-iL_{\mathrm{S}}\left|p^{\mathrm{L}}\right|/2}\left(1-\chi\right)^{iL_{\mathrm{S}}\left|p^{\mathrm{R}}\right|/2}f\left(\chi\right)\,,$
(81)

allows us to express Eq. (6) in the same form as the differential equation for
the Gauss Hypergeometric Function [32],

$\chi\left(1-\chi\right)f^{\prime\prime}+\left[c-\left(a+b+1\right)\chi\right]f^{\prime}-abf=0\,,$
(82)

provided

$\displaystyle
a=\frac{1}{2}\left[iL_{\mathrm{S}}\left(\left|p^{\mathrm{R}}\right|-\left|p^{\mathrm{L}}\right|\right)+1+i\sqrt{\left(\frac{L_{\mathrm{S}}\delta
U}{v_{s}}\right)^{2}-1}\right]\,,$ $\displaystyle
b=\frac{1}{2}\left[iL_{\mathrm{S}}\left(\left|p^{\mathrm{R}}\right|-\left|p^{\mathrm{L}}\right|\right)+1-i\sqrt{\left(\frac{L_{\mathrm{S}}\delta
U}{v_{s}}\right)^{2}-1}\right]\,,$
$\displaystyle\;c=1-iL_{\mathrm{S}}\left|p^{\mathrm{L}}\right|\,.$ (83)

Using the exact solutions (158) and (159) and the coefficient (161) discussed
in Appendix B, we find that

$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}=\frac{\sinh\left(\pi
L_{\mathrm{S}}\left|p^{\mathrm{R}}\right|\right)\sinh\left(\pi
L_{\mathrm{S}}\left|p^{\mathrm{L}}\right|\right)}{\sinh^{2}\left[\pi
L_{\mathrm{S}}\left(\left|p^{\mathrm{R}}\right|-\left|p^{\mathrm{L}}\right|\right)/2\right]+\cosh^{2}\left(\frac{\pi}{2}\sqrt{\left(L_{\mathrm{S}}\delta
U/v_{s}\right)^{2}-1}\right)}\,.$ (84)

Result (84) holds in the ranges $\Omega_{1}$, $\Omega_{5}$, and $\Omega_{3}$.
Taking into account relation (19), one finds
$\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{2}$. In the ranges
$\Omega_{1}$ and $\Omega_{5}$ use of these coefficients in Eq. (42) gives the
relative probabilities of the reflection,
$\left|R_{m}\right|^{2}=1-\left|T_{m}\right|^{2}$, and the transmission,
$\left|T_{m}\right|^{2}=\left[1+\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{2}\right]^{-1}$.
In the range $\Omega_{3}$ according to relation (43) the coefficient
$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}$ gives the
differential mean numbers of the magnon-antimagnon pairs created from the
vacuum, $N_{m}^{\mathrm{cr}}$. In particular, for any $\pi_{\bot}\neq 0$, one
of the following limits holds true:

$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}\sim\left|L_{\mathrm{S}}p^{\mathrm{R}}\right|\rightarrow
0,\ \
\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}\sim\left|L_{\mathrm{S}}p^{\mathrm{L}}\right|\rightarrow
0.$ (85)

This means that in the range $\Omega_{3}$, the mean nambers tend to zero,
$N_{m}^{\mathrm{cr}}\rightarrow 0$, while in the ranges $\Omega_{1}$ and
$\Omega_{5}$ the relative probability of the transmission reads
$\left|T_{m}\right|^{2}\rightarrow 0$ if $m$ tends to the boundary with either
the range $\Omega_{2}$ ($\left|p^{\mathrm{L}}\right|\rightarrow 0$) or the
range $\Omega_{4}$ ($\left|p^{\mathrm{R}}\right|\rightarrow 0$).

In cases where the magnetic step is high enough and stretches over a wide
region of the space, such that $L_{\mathrm{S}}\delta U/v_{s}\gg 1,$ the mean
number of created pairs can be approximated as

$N_{m}^{\mathrm{cr}}\approx N_{m}^{\mathrm{as}}=e^{-\pi\tau}\,,\ \
\tau=L_{\mathrm{S}}\left(2\left|\mu
B^{\prime}\right|L_{\mathrm{S}}/v_{s}-\left|p^{\mathrm{R}}\right|-\left|p^{\mathrm{L}}\right|\right)\,.$
(86)

When $L_{\mathrm{S}}\delta U/v_{s}\rightarrow\infty$, one obtains
$N_{m}^{\mathrm{cr}}\rightarrow N_{m}^{\mathrm{uni}}$ where
$N_{m}^{\mathrm{uni}}$ is given by Eq. (74).

Sharp-gradient configuration

$\delta U=\left|\mu B^{\prime}\right|L_{\mathrm{S}}=\mathrm{const}>2\Delta\,,\
\ \delta UL_{\mathrm{S}}/v_{s}\ll 1\,,$ (87)

corresponds to a very sharp field derivative $\partial_{x}U$, highly
concentrated near the origin $x=0$, described by a very “steep” potential
step. This configuration has a special interest because it corresponds to a
regularization of the Klein step (originally an electric step potential)

$U\left(x\right)=\left\\{\begin{array}[]{l}U_{\mathrm{L}}\;\mathrm{if}\;\
x<0\\\ U_{\mathrm{R}}\;\mathrm{if}\ x>0\end{array}\right.,$ (88)

where $U_{\mathrm{R}}$ and $U_{\mathrm{L}}$ are constants, and may be useful
in a discussion of the Klein paradox. In the ranges $\Omega_{1}$ and
$\Omega_{5}$ the energy $\left|\varepsilon\right|$ is not restricted from the
above, that is why in what follows we consider only the subranges, where
$\max\left\\{L_{\mathrm{S}}\left|p^{\mathrm{L}}\right|,L_{\mathrm{S}}\left|p^{\mathrm{R}}\right|\right\\}\ll
1.$ Note that in these ranges
$\left|\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|\right|>\delta
U$ then the parameter
$k=\left|p^{\mathrm{R}}\right|\left/\left|p^{\mathrm{L}}\right|\right.$ does
not achieve the unit value, $k\neq 1$. Then one has:

