# Perturbative Understanding of Non-Perturbative Processes
and Quantumization versus Classicalization

Gia Dvali  Lukas Eisemann<EMAIL_ADDRESS>Arnold-Sommerfeld-
Center, Ludwig-Maximilians-Universität, Theresienstraße 37, 80333 München,
Germany Max-Planck-Institut für Physik, Föhringer Ring 6, 80805 München,
Germany

###### Abstract

In some instances of study of quantum evolution of classical backgrounds it is
considered inevitable to resort to non-perturbative methods at the price of
treating the system semiclassically. We show that a fully quantum perturbative
treatment, in which the background is resolved as a multi-particle state,
recovers the semiclassical non-perturbative results and allows going beyond.
We reproduce particle-creation by a classical field in a theory of two scalars
as well as in scalar QED in terms of scattering processes of high
multiplicity. The multi-particle treatment also gives a transparent picture of
why a single-process transition from a classical to a quantum state, which we
call quantumization, is exponentially suppressed, whereas the opposite
process, classicalization, can take place swiftly if the microstate degeneracy
of the classical state is high. An example is provided by the $N$-graviton
portrait of a black hole: a black hole can form efficiently via a $2\to N$
classicalization process in the collision of high-energy particles but its
quantumization via a decay $N\to 2$ is exponentially suppressed.

###### Contents

  1. I Introduction
  2. II Example 1: $\phi^{2}\chi^{2}$
    1. II.1 Coherent states and background fields
    2. II.2 Quantum calculation
    3. II.3 Semiclassical calculation
  3. III Example 2: Scalar QED
    1. III.1 Quantum calculation
    2. III.2 Semiclassical calculation
  4. IV Example 3: Scalar QED with massive photon
    1. IV.1 Semiclassical result
    2. IV.2 Quantum calculation
  5. V Parameter regimes
    1. V.1 Semiclassical limit
    2. V.2 Finite $g^{2}$
      1. V.2.1 Effective potential
      2. V.2.2 Diagrams with different $n$-scaling
  6. VI Quantumization and classicalization
    1. VI.1 Quantumization
    2. VI.2 Classicalization
  7. VII Summary and outlook

##  I Introduction

It is common to classify quantum field theoretic phenomena into the sub-
categories of perturbative and non-perturbative effects. A given effect is
attributed to the non-perturbative category when its physical characteristics,
such as a cross section or a transition rate, are not representable in the
form of a perturbative power series in a relevant coupling constant.

A classic example is provided by the Hawking evaporation rate of a black hole.
This rate is proportional to a negative power of the Newtonian gravitational
coupling. It therefore creates the impression that the process is non-
perturbative in Newton’s constant. The same reasoning applies to many other
examples of particle-creation in a background classical field.

In reality, in many instances, the underlying physics is fully perturbative in
the coupling. This perturbativity becomes visible only upon the resolution of
the classical background in form of a multi-particle state, such as a
condensate, which may be realized in form of a number state or a coherent
state. The corpuscular resolution makes transparent that a seeming non-
perturbativity is the result of an interplay between the relevant coupling
constant and the occupation number of quanta. This has been explicitly
demonstrated for a number of systems.

For example, the theory of [1, *BHQPT] offers a resolution of a black hole in
form of a condensate (or a coherent state) of gravitons with occupation number
$N$. This number is proportional to the inverse of the gravitational coupling.
In this picture, Hawking evaporation is described as the depletion of the
condensate due to re-scattering of constituent gravitons. The process is
perturbative in Newton’s coupling. The appearance of its negative power in the
final expression of the evaporation rate is due to the Bose-enhancement of the
process by the combinatorics of the occupation number $N$. Since $N$ scales as
the inverse of the Newtonian coupling, the rate also comes out to be inversely
proportional to it. Hence the apparent non-perturbativity of the process.

In general, an understanding of non-perturbative mean-field effects as
perturbative $N$-particle processes carries obvious advantages. In particular,
it allows to go beyond the leading order approximation and to capture the
$1/N$-corrections coming from quantum effects of the individual particles.
Such effects are extremely important for properly accounting for the quantum
backreaction, which is due to particle-creation on a would-be classical
background.

In certain cases this back reaction leads to a complete breakdown of the
classical approximation. The concept was originally introduced within the
framework of the black hole $N$-portrait [1], with the first explicit
simulation of a prototype model conducted in [3], where the effect was
referred to as quantum breaking.

Within the black hole $N$-portrait [1], it has been argued [4, 5, 6, 7] that
quantum breaking takes place at the latest by the time of half-decay.

