aainstitutetext: Niels Bohr International Academy, Niels Bohr Institute,
Blegdamsvej 17, Copenhagen, DK 2100, Denmark
bbinstitutetext: Nordita, KTH Royal Institute of Technology and Stockholm
University,
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
ccinstitutetext: Instituut-Lorentz for Theoretical Physics, Leiden University,
2333 CA Leiden, The Netherlands
ddinstitutetext: Kavli Institute for the Physics and Mathematics of the
Universe (WPI), UTIAS,
The University of Tokyo, Chiba 277-8583, Japan

# Backreaction of Schwinger pair creation in massive QED2

Gregory Gold b David A. McGady a, c Subodh P. Patil d and Valeri Vardanyan
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Particle-antiparticle pairs can be produced by background electric fields via
the Schwinger mechanism provided they are unconfined. If, as in QED in
(3+1)-$d$ these particles are massive, the particle production rate is
exponentially suppressed below a threshold field strength. Above this
threshold, the energy for pair creation must come from the electric field
itself which ought to eventually relax to the threshold strength. Calculating
this relaxation in a self-consistent manner, however, is difficult. Chu and
Vachaspati addressed this problem in the context of capacitor discharge in
massless QED2 Chu:2010xc  by utilizing bosonization in two-dimensions. When
the bare fermions are massless, the dual bosonized theory is free and
capacitor discharge can be analyzed exactly Chu:2010xc , however, special care
is required in its interpretation given that the theory exhibits confinement.
In this paper we reinterpret the findings of Chu:2010xc , where the capacitors
Schwinger-discharge via electrically neutral dipolar meson-production, and
generalize this to the case where the fermions have bare masses. Crucially, we
note that when the initial charge of the capacitor is large compared to the
charge of the fermions, $Q\gg e$, the classical equation of motion for the
bosonized model accurately characterizes the dynamics of discharge. For
massless QED2, we find that the discharge is suppressed below a critical plate
separation that is commensurate with the length scale associated with the
meson dipole moment. For massive QED2, we find in addition, a mass threshold
familiar from (3+1)-$d$, and show the electric field relaxes to a final steady
state with a magnitude proportional to the initial charge. We discuss the
wider implications of our findings and identify challenges in extending this
treatment to higher dimensions.

##  1 Preliminary remarks

The most remarkable thing about the vacuum is that it conducts whenever the
applied potential difference across a given region exceeds a critical value –
an effect first identified in Sauter:1931zz ; Heisenberg:1935qt  that now
bears the name of Schwinger Schwinger:1951nm . Schwinger gave a complete
characterization of how positrons and electrons can be pair produced once the
applied electric field in a given region exceeds the critical strength

$E_{c}=\frac{m_{e}^{2}c^{3}}{e\hbar}\approx 10^{18}\rm{V/m}.$ (1)

This is also the field strength above which quantum effects render
electromagnetism to be non-linear. This raises the immediate question, where
does the energy required to produce these particle-antiparticle pairs come
from? Clearly, it must be drawn from the electric field itself, which must
subsequently decay until it reaches the threshold strength Eq. (1) when pair
production becomes exponentially suppressed again. However, quantifying how
this occurs is a challenging problem to say the least.

Schwinger’s original calculation relied on background field quantization,
wherein the electric field is taken to be an external field and the 1PI
effective action is evaluated in the proper time formalism by integrating out
the fermion fields. On a background with a constant electric field, this
results in an imaginary contribution to the effective action (see for instance
Dunne:1998ni )

${\rm
Im\,S}=\frac{(eE)^{2}}{8\pi^{3}}V\sum_{n=1}\frac{1}{n^{2}}e^{-\frac{m^{2}\pi
n}{eE}},$ (2)

where we work in natural units, and where $V=L^{3}T$ is the volume of
spacetime.

In this context, the imaginary part of the effective action corresponds to a
breakdown of the vacuum state which contains neither electrons nor positrons.
Put differently, the imaginary component of the one-loop effective action
corresponds to a breakdown of the fermionic vacuum state via electron-positron
pair creation. Nikishov Nikishov:1970br  and others, e.g. Cohen:2008wz , have
emphasized that the actual rate of pair creation per unit volume of spacetime
is just (twice) the first term in this infinite sum:

$\Gamma=\frac{dN}{dtdV}=\frac{(eE)^{2}}{4\pi^{3}}e^{-\frac{\pi
m^{2}}{eE}}\leavevmode\nobreak\ .$ (3)

These computations are done in the strict semiclassical limit using background
field quantization, where the charged particles produced by the externally
applied electromagnetic fields have negligible effect on the external fields
themselves. In other words, these computations explicitly ignore the dynamics
of backreaction.

However, it is very important to understand how this backreaction process
works in detail, both in the particular context of Schwinger-dissipation and
in the much more general context of quantum back-reactions induced by
classical field configurations dissipating the initial classical solutions
themselves Fradkin:1991zq . In this context, the Schwinger mechanism, where an
initial external electric field causes back-reaction against itself, is a
comparatively simpler process than analogous back-reaction processes in
(quantum) gravity such as black-hole evaporation. Further, as Schwinger pair
creation may actually be observed in laboratory settings such as single-layer
graphene Allor:2007ei , it is important to have a realistic, concrete,
theoretical understanding for how such systems will evolve in time.

There are a number of ways one could imagine calculating the backreaction on
the electric field of the produced particles. The brute force method would be
to interpret Eq. (3) as the energy extracted per unit spacetime volume from a
spatially uniform electric field and make an ansatz for a new time dependent
but spatially uniform electric field, which would also source a time dependent
magnetic field, the sum of whose energy densities are required to dissipate
according to Eq. (3). The vector potential one calculates from this ansatz
would be the new background which one would background field quantize around
to calculate the imaginary part of the effective action. One then repeats this
process iteratively in the hope that this procedure converges on a self
consistent quantum corrected background – a process that can quickly become
cumbersome to the point of being impractical.

Evidently, the problem of completely characterizing the dissipation of the
background electric field from particles produced via the Schwinger effect
lies within the ambit of non-equilibrium techniques, however the precise
manner in which one should proceed is unclear.111For instance, 2PI methods and
their possible extensions (see Berges:2004yj  for an excellent review) are
problematic from the perspective of maintaining gauge invariance in QED
Mottola:2003vx . In spite of this, some notable attempts have been made using
a variety of approximations. In Akhmedov:2009vs , the authors made a
truncation of the recursive procedure discussed above in the context of the
Euler-Heisenberg effective action for scalar QED. Notable also is the mean
field approach in the context of scalar QED of Habib:1995ee . Earlier, the
authors of Cooper:1989kf ; Kluger:1991ib ; Kluger:1992gb  adopted a semi-
classical approach that replaced currents with their expectation values and
attempted a numerical study of the backreaction problem; here particle
production and back-reaction is studied with the use of Boltzmann-Vlasov
equations (see also Kluger:1998bm ). The effects of time varying external
fields in different spacetime dimensions was studied in Gavrilov:1996pz , and
the study of secular growth of loop corrections in the proper time formalism
was studied and carefully interpreted in Akhmedov:2014hfa ; Akhmedov:2014doa .
However a fully self-consistent background at the quantum level remains as of
yet elusive. In a remarkable insight, Chu and Vachaspati Chu:2010xc  realized
the latter could in fact be addressed by considering the problem of vacuum
capacitor discharge, albeit in the context of 1+1-dimensions in a theory with
massless fermions – massless QED2– by considering its bosonized version, an
approach we first review before reinterpreting and extending to the case of
massive bare fermions.

The first obstacle one must address in the interpretation of Chu:2010xc  is
that in 1+1-dimensions, both massless and massive QED2 exhibit confinement,
and so any interpretation of the electric field discharge in terms of currents
as put forward in Chu:2010xc  must be treated with care. We find by
considering the duality between the massless Schwinger model Schwinger:1962tp
and the free (gapped) dual scalar field theory, the electric field discharge
found in Chu:2010xc  should be interpreted in terms of meson production,
whereby bound particle-antiparticle pairs are produced whenever the plate
separation surpasses the threshold length scale defined by the electric dipole
moment of the produced mesons. Capacitor discharge then proceeds via dipole
screening by the induced meson cloud. In this way, we can qualify one of the
caveats usually applied in applying lessons learned from studying Schwinger
pair production towards Hawking evaporation or the semi-classical stability of
de Sitter space dS1 ; Anderson:2017hts  – namely that such interpretations are
limited by the fact that there are no negative gravitational charges (i.e.
masses). Here, we show in the context of a toy model that electric field
discharge can proceed even with the production of neutral
particles.222However, this qualification is immediately tempered by the fact
that gravitational dipoles do not exist either.

The structure of this paper is as follows. In section 2, we first describe the
mapping between massless QED2 and the integrable bosonized theory and then
revisit the analysis of Chu:2010xc  applied to the problem of electric field
discharge within a capacitor. Importantly, we identify additional
characteristic timescales to that noted in Chu:2010xc . In section 3, we
physically interpret this discharge. Our analysis tracks the content of
Coleman:1975pw  and Coleman:1976uz  closely by focusing on the properties of
the physical excitations of the system, i.e. the massive and electrically
neutral $\varphi$-mesons which carry electric dipole moments, identifying a
threshold plate separation below which discharge does not occur. We further
identify additional timescales associated with dynamical capacitor discharge
in massless QED2. Then, in section 4, we explicitly show how bare fermion
masses change the relevant equations of motion. Crucially, adding a fermion-
mass induces a non-zero cosine-potential within the dual bosonic theory. In
section 5, we study the equation of motion for the bosonic configuration dual
to capacitor-discharge in massive QED2. When the charge of the capacitor plate
is large compared to the charge of the massive fermion in QED2, i.e. when
$Q\gg e$, occupation numbers are large and we can self-consistently integrate
the classical non-linear equations of motion on the bosonic side, recovering a
mass threshold at the critical field
$E_{c}=\frac{e^{\gamma_{E}}m_{\psi}}{2\sqrt{\pi}}$, where $\gamma_{E}$ is the
Euler gamma constant and $m_{\psi}$ is the bare fermion mass in the notation
of section 4. As we discuss in section 5, the bare mass $m_{\psi}$ is not to
be taken as the mass of any physical state in the massive Schwinger model,
rather it is to be interpreted as a parameter that determines the spectrum of
states in the theory. We conclude by comparing the final discharge profile in
massive and massless QED2, highlighting the role of the two independent
thresholds as we dial the plate separation and initial charge on the
capacitors.