$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}\approx\frac{4k}{\left(1-k\right)^{2}},$
(89)

and obtains the transmission coefficient as

$\left|T_{m}\right|^{2}\approx\frac{4k}{\left(1+k\right)^{2}},$ (90)

that is in agreement with results of the nonrelativistic consideration
obtained in any textbook for one dimensional quantum motion. In the range
$\Omega_{3}$ for any given $\pi_{\bot}$ the absolute values of
$\left|p^{\mathrm{R}}\right|$ and $\left|p^{\mathrm{L}}\right|$ are restricted
from above,

$0\leq\left|\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|\right|\leq\sqrt{\delta
U\left(\delta U-2\pi_{\bot}\right)}.$ (91)

As it follows from Eq. (84), in the range $\Omega_{3}$ the differential mean
numbers of created magnon-antimagnon pairs read:

$N_{m}^{\mathrm{cr}}=\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}\approx\frac{4\left|p^{\mathrm{L}}\right|\left|p^{\mathrm{R}}\right|}{\left(\frac{\delta
U^{2}L_{\mathrm{S}}}{2v_{s}^{2}}\right)^{2}+\left(\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|\right)^{2}}.$
(92)

They have a maximum at $k=1$ that can be quite large,

$\max N_{m}^{\mathrm{cr}}=\frac{4}{\left(L_{\mathrm{S}}\delta
U/v_{s}\right)^{2}}\left[1-\left(\frac{2\pi_{\bot}}{\delta
U}\right)^{2}\right].$ (93)

The limit $L_{\mathrm{S}}\rightarrow 0$ in Eq. (92) is possible only when the
difference $\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|$ is not
very small, namely when

$\left(\frac{\delta
U^{2}L_{\mathrm{S}}}{2v_{s}^{2}}\right)^{2}\ll\left(\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|\right)^{2}.$

Only under the latter condition one can neglect an $L_{\mathrm{S}}$-depending
term in Eq. (92) to obtain the form given by Eq. (78). Thus, we have an
another example of the regularization of the Klein step.

####  IV.1.3 Exponential step

We present here an example of the magnetic field inhomogeneity whose
analytical form is a piecewise, continuous exponential functions of $x$. In
this case we have a possibility to consider various asymmetric peak
configurations. The electric-analog of this field in QED was considered in
Ref. [33]. The magnetic step has the form:

$B\left(x\right)=B^{\prime}\left\\{\begin{array}[]{ll}k_{1}^{-1}\left(1-e^{k_{1}x}\right)\,,&x\in\mathrm{I}=\left(-\infty,0\right]\\\
k_{2}^{-1}\left(e^{-k_{2}x}-1\right)\,,&x\in\mathrm{II}=\left(0,+\infty\right)\end{array}\right.,$
(94)

where $k_{j}$, $j=1,2$, are positive constants that characterizes how steep or
smooth the field decays from $x=-\infty$ to $x=+\infty$. Similarly to the
preceding examples, the exponential magnetic step (94) reaches constant values
at remote regions which means that $U_{\mathrm{L}}=\mu B^{\prime}k_{1}^{-1}$
and $U_{\mathrm{R}}=-\mu B^{\prime}k_{2}^{-1}$. As a result, the magnitude of
the potential step is $\delta U=U_{\mathrm{L}}-U_{\mathrm{R}}=\mu
B^{\prime}\left(k_{1}^{-1}+k_{2}^{-1}\right)$.

To solve Eq. (6) with this field, we perform the change of variables

$\displaystyle\eta_{1}$ $\displaystyle=$ $\displaystyle ih_{1}e^{k_{1}x}\,,\ \
h_{1}=\frac{2\mu B^{\prime}}{k_{1}^{2}v_{s}}\,,\ \ x\in\mathrm{I}\,,$
$\displaystyle\eta_{2}$ $\displaystyle=$ $\displaystyle ih_{2}e^{-k_{2}x}\,,\
\ h_{2}=\frac{2\mu B^{\prime}}{k_{1}^{2}v_{s}}\,,\ \ x\in\mathrm{II}\,,$ (95)

and represent the scalar functions in the form

$\displaystyle\varphi_{m}\left(x\right)=e^{-\eta_{j}/2}\eta_{j}^{\nu_{j}}R_{j}\left(\eta_{j}\right)\,,$
$\displaystyle\nu_{1}=\frac{i\left|p^{\mathrm{L}}\right|}{k_{1}}\,,\ \
\nu_{2}=\frac{i\left|p^{\mathrm{R}}\right|}{k_{2}}\,,$ (96)

to learn that the functions $R_{j}\left(\eta_{j}\right)$ obey the Confluent
Hypergeometric equations

$\eta_{j}R_{j}^{\prime\prime}+\left(c_{j}-\eta_{j}\right)R_{j}^{\prime}-a_{j}R_{j}=0\,,$
(97)

provided

$c_{j}=2\nu_{j}+1\,,\ \
a_{1}=\nu_{1}+\frac{1}{2}+\frac{i\pi_{0}\left(\mathrm{L}\right)}{k_{1}v_{s}}\,,\
\
a_{2}=\nu_{2}+\frac{1}{2}-\frac{i\pi_{0}\left(\mathrm{R}\right)}{k_{2}v_{s}}\,.$
(98)

Fundamental pairs of solutions to Eq. (97) with special asymptotic properties
at remote regions are proportional to Confluent Hypergeometric Functions (CHF)
$\Phi\left(a_{j},c_{j};\eta_{j}\right)$ and
$\eta_{j}^{1-c_{j}}e^{\eta_{j}}\Phi\left(1-a_{j},2-c_{j};-\eta_{j}\right)$.