Beyond black holes, the corpuscular approach to particle-creation has been
applied to a number of other systems. In [6, 8, 9, 10, 11, 12, 13] the
classical de Sitter metric has been resolved as the coherent state of
gravitons on the Minkowski vacuum. In particular, such a resolution is
mandatory within the $S$-matrix formulation of gravity, since de Sitter cannot
serve as a valid $S$-matrix vacuum [11]. In this picture, the Gibbons-Hawking
particle-creation [14] is described as re-scattering of coherent state
gravitons into all possible particle species. The backreaction due to $1/N$
effects leads to a gradual breakdown of the classical approximation. In
particular, this is due to the generation of entanglement [6, 8, 9, 11, 12].

Likewise, an analogous corpuscular study of the decay of the coherently
oscillating axion field was given in [15]. Some further studies on quantum
breaking of coherent states using background field methods can be found in
[16, 17]. For other aspects of the corpuscular resolution of de Sitter, cf.
[18, 19, 20].

In all these examples, the non-perturbative semiclassical effect is recovered
as the infinite-$N$ limit of the quantum picture.

The present paper represents a continuation of the above program but with some
important novelties that allow to more cleanly extract the specific multi-
particle effects in simple examples of basic importance. We study systems that
represent the simplest prototype models describing the process of particle-
creation by a background field. We do this both in the semiclassical
approximation as well as in a fully quantum treatment. We show that the
perturbative computation in the fundamental quantum theory captures the
seemingly non-perturbative phenomena obtained in the semiclassical treatment
of the same system.

In particular, we make a special focus on the regime when the energies of the
produced quanta exceed the oscillation frequency of the background field. The
non-perturbative semiclassical analysis gives a very specific suppression of
particle-creating instabilities. We show that the fully quantum treatment,
perturbative in coupling $g^{2}$, reproduces these semiclassical results in
large-$N$.

To be more precise, this correspondence becomes exact in the following double-
scaling limit,

$g^{2}\to 0,\quad\frac{N}{Vm^{3}}\to\infty,\quad
g^{2}\frac{N}{Vm^{3}}=\text{fixed}\,,$ (1)

where $m$ is the mass of the particles composing the would-be classical field
and $V$ is the volume. However, for finite values of the coupling $g^{2}$ and
the particle number density $N/V$, the fully quantum perturbative analysis
allows to go beyond the semiclassical approximation and capture effects that
are higher order in $1/N$.

We consider three examples. In the first example, the quanta of a scalar field
$\chi$ are created in the background of an oscillating classical scalar field
$\phi$. The coupling between the two fields is $g^{2}$. At the level of
semiclassical analysis, in which $\phi$ is treated classically, such systems
have been widely considered and have many applications, e.g., for reheating
after inflation [21]. In this approximation, the creation of $\chi$-quanta is
accounted for by certain instabilities in the background mode equation. It is
usually said that these effects are non-perturbative. This may create the
false impression that they cannot be captured by a perturbative analysis. We
show that this is not the case.

We achieve this by giving a fully quantum resolution of the system. Namely, we
represent the classical $\phi$ field as a quantum state of high occupation
number $N$. In this language, the particle-creation can be understood as a
scattering process in which a number $n$ of $\phi$-quanta is converted into a
pair of $\chi$s. We show that the perturbative treatment fully captures the
seemingly non-perturbative effect obtained in the semiclassical theory.

In two further examples, we generalize the effect to systems with gauge
symmetry. One of these examples considers production of photons by a time-
dependent charged scalar field. Here, too, the semiclassical picture is fully
reproduced by perturbative quantum re-scattering of the charged constituent
quanta of the condensate.

The final example is concerned with the inverted situation, i.e., production
of a pair of scalar electrons in an oscillating electric field. In quantum
language we describe this process as the creation of a particle-antiparticle
pair in the annihilation of many photons. These photons represent the quantum
constituents of the background electric field.

Our study has implications for fundamental questions regarding classical-to-
quantum transitions and vice versa. In particular, it shines light on the
question how fast a classical system can transit to a quantum state in which
the classical approximation ceases to be valid. In [3], the timescale of such
a breakdown was called the quantum break-time, which we denote by $t_{Q}$. The
physically relevant timescale to which it must be compared is the
characteristic inverse frequency of the constituents of the classical state.
In our examples, this is the frequency of the coherent oscillations of the
classical field or a condensate. In the case of the $N$-portraits of black
holes [1] and of de Sitter [6, 1], the basic frequency of the constituent
gravitons is given by the inverse of the classical curvature radius.