##  2 Integrability of capacitor discharge in massless QED2

In this section, we review the treatment of Chu:2010xc  which analytically
computed capacitor discharge in massless QED2. Our purpose is twofold. First,
so that we can revisit the physical interpretation of Ref. Chu:2010xc ’s
results for capacitor discharge, in light of the mass-gap due to electron
confinement within the Schwinger model Schwinger:1962tp ; Coleman:1975pw .
Second, to lay the groundwork for our study of capacitor discharge in
1+1-dimensions with massive charged fermions in sections 4 and 5.

To begin we note that the insight of Chu:2010xc  was to point out that the
action for massless QED in 1+1-dimensions,

$\displaystyle S_{\psi,A}:=\int
d^{2}x\bigg{(}\bar{\psi}(i\not{D}-e\not{A})\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\bigg{)}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (4)

and the action tying a putatively massless scalar to the same gauge-field and
its corresponding field strength, $A_{\mu}$ and
$F_{\mu\nu}:=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$,

$\displaystyle S_{\varphi,A}:=\int
d^{2}x\bigg{(}(\partial^{\mu}\varphi)(\partial_{\mu}\varphi)+g\varphi\big{(}\epsilon^{\mu\nu}F_{\mu\nu}\big{)}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\bigg{)}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (5)

where $g:=\frac{e}{\sqrt{\pi}}$, describe the same theory at the quantum
mechanical level via the following mapping between normal-ordered (:$AB$:)
operators in $S_{\varphi,A}\leftrightarrow S_{\psi,A}$:

$\displaystyle:\bar{\psi}i\gamma^{\mu}\psi:\leftrightarrow\epsilon^{\mu\nu}\partial_{\nu}\varphi\leavevmode\nobreak\
\implies S_{\psi,A}\leftrightarrow S_{\varphi,A}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (6)

This operator correspondence, an important ingredient in the duality between
the massive-Thirring model and the sine-Gordon model Coleman:1974bu , is the
principle ingredient that allowed Ref. Chu:2010xc  to analytically solve for
the dynamics of capacitor discharge in massless QED in 1+1-dimensions.

Fundamentally, this operator map allows the trilinear coupling in ${\rm
QED}_{2}$ to be converted into a quadratic coupling in the dual theory. Up to
topological boundary terms, we have
:$\bar{\psi}\not{A}\psi$:$\leftrightarrow\\!\\!\varphi\epsilon^{\mu\nu}F_{\mu\nu}$.
This, in turn, converts equations of motion that are quadratic in $\psi$ into
linear equations of motion for $\varphi$. We then consider the $\varphi$-field
profile that corresponds to an initial state corresponding to a parallel-plate
capacitor in QED. Integrating the equations of motion for $\varphi$ Chu:2010xc
gives closed form expressions for the charge density, i.e.
$:\bar{\psi}\gamma^{\mu}\psi:$, and thus yields an exact description of
capacitor discharge.

However, one must be mindful of a number of conceptual points in the mapping
between the actions $S_{\psi,A}$ and $S_{\varphi,A}$, relevant to both
massless QED2 Chu:2010xc  and our extension to massive QED2. Primarily, as
Schwinger showed in 1962 Schwinger:1962tp , QED in 1+1-dimensions with
massless fermions is known to have a gapped spectrum. Further, there is no
“photon”, that is, electromagnetic fields are non-dynamical in 1+1-dimensions.
Thus any putatively dynamical gauge-field $A^{\mu}$ may be integrated out of
any on-shell description of the gapped system, whose Hamiltonian is
Coleman:1975pw :

$\displaystyle H_{\varphi,D}$
$\displaystyle=\frac{1}{2}\\!:\\!\\!\bigg{(}\Pi_{\varphi}^{2}+(\partial_{x}\varphi)^{2}+g^{2}\left(\varphi+\frac{D_{\rm
ext}}{g}\right)^{2}\bigg{)}\\!\\!:\leavevmode\nobreak\ \leavevmode\nobreak\ ,$
(7)

where again $g^{2}=e^{2}/\pi$, $D_{\rm ext}$ is the scalar potential that
sources the external fields $F^{\mu\nu}_{\rm ext}=\epsilon^{\mu\nu}D_{\rm
ext}$, $\Pi_{\varphi}$ denotes the momentum canonically conjugate to the
$\varphi$-field, and :$(A...Z)$: denotes the normal ordered operators $A...Z$.
As is discussed in detail in Coleman:1975pw , the on-shell excitations of the
system, i.e. single-particle excitations of the $\varphi$-field, are massive
and correspond to neutral mesons, i.e. massive bound states of massless
charged fermionic fields $\varphi\sim\psi^{+}\psi^{-}$.

Following Chu:2010xc , we consider a configuration of external charges $\pm Q$
a distance of $L$ apart. The corresponding field strength tensor and the
current are given by

$\displaystyle F^{\mu\nu}_{\rm ext}(x,t)\coloneqq\epsilon^{\mu\nu}D_{\rm
ext}(x,t),$ $\displaystyle j^{\mu}_{\rm
ext}\coloneqq\partial_{\nu}F^{\mu\nu}_{\rm
ext}=Qu^{\mu}\left(\delta\left(x+\tfrac{L}{2}\right)-\delta\left(x-\tfrac{L}{2}\right)\right)\leavevmode\nobreak\
\\!.$ (8)

where $D_{\rm ext}(x,t):=Q\left[\theta(x+L/2)-\theta(x-L/2)\right]$ with
$\theta(x)$ is the usual step-function,
$\theta(x)\coloneqq\int^{x}_{-\infty}dy\,\delta(y)$, and
$u^{\mu}\coloneqq(1,0)$. The equations of motion (EOMs) derived from
$S_{\varphi,A}$ for $\varphi$ and $A_{\mu}$ (which is non-dynamical in
$(1+1)$-dimensions) are given by

$\displaystyle{\rm EOMs}:\leavevmode\nobreak\ \leavevmode\nobreak\
\begin{cases}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\partial^{2}\varphi&\\!\\!\\!\\!=g\epsilon^{\mu\nu}\leavevmode\nobreak\
F_{\mu\nu}\,,\\\
\partial_{\mu}F^{\mu\nu}&\\!\\!\\!\\!=g\epsilon^{\mu\nu}\leavevmode\nobreak\
\partial_{\nu}\varphi+j^{\nu}_{\rm ext}\leavevmode\nobreak\
\leavevmode\nobreak\ .\end{cases}$ (9)

Crucially, these equations of motion are linear in the fields. We can thus
straightforwardly integrate the non-dynamical $A^{\mu}$ field out of the
equations of motion, which effectively reparametrizes the energy density
stored in the electric field in terms of the value of the $\varphi$-field at a
point. In terms of this $\varphi$-scalar field, the full space-time dependence
of the E-field is given by Chu:2010xc

$F^{\mu\nu}=\epsilon^{\mu\nu}(g\varphi+F)+F_{\rm
ext}^{\mu\nu}=\epsilon^{\mu\nu}\big{(}g\varphi+D_{\rm
ext}+F\big{)}\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (10)

where we have re-expressed the electromagnetic field-strength in terms of a
single scalar potential, $D_{\rm ext}$, and $F$ is an additive integration-
constant acquired when one moves from the EOM in Eq. (9) to the field-
strength, and is further directly proportional to the $\theta$-term seen in
the massive Schwinger model Coleman:1976uz : $\theta:=2\pi F/e$.333Explicitly,
this $F$ (or, equivalently, $\theta$)-offset in the relationship between the
electromagnetic field-strength and the $\varphi$-scalar meson field is:
$\displaystyle F^{\mu\nu}=\epsilon^{\mu\nu}\big{(}g\varphi+D_{\rm
ext}+F\big{)}=\epsilon^{\mu\nu}\bigg{(}g\bigg{(}\varphi+\frac{\theta}{2\sqrt{\pi}}\bigg{)}+D_{\rm
ext}\bigg{)}\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (11) For the massless
Schwinger model, we may re-define $\varphi\to\varphi-\theta/(2\sqrt{\pi})$ to
eliminate this $\theta$-term. However, as we shall see in sections 4 and 5,
because finite fermion masses introduce $\cos(2\sqrt{\pi}\varphi)$-terms in
the Hamiltonian, this shift of $\varphi\to\varphi-\theta/(2\sqrt{\pi})$
induces a $\theta$-dependence into the cosine-potential in Eq. (55)
Coleman:1975pw ; Coleman:1976uz , via
$\cos(2\sqrt{\pi}\varphi)\to\cos(2\sqrt{\pi}\varphi+\theta)$.  Rewriting
$F^{\mu\nu}_{\rm ext}$ in this way will, later, allow us to explicitly map
between the Hamiltonian-based and the action-based frameworks in,
respectively, Refs. Chu:2010xc  and Coleman:1975pw .