Using exact solutions and their connection via $g$-coefficients discussed in
Appendix B, we find:

$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}=\frac{4\left|p^{\mathrm{L}}\right|\left|p^{\mathrm{R}}\right|}{\exp\left[-\pi\left(k_{1}^{-1}\left|p^{\mathrm{L}}\right|-k_{2}^{-1}\left|p^{\mathrm{R}}\right|\right)\right]}\left|\left.\left(k_{1}h_{1}y_{1}^{2}\frac{d}{d\eta_{1}}y_{2}^{1}+k_{2}h_{2}y_{2}^{1}\frac{d}{d\eta_{2}}y_{1}^{2}\right)\right|_{x=0}\right|^{-2}\,.$
(99)

Taking into account relation (19) one finds
$\left|g\left({}_{+}\left|{}^{+}\right.\right)\right|^{2}$. In the ranges
$\Omega_{1}$ and $\Omega_{5}$ use of these coefficients in Eq. (42) gives the
relative probabilities of the reflection, $\left|R_{m}\right|^{2}$, and the
transmission, $\left|T_{m}\right|^{2}$. In the range $\Omega_{3}$ according to
relation (43) the coefficient (99) gives the differential mean numbers of
pairs created from the vacuum, $N_{m}^{\mathrm{cr}}$. In particular, if either
$\left|p^{\mathrm{R}}\right|$ or $\left|p^{\mathrm{L}}\right|$ tends to zero
for any $\pi_{\bot}\neq 0$, one of the following limits holds true:

$\left|g\left({}_{-}|^{+}\right)\right|^{-2}\sim\left|p^{\mathrm{R}}\right|\rightarrow
0,\ \
\left|g\left({}^{+}|_{-}\right)\right|^{-2}\sim\left|p^{\mathrm{L}}\right|\rightarrow
0.$ (100)

If the step is high enough and stretches over a wide region of the space, so
that the conditions are satisfied

$\min\left(h_{1},h_{2}\right)\gg\max\left\\{1,\frac{\Delta^{2}}{v_{s}\mu
B^{\prime}}\right\\}\,,\ \ \frac{k_{1}}{k_{2}}=\mathrm{fixed}\,,$ (101)

it is possible to show based on the results of Ref. [33] that the mean numbers
of created pairs, given by Eq. (99), admit simpler forms,

$\displaystyle
N_{m}^{\mathrm{cr}}=\left|g\left({}_{-}|^{+}\right)\right|^{-2}$
$\displaystyle\approx\left\\{\begin{array}[]{ll}\exp\left\\{-\frac{2\pi}{k_{1}}\left[\left|\pi_{0}\left(\mathrm{L}\right)\right|-\left|p^{\mathrm{L}}\right|\right]\right\\}\,,&0\leq\varepsilon<U_{\mathrm{L}}-\pi_{\bot}\,,\\\
\exp\left\\{-\frac{2\pi}{k_{2}}\left[\pi_{0}\left(\mathrm{R}\right)-\left|p^{\mathrm{R}}\right|\right]\right\\}\,,&U_{\mathrm{R}}+\pi_{\perp}\leq\varepsilon<0\,.\end{array}\right.$
(104)

When $h_{1},h_{2}\rightarrow\infty$, one obtains
$N_{m}^{\mathrm{cr}}\rightarrow N_{m}^{\mathrm{uni}}$.

One can consider an essentially asymmetric configuration with
$k_{\mathrm{1}}\gg k_{\mathrm{2}}$.

By choosing large parameters $k_{\mathrm{1}}$,
$k_{\mathrm{2}}\rightarrow\infty$ with a fixed ratio
$k_{\mathrm{1}}/k_{\mathrm{2}}$, one can obtain sharp gradient fields that
exist only in a small area in a vicinity of the origin $x=0$. We assume that
the corresponding asymptotic potential energies $U_{\mathrm{R}}$ and
$U_{\mathrm{L}}$ define finite magnitudes of the potential steps $\Delta
U_{\mathrm{1}}$ and $\Delta U_{\mathrm{2}}$ for increasing and decreasing
parts,

$U_{\mathrm{L}}=\Delta U_{\mathrm{1}},\;\;U_{\mathrm{R}}=-\Delta
U_{\mathrm{2}},$ (105)

respectively, and satisfy the following inequalities:

$\Delta U_{\mathrm{1}}/k_{\mathrm{1}}\ll 1,\;\;\Delta
U_{\mathrm{2}}/k_{\mathrm{2}}\ll 1.$ (106)

This case corresponds to a very sharp peak with a given step magnitude $\delta
U=\Delta U_{\mathrm{1}}+\Delta U_{\mathrm{2}}$. In the ranges $\Omega_{1}$ and
$\Omega_{5}$ the energy $\left|\varepsilon\right|$ is not restricted from the
above, that is why in what follows we consider only the subranges, where

$\max\left(\left|\pi_{0}\left(\mathrm{L}\right)\right|/k_{\mathrm{1}},\left|\pi_{0}\left(\mathrm{R}\right)\right|/k_{\mathrm{2}}\right)\ll
1.$ (107)

In these ranges taking into account that
$\left|\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|\right|>\delta
U$ we obtain that the
$\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}$ and the
transmission coefficient can be presented by the functions (89) and (90) of
the parameter
$k=\left|p^{\mathrm{R}}\right|\left/\left|p^{\mathrm{L}}\right|\right.$.