The study performed in [3] shows that in a classically unstable system (i.e.,
a system with Lyapunov instability), $t_{Q}$ can scale logarithmically in the
number $N$ of the system’s constituents. Thus, such a system can quantum break
swiftly.

On the other hand, in generic classically-stable or stationary systems, the
quantum break-time was argued to be macroscopic in $N$ [6, 8, 9]. In such
systems the transition to a quantum state is gradual and quantum breaking is a
cumulative effect of a large number of elementary processes, each with
participation of a small number of quanta. In particular, this was argued to
be the case for black holes [1, 4, 5, 6, 7], as well as for de Sitter [6, 8,
9, 10, 11, 12].

In contrast, the quantum breaking driven by single-process transitions, with
participation of order $N$ constituents, are expected to be highly suppressed.
Our analysis contributes to this understanding substantially. Due to the
physical importance of this regime, we shall introduce a special term for such
a process and refer to it as quantumization.

The flip side of the coin is the process called classicalization [22]. It
represents a process of an inverse transition from a quantum state of few
(say, two) energetic quanta into a classical state of soft quanta of high
occupation number $N$ [23, 24, 25, 26, 27]. Thus, the basis for both
processes, quantumization as well as classicalization, is the $S$-matrix
element between $2$-particle and $N$-particle number eigenstates.

On very general physics grounds one can argue [27] that the square of such a
matrix element is limited from above by an exponentially small factor
$e^{-N}$. This has been confirmed by explicit computations, for example, by
the computation of $2\to N$ graviton scattering [25]. In fact, for the systems
we consider in the present paper, in certain regimes we shall obtain an even
stronger suppression.

Although the basic matrix elements for quantumization and classicalization
processes are the same, their physical manifestations are very different. The
significance of $2\to N$ versus $N\to 2$ has already been appreciated within
the black hole $N$-portrait [1] and is generic for any classicalizing theory
[26, 27].

The reason is the exponential difference between the degeneracies of
$N$-particle and $2$-particle microstates connected by one and the same matrix
element. Correspondingly, it makes all the difference whether $N$ appears in
the initial or the final state of the process. The classicalization processes,
$2\to N$, can take place with order-one probability provided the $N$-particle
classical state has a sufficient microstate entropy [1, 26, 27]. Such states
in [27] were called “saturons”, as their entropy is close to saturating the
upper bound imposed by unitarity.

In contrast, the quantumization processes, $N\to 2$, are always exponentially
suppressed. This is due to the fact that a valid $2$-particle state has
insufficient degeneracy for compensating the exponentially suppressed
$S$-matrix element.

A good example of the difference between classicalization and quantumization
processes is provided by black hole formation and decay as described by its
$N$-portrait [1]. Since in this theory a black hole is a state of $N$ soft
gravitons, its formation in a collision of two high energy particles
represents a process of classicalization, $2\to N$.

The computation of the $2\to N$ graviton process [25, 28] shows that the
probability of formation for each microstate is $e^{-N}$. However the
suppression is compensated by the microstate degeneracy factor $e^{N}$. In
other words, the black hole has sufficient degeneracy for compensating the
exponentially suppressed matrix element, i.e., it is a saturon [27].

In contrast, a decay of a black hole into two very energetic quanta,
represents a quantumization process of the type $N\to 2$. The microstate
degeneracy of the final $2$ particle state is negligible as compared to the
exponential suppression. This explains the well-known property of black holes
that, while they can form very fast, they always decay very slowly. Due to
suppressed quantumization the decay happens via gradual emission of quanta
through Hawking radiation, as opposed to an explosive decay into a few
particles.

Also, one more physical consequence of the suppressed quantumization is an
inability of the classical system to evolve into a highly entangled quantum
state via a single-step process.

The examples studied in the present paper make these generic features of
fundamental importance very transparent.

##  II Example 1: $\phi^{2}\chi^{2}$

For the first example, we consider the model

$L=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}+\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi-\frac{1}{2}m_{\chi}^{2}\chi^{2}-g^{2}\phi^{2}\chi^{2}\,.$
(2)

In a semiclassical treatment, an initial $\phi$-condensate is described by the
classical solution

$\phi_{B}(t)=\phi_{0}\cos\left(mt\right)\,,\quad\chi_{B}=0\,,$ (3)

and the fluctuations around the background (3) are quantized:

$\hat{\phi}=\phi_{B}+\hat{\delta\phi}\,,\quad\hat{\chi}=\hat{\delta\chi}\,.$
(4)

The initial state we would like to consider is then

$|t_{0}\rangle=|0\rangle_{\delta\phi}|0\rangle_{\chi}\,.$ (5)