Returning to capacitor discharge, we see that inserting $F_{01}=D_{\rm
ext}+g\varphi$ back into the equation of motion for $\varphi$, we recover the
well-known fact that the coupling between the non-dynamical electromagnetic
field and the dynamical scalar field $\varphi$, which does not have a mass-
term within the action $S_{\varphi,A}$, will source a mass-term for on-shell
excitations of this scalar field. We therefore find

$\displaystyle(\partial^{2}+m_{\varphi}^{2})\varphi=-gD_{\rm
ext}\leavevmode\nobreak\ \leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\
m_{\varphi}^{2}:=g^{2}=\frac{e^{2}}{\pi}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (12)

In this way, we see explicitly that the bosonization map
$:\bar{\psi}\gamma^{\mu}\psi:\leftrightarrow\partial^{\mu}\varphi$, which we
used to naively bosonize the action for massless QED in 1$+1$ \- dimensions
reproduces the well-known gap in the Schwinger model Schwinger:1962tp ;
Coleman:1975pw .

Because the equations of motion are linear in the $\varphi$-basis, Ref.
Chu:2010xc  exploited linearity to solve for the dynamics of the scalar field
profile due to the presence of a single charge $Q$ located at a definite
position on the $x$-axis. The static solution $\varphi(x)$ due to this single
point-charge,

$\displaystyle(\partial^{2}+m_{\varphi}^{2})\varphi=(gQ)\leavevmode\nobreak\
{\rm sign}(x)\leavevmode\nobreak\ ,$ (13)

is, by using retarded Greens functions, found to simply be

$\displaystyle\varphi_{I}(x)$
$\displaystyle=-gQ\int\frac{dk}{2\pi}\frac{\sin(kx)}{k(k^{2}+m_{\varphi}^{2})}$
$\displaystyle=-\frac{1}{2}\frac{gQ}{m_{\varphi}^{2}}\leavevmode\nobreak\ {\rm
sign}(x)\leavevmode\nobreak\
\bigg{(}1-e^{-m_{\varphi}|x|}\bigg{)}\leavevmode\nobreak\ .$ (14)

Note that the energy of this configuration diverges linearly with the system
size $\lambda$, i.e. when $|x|\leq\lambda$ and $\lambda\gg L$,
$\int_{\lambda}dxH_{\varphi_{I},D}(x)\sim\lambda Q^{2}g^{4}/m_{\varphi}^{4}$.
This is another reflection of the fact that electric charges confine in
(1+1)-dimensions, due to the linear growth of the scalar potential from a
point-charge.

Linearity of the fields themselves, and of the equations of motion, gives the
total $\varphi$ field-profile from the two initial capacitors with $\pm Q$ at
$x=\pm L/2$ in terms of a difference of two such profiles:

$\displaystyle\varphi_{s}(x)$
$\displaystyle=\varphi_{I}\left(x-\tfrac{L}{2}\right)-\varphi_{I}\left(x+\tfrac{L}{2}\right)$
$\displaystyle=-\frac{gQ}{m_{\varphi}^{2}}\times\begin{cases}\qquad\leavevmode\nobreak\
\\!e^{-m_{\varphi}|x|}\sinh(m_{\varphi}L/2)\\!\\!\\!&;\leavevmode\nobreak\
|x|>\tfrac{L}{2}\leavevmode\nobreak\ ,\\\
(1-e^{-m_{\varphi}|L|/2}\cosh(m_{\varphi}x))\\!\\!\\!&;\leavevmode\nobreak\
|x|<\tfrac{L}{2}\leavevmode\nobreak\ .\end{cases}$ (15)

Note that these equal and opposite charges render the previously divergent
energy of each individual point charge, now, finite. The total energy is sub-
extensive in their separation.

The complete solution for the classical $\varphi$-field profile that
corresponds to the end-point of the capacitor’s discharge, though, comes from
imposing the initial-value condition that the only charged fermions at $t=0$
comes from these two piles of charge $+Q$ and $-Q$, separated by a distance of
$L$. Crucially, the initial conditions imposed by the starting capacitor
configuration are:

$\displaystyle\varphi(x,t)|_{t=0}=\partial_{t}\varphi(x,t)|_{t=0}=0\leavevmode\nobreak\
.$ (16)

At this point, we pause to note that the above initial-value condition is
_two_ -fold. I.e., the full solution to the EOM, $\varphi(x,t)$ must have both
a vanishing profile at $t=0$ _and_ a vanishing time-derivative at $t=0$.
Crucially, as the static solution in Eq. (15) is generically non-zero, it does
not satisfy both conditions laid-out in Eq. (16). Precisely because of this,
we see that the system must dynamically evolve from a more complex
$\varphi$-profile before finally relaxing to the late-stage, static profile in
Eq. (15).

It is straightforward to see that the total solution consistent with the
initial conditions (16) and the static final-state field configuration
$\varphi_{s}(x)$ in (15) is simply,

$\displaystyle\varphi(x,t)=-2gQ\int\frac{dk}{2\pi}\frac{\cos(kx)\sin(kL/2)}{k(k^{2}+m_{\varphi}^{2})}\bigg{(}1-\cos\left(t\sqrt{k^{2}+m_{\varphi}^{2}}\leavevmode\nobreak\
\right)\bigg{)}\leavevmode\nobreak\ .$ (17)

These integrals are closely related to Bessel-J functions. For instance, as
pointed-out in Chu:2010xc , we have

$\displaystyle\partial_{x}\partial_{t}\varphi(x,t)$
$\displaystyle=gQ\bigg{[}J_{0}\left(m_{\varphi}\sqrt{t^{2}-x_{-}^{2}}\leavevmode\nobreak\
\right)\theta(t-x_{-})-J_{0}\left(m_{\varphi}\sqrt{t^{2}-x_{+}^{2}}\leavevmode\nobreak\
\right)\theta(t-x_{+})\bigg{]}\leavevmode\nobreak\ ,$ (18)

where $x_{\pm}:=x\pm L/2$, and the two terms correspond to the dynamical
charges due to the left/right capacitor plates. We can find the late-time
behavior of the electromagnetic decay-response prompted by the initial,
unstable, charged capacitor conditions by manipulating these exact
expressions. Fig. 1 demonstrates the capacitor-discharge as a function of time
$\hat{t}\equiv t/m_{\varphi}$ at $x=0$ in massless $\mathrm{QED}_{2}$ for
plate separations $\hat{L}\equiv L/m_{\varphi}=\\{2,10,20\\}$.

Figure 1: Capacitor-discharge in massless QED2 via dipole discharge at plate
separations $\hat{L}\equiv L/m_{\varphi}=\\{2,10,20\\}$, at $x=0$. The initial
electric field discharges to the final field configuration (solid black line)
undergoing oscillations along the way. Evidently, as one approaches the
threshold separation at $L=m^{-1}_{\varphi}$, the final electric field never
completely discharges within the capacitor.

Immediately evident from both these figures and/or the related content in Eq.
(15), is the appearance of a threshold plate separation below which discharge
becomes inefficient. As we shall elaborate upon in the next section, this is
consistent with our proposed physical interpretation of this process as being
due to dipole discharge.

##  3 Physical interpretation of Schwinger production in (1+1)-$d$

The goal of this investigation is to extend the analysis of Chu:2010xc  to the
case of massive QED2. Before we do so, it is important to arrive at the
correct physical interpretation of the results of Chu:2010xc  reviewed in the
previous section. We do this by clarifying the following four points: Firstly,
we show that the Hamiltonian and Lagrangian formulations of Refs. Chu:2010xc
and Coleman:1975pw  match. This is slightly nontrivial, but of course had to
happen.

Secondly, we explicitly discuss how the two charges $\pm Q$, separated by a
distance $L$, discharge via production of clouds of $\varphi$-mesons. As we
are in $d=2$, electric charges confine. Thus the produced degrees of freedom
cannot themselves be charged. However, they can and do have electric dipole
moments. Thus the only way to discharge the initial capacitor is by producing
clouds of polarized $\varphi$ mesons. Producing such clouds requires clear
threshold effects, somewhat different to the $e^{-\pi m^{2}/(qE)}$ threshold
that appears in Schwinger’s classic result. Indeed, there are three
fundamental length-scales in this system, given by the separation between the
capacitor plates $L$, the length-scale of the electric field, and the Compton
wavelength set by the physical degrees of freedom of this system, i.e. the
massive neutral $\varphi$-mesons, $\ell_{\varphi}$. This third length-scale,
which is ultimately due to the fact that QED is a relevant interaction in
$d<4$, is given in terms of the charge of the fundamental fermions of QED2:

$\displaystyle m_{\varphi}=\frac{e}{\sqrt{\pi}}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\ell_{\varphi}=\frac{1}{m_{\varphi}}=\frac{\sqrt{\pi}}{e}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (19)

This characteristic length-scale gives an interesting wrinkle on how one may
envision the initial conditions of highly-charged and well-separated capacitor
plates—and for their subsequent (exact) time-evolution and eventual discharge.

Thirdly, we note that these scales allow us to study capacitor discharge
within a self consistent approximation in various limits. There are two
independent dimensionless ratios of these length-scales. There is the ratio of
the Compton wavelength to the separation between the capacitor plates, $L$,
and thus

$L/\ell_{\varphi}:=qL=eL/\sqrt{\pi}.$ (20)

Furthermore, there is the ratio of the Compton wavelength to the length-scale
set by the electric field strength of the undischarged capacitor,

$\ell_{F}:=1/Q,$ (21)

and so $\ell_{F}/\ell_{\varphi}=1/(Q\ell_{\varphi})=m_{\varphi}/Q=q/Q$.

Fourth and finally, we would like to discuss the extent to which solutions to
the classical equations of motion capture the full quantum mechanical
evolution of capacitor discharge in massless QED2. That is, when do the
solutions for the classical equations of motion solve for the full quantum
dynamics of a given system? We address each of these points in the following
subsections.