In the range $\Omega_{3}$ the difference
$\left|\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|\right|$ are
restricted from above by Eq. (91) and can tend to zero. That is why the
differential mean number of pairs created is

$\displaystyle
N_{m}^{\mathrm{cr}}\approx\frac{4\left|p^{\mathrm{L}}\right|\left|p^{\mathrm{R}}\right|}{\left(\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|\right)^{2}+b^{2}},$
$\displaystyle b=\frac{2\Delta
U_{\mathrm{1}}}{k_{\mathrm{1}}}\left[\frac{\Delta
U_{\mathrm{1}}}{4}+\left|\pi_{0}\left(\mathrm{L}\right)\right|\right]+\frac{2\Delta
U_{\mathrm{2}}}{k_{\mathrm{2}}}\left[\frac{\Delta
U_{\mathrm{2}}}{4}+\pi_{0}\left(\mathrm{R}\right)\right].$ (108)

It has a maximum at $k=1$ that can be large,
$N_{m}^{\mathrm{cr}}=4\left|p^{\mathrm{L}}\right|\left|p^{\mathrm{R}}\right|/b^{2}$.
Under the condition
$b^{2}\ll\left(\left|p^{\mathrm{L}}\right|-\left|p^{\mathrm{R}}\right|\right)^{2}$
one can verify that the mean number $N_{m}^{\mathrm{cr}}$ is given by Eq.
(78). Thus, we have additional example of the regularization of the Klein
step.

####  IV.1.4 Inverse-square step

As a last example, we present below a model of the magnetic field
inhomogeneity that is also a piecewise and a continuous function of $x$. In
QED, we call the electric field corresponding to the potential step inverse-
square electric field, whose solutions to relativistic wave equations were
found by us recently in [35]. The magnetic step has the form:

$B\left(x\right)=B^{\prime}\left\\{\begin{array}[]{ll}\varrho_{1}\left[1-\left(1-x/\varrho_{1}\right)^{-1}\right]\,,&x\in\mathrm{I}=\left(-\infty,0\right]\\\
\varrho_{2}\left[\left(1+x/\varrho_{2}\right)^{-1}-1\right]\,,&x\in\mathrm{II}=\left(0,+\infty\right)\end{array}\right.,$
(109)

where $\varrho_{j}$, $j=1,2$, are positive constants that characterize how the
field grows/decays along the $x$-axis. The magnitude of the potential step for
this field is $\delta U=U_{\mathrm{L}}-U_{\mathrm{R}}=\mu
B^{\prime}\left(\varrho_{1}+\varrho_{2}\right)$.

By performing the change of variables

$\displaystyle z_{1}\left(x\right)$ $\displaystyle=$ $\displaystyle
2i\left|p^{\mathrm{L}}\right|\varrho_{1}\left(1-x/\varrho_{1}\right)\,,\ \
x\in\mathrm{I}\,,$ $\displaystyle z_{2}\left(x\right)$ $\displaystyle=$
$\displaystyle
2i\left|p^{\mathrm{R}}\right|\varrho_{2}\left(1+x/\varrho_{2}\right)\,,\ \
x\in\mathrm{II}\,,$ (110)

we may write the second-order differential equation (6) as

$\left(\frac{d^{2}}{dz_{j}^{2}}-\frac{1}{4}+\frac{\kappa_{j}}{z_{j}}+\frac{1/4-\mu_{j}^{2}}{z_{j}^{2}}\right)\varphi_{m}\left(x\right)=0\,,$
(111)

whose solutions are proportional to the Whittaker functions
$W_{\kappa_{j},\mu_{j}}\left(z_{j}\right)$ and
$W_{-\kappa_{j},\mu_{j}}\left(e^{-i\pi}z_{j}\right)$, provided the parameters
$\kappa_{j}$ and $\mu_{j}$ have the form:

$\displaystyle\kappa_{1}$ $\displaystyle=$ $\displaystyle-i\frac{\mu
B^{\prime}\varrho_{1}^{2}}{v_{s}}\frac{\pi_{0}\left(\mathrm{L}\right)/v_{s}}{\left|p^{\mathrm{L}}\right|}\,,\
\ \kappa_{2}=i\frac{\mu
B^{\prime}\varrho_{2}^{2}}{v_{s}}\frac{\pi_{0}\left(\mathrm{R}\right)/v_{s}}{\left|p^{\mathrm{R}}\right|}\,,$
$\displaystyle\mu_{1}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{1}{4}-\left(\frac{\mu
B^{\prime}\varrho_{1}^{2}}{v_{s}}\right)^{2}}\,,\ \
\mu_{2}=\sqrt{\frac{1}{4}-\left(\frac{\mu
B^{\prime}\varrho_{2}^{2}}{v_{s}}\right)^{2}}\,.$ (112)

Using the exact solutions (172) and (173) and coefficient (180) that connects
different solutions as is discussed in Appendix B, we obtain:

$N_{m}^{\mathrm{cr}}=\left|p^{\mathrm{L}}\right|\left|p^{\mathrm{R}}\right|\left|\left.\left[\left|p^{\mathrm{L}}\right|w_{1}^{2}\left(z_{2}\right)\frac{d}{dz_{1}}w_{2}^{1}\left(z_{1}\right)+w_{2}^{1}\left(z_{1}\right)\left|p^{\mathrm{R}}\right|\frac{d}{dz_{2}}w_{1}^{2}\left(z_{2}\right)\right]\right|_{x=0}\right|^{-2}\,.$
(113)

Finally, considering cases where the step is sufficiently high and stretches
over a wide region of the space, specified by the conditions

$\min\left(U_{\mathrm{L}}\varrho_{1},\left|U_{\mathrm{R}}\right|\varrho_{2}\right)\gg\max\left\\{1,\frac{\Delta^{2}}{v_{s}\mu
B^{\prime}}\right\\},\ \ \frac{\varrho_{1}}{\varrho_{2}}=\mathrm{fixed}\,,$
(114)

it is possible to demonstrate based on our previous results [35] that the mean
numbers (113) admit the asymptotic forms