The background alters the propagation of fluctuations to allow for particle-
creation out of the vacuum, for example

$0\to 2\chi\,.$ (6)

By contrast, in a fully quantum treatment, the condensate is described by an
initial state

$|t_{0}\rangle=|N\rangle_{\phi}\,|0\rangle_{\chi}\,,$ (7)

where $|N\rangle_{\phi}$ denotes a state of $\phi$-quanta in the mode of
vanishing 3-momentum, $\bm{p}=0$, in a superposition centered around the mean
occupation $N$. This could, in particular, be a coherent state or simply a
number state, and the consideration in the limit (1) is independent of the
choice. For definiteness, we are going to assume a number state for now. The
quantum processes giving rise to creation of a pair of $\chi$s are then

$N\phi\to(N-n)\phi+2\chi\,.$ (8)

Here, the final $\phi$s are understood to be also in the $\bm{p}=0$ mode. Of
course there are also processes involving scattered $\phi$s. As compared to
(8), these are however suppressed by extra powers of the coupling $g^{2}$ and
vanish in the limit (1). Such suppression is not accompanying diagrams with
only forward-scattered $\phi$s, but those contribute to the same process (8)
(see also sec. V). In the absence of an elementary self-interaction of $\phi$,
and in the regime of negligible Coleman-Weinberg correction to the potential,
the energetics of the condensate are well approximated by

$\frac{N}{V}=\frac{m\phi_{0}^{2}}{2}\,.$ (9)

This relates the quantum and classical parameters. Thus, in semiclassical
terms, the limit (1) reads

$g^{2}\to 0\,,\quad\frac{\phi_{0}^{2}}{m^{2}}\to\infty\,,\quad
g^{2}\frac{\phi_{0}^{2}}{m^{2}}=\text{fixed}\,.$ (10)

In the following, we are going to compare the quantum-perturbative and
semiclassical-non-perturbative prediction for creation of $\chi$-pairs with
momenta $k\equiv|\bm{k}|$ corresponding to

$n\gg\frac{g\phi_{0}}{m}\,.$ (11)

###  II.1 Coherent states and background fields

Before going to the calculations of the rate, we would like to recall a
general correspondence between calculations in a semiclassical approximation
and a fully quantum one that involves coherent states [29]. In the notation of
the example introduced above, consider a coherent state $|c\rangle$ with the
property $\langle c|\hat{\phi}(t)|c\rangle=\phi_{B}(t)$ and $\langle
c|\hat{\chi}(t)|c\rangle=\chi_{B}(t)$, where $\hat{\phi}(t)$ and
$\hat{\chi}(t)$ are evolving according to the free Hamiltonian. For such a
state, the S-matrix operator in the presence of a background field agrees with
the one of the fully quantum theory between $|c\rangle$,

$\langle c,B|\hat{S}|c,A\rangle_{\phi.\chi}=\langle
B|\hat{S}[\phi_{B},\chi_{B}]|A\rangle_{\delta\phi,\delta\chi}\,,$ (12)

where the states $|A\rangle$ and $|B\rangle$ denote arbitrary number
eigenstates of modes that are not contained in the background. As we shall
see, the convergence of a perturbation series in the field strength depends
also on the multiplicity $n$ of the process under consideration. For the
condensate decay (6) and (8) in question, identity (12) means that in the
perturbative regime we have to find agreement of the perturbative fully
quantum and non-perturbative semiclassical creation rates with any differences
arising from deviations of the initial and final $\phi$-state from
$|c\rangle$. Such deviations are model independent and generically vanish in
the limit (1) as $\sim N^{-1}$. The following calculations for
$\phi^{2}\chi^{2}$ as well as those in subsequent sections thus represent
explicit examples of the correspondence (12).

###  II.2 Quantum calculation

There is only a single diagram (see fig. 1) entering the calculation of the
rate for the process (8) at leading order in $g^{2}N$.

Figure 1: Diagrammatic representation of the leading order terms in
perturbation theory contributing to the amplitude of the process (8).