###  3.1 Actions and Hamiltonians

To begin with, Ref. Chu:2010xc ’s computations were done using the equations
of motion derived from the action $S_{\varphi,A}$. When one evaluates the
equations of motion for the naively massless scalar field $\varphi$, one sees
that integrating-out the non-dynamical electromagnetic field induces a non-
zero mass, $m_{\varphi}^{2}=e^{2}/\pi$. As Ref. Chu:2010xc  notes, this is
consistent with the gap in the Schwinger model Schwinger:1962tp . Yet, as the
field content in $S_{\varphi,F}$ and $H_{\varphi,D}$ seem to be fundamentally
different, one may worry whether the descriptions match exactly. To see that
they do, recall the expression for the 00-component of the stress-energy
tensor derived from the action, $S_{\varphi,F}\mapsto T^{\mu\nu}:=\delta
S_{\varphi,F}/\delta g_{\mu\nu}$, in Chu:2010xc :

$\displaystyle
T^{00}=\frac{1}{2}\phantom{}:\\!\\!\bigg{(}\Pi_{\varphi}^{2}+(\partial_{x}\varphi)^{2}+m_{\varphi}^{2}\left(\varphi+\frac{D_{\rm
ext}}{m_{\varphi}}\right)^{2}\bigg{)}\\!\\!:$ (22)

Here we made free use of the normal ordering prescription as these two
functionals are implicitly, for us, evaluated in reference to specific field
profiles for $\varphi(x,t)$ and $D_{\rm ext}(x,t)$ given in section 2.

Thus, we see that the local expression for the 00-component of $T^{\mu\nu}$
derived from $S_{\varphi,F}$ from Chu:2010xc  and the Hamiltonian
$H_{\varphi,D}$ from Coleman:1975pw , when evaluated on specific, shared,
profiles for both $\varphi$ and $\epsilon^{\mu\nu}D_{\rm ext}=F^{\mu\nu}_{\rm
ext}$, indeed match exactly:

$\displaystyle T^{00}(x,t)\\!=\\!H_{\varphi,D}(x,t)\leavevmode\nobreak\ .$
(23)

In section 5, we use this and the explicit results for the massive Schwinger
model’s Hamiltonian in (1+1)-dimensions in Coleman:1975pw  to extend the
results of Ref. Chu:2010xc  to capacitor discharge in massive QED2. We do so
by first studying how the various length-scales in the problem interplay to
determine the late-time, discharged, field profile. We then study the time-
scales for this dynamical relaxation.

###  3.2 Capacitor discharge and $\varphi$-meson clouds: Length-scales

The late-time, static, E-field profile is the sum of two terms, $E_{\rm
static}(x):=D_{\rm ext}+g\varphi_{s}(x)$, which are respectively given in Eqs.
(10) and (15). It takes the following explicit form:

$\displaystyle
F_{01}(x,t)\bigg{|}_{t\to\infty}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=E_{\rm
static}(x)=\begin{cases}-\frac{Q}{2}\times\left({\rm
Exp}\left(-\tfrac{2|x|-L}{2\ell_{\varphi}}\right)-{\rm
Exp}\left(-\tfrac{2|x|+L}{2\ell_{\varphi}}\right)\right),&\\!\\!|x|>L/2\leavevmode\nobreak\
,\\\ +\frac{Q}{2}\times\left(1-{\rm
Exp}\left(-\tfrac{L-2|x|}{2\ell_{\varphi}}\right)+1-{\rm
Exp}\left(-\tfrac{L+2|x|}{2\ell_{\varphi}}\right)\leavevmode\nobreak\
\right),&\\!\\!|x|<L/2\leavevmode\nobreak\ .\end{cases}$ (24)

The energy stored in the electric field between the two capacitor plates,
${\cal H}(t)$, is given by

$\displaystyle{\cal
H}(t):=\frac{1}{2}\int_{-L/2}^{L/2}dx\bigg{(}F_{01}(x,t)^{2}\bigg{)}=\frac{1}{2}\int_{-L/2}^{L/2}dx\bigg{(}D_{\rm
ext}(x)+g\varphi(x,t)\bigg{)}^{2}\leavevmode\nobreak\ ,$ (25)

where $\varphi(x,t)$ is as given in (17) and $D_{\rm ext}(x)$ is, again, the
field profile in Eq. (10).

Now, to our first point: How the characteristic length-scale for the size of
the $\varphi$-meson enters into threshold effects associated with capacitor
discharge. One of the most clear instances of this comes from the energy
stored in the electric field after the discharge has completed, at
$t\to\infty$. Noting that the transient solutions eventually damp-out to the
static solution, $\varphi_{s}(x)$ in Eq. (15), we can compute the energy that
remains in the capacitor plate geometry after discharge to be Chu:2010xc ,

$\displaystyle\begin{cases}{\cal H}(0)\leavevmode\nobreak\
=\int_{-L/2}^{L/2}dx\big{(}T^{00}(x,t)\big{|}_{t\to
0}\big{)}\leavevmode\nobreak\ \leavevmode\nobreak\
\\!\\!=\frac{Q^{2}L}{2}\leavevmode\nobreak\ ,\\\ {\cal
H}(\infty)=\int_{-L/2}^{L/2}dx\big{(}T^{00}(x,t)\big{|}_{t\to\infty}\big{)}\\!=\frac{Q^{2}L}{2}\left(\frac{1-e^{-2Lm_{\varphi}}}{2Lm_{\varphi}}\right)={\cal
H}(0)\times\bigg{(}1-\tfrac{L}{\ell_{\varphi}}+{\cal
O}\big{(}\tfrac{L^{2}}{\ell_{\varphi}^{2}}\big{)}\bigg{)}\leavevmode\nobreak\
.\end{cases}$ (26)

This result for the residual energy stored in the E-field of the fully
discharged capacitor has several interesting features. Note that this shows us
that the final-state energy of the discharged capacitor is sub-extensive in
the the ratio between the separation of the two capacitor plates and the
length-scale associated with producing a pair of neutral $\varphi$-mesons.
Running this exponential suppression in the other direction we see that this
guides our interpretation of how a capacitor could discharge in massless QED2
at all: Namely, capacitor discharge can only happen via emission/creation of a
pair of (clouds of) $\varphi$-mesons whose electric dipole moments are aligned
to cancel and screen the dipole generated by the external field itself, in
effect screening the original charges of $\pm Q$.

Importantly, if the separation between these capacitor plates is insufficient
to physically fit any such mesons between them, then the external fields set
by the capacitor cannot adequately discharge. Furthermore, we expect the
discharge to become inefficient as the plate separation $L$ approaches
$\ell_{\varphi}$, as evident from the leftmost panel in Fig. 1. Indeed,
whenever $L\lesssim\ell_{\varphi}$, then it is simply not possible to create
any static capacitor-plate geometry out of a configuration of these neutral
$\varphi$-mesons within QED2. (We comment further on the feasibility of
creating a capacitor of charges $\pm Q$, separated by a distance $L$, directly
out of these $\varphi$-mesons—in a manner analogous to how actual capacitor
plates are constructed with actual electrons deposited on physical metal
plates in a transistor radio —in our concluding remarks in section 7.)
Furthermore, the final electric field configuration in Eq. (24) has exactly
the interpretation of a pair of clouds of neutral $\varphi$-mesons screening
each capacitor plate of charge $\pm Q$. The difference between the magnitude
of the net electric field across each plate is given by $\pm Q$ while the sum
of the two fields just across each plate is $Qe^{-L/\ell_{\varphi}}$. The net
polarization of the $\varphi$-meson fields acts to dipole screen the initial
charges $\pm Q$, and the penetration-depth of each cloud is given by the
Compton wavelength of the $\varphi$-mesons themselves: $\lambda_{\rm
depth}=\ell_{\varphi}$.

###  3.3 Capacitor discharge and $\varphi$-meson clouds: Time-scales and late
time tails

Next, we turn to the important question of timescales associated with
producing the $\varphi$-meson clouds that discharge the electric field (see
also Kukuljan:2021ddg  for a more detailed study). In this context, the
evolution of the energy stored in the E-field between the two plates, ${\cal
H}(t)$, plays a central role. We identify three time-scales: One associated
with the speed of the initial capacitor discharge at $t=0$ which we call
$\tau_{0}$, and two associated with the two time-varying terms in the
expression for ${\cal H}(t)$ itself which we denote $\tau_{1}$ and $\tau_{2}$.
Before we determine these scales, we again stress that although massless QED2
has $g=m_{\varphi}$ (cf. eq (12)), this degeneracy will turn out to be broken
in massive QED2. For this reason, in our analysis of the relevant timescales,
we carefully distinguish between factors of $g$ and factors of $m_{\varphi}$.

We define the first timescale $\tau_{0}$, associated with instantaneous
capacitor discharge at $t=0$:

$\displaystyle\frac{1}{\tau_{0}^{2}}:=\frac{{\cal H}^{\prime\prime}(t)}{{\cal
H}(t)}\bigg{|}_{t\to 0}\leavevmode\nobreak\ .$ (27)

The time-variation within $F_{01}(x,t)$ is isolated in the cosine-term in Eq.
(17), we have

$\displaystyle F_{01}(x,t)=E_{\rm static}(x)+E_{\rm vary}(x,t),$ (28)

where

$\displaystyle E_{\rm
vary}(x,t)=Qg^{2}\\!\int\frac{dk}{2\pi}\frac{\cos(kx)\sin(kL/2)}{k(k^{2}+m_{\varphi}^{2})}\cos(t\sqrt{k^{2}+m_{\varphi}^{2}}).$
(29)

It is straightforward to see that ${\cal H}^{\prime\prime}(0)=g^{2}Q^{2}L/2$,
while ${\cal H}(0)=Q^{2}L/2$, and thus

$\displaystyle\frac{1}{\tau_{0}^{2}}=g^{2}\implies\tau_{0}=\frac{1}{g}=\frac{\sqrt{\pi}}{e}\leavevmode\nobreak\
.$ (30)

Now, the timescale $\tau_{0}$ characterizes the initial rate of capacitor
discharge. Further, it depends on the charge of the electron in QED2, but does
not depend on the mass of the $\varphi$-boson itself. Thus, this timescale for
capacitor discharge is common to both massless and massive “electrons” of
charge $e=g\sqrt{\pi}$. Put differently, the initial rate of discharge of the
capacitor’s E-field is independent of the mass, $m_{\psi}$, for the degrees-
of-freedom in the Lagrangian for either massless or massive QED2.444We can
qualitatively and quantitatively tie this independence to the fact that, as
highlighted by the expressions for ${\cal H}(t)$ in Eq. (25) and by the
expressions/initial conditions for $\varphi(x,t)$ in Eqs. (16) and (17), all
aspects of the $\varphi$-boson’s mass explicitly drop out of the expressions
for ${\cal H}(t)$ and its time derivatives, when evaluated at $t=0$.