$N_{m}^{\mathrm{cr}}\approx\left\\{\begin{array}[]{ll}\exp\left(2\pi\omega_{1}^{+}\right)\,,&0\leq\varepsilon<U_{\mathrm{L}}-\pi_{\bot}\,,\\\
\exp\left(2\pi\omega_{2}^{-}\right)\,,&U_{\mathrm{R}}+\pi_{\perp}\leq\varepsilon<0\,.\end{array}\right.$
(115)

where

$\displaystyle\omega_{1}^{\pm}$ $\displaystyle=$ $\displaystyle\pm
i\left(\kappa_{1}\pm\mu_{1}\right)$ $\displaystyle=$
$\displaystyle\frac{U_{\mathrm{L}}\varrho_{1}}{v_{s}}\left[\sqrt{1-\left(\frac{2U_{\mathrm{L}}\varrho_{1}}{v_{s}}\right)^{-2}}\pm\frac{\pi_{0}\left(\mathrm{L}\right)/v_{s}}{\left|p^{\mathrm{L}}\right|}\right]\,,$
$\displaystyle\omega_{2}^{\pm}$ $\displaystyle=$ $\displaystyle\mp
i\left(\kappa_{2}\pm\mu_{2}\right)$ (116) $\displaystyle=$
$\displaystyle\frac{\left|U_{\mathrm{R}}\right|\varrho_{2}}{v_{s}}\left[\sqrt{1-\left(\frac{2\left|U_{\mathrm{R}}\right|\varrho_{2}}{v_{s}}\right)^{-2}}\pm\frac{\pi_{0}\left(\mathrm{R}\right)/v_{s}}{\left|p^{\mathrm{R}}\right|}\right]\,.$

When
$U_{\mathrm{L}}\varrho_{1},\left|U_{\mathrm{R}}\right|\varrho_{2}\rightarrow\infty$,
we have $N_{m}^{\mathrm{cr}}\rightarrow N_{m}^{\mathrm{uni}}$ where
$N_{m}^{\mathrm{uni}}$. One can see that all of the above presented examples
in the model of smooth-gradient step can be seen as regularizations of
linearly growing magnetic field.

The choosing small parameters $\varrho_{1}$ and $\varrho_{2}$,
$\max\left(U_{\mathrm{L}}\varrho_{1},\left|U_{\mathrm{R}}\right|\varrho_{2}\right)\ll
1$, with a fixed ratio $\varrho_{1}/\varrho_{2}$, one can obtain the mean
number of magnon-antimagnon created from the vacuum by steep inversely-square
steps. These inequalities imply that the arguments of the Whittaker functions
involved in (113) are small. In this case, the mean number
$N_{m}^{\mathrm{cr}}$ is smaller than unity due to the logarithmic
contributions in the denominator; see Ref. [35] for a more comprehensive
discussion. Despite the similarities of the inverse-square step (109) with the
external fields discussed in the preceding sections, the mean number (113)
does not admit the representation (78), i.e., this field does not represent a
regularization of the Klein step.

###  IV.2 Integral quantities

According to the discussion in preceding section, the total number of the
magnon-antimagnon pairs produced from the vacuum by magnetic steps is a sum of
the differential numbers over the quantum numbers within the Klein zone
$\Omega_{3}$, given by Eq. (46). The created magnons continue to move in the
$x$ direction and form the flux density $j_{x}$, given by Eq. (48), in the
area $S_{\mathrm{R}}$, while the antimagnon motion in the opposite direction
form the flux density $-j_{x}$ in the area $S_{\mathrm{L}}$. These flux
densities can be represented in an integral form as:

$j_{x}=\frac{1}{\left(2\pi\right)^{3}}\int_{U_{\mathrm{R}}+\pi_{\perp}}^{U_{\mathrm{L}}-\pi_{\bot}}d\varepsilon\int
d\mathbf{p}_{\perp}N_{m}^{\mathrm{cr}}\,.$ (117)

In general, analytical integration in Eq. (117) is impossible. Nevertheless,
due to the analogy between QED with $x$-steps and the present study, we may
use universal expressions in a strong electric field from Ref. [21] to
conclude that if the magnetic step is sufficiently high but evolves gradually
along the $x$-axis (smooth-gradient step), one can approximate Eq. (117) as

$\displaystyle
j_{x}\approx\tilde{j}_{x}=\frac{1}{\left(2\pi\right)^{3}}\int_{x_{\mathrm{L}}}^{x_{\mathrm{R}}}dx\
U^{\prime}\left(x\right)\int
d\mathbf{p}_{\bot}N_{n}^{\mathrm{uni}}\left(x\right),$ $\displaystyle
N_{n}^{\mathrm{uni}}\left(x\right)=\exp\left[-\pi\frac{\pi_{\bot}^{2}}{v_{s}U^{\prime}\left(x\right)}\right],$
(118)

where the prime over the potential denotes differentiation with respect to
$x$, $U^{\prime}\left(x\right)=\mu dB\left(x\right)/dx$. The quantity
$N_{n}^{\mathrm{uni}}\left(x\right)$ has a universal form which can be used to
calculate any total characteristic of the pair creation effect. Integrating
the latter expression over $d\mathbf{p}_{\bot}$ one obtain the final form,

$j_{x}\approx\tilde{j}_{x}=\frac{1}{(2\pi)^{3}}\int_{x_{\mathrm{L}}}^{x_{\mathrm{R}}}dxU^{\prime}\left(x\right)^{2}\exp\left[-\pi\frac{\Delta^{2}}{v_{s}U^{\prime}\left(x\right)}\right]\,.$
(119)

For the examples discussed in the preceding subsections IV.1.1-IV.1.4, the
total density of the magnon pair production has the following form:

$\tilde{j}_{x}=r^{\mathrm{cr}}\frac{\delta U}{\left|\mu
B^{\prime}\right|}\left\\{\begin{array}[]{l}1\,,\ \text{{for}
}L\text{{-constant step}}\\\ \tilde{\delta}/2\,,\ \text{{for Sauter-like}
{step}}\\\ G\left(2,\pi\frac{\Delta^{2}}{v_{s}\left|\mu
B^{\prime}\right|}\right),\ \text{{for exponential step}}\\\
\frac{1}{2}G\left(\frac{3}{2},\pi\frac{\Delta^{2}}{v_{s}\left|\mu
B^{\prime}\right|}\right),\ \text{{for inverse-square
step}}\end{array}\right.,$ (120)

where $\delta U=U_{\mathrm{L}}-U_{\mathrm{R}}$ is the magnitude of the step in
all the examples,