The number of Wick contractions is canceled by the Taylor-coefficient of the
relevant term of the S-matrix operator up to a factor of $2^{n/2}$. The
diagram’s multiplicity is $n!$ and there is a factor of $1/\sqrt{n!}$
accompanying it because of the initial $n-$fold occupation of the zero mode of
$\phi$. Thus the squared amplitude is given by $2^{n}n!|d|^{2}$, with $d$ the
value of the diagram. If $q_{l}$ denotes the the virtual momentum in the
propagator following the $l$th insertion of a pair of $\phi$-legs, then
$q_{l}^{\mu}=2lm\delta^{\mu}_{0}-k^{\mu}$ and the propagators contribute to
$d$ a factor of

$\prod_{l=1}^{n/2-1}\left(q^{2}_{l}-m_{\chi}^{2}\right)^{-1}=(-1)^{n/2-1}m^{2-n}\,n^{2}\,2^{-n}\,\left(n/2\right)!^{-2}\,.$
(13)

Since the initial state does not carry angular momentum, the phase space
integration is trivial. Perturbatively, the kinematics are

$nm=2\left(m_{\chi}^{2}+k^{2}\right)^{1/2}\,,$ (14)

which corresponds to a kinematic threshold for the multiplicity $n$ that is
given by

$n>n_{0}\equiv\frac{2m_{\chi}}{m}\,.$ (15)

The resulting tree-level rate for $n\phi\to 2\chi$ is

$\Gamma_{n\phi\to
2\chi}=\frac{1}{4\pi}Vm^{4}\sqrt{1-\frac{n_{0}^{2}}{n^{2}}}n^{4}\left(\frac{g^{2}}{4Vm^{3}}\right)^{n}\frac{n!}{(n/2)!^{4}}\,.$
(16)

Perturbation theory can be seen not to break down regardless of the values of
$n$. Including the combinatoric enhancement factor due to the initial $N$
quanta,

$C_{Nn}=\begin{pmatrix}N\\\
n\end{pmatrix}=\frac{N^{n}}{n!}\left(1+\mathcal{O}\left(\frac{n^{2}}{N}\right)\right)\,,$
(17)

the rate for the process (8) to leading order is given by

$\Gamma\equiv C_{Nn}\Gamma_{n\phi\to
2\chi}\sim\left(\frac{e^{2}}{2n^{2}}\frac{g^{2}\phi_{0}^{2}}{m^{2}}\right)^{n}\,,$
(18)

where we have used relation (9), the Stirling approximation for the factorial,
and omitted factors that scale less strongly with $n$ than exponentially. The
time evolution of $n_{k}$, the expected occupation number density of $\chi$s
per momentum $\bm{k}$, is given for early times by $n_{k}(t)\sim\Gamma\,t$.
For later times, however, $\chi$-Bose-enhancement becomes important, leading
to an enhanced effective rate

$\Gamma_{\text{eff}}\sim\left(1+2n_{k}+\mathcal{O}\left(\frac{n_{k}^{2}n^{2}}{N}\right)\right)\Gamma\,,$
(19)

and thus

$n_{k}(t)\sim\exp\left(2\Gamma t\right)\,,\quad n_{k}\gtrsim 1\,.$ (20)

Therefore, the connection between the quantum rate and the semiclassical
prediction is provided by

$\Gamma\sim\frac{\dot{n}_{k}}{n_{k}}\,.$ (21)

The evolution (20) is of course neglecting modification due to sources of
quantum breaking, such as depletion or evolution of entanglement. As long as
the number of depleted quanta is much smaller than $N$, it is justified to
neglect those effects.

###  II.3 Semiclassical calculation

In the limit (10), the equations of motion simplify. The equation for
$\delta\phi$ reads

$\displaystyle\left(\partial^{2}+m^{2}\right)\delta\phi$
$\displaystyle=-2\left(d_{t}^{2}+m^{2}\right)\phi_{B}-2g^{2}\chi^{2}\left(\phi_{B}+\delta\phi\right)$
$\displaystyle=\mathcal{O}\left(g\right)\,.$ (22)

The terms involving only the background vanish due to (3). The equation for
$\chi$ reads

$\displaystyle\left(\partial^{2}+m_{\chi}^{2}\right)\chi$
$\displaystyle=-2g^{2}\chi\left(\phi_{B}+\delta\phi\right)^{2}$
$\displaystyle=-2g^{2}\phi^{2}_{B}\chi+\mathcal{O}\left(g\right)\,.$ (23)

Thus, in the limit (10), $\delta\phi$ decouples as a free field and $\chi$ has
a time dependent contribution to its mass due to the background. For the non-
perturbative solution, we can follow for example [30, *Mukhanov:2007zz]. In
the case of a linear equation of the form (II.3), the prediction for
$\chi$-creation can be given in real time in terms of the mode function
$v_{k}$, which is defined through the mode expansion of the field operator as

$\hat{\chi}(x)=\int
d^{3}k\left(v_{k}(t)\hat{a}_{0}(\bm{k})e^{i\bm{k}\cdot\bm{x}}+\textit{h.c.}\right)\,.$
(24)

Here, $\hat{a}_{0}(\bm{k})$ and its Hermitean conjugate are the constant
annihilation and creation operators. The time evolution is contained entirely
in the mode function, which obeys the equation