The other two important timescales we derive come from the late-time behavior
of ${\cal H}(t)$, e.g. when
$t\gg|x\pm\tfrac{L}{2}|,1/m_{\varphi},m_{\varphi}L^{2}$. As pointed out in
Chu:2010xc , the spatial current can be evaluated as

$\displaystyle j^{x}(x,t)$
$\displaystyle=g\partial_{t}\varphi(x,t)=-2g^{2}Q\int\frac{dk}{2\pi}\frac{\cos(kx)\sin(kL/2)}{k(k^{2}+m_{\varphi}^{2})^{1/2}}\sin(t(k^{2}+m_{\varphi}^{2})^{1/2})$
(31)
$\displaystyle=-\frac{g^{2}Q}{2}\int_{0}^{L/2}d\ell\bigg{[}J_{0}\left(m_{\varphi}\sqrt{t^{2}-x_{\ell-}^{2}}\right)\theta(t-x_{\ell+})+J_{0}\left(m_{\varphi}\sqrt{t^{2}-x_{\ell+}^{2}}\right)\theta(t-x_{\ell+})\bigg{]}\leavevmode\nobreak\
,\\!\\!\\!\\!$ (32)

where, here, $x_{\ell\pm}:=x\pm\tfrac{L}{2}$. At late times $t\gg|x\pm L/2|$,
this expression can be expanded in terms of $J_{0}(m_{\varphi}t)$-type Bessel
functions which, when $t\gg 1/m_{\varphi},m_{\varphi}L^{2}$, can be further
expanded as

$\displaystyle
j^{x}(x,t)\bigg{|}_{t\gg\tfrac{1}{m_{\varphi}},\leavevmode\nobreak\
m_{\varphi}L^{2},\leavevmode\nobreak\ |x_{\pm}|}=-\frac{Qg^{2}L}{\sqrt{2\pi
m_{\varphi}t}}\bigg{(}\cos\left(m_{\varphi}t-\frac{\pi}{4}\right)+{\cal
O}\left(\frac{m_{\varphi}L^{2}}{t}\right)\bigg{)}\leavevmode\nobreak\ .$ (33)

Since $\partial_{t}F^{tx}(x,t)=\partial_{t}E(x,t)=j^{x}(x,t)$, for the
electric field we obtain

$\displaystyle E(x,t)=E_{\rm static}(x)+\int
dt\big{(}j^{x}(x,t)\big{)}\leavevmode\nobreak\ .$ (34)

Thus, at late times, the E-field between the capacitor plates approaches

$\displaystyle\\!\\!\\!E_{\mathrm{vary}}(t)\bigg{|}_{t\gg
t_{\star}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=-\\!\\!\int
dt\frac{Qg^{2}L}{\sqrt{2\pi
m_{\varphi}t}}\bigg{(}\cos\left(m_{\varphi}t-\frac{\pi}{4}\right)+{\cal
O}\left(\frac{m_{\varphi}L^{2}}{t}\right)\bigg{)},\\!\\!\\!\\!$ (35)

where $t\gg t_{\star}$ is a proxy for the timescales such as
$|x_{\pm}|,1/m_{\varphi},m_{\varphi}L^{2}$. This implicitly defines $E_{\rm
vary}(t)$ at late times.

Using this late-time expression, we see that at late times the electric field
between the two capacitor plates, i.e. for $|x|\leq L/2$, is the sum of a term
that varies in space but not in time and a term that varies in time but not in
space. Consequently, the energy stored between the capacitor plates, ${\cal
H}(t)$, is the sum of three terms:

$\displaystyle{\cal H}(t)|_{t\gg
t_{\star}}=\frac{1}{2}\int_{-L/2}^{L/2}dx\bigg{\\{}E_{\rm vary}(t)+E_{\rm
static}(x)\bigg{\\}}^{2}\leavevmode\nobreak\ ,$ (36)

and, again, $t\gg t_{\star}$ is a proxy for the various timescales discussed
above. As the oscillatory envelope of $E_{\rm vary}(t)$ decays as
$1/\sqrt{t}$, at extremely late times only the spatially varying term matters,
i.e. when $t\to\infty$.

We are interested in the characteristic time scales of the decay of the second
two terms in ${\cal H}(t)$. To render the expressions appropriately
dimensionless (and to follow Chu:2010xc ), we consider the ratio

$\displaystyle\frac{{\cal H}(t)}{{\cal H}(0)}\bigg{|}_{t\gg t_{\star}}^{\rm
RMS}:=\frac{T_{1}}{t}+\frac{T_{2}}{\sqrt{t}}+T_{3}\leavevmode\nobreak\ ,$ (37)

where “RMS” refers to replacing the oscillatory parts of $E_{\rm vary}(t)$
with their root-mean-square averages. Now, the term $T_{3}$ is recognizable as
the ratio of the energy stored in the final-state of the capacitor to the
energy stored in the initial state, which was computed in Eq. (26).

The first two terms, $T_{1}/t$ and $T_{2}/\sqrt{t}$, strike at the heart of
this late-time tail of the discharge. They define characteristic time-scales
associated to the discharge of the capacitor which we denote $\tau_{1}$ and
$\tau_{2}$, respectively. Explicitly, these are defined as:

$\displaystyle\tau_{1}:=T_{1}\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\tau_{2}:=T_{2}^{2}\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (38)

After straightforward computations, and carefully distinguishing between $g$
and $m_{\varphi}$, we find

$\displaystyle\bigg{(}\frac{T_{1}}{t},\frac{T_{2}}{\sqrt{t}}\bigg{)}\mapsto(\tau_{1},\tau_{2})=\bigg{(}\frac{1}{g}\leavevmode\nobreak\
\frac{g^{2}L^{2}}{4\pi}\bigg{(}\frac{g}{m_{\varphi}}\bigg{)}^{3}\leavevmode\nobreak\
,\leavevmode\nobreak\ \frac{1}{m_{\varphi}}\leavevmode\nobreak\
\frac{Q^{2}L^{2}}{\pi}\bigg{(}\frac{g}{m_{\varphi}}\bigg{)}^{6}\bigg{(}1-\frac{1-e^{-m_{\varphi}L}}{m_{\varphi}L}\bigg{)}^{2}\bigg{)}$
(39)

Now, because $g=m_{\varphi}$ for massless QED2, we see that $\tau_{1}$ matches
the timescale Ref. Chu:2010xc  associated with capacitor discharge. Crucially,
$\tau_{2}$ represents a new timescale for the tail of the discharge:

$\displaystyle(\tau_{1},\tau_{2})\bigg{|}_{g\to
m_{\varphi}}=\bigg{(}\frac{1}{g}\leavevmode\nobreak\
\frac{g^{2}L^{2}}{4\pi}\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \frac{1}{g}\leavevmode\nobreak\
\frac{Q^{2}L^{2}}{\pi}\bigg{(}1-\frac{1-e^{-gL}}{gL}\bigg{)}^{2}\bigg{)}$ (40)

Note that between the two terms in the expression for the decay of the energy
stored in the capacitor with respect to time, $T_{2}/\sqrt{t}$ and $T_{1}/t$,
the $T_{2}$-term dominates at late-times, $t\gg t_{\star}$. (Further, because
we are at late-times, both terms should be small.) Thus the timescale
$\tau_{2}$ plays a vital role in capacitor discharge dynamics. There are
several interesting features of these timescales.

First, as noted above, the initial rate of capacitor discharge set by
$\tau_{0}$ is independent of all details of the system save for the charge of
the fermions in the QED2 action. This is physically sensible. At the very
onset of the discharge, the response of the system to any applied field should
be governed by how the fundamental, short-distance, degrees of freedom couple
to this field. It does not have time to “see” the size of the system.
Fundamentally, ${\cal H}^{\prime\prime}(0)/{\cal H}(0)$ equals $g^{2}$ because
the $\varphi$-mesons themselves source electric fields via, effectively, an
overall factor of $g^{2}$.

Second, the two timescales that dictate how energy stored in the discharging
capacitor dissipates are each (super)-extensive in terms of the physical size
of the capacitor. Put differently, each timescale is proportional to $L^{2}$:
$\tau_{2}\propto L^{2}$ and $\tau_{1}\propto L^{2}$. This seems sensible as
the final energy in the capacitor is sub-extensive in $L$. Further, the
latest-time tails in the energy stored in the capacitor are governed by the
$\sqrt{\tau_{2}}/\sqrt{t}=T_{2}/\sqrt{t}$-term in the RMS-averaged ratio
${\cal H}(t)/{\cal H}(0)$. Crucially, this timescale is also proportional to
the initial energy stored in the capacitor, namely $\tau_{2}\propto Q^{2}$.

###  3.4 Quantum evolution from classical equations of motion?

Finally, we turn to the question of when knowledge of the classical dynamics
of a system is sufficient to capture the full quantum dynamics. There are
three possibilities. For a free theory, one can construct coherent states that
form an over-complete basis of states. Any expectation value of field
operators in a coherent state will satisfy the classical equations of motion.
However, whether the specific initial conditions of charged capacitor plates
can be realized this way is a non-trivial question. A second possibility also
presents itself when the theory is integrable (in other words, fully
solvable), as for example is the case for the sine-Gordon model in (1+1)-$d$.
The latter would be necessary were we interested in the full range of the
dynamics of the system, particularly in states with low occupation numbers,
and is an interesting exercise in principle worthy of a followup investigation
that we will not pursue here.