$r^{\mathrm{cr}}=\frac{\left|\mu
B^{\prime}\right|^{2}}{(2\pi)^{3}}\exp\left(-\pi\frac{\Delta^{2}}{v_{s}\left|\mu
B^{\prime}\right|}\right)\,,$ (121)

and

$\displaystyle\tilde{\delta}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dtt^{-1/2}\left(1+t\right)^{-5/2}\exp\left(-\frac{\pi
t\Delta^{2}}{v_{s}\left|\mu B^{\prime}\right|}\right)$ $\displaystyle=$
$\displaystyle\sqrt{\pi}\Psi\left(\frac{1}{2},-2;\pi\frac{\Delta^{2}}{v_{s}\left|\mu
B^{\prime}\right|}\right)\,,$ $\displaystyle G\left(\alpha,z\right)$
$\displaystyle=$
$\displaystyle\int_{1}^{\infty}\frac{ds}{s^{\alpha+1}}e^{-x\left(s-1\right)}=e^{x}x^{\alpha}\Gamma\left(-\alpha,x\right)\,.$
(122)

Here $\Psi\left(\alpha,\beta;z\right)$ and $\Gamma\left(\beta,z\right)$ denote
a Confluent Hypergeometric Function and an incomplete Gamma function,
respectively.

From the general expression for the vacuum-to-vacuum transition probability,
given by Eq. (37), the universal form of the vacuum-to-vacuum transition
probability for the present case reads:

$P_{\mathrm{v}}\approx\exp\left\\{-\frac{V_{\bot}T}{(2\pi)^{3}}\sum_{l=1}^{\infty}\left(-1\right)^{l-1}\int_{x_{\mathrm{L}}}^{x_{\mathrm{R}}}dx\frac{U^{\prime}\left(x\right)^{2}}{l^{2}}\exp\left[-\pi\frac{l\Delta^{2}}{v_{s}U^{\prime}\left(x\right)}\right]\right\\}.$
(123)

Representation (123) coincide with the vacuum-to-vacuum transition
probabilities obtained from the imaginary part of an effective action in the
locally constant field approximation [22, 23]. In this approximation, the
effective action $S$ is expanded about the constant field case, in terms of
derivatives of the background field strength $F_{\mu\nu}$,

$S=S^{\left(0\right)}[F_{\mu\nu}]+S^{\left(2\right)}[F_{\mu\nu},\partial_{\mu}F_{\nu\rho}]+...$
(124)

where $S^{\left(0\right)}$ involves no derivatives of the background field
strength $F_{\mu\nu}$ (that is, $S^{\left(0\right)}$ is a locally constant
field approximation for $S$ that has a form of the Heisenberg-Euler action),
while the first correction $S^{\left(2\right)}$ involves two derivatives of
the field strength, and so on; see Ref. [24] for a review. Using
representation (44), one can see that in the locally constant field
approximation the probability $P_{\mathrm{v}}$ is given by Eq. (44) where the
action $S$ is replaced by $S^{\left(0\right)}$ .

##  V Conclusion

In the present article we present a Fock space realization of the effective
field model describing low-energy dynamics of the magnons. Mapping the model
to the theory of a charged scalar field interacting with an external constant
electric field we apply recently developed approach of the strong-field QED
with step potentials to study the Schwinger effect of the magnon-antimagnon
pair production on magnetic field inhomogeneities. Initial and final one-
particle states are constructed from stationary plane waves satisfied the
Klein-Gordon equation. Initial and final vacua are defined and initial and
final states of the Fock space are constructed. Mean numbers of magnons and
antimagnons created from the vacuum are expressed via overlap amplitudes of
the stationary plane waves. Observable physical quantities specifying the
vacuum instability are determined. The fluxes of energy and magnetic moments
of created magnons are analyzed. Characteristics of the vacuum instability
obtained for the number of magnetic steps that allows exact solving the Klein-
Gordon equation are presented. In the case of a smooth-gradient step,
universal behavior of the flux density of created pairs is described and the
relation to the imaginary part of a one-loop effective action $S$ in a locally
constant field approximation is established. The results are quite general and
are not limited to the simple cubic-type lattice and the G-type
antiferromagnet. The presented study demonstrates a consistent application of
strong field QED methods in magnonics avoiding contradictions and non-existent
paradoxes in the interpretation of the obtained theoretical results. As a
result, one can now apply many of the results known from strong-field QED in
magnonics.

Condensed matter systems provide a possibility for experimental laboratory
study of quantum wacuum effects in strong fields, in particular, a laboratory
observation the Schwinger effect of the violation of the vacuum instability
due to particle-antiparticle pair production, previously considered possible
only in supercritical fields in astrophysical situations. This is due to the
fact that in many cases low energy dynamics of quasiparticle excitations in
condensed matter systems can be described by the Dirac model in which
quasiparticles are virtually massless. That is why external field intensities
needed for the vacuum violation are relatively small and can be observed in
the laboratory conditions. Usually the literature discusses the possibility to
observe effects related to the electron-hole creation from vacuum in the case
of Dirac and Weyl semimetals, such as the graphene and topological insulators,
where the low energy exitations are fermions; see, e.g., Refs. [25, 27, 28]
for the review. Since, the low-energy magnons are bosons with small effective
mass, then for the first time it becomes possible to observe the Schwinger
effect in the case of the Bose statistics, in particular, the bosonic Klein
effect in laboratory conditions. As was already mentioned above, in the case
of the Bose statistics appears a new mechanism for amplifying the effect of
the pair creation, which we call statistically-assisted Schwinger effect.

###### Acknowledgements.

The work is supported by Russian Science Foundation (Grant no. 19-12-00042).
Gitman is grateful to CNPq for continued support.