$\left(d_{t}^{2}+\omega^{2}_{k}(t)\right)v_{k}=0\,,$ (25)

with

$\omega^{2}_{k}(t)\equiv m_{\chi}^{2}+k^{2}+2g^{2}\phi_{0}^{2}\,{\rm
cos}^{2}(mt)\,.$ (26)

The expected particle-creation is then given as

$n_{k}(t)\equiv\langle
0|\hat{n}_{k}(t)|0\rangle=\frac{1}{2\omega_{k}}\left(|\dot{v}_{k}|^{2}+\omega^{2}_{k}|v_{k}|^{2}\right)-\frac{1}{2}\,,$
(27)

where $\hat{n}_{k}$ is the time evolved operator of the number density of
$\chi$-particles per mode $\bm{k}$. The initial conditions of $v_{k}$ are
constrained by being consistent with the commutation relations of the
operators as well as by initially defining the lowest energy state. Equations
(25) and (26) imply that $v_{k}$ obeys a Mathieu equation. In order to find
(27) in the high multiplicity regime (11), we can refer to the treatment of
the Mathieu equation in [32]. Equations (25) and (26) can be parametrized as

$d_{t}^{2}x+\omega_{0}^{2}\left(1+h\cos\left(\gamma t\right)\right)\,x=0\,,$
(28)

where the correspondence of parameters is

$\displaystyle\omega_{0}^{2}$
$\displaystyle\leftrightarrow\overline{\omega_{k}}^{2}\equiv
m_{\chi}^{2}+k^{2}+g^{2}\phi_{0}^{2}\,,$ $\displaystyle h$
$\displaystyle\leftrightarrow\frac{g^{2}\phi_{0}^{2}}{\overline{\omega_{k}}^{2}}\,,$
$\displaystyle\gamma$ $\displaystyle\leftrightarrow 2m\,.$ (29)

Equation (28) can exhibit so called parametric resonance, i.e., has solutions
that for certain parameter combinations exhibit exponential growth between the
cycles of period $\tau\equiv 2\pi/\gamma$:

$x(t+\tau)=e^{s\tau}x(t)\,,$ (30)

with a parameter of instability $s>0$. For the particle number one thus has

$\frac{n_{k}(t+\tau)}{n_{k}(t)}\sim\exp(2s\tau)\,,$ (31)

and, coarse graining over several periods, one has

$\frac{\dot{n}_{k}}{n_{k}}\sim s\,.$ (32)

In [32], it is shown that for $h\ll 1$, there is parametric resonance in the
bands

$\gamma=\frac{2\omega_{0}}{l}+\epsilon\,,\quad l\in\mathbb{N}\,.$ (33)

The maximal value of the exponent within these bands scales as

$s\sim h^{l}\,,$ (34)

as does the width $\epsilon$. Comparing (14) and (33), one has the
correspondence $l\leftrightarrow n/2$. Therefore, from (21) and (32), we see
that the quantum rate (18) is to be compared with $s$ for $2l$:

$s\sim
h^{2l}\leftrightarrow\left(\frac{g^{2}\phi_{0}^{2}}{m^{2}}\frac{1}{n^{2}}\right)^{n}\,,$
(35)

where we have used (II.3) and $h\ll 1$. The agreement with the parametric
scaling of (18) can be seen to be complete.

##  III Example 2: Scalar QED

A second example is provided by scalar QED with vanishing fundamental self-
coupling of the scalar,

$L=D_{\mu}\phi(D^{\mu}\phi)^{\dagger}-m^{2}\phi^{\dagger}\phi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,$
(36)

where $D_{\mu}\equiv\partial_{\mu}-igA_{\mu}$. Let us consider the same
questions as in the previous example with now $\phi$ and $A$ in the roles of
$\phi$ and $\chi$. That is, semiclassically, we consider out of the vacuum
production of a photon pair in the background

$\phi_{B}=\phi_{B}^{\dagger}=\phi_{0}\cos(mt)\,,\quad A_{B}^{\mu}=0\,,$ (37)

whereas in the fully quantum picture, we consider the many-particle
annihilation processes

$\frac{N}{2}s^{-}+\frac{N}{2}s^{+}\to\frac{N-n}{2}s^{-}+\frac{N-n}{2}s^{+}+2\gamma\,.$
(38)

quantum and classical parameters are now related by

$\frac{N}{V}=m\phi_{0}^{2}\,.$ (39)

and the double scaling limit (1) in semiclassical terms takes the same form as
(10).