Instead, we focus on the third and simplest possibility – that one can work
within a self consistent approximation in the context of initial states with
large occupation numbers, a correspondence can also be shown to hold in out of
equilibrium and interacting systems Mueller:2002gd . In the present context,
this requires us to work in the limit that the charge of the capacitors be
vastly larger than the fundamental quanta of charge. That is:

$\displaystyle N:=\frac{Q}{e}\gg 1.$ (41)

In this limit, we can be sure to recover the fully quantum evolution of the
discharging capacitor from the classical equations of motion even if the
classical equations of motion are non-linear. To see this in detail, we adapt
the treatment reviewed in Mueller:2002gd  and consider a generic scalar field
theory with canonical kinetic term and arbitrary potential.

Consider the finite time correlation functions of some observable ${\cal
O}(t)$ consisting of a string of $\varphi$ operators evaluated at time $t$
(but not necessarily at spatially coincident points)

$\langle{\cal O}(t)\rangle=\int\mathcal{D}[\varphi_{t}]{\cal
O}[\varphi_{t}]\int\mathcal{D}\left[\varphi_{0}\right]\rho\left[\varphi_{0}\right]U^{*}\left[\varphi_{t},\varphi_{0},t-t_{0}\right]U\left[\varphi_{t},\varphi_{0},t-t_{0}\right],$
(42)

where $\rho[\varphi_{0}]$ is the initial density matrix corresponding to some
initial field profile $\varphi(t_{0},x):=\varphi_{0}$, and where

$U\left[\varphi_{t},\varphi_{0},t-t_{0}\right]=\int\mathcal{D}[\varphi(t^{\prime})]e^{i\int_{t_{0}}^{t}L[\phi(t^{\prime})]dt^{\prime}}$
(43)

integrates over all field profiles that interpolate between $\varphi_{0}$ and
$\varphi(t,x):=\varphi_{t}$. Instead of viewing $U$ and $U^{*}$ as containing
separate functional integrals over the same field $\varphi$, a standard trick
employed in real-time perturbation theory is to view the fields in the
separate integrations as being distinct fields, $\varphi_{-}$ and
$\varphi_{+}$, and writing the Lagrangian as

$L=\int
dx\left[\frac{1}{2}\left(\partial_{\mu}\varphi_{-}\right)^{2}-\frac{1}{2}m^{2}\varphi_{-}^{2}-\sum_{n=3}\frac{c_{n}}{n!}\varphi_{-}^{n}\right]-\int
dx\left[\frac{1}{2}\left(\partial_{\mu}\varphi_{+}\right)^{2}-\frac{1}{2}m^{2}\varphi_{+}^{2}-\sum_{n=3}\frac{c_{n}}{n!}\varphi_{+}^{n}\right]$
(44)

so that

$U^{*}U=\int\mathcal{D}\left[\varphi_{-}\right]\mathcal{D}\left[\varphi_{+}\right]e^{i\int_{t_{0}}^{t}L\left[\varphi_{-},\varphi_{+}\right]dt^{\prime}},$
(45)

where $\varphi_{-},\varphi_{+}$ are both equal to $\varphi(t,x)$ and
$\varphi(t_{0},x)$ at $t$ and $t_{0}$ respectively. Making the field
redefinitions (where the meaning of the subscripts will become clear shortly)

$\varphi_{c}:=\frac{1}{2}\left(\varphi_{-}+\varphi_{+}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\varphi_{q}:=\varphi_{-}-\varphi_{+},$ (46)

and working with systems where both $\varphi_{+}$ and $\varphi_{-}$ correspond
to field profiles with large initial occupation, and so are themselves large,
one can show under certain assumptions on $\rho[\varphi_{0}]$ that
$\varphi_{c}$ can also naturally be taken to be large whereas $\varphi_{q}$ is
small Mueller:2002gd . Therefore, rewriting the action (44) in terms of
$\varphi_{c}$ and $\varphi_{q}$ and expanding to first order in $\varphi_{q}$
one finds

$L[\varphi_{c},\varphi_{q}]=\int
dx\left[\partial^{\mu}\varphi_{c}\partial_{\mu}\varphi_{q}-m^{2}\varphi_{c}\varphi_{q}-\varphi_{q}\sum_{n=4}\frac{c_{n}\varphi_{c}^{n-1}}{(n-1)!}\right]+...$
(47)

where the ellipses denote terms of order $\varphi_{q}^{2}$ and higher. The
final term in the square parentheses is simply the Taylor expansion of
$\varphi_{q}V^{\prime}(\varphi_{c})$. Recalling that our boundary conditions
are such that $\varphi_{q}\equiv 0$ at $t$ and $t_{0}$ and
$\varphi_{t}=\varphi_{c}(t,x)$ and $\varphi_{0}=\varphi_{c}(t_{0},x)$, so that
Eq. (42) becomes

$\langle{\cal O}(t)\rangle=\int\mathcal{D}[\varphi_{c}(t)]{\cal
O}[\varphi_{c}(t)]\int\mathcal{D}\left[\varphi_{c}(t_{0})\right]\rho\left[\varphi_{c}(t_{0})\right]\int\mathcal{D}\left[\varphi_{c}\right(t^{\prime})]\mathcal{D}\left[\varphi_{q}\right(t^{\prime})]e^{i\int_{t_{0}}^{t}L\left[\varphi_{c},\varphi_{q}\right]dt^{\prime}}.$
(48)

Because the assumption of large initial occupation forces $\varphi_{q}$ to be
small enough so that higher order terms in the expansion Eq. (48) can be
neglected, we see that its functional integral straightforwardly enforces the
constraint on the measure $\mathcal{D}[\varphi_{c}(t^{\prime})]$:

$\int\mathcal{D}\left[\varphi_{c}\right(t^{\prime})]\mathcal{D}\left[\varphi_{q}\right(t^{\prime})]e^{i\int_{t_{0}}^{t}L\left[\varphi_{c},\varphi_{q}\right]dt^{\prime}}=\int\mathcal{D}\left[\varphi_{c}\right(t^{\prime})]\prod_{x,t^{\prime}}2\pi\delta\left[\left(\square+m^{2}\right)\varphi_{c}({x},t^{\prime})+V^{\prime}[\varphi_{c}({x},t^{\prime})]\right],$
(49)

meaning that only fields that satisfy the classical equations of motion,

$\left(\square+m^{2}\right)\varphi_{c}+V^{\prime}[\varphi_{c}]=0$ (50)

contribute to the functional integral in Eq. (48). Therefore, characterizing
the full quantum of the system in the limit of large occupation number boils
down to knowledge of the classical solution to the equations of motion Eq.
(50). Defining the kernel

$K\left[\varphi_{t},\varphi_{0},t-t_{0}\right]:=\int\mathcal{D}[\varphi_{c}(t^{\prime})]\mathcal{D}[\varphi_{q}(t^{\prime})]e^{i\int_{t_{0}}^{t}L[\varphi_{c},\varphi_{q}]dt^{\prime}},$
(51)

we see that it is annihilated by the classical equations of motion, and at the
initial time satisfies

$\lim_{t\to
t_{0}}K[\varphi(t),\varphi(t_{0}),t-t_{0}]=\prod_{x}\delta[\varphi(x,0)-\varphi_{0}(x,0)],$
(52)

which manifests the classical nature of the system on all expectation values

$\langle{\cal
O}(t)\rangle=\int\mathcal{D}[\varphi_{t}]\mathcal{D}\left[\varphi_{0}\right]{\cal
O}[\varphi_{t}]K\left[\varphi_{t},\varphi_{0},t-t_{0}\right]\rho\left[\varphi_{0}\right].$
(53)

A direct corollary of the above is that an initial density matrix sharply
peaked around the profile $\varphi_{0}(t_{0},x)$ returns the expectation value
$\langle\varphi^{n}(t,x)\rangle=\varphi_{c}^{n}(t,x)$, with $\varphi_{c}$
satisfying Eq. (50) with initial conditions
$\varphi_{c}(t_{0},x)\equiv\varphi_{0}(t_{0},x)$.

##  4 Bare fermion masses in the Schwinger model

We now turn our attention towards the addition of bare fermion masses to the
Schwinger model. When the bare mass of the fermion in QED2 vanishes, the mass
of the neutral pion equals the mass-scale set by the effective coupling
constant $g$ in the term $g\varphi\epsilon^{\mu\nu}F_{\mu\nu}$ within the
action $S_{\varphi,A}$, i.e. $g=e/\sqrt{\pi}=m_{\varphi}$. However, when this
fermion has a bare mass, $m_{\psi}\neq 0$, the spectrum of stable excitations
in the theory becomes much more rich Coleman:1976uz . That is, introducing a
bare fermion mass distinguishes between the two mass-scales $g$ and
$m_{\varphi}$ that characterize the physical excitations of massive QED2.