##  Appendix A Some details of scalar field quantization in the presence of
critical potential steps

The Heisenberg operator of the Klein-Gordon field $\hat{\Phi}\left(X\right)$
is assigned to the scalar field $\Phi\left(X\right).$ It is convenient to
consider the canonical pair of the field operator $\hat{\Phi}\left(X\right)$
and its canonical momentum $\hat{\Pi}\left(X\right)$ as a column
$\hat{\Psi}\left(X\right),$

$\hat{\Psi}\left(X\right)=\left(\begin{array}[]{c}i\hat{\Pi}^{{\dagger}}\left(X\right)\\\
\hat{\Phi}\left(X\right)\end{array}\right).$

The latter satisfies both the Klein-Gordon equation (4) given in the
Hamiltonian form,

$\displaystyle\left[i\partial_{0}-U\left(x\right)\right]\hat{\Psi}\left(X\right)=H^{\mathrm{kin}}\hat{\Psi}\left(X\right)\,,$
$\displaystyle
H^{\mathrm{kin}}=\left(\begin{array}[]{cc}0&-v_{s}^{2}\delta^{ij}\partial_{i}\partial_{j}+\Delta^{2}\\\
1&0\end{array}\right),$ (127)

and the equal time canonical commutation relations

$\displaystyle\left.\left[\hat{\Psi}\left(X\right),\hat{\Psi}\left(X^{\prime}\right)\right]_{-}\right|_{t=t^{\prime}}=0,$
$\displaystyle\left.\left[\hat{\Psi}\left(X\right),\hat{\Psi}^{{\dagger}}\left(X^{\prime}\right)\right]_{-}\right|_{t=t^{\prime}}=\delta\left(\mathbf{r-r}^{\prime}\right)\sigma_{1}\
,$ (128)

where $H^{\mathrm{kin}}$ is the one-particle kinetic energy operator, see e.g.
[29, 30]. It follows from Eq. (127) that

$i\hat{\Pi}^{{\dagger}}\left(X\right)=\left[i\partial_{0}-U\left(x\right)\right]\hat{\Phi}\left(X\right),$

The Hamiltonian $\widehat{\mathbb{H}}$ of the quantized scalar field has the
form:

$\displaystyle\widehat{\mathbb{H}}=\widehat{\mathbb{H}}^{\mathrm{kin}}-\frac{1}{v_{s}^{2}}\int\hat{\Psi}^{\dagger}\left(X\right)\sigma_{1}U\left(x\right)\hat{\Psi}\left(X\right)d\mathbf{r},$
$\displaystyle\widehat{\mathbb{H}}^{\mathrm{kin}}=\int\hat{T}^{00}d\mathbf{r}-\mathbb{H}_{0}\mathbf{,\;}$
$\displaystyle\hat{T}^{00}=\frac{1}{v_{s}^{2}}\hat{\Psi}^{\dagger}\left(X\right)\sigma_{1}\left[i\partial_{0}-U\left(x\right)\right]\hat{\Psi}\left(X\right)\mathbf{\
,}$ (129)

where $\widehat{\mathbb{H}}^{\mathrm{kin}}$ is a kinetic energy operator,
which, just from the beginning, we write in a renormalized form with a
constant (in general, infinite) term $\mathbb{H}_{0}$ that corresponds to the
kinetic energy of vacuum fluctuations. In the case under consideration, in the
similar manner as discussing the inner product (22), it is possible to
evaluate integrals (129) for arbitrary field $U\left(x\right)$, using only the
asymptotic behavior (8) of functions in the regions $S_{\mathrm{L}}$and
$S_{\mathrm{R}}$ where particles are free. Decomposing the field
$\hat{\Psi}\left(X\right)$ over the complete set (5), and dividing integral
(129) in three integrals within the regions $S_{\mathrm{L}}$,
$S_{\mathrm{int}}$, and $S_{\mathrm{R}}$, we reduce calculating the quantity
(129) to calculating its one-particle matrix elements in the regions
$S_{\mathrm{L}}$and $S_{\mathrm{R}}$; see Sect. IVB.1 and Appendix B in Ref.
[14] for details. One can see that the matrix elements of the Hamiltonian
$\widehat{\mathbb{H}}$ depends on the total energy of a particle
$\varepsilon$. However, probabilities of the magnon scattering, reflection,
and the magnon-antimagnon pair production depend on the kinetic energy terms
$\pi_{0}\left(\mathrm{L/R}\right)=\varepsilon-U_{\mathrm{L/R}}$, which is a
combination of $\varepsilon$ and $U_{\mathrm{L/R}}$. This is due to the fact
that not the field $U\left(x\right)$ itself, acting on the magnons, does the
work, but its derivative $\partial_{x}U\left(x\right)$. That is why in the
field-theoretical description of magnons embedded into an external
inhomogeneous magnetic field namely eigenvalues of the operator
$\widehat{\mathbb{H}}^{\mathrm{kin}}$ define vacuum states and other state
vectors in the Fock space 555In QED, where $U_{\mathrm{L/R}}$ are potentials
of an electromagnetic field, the operator $\hat{\mathbb{H}}^{\mathrm{kin}}$ is
gauge invariant, which implies that it is an observable physical quantity in
contrast with the Hamiltonian $\hat{\mathbb{H}}$. In the case under
consideration the constant values $U_{\mathrm{L/R}}$ are physical quantities
and are used to tune the collinear ground state within the regions
$S_{\mathrm{L}}$ and $S_{\mathrm{R}}$.

The formal expression of the effective charge (magnetic moment) operator
$\hat{Q}$ is

$\hat{Q}=\int\hat{\rho}d\mathbf{r,\;\hat{\rho}=}\frac{\mu}{2v_{s}^{2}}\left[\hat{\Psi}^{\dagger}\left(X\right)\sigma_{1},\hat{\Psi}\left(X\right)\right]_{+},$
(130)

where $\left[A,B\right]_{+}=AB+BA$ stands for the anticommutator. The
eigenvalues of the operator $\widehat{\mathbb{H}}^{\mathrm{kin}}$, together
with one-particle mean values of the effective charge, allow one to
distinguish particles and antiparticles.