###  III.1 Quantum calculation

The presence of the 3-point vertex,
$igA^{\mu}\phi^{\dagger}\partial_{\mu}\phi+h.c.$, in principle opens up a
large variety of Wick contractions contributing to the process (38) at leading
order $\left(Ng^{2}\right)^{n}$. However, all but one of the corresponding
diagrams are vanishing due to the combination of gauge redundancy and the
special condensate kinematics. This can be seen as follows. All diagrams
involving an incoming scalar pair connected by a single 3-point vertex vanish
because the derivative yields a factor of zero in the case of the initial
condensate momenta. That leaves only diagrams of the type fig. 1, where the
dotted lines represent a photon and the 4-point vertices are either elementary
or effective ones consisting of two 3-point vertices with one internal
$\phi$-line:

$-ig^{2}\frac{q^{\mu}_{2l+2}q^{\nu}_{2l}}{q^{2}_{2l+1}-m^{2}}\,.$ (40)

Here, $l$ is the number of vertices preceding the vertex. In every diagram
with one or more vertices like (40), two momenta $q_{l}$ are contracted only
with photon polarization vectors that are orthogonal:

$q_{l}^{\mu}\,\epsilon_{\mu}\left(q_{l^{\prime}},r\right)=0\,,\quad\forall\,\,l,l^{\prime}\,.$
(41)

This relation holds for transverse polarizations $r$, which either belong to
an outgoing photon or a neighbouring photon propagator, whose non-transverse
part has been projected out by an outgoing photon. Thus the only non-vanishing
diagram is the one constructed solely out of the elementary 4-point vertex,
$g^{2}A_{\mu}A^{\mu}\phi^{\dagger}\phi$. The value of the diagram is different
from the 2-scalar case only by a factor

$\epsilon^{*}_{\mu}(k,r)\epsilon^{*\mu}(k^{\prime},r^{\prime})=\delta_{r,r^{\prime}}\,.$
(42)

The resulting tree-level rate for $\frac{n}{2}s^{-}+\frac{n}{2}s^{+}\to
2\gamma$ for each of the two polarizations is thus identical with (16). The
combinatoric enhancement factor due to the initial $N$ quanta (using the
Stirling approximation) is

$C_{Nn}=\begin{pmatrix}N/2\\\ n/2\end{pmatrix}^{2}\sim
2^{-n}\frac{N^{n}}{(n/2)!^{2}}\left(1+\mathcal{O}\left(\frac{n^{2}}{N}\right)\right)\,.$
(43)

With this and (39), the rate for the process (38) to leading order is given by

$\Gamma\equiv C_{Nn}\Gamma_{\frac{n}{2},\frac{n}{2}\to
2\gamma}\sim\left(\frac{e^{2}}{2n^{2}}\frac{g^{2}\phi_{0}^{2}}{m^{2}}\right)^{n}\,,$
(44)

where again we have used the Stirling approximation and omitted factors that
scale less strongly with $n$ than exponentially.

###  III.2 Semiclassical calculation

In the limit (10), several terms in the equations of motion are suppressed.
Furthermore, the background $\phi_{B}$ obeys a harmonic equation and has
vanishing current. Thus, with the notation

$x\equiv\text{Re}\,\delta\phi\,,\quad y\equiv\text{Im}\,\delta\phi\,,$ (45)

the equations for $\delta\phi$ read

$\left(\partial^{2}+m^{2}\right)x=\mathcal{O}\left(g\right)$ (46)

and

$\left(\partial^{2}+m^{2}\right)y=gA^{\mu}\partial_{\mu}\phi_{B}+g\partial_{\mu}\left(A^{\mu}\phi_{B}\right)+\mathcal{O}\left(g\right)\,.$
(47)

The equation for $A_{\mu}$ reads

$\partial_{\mu}F^{\mu\nu}=2g^{2}\phi^{2}_{B}A^{\nu}-2g\left(\phi_{B}\partial^{\nu}y-y\partial^{\nu}\phi_{B}\right)+\mathcal{O}\left(g\right)\,.$
(48)

One sees that in the limit, $x$ decouples as a free field. Projecting out the
transverse polarizations of $A_{\mu}$, their equation becomes

$\displaystyle\left(\partial^{2}-2g^{2}\phi^{2}_{B}\right)A^{j}_{T}=\mathcal{O}\left(g\right)\,.$
(49)

Thus, like in the previous example, the propagating photon degrees of freedom
decouple from the fluctuations $\delta\phi$. The creation of $s^{+}s^{-}$ is a
result of the interaction of $y$ and the Coulomb degree of freedom encoded in
$A_{0}$ and $A^{j}_{L}$. Restricting the attention to processes of photon
creation, (49) implies that the mode functions of the two transverse photon
polarizations likewise obey a Mathieu equation,