As discussed in Refs. Coleman:1975pw ; Coleman:1976uz , when the fermions in
QED2 acquire a bare mass, the Hamiltonian for the dual $\varphi$-meson field,
is modified in the following way:

$\displaystyle H_{\varphi,D}\to H_{\varphi,D}(m_{\psi})$
$\displaystyle=H_{\varphi,D}+\Delta H_{\varphi,D}\leavevmode\nobreak\ $ (54)

where

$\displaystyle\Delta H_{\varphi,D}\equiv
m_{\psi}eC:\\!\cos\left(2\sqrt{\pi}\varphi+\theta\right)\\!:.$ (55)

Here $C=e^{\gamma_{E}}/(2\pi^{3/2})\simeq 0.16$ is a numerical constant
Gross:1995bp , $m_{\psi}$ is the bare mass of the electron Coleman:1975pw ,
and $\theta$ is a non-dynamical real parameter that lies within the range
$\theta\in[-\pi,\pi]$ that, ultimately, originates from the derivative acting
on $\varphi$ in the duality/mapping
$J^{\mu}\leftrightarrow\epsilon^{\mu\nu}\partial_{\nu}\varphi$ (as seen in
e.g. Eq. (9)). It is somewhat striking that this new Hamiltonian explicitly
depends on the parameter $\theta$ that characterizes the one-parameter family
of vacua of the Schwinger model Coleman:1975pw . Crucially, when $m_{\psi}=0$,
the vacua are degenerate. However, we see that when the fermion has a bare
mass within the QED2 action, this degeneracy is lifted. Equivalently, the
action for the dual scalar is,

$\displaystyle S_{\varphi,D}\to S_{\varphi,D}(m_{\psi})=S_{\varphi,D}+\Delta
S_{\varphi,D}\leavevmode\nobreak\ .$ (56)

where

$\displaystyle\Delta S_{\varphi,D}\equiv-C\bigg{(}\int
d^{2}x\leavevmode\nobreak\
(m_{\psi}e):\\!\cos\left(2\sqrt{\pi}\varphi+\theta\right)\\!:\bigg{)}\leavevmode\nobreak\
.$ (57)

At this point, one could attempt to use the fact that for small values of
$\varphi$ the cosine-potential, $\cos(2\sqrt{\pi}\varphi)$, the equation of
motion is approximately linear. In this context, the equations of motion that
dictate capacitor discharge in both massive and massless QED2 would continue
to be linear and can straightforwardly be shown to be solved by a similar
integral to Eq. (17), where now

$m_{\varphi}=\sqrt{g(g+2e^{\gamma_{E}}m_{\psi})},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
g=\sqrt{e^{2}/\pi}\leavevmode\nobreak\ .$ (58)

Self-consistency of this approximation would thus require that $\varphi(x,t)$,
which solves the linearized equation of motion, remains bounded and small at
all times; if $\varphi(x,t)$ is ever not-small, then we must consider the non-
linear higher-order terms in the cosine-potential. Put differently, even
though the two scales $g$ and $m_{\varphi}$ are split when the fermion has a
bare mass, $m_{\psi}\neq 0$, if it is consistent to both fix $\theta\mapsto 0$
and to remain in the small-$\varphi$ regime, then capacitor discharge in
massive QED2 would still be described by the dynamics found in Chu:2010xc .

However, there can be no appreciable capacitor discharge in this limit. The
reason is as follows. As commented in section 3.4, we cannot trust the
equation of motion to describe capacitor discharge unless we can guarantee
large-occupation numbers Mueller:2002gd . In this context, we must have $N\gg
1$. Now, note that $\varphi$ scales as
$\varphi\sim-2(g^{2}/m_{\varphi}^{2})N\times I_{L}(x,t)$, where $I_{L}(x,t)$
is an integral that depends only on the separation between the capacitor
plates, $L$, and on $x$ and $t$. (Crucially, $I_{L}(x,t)$ does not depend on
$N$.) When written in this way, we see that the range of capacitor
configurations where $|\varphi|\ll 1$ holds throughout the discharge is
dramatically restricted at large-$N$. Further, as the electric field evolves
as

$\displaystyle E(x,t)$
$\displaystyle=D_{\mathrm{ext}}-g\varphi(x,t)=D_{\mathrm{ext}}-\frac{Q}{N}\varphi(x,t)\leavevmode\nobreak\
,$ (59)

we see that even if $|\varphi|$ is bounded by a fixed small number (i.e.
$|\varphi|\ll 1$), then the discharge is a $1/N$ effect. Indeed, after all is
said and done, it is possible to show that ${\cal H}(0)-{\cal
H}(\infty)\lesssim m_{\varphi}$ in this scenario. In the next section, we
study the numerical solutions to the full non-linear equations of motion for
this massive bosonic model, which resembles a mass-deformed sine-Gordon model.
This analysis is at once more general, in the sense that it only requires the
large occupation approximation so that integrating the classical equations of
motion is equivalent to solving for the full quantum dynamics of the system,
and less analytically tractable. That said, the numerics speak for themselves.

##  5 Integrability of capacitor discharge in massive QED2

As discussed above, in the limit where $N=Q/e$ is large, we are in a regime
where the system is at large initial occupation, which as detailed in Sec.
3.4, implies that the expectation value of the operator equation of motion is
equivalent to solving the classical equation of motion

$\displaystyle(\partial^{2}+g^{2})\varphi_{c}=gD_{\mathrm{ext}}+\bigg{(}\frac{e^{\gamma_{E}}}{\pi}\leavevmode\nobreak\
m_{\psi}g\bigg{)}\sin(2\sqrt{\pi}\varphi_{c}+\theta).$ (60)

That is, the quantum dynamics of capacitor discharge is completely
characterized by solutions to the above equation of motion for large enough
initial charge. In order to facilitate numerical integration, we consider the
dimensionless variables,

$\displaystyle\hat{N}:=\frac{Q}{g}\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \hat{x}:=xg\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\hat{t}:=tg\leavevmode\nobreak\ \leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \hat{L}:=Lg\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \hat{m}:=\frac{e^{\gamma_{E}}}{\pi
g}m_{\psi}\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (61)

so that the equations of motion become

$\displaystyle(\hat{\partial}^{2}+1)\varphi_{c}$
$\displaystyle=\hat{N}(\theta(\hat{x}-\hat{L}/2)-\theta(\hat{x}+\hat{L}/2)+\hat{m}\sin(2\sqrt{\pi}\varphi_{c}+\theta).$
(62)

Fig. 2 depicts final fields profile for $\varphi_{c}$ that solve the above
equation of motion at $\theta=0$ and $\hat{N}=200$, for a range of values of
$\hat{L}$ and $\hat{m}$.

Before discussing the obtained solutions we briefly summarize our numerical
algorithm.555Our codes are available at
https://github.com/valerivardanyan/QED-discharge. The final field profile is a
solution of an elliptical boundary value problem. The equation of motion in
our case is non-linear for non-zero values of $\hat{m}$, and we are using an
iterative relaxation method to find its solution. We start with a proposal
profile, linearize the equation of motion in terms of deviation from the
proposal profile, and solve for linear deviations. We then iterate this
procedure until the updated profile solves the non-linear equation with
sufficient precision. Such an algorithm is commonly used for handling highly
non-linear boundary value problems. More details can be found in e.g.
symmetron_numerics . The results of the numerical integration is plotted in
Fig. 2, and depicts the fully discharged electric-field for various values of
$\hat{m}$, and taking $\hat{N}=200$ for concreteness. We see that the final
discharged E-field profile for massless QED2, and those for massive QED2, are
very close to each-other for $\hat{m}\ll\hat{N}$.

Figure 2:  Numerically obtained asymptotic electric field profiles at the end
of capacitor discharge in massive QED2 for three different plate separations
(corresponding to separate panels), and for a range of mass parameter
$\hat{m}$ (corresponding to the color-coded dashed curved), with
$\hat{N}=200$. The electric field profile prior to discharge is shown in solid
red line. The black solid lines underneath the yellow dashed curves correspond
to the analytical result Eq. (24) in the massless situation, which agrees with
our numerical solution. The existence of the mass threshold is visible in the
fact that, at finite bare fermion mass ($g<m_{\varphi}$), the electric field
never completely discharges.

As $\hat{m}/\hat{N}$ increases, the source and the potential terms in the
classical equation of motion start to become comparable to each other,
culminating in fact that when $\hat{m}/\hat{N}=2$, the maximally discharged
state of the capacitor is very close to the initial state of the system, as
can be seen from Fig. 2. This is evidently the (1+1)-$d$ manifestation of the
mass threshold seen in (3+1)-$d$, which exponentially cuts off as the mass of
the charged quanta increases according to Eq. (3): $\Gamma\propto
e^{-\frac{\pi m^{2}}{eE}}$. Here, the approach to the threshold evidently
proceeds linearly as $\hat{m}/\hat{N}\to 2$ and cuts off at
$\hat{m}/\hat{N}=e^{\gamma_{E}}m_{\psi}/(\sqrt{\pi}Q)=2$, implying a critical
electric field of

$E_{c}=\frac{e^{\gamma_{E}}m_{\psi}}{2\sqrt{\pi}}\approx\frac{m_{\psi}}{2}$
(63)

below which discharge effectively does not occur.666Note that in (1+1)-$d$,
electric fields and charge both have mass dimension one, to be contrasted to
(3+1)-$d$ where $E$ has dimension mass squared and charge is dimensionless.
The mass threshold in Eq. (63) is only valid within our approximation of large
occupation number, i.e. when the initial electric field $E_{0}\gg e$. For much
smaller values of $E_{0}$ beyond the remit of our approximation, one can still
show that discharge does not occur unless $E_{0}>\frac{e}{2}$ Coleman:1976uz .
It would be a mistake, however, to interpret the bare $m_{\psi}$ as the mass
of any physical states of the massive Schwinger model. To understand why, we
need to understand the spectrum of states in the theory.