Before proceeding to the definition of the operators of interest (operators of
fluxes), which usually are not considered in QFT, it is useful to note that
even in the framework of the corresponding classical field theory an
observable $\mathcal{F}$ can be realized as an inner product of type (22) of
localizable wave packets $\Phi\left(X\right)$ and $\hat{F}\Phi^{\prime}(X)$,

$\mathcal{F}\left(\Phi,\Phi^{\prime}\right)=\left(\Phi,\mathcal{\hat{F}}\Phi^{\prime}\right),$
(131)

where $\mathcal{\hat{F}}$ is a differential operator, whereas $\Phi(X)$ and
$\Phi^{\prime}(X)$ are solutions of the Klein-Gordon equation. Assuming that
an observable $\mathcal{F}\left(\Phi,\Phi^{\prime}\right)$ is time-independent
during the time $T$ one can represent it in the form of an average over the
period $T$,

$\left\langle\mathcal{F}\right\rangle=\frac{1}{T}\int_{-T/2}^{+T/2}\mathcal{F}\left(\phi,\phi^{\prime}\right)dt.$
(132)

In general the wave packets $\Phi(X)$ and $\Phi^{\prime}(X)$ can be decomposed
into plane waves $\phi_{m}(X)$ and $\phi_{m}^{\prime}(X)$,

$\Phi(X)=\sum_{m}\alpha_{m}\phi_{m}(X),\;\Phi^{\prime}(X)=\sum_{m}\alpha_{m}^{\prime}\phi_{m}^{\prime}(X),$

where $\phi_{m}(X)$ and $\phi_{m}^{\prime}(X)$ are superpositions of the
solutions${}_{\;\zeta}\phi_{m}\left(X\right)$ and ${}^{\;\zeta}\phi_{m}(X)$.
Taking into account the orthogonality relation (13), one finds that the
corresponding decomposition of $\left\langle\mathcal{F}\right\rangle$ does not
contain interference terms,

$\left\langle\mathcal{F}\right\rangle=\sum_{m}\mathcal{F}\left(\alpha_{m}\phi_{m},\alpha_{m}^{\prime}\phi_{m}^{\prime}\right).$

A physical quantity useful for the further analysis is the average over the
period $T$ of the effective charge current through the surface
$x=\mathrm{const}$. Since the regions $S_{\mathrm{L}}$and $S_{\mathrm{R}}$
supposed to be macroscopic and the particles that come there are free, then
such a semiclassical statement of the problem seems to be justified. Moreover,
this is how the problem statement is formulated in the theory of the potential
scattering. The corresponding QFT operator of the effective charge (magnetic
moment) current is proportional to the inner product on this surface given by
Eq. (10),

$\displaystyle\widehat{\mathbb{J}}=\frac{1}{T}\int\hat{J}_{x}dtd\mathbf{r}_{\bot},$
$\displaystyle\hat{J}_{x}=\frac{\mu
i}{2}\left\\{\left[\hat{\Phi}\left(X\right),\partial_{x}\hat{\Phi}^{{\dagger}}\left(X\right)\right]_{+}-\left[\hat{\Phi}^{{\dagger}}\left(X\right),\partial_{x}\hat{\Phi}\left(X\right)\right]_{+}\right\\}$
(133)

Here $\hat{J}_{x}$ is the longitudinal component of the operator of the
effective current density $\mathbf{\hat{J}}$. Note that by virtue of Eq. (4)
the latter operator and the operator of the effective charge density
$\hat{\rho}$ satisfy the continuity equation
$\nabla\mathbf{\hat{J}}+\partial_{t}\hat{\rho}=0$.

In what follows, we consider the energy flux of the scalar field through the
surface $x=\mathrm{const}$. Its QFT operator has the form

$\displaystyle\widehat{\mathbb{F}}\left(x\right)=\frac{1}{T}\int
v_{s}\hat{T}^{10}dtd\mathbf{r}_{\bot}\ ,$
$\displaystyle\hat{T}^{10}\left(x\right)=-\left[\partial_{x}\hat{\Phi}^{{\dagger}}\left(X\right)\right]\hat{\Pi}^{{\dagger}}\left(X\right)-\hat{\Pi}\left(X\right)\partial_{x}\hat{\Phi}\left(X\right),$
(134)

where $\hat{T}^{10}$ is the component of the operator of the energy momentum
tensor; see Refs. [14, 15] for more details.

One can decompose the operator $\hat{\Phi}\left(X\right)$ into solutions of
the either initial or final complete sets (31) to construct in- and out-states
in an adequate Fock space:

$\displaystyle\hat{\Phi}\left(X\right)$ $\displaystyle=$
$\displaystyle\sum_{m}\mathcal{M}_{m}^{-1/2}\left[A_{m}(\mathrm{in})\phi_{m}^{\left(\mathrm{in},+\right)}\left(X\right)\right.$
(135) $\displaystyle+$
$\displaystyle\left.B_{m}^{\dagger}(\mathrm{in})\phi_{m}^{\left(\mathrm{in},-\right)}\left(X\right)\right]$
$\displaystyle=$
$\displaystyle\sum_{m}\mathcal{M}_{m}^{-1/2}\left[A_{m}(\mathrm{out})\phi_{m}^{\left(\mathrm{out},+\right)}\left(X\right)\right.$
$\displaystyle+$ $\displaystyle\left.B_{m}^{\dagger}(\mathrm{out})\
\phi_{m}^{\left(\mathrm{out},-\right)}\left(X\right)\right]$

The operator-valued coefficients can be determined with the help of the inner
product (22),

$\displaystyle
A_{m}(\mathrm{in})=\frac{\mathcal{M}_{m}^{1/2}\left(\phi_{m}^{\left(\mathrm{in},+\right)},\hat{\Phi}\right)}{\left(\phi_{m}^{\left(\mathrm{in},+\right)},\phi_{m}^{\left(\mathrm{in},+\right)}\right)},$
$\displaystyle