The bosonized massive Schwinger model is qualitatively distinct from the sine-
Gordon model in that there are no degenerate vacua for non-vanishing electric
charge with the choice $\theta=0$. This is due to the presence of the ‘mass’
term in Eq. (60) proportional to $g^{2}$ which gaps the spectrum even for the
bosonized version of the massless Schwinger model. This makes sense, as there
are no free charges in the theory, and the kink/ anti-kink solutions in the
sine-Gordon model are the solitonic dual of free fermions. Since the Schwinger
model can only have bound states of these fermions in its spectrum, their
bosonic dual states must be more complicated. Recalling the result for the
spectrum of stable states of the massive Schwinger model in the regime where
$m_{\psi}/g=\hat{m}\gg 1$ Coleman:1976uz , we see that when $\theta$ is set
equal to zero, there are in fact $N_{s}\approx\hat{m}^{2}$ stable bound-states
with masses $m\leq 4m_{\psi}$ that scale as

$\displaystyle m_{n}\approx\Delta\sqrt{n}\leavevmode\nobreak\
,\leavevmode\nobreak\ {\rm for}\leavevmode\nobreak\
\Delta=4g\leavevmode\nobreak\ ,\leavevmode\nobreak\ {\rm
and}\leavevmode\nobreak\ 1\leq n\leq
N_{s}\approx\hat{m}^{2}=\tfrac{m_{\psi}^{2}}{g^{2}}\implies\begin{cases}m_{1}\leavevmode\nobreak\
\leavevmode\nobreak\ =m_{\rm min}&\\!\\!\\!\approx 4g\leavevmode\nobreak\ ,\\\
m_{N_{s}}=m_{\rm max}&\\!\\!\\!\approx 4m_{\psi}\leavevmode\nobreak\
.\end{cases}$ (64)

Therefore the $\varphi$ field in the bosonized theory must correspond to a
collective excitation of this finite tower of meson states, with lightest mass
$4g$ and heaviest mass $4m_{\psi}$ and as such, cannot cleanly be identified
with a single mass scale.

Now, recall the dimensional analysis in section 3.2. This analysis tied the
exponential nature of the small-tails in the fully discharged E-field in
massless QED2 rigidly to the Compton wavelength of the massive
$\varphi$-boson, $\ell_{\varphi}=1/g$; this is well-motivated, as the
$\varphi$-meson is the only on-shell degree of freedom in massless QED2.
Naively, were we to apply this same logic to massive QED2, we would expect
that shape of the final-state E-fields for large $\hat{m}$ is similarly
dictated by the Compton wavelengths of these stable particles. In particular,
because the lightest stable meson has a mass $m\approx 4g$, one might expect
roughly similar field profiles.

However, as is clearly seen in Fig. 2, this analysis does not describe the
shape of the fully discharged E-fields for massive QED2 whenever
$\hat{m}=m_{\psi}/g\gg 1$, particularly for the plate separation threshold
evident on the leftmost panel. One possible interpretation for this, is that
pair producing mesons from the vacuum still requires overcoming the mass
thresholds for the constituent fermions, before flux tubes form and compensate
the bound state energy, resulting in the composite state having a mass
significantly less than the sum of its constituents. In this way, one still
has to overcome a large mass threshold to produce these bound states, even if
the resulting condensate consists of parametrically lighter states. The dipole
length commensurate with the lightest of these states with mass $m_{1}$ may be
parametrically quite different from $m^{-1}_{1}$ given the large binding
energies involved for larger values of the bare mass, resulting in
$\ell_{\varphi}\ll 1/m_{1}$ for larger values of the bare mass and approaching
$\ell_{\varphi}\sim 1/m_{1}\sim 1/g$ for smaller values of the bare mass,
which would simultaneously account for the mass and plate separation
thresholds depicted in Fig. 2. However, a quantitative understanding of this
is beyond the scope of this paper and necessitates further investigation.
Here, we merely wish to point out the existence of the independent mass and
plate separation thresholds in massive QED2, a conclusion we can confidently
draw from the accuracy of integrating the classical equations of motion as an
approximation of the bosonized description at large occupation number.

Finally, it is also interesting to note that for fermion masses in the lighter
range ($m_{\psi}/g\lesssim{\cal O}(1)$), the consistency of our approximation
results in the source term dominating the dynamics, and so the discharge
proceeds almost as in the massless case, in which care we can consider the
analogs of the timescale in the massless QED2, but now with the degeneracy
between $g$ and $m_{\varphi}$ broken according to Eq. (58), so that the three
timescales associated with discharge from Eqs. (30) and (39) become

$\displaystyle\tau_{0}$ $\displaystyle=\frac{1}{g},$ (65)
$\displaystyle\tau_{1}$
$\displaystyle=\frac{1}{m_{\varphi}}\leavevmode\nobreak\
\frac{Q^{2}L^{2}}{\pi}\left(\frac{g}{m_{\varphi}}\right)^{6}\left(1-\frac{1-e^{-m_{\varphi}L}}{m_{\varphi}L}\right)^{2},$
(66) $\displaystyle\tau_{2}$ $\displaystyle=\frac{1}{g}\leavevmode\nobreak\
\frac{g^{2}L^{2}}{4\pi}\left(\frac{g}{m_{\varphi}}\right)^{3},$ (67)

allowing for parametrically different behavior for the late vs. early time
dynamics of the discharge relative to the massless case. A complete
understanding of the dynamics of capacitor discharge (in particular with small
initial charge) is not possible within our approximation.777A notable attempt
to study electric field discharge in the massive Schwinger model was made
using lattice simulations in Hebenstreit:2013qxa  building on the formalism
introduced in Aarts:1998td , however the computational cost of these
simulations meant that only the initial stages of this discharge could be
tracked.

##  6 Discharge in higher dimensions ($d>2$)

When viewed as a tunneling process, the salient aspects of Schwinger pair
production in (3+1)-dimensions are often well-captured by computing the
probability of filled, negative energy states tunneling through an effective
potential-barrier in (1+1)-dimensions, as formulated in e.g. Nikishov:1970br ;
Cohen:2008wz ; Allor:2007ei . It is thus tempting to use our fully back-
reacted discharge in (1+1)-dimensions to model capacitor-discharge in higher-
dimensions. However, there are prominent obstacles between using our present
understanding of capacitor discharge in massless QED2 and any full
understanding of capacitor discharge in higher-dimensions, whose root cause
come from the computation of the tunneling process to begin with.

The main, essential, difficulty here is the notion of scattering states
themselves. The simpler case of electron-positron pairs creation due to a
static electric field between two charged capacitor plates can easily be
reframed in terms of an integral over a continuum of scattering states which
tunnel through an effective potential. Explicitly, as the capacitor plates are
invariant in the $d-2$ transverse directions, then the problem becomes
(1+1)-dimensional. Further, because the E-field is static in time, then by
choosing the gauge where $\vec{E}=-\vec{\nabla}\phi$, the equations of motion
are time-translation invariant. Thus, here, one can define scattering states
with well-defined energy $E$ and transverse momentum $\vec{p}_{T}$, incident
on the effective potential set-up by the E-field between the capacitor plates
Nikishov:1970br ; Wang:1988ct . It is then straightforward to find the
tunneling amplitudes and thus the rate of pair production as discussed in e.g.
Cohen:2008wz ; Allor:2007ei ; McGady:2017slr .

Unfortunately, the very fact that the E-field is static at once makes this
set-up tractable while also excising any appreciable back-reaction. (Formally,
this setup describes Schwinger pair-creation in the limit where the electron
charge is sent to zero whilst the field-strength is held fixed, i.e. where
$e\to 0$ while $eE$-fixed.) Relaxing this condition and allowing the E-field
to vary in time, i.e. to allow it to discharge, ruins the time-translation
invariance needed to label the scattering states by their energy and, further,
causes the “effective potential” to vary in time. Together, these two
technical problems obfuscate the most obvious paths between our exact
treatment for (1+1)-dimensional discharge and understanding discharge in
higher dimensions.

Related to this, in $d>2$ there are nontrivial magnetic fields. Thus, the
time-variation of the electric field induces magnetic fields. These magnetic
fields deflect the paths of charged particles and paths between our exact
solution in (1+1)-dimensions and exact understandings of capacitor discharge
in $d$-dimensions alike. For these reasons, we expect the results/techniques
in this paper may be of limited use in understanding detailed dynamics of
capacitor discharge above $d=2$.

##  7 Discussion

Having reinterpreted and generalized the findings of Chu:2010xc  to
incorporate bare fermion masses within the Schwinger model, the main findings
of our investigation can be summarized as:

  * •

Electric fields can discharge even in theories exhibiting confinement via
dipole production. These correspond to meson bound states in the context of
the Schwinger model.

  * •

This dipole production will necessarily be suppressed below the threshold
plate separation corresponding to the characteristic size of the dipole
$L=\ell_{\varphi}\sim g^{-1}$ in the massless case.

  * •

When generalized to the massive case, the mesons correspond to a finite tower
bound states, and the threshold separation becomes a more complicated function
of the bare fermion mass and $g$.

  * •

The introduction of bare fermion masses to the Schwinger model results in the
appearance of an additional threshold at
$E_{c}=\frac{e^{\gamma_{E}}m_{\psi}}{2\sqrt{\pi}}\approx\frac{m_{\psi}}{2}$,
below which electric field discharge is suppressed.

There are a number of conceptual issues, however, that remain to be discussed.
The issue of how to realize the initial conditions of two charges of $\pm Q$
separated by a distance $L$ within massless QED2 — i.e. how to construct the
capacitor itself directly in terms of polar $\varphi$-mesons themselves — is a
very interesting and, to our knowledge, open question. It should be possible
to directly construct this initial condition as coming from a specific initial
state of free, non-interacting, $\varphi$-mesons as the bosonized theory is
free and integrable. Furthermore, it would be useful to understand whether
quantitative progress can be made away from our semi-classical approximation
which relied on large initial occupation, and therefore large initial electric
fields. However carrying out these exercises is beyond the scope of the
current work. Here, we content ourselves with the observation that at large
occupation number, provided the initial capacitor state can be constructed
(possibly involving operators in a larger theory, a consistent truncation of
which is the Schwinger model), charge conservation and the unitarity of the
Schwinger model renders our treatment self-consistent.

## Acknowledgements

We’d like to thank Konstantin Zarembo, Tanmay Vachaspati, and Poul Henrik
Daamgard for insightful and helpful comments on the draft. DM was supported by
the grant “Exact Results in Gauge and String Theories” from the Knut and Alice
Wallenberg foundation. V.V. is supported by the WPI Research Center
Initiative, MEXT, Japan, and by JSPS KAKENHI Grant Number 20K22348.

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