$\left(d_{t}^{2}+\omega^{2}_{k}(t)\right)v_{k,r}=0\,,$ (50)

where now

$\omega^{2}_{k}(t)\equiv k^{2}+2g^{2}\phi_{0}^{2}\,{\rm cos}^{2}(mt)\,.$ (51)

This can again be parametrized as in (28) and the analogous version of (II.3).
Therefore, the semiclassical prediction for the rate is again

$\frac{\dot{n}_{k,r}}{n_{k,r}}\sim\left(\frac{g^{2}\phi_{0}^{2}}{m^{2}}\frac{1}{n^{2}}\right)^{n}\,.$
(52)

Thus, in this example, too, the parametric agreement of (44) and (52) is
complete.

##  IV Example 3: Scalar QED with massive photon

Let us consider again scalar QED, but now with a non-zero Proca mass $m$:

$L=D_{\mu}\phi(D^{\mu}\phi)^{\dagger}-m_{e}^{2}\phi^{\dagger}\phi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}A_{\mu}A^{\mu}\,.$
(53)

If photon and electron in the preceding example interchange roles, we are
dealing with pair-creation in an electric field. Semiclassically, we are
looking at an out of the vacuum creation of $s^{+}s^{-}$ in a theory with
fields quantized around the background

$A_{B}^{\mu}=\delta^{\mu}_{z}\frac{E_{0}}{m}\cos\left(mt\right)\,,\quad\phi_{B}=0\,.$
(54)

The relevant processes in the fully quantum treatment on the other hand are

$N\gamma\to\left(N-n\right)\gamma+s^{+}s^{-}\,.$ (55)

quantum and classical parameters are approximately related through

$\frac{N}{V}=\frac{1}{2m}E_{0}^{2}\,.$ (56)

The double scaling limit (1) in semiclassical terms then takes the same form
as (10) with $E_{0}/m$ in place of $\phi_{0}$. Equation (54) corresponds to a
background electromagnetic field of the form

$F_{0j}=\delta_{j}^{z}E_{0}\sin\left(\omega t\right)\,,\quad F_{kj}=0\,,$ (57)

with frequency $\omega=m$. This field only has an electric component pointing
in $z$-direction. For example, it may serve as an approximate description of
the field created in the antinodes of superposing laser light, on length
scales short compared to the wavelength, $2\pi/\omega$. For an optical or
X-ray laser, the kinematic threshold is necessarily high, $n_{0}\equiv
2m_{e}/m\gg 1$. This is in contrast to the previous examples with $m_{\chi}$
arbitrary and $m_{\gamma}=0$, respectively. For such a hierarchy, the dominant
process is the one closest to the threshold,

$n=n_{0}+\delta\,,\quad 0<\delta\leq 1\,,$ (58)

as follows from the scaling of (18) and (44) and will turn out to be the case
here, too. The regime (11) for $n\sim n_{0}$ corresponds to

$\ \frac{gE_{0}}{mm_{e}}\ll 1\,.$ (59)

###  IV.1 Semiclassical result

The semiclassical rate of pair-creation in the background field (57) averaged
over a period of oscillation has been found in [33, 34] (see also [35]) in the
regime of $n_{0}\gg 1$ and $E_{0}\ll m_{e}^{2}/g$. The full result
interpolates between the following two asymptotic expressions. In the regime
of $\frac{gE_{0}}{m_{e}\omega}\gg 1$, the rate asymptotes to

$\displaystyle\Gamma\sim$
$\displaystyle\frac{Vm_{e}^{4}}{2\sqrt{2}\pi^{4}}\left(\frac{gE_{0}}{m_{e}2}\right)^{5/2}\exp\left(-\pi\frac{m_{e}^{2}}{gE_{0}}\right)\,,$
(60)

which is essentially the suppression obtained by Schwinger for the case of a
constant electric field [36]. For the opposite case,
$\frac{gE_{0}}{m_{e}\omega}\ll 1$, the result is asymptotic to

$\displaystyle\Gamma\sim\frac{Vm_{e}^{4}}{(2\pi)^{5/2}}e^{-2\delta}\left(\frac{\omega}{m_{e}}\right)^{5/2}\left(\frac{e}{4}\frac{gE_{0}}{m_{e}\omega}\right)^{2(n_{0}+\delta)}\text{Erfi}\left(\sqrt{2\delta}\right)\,.$
(61)

The latter regime is sometimes referred to as the multi-photon regime and
coincides with (59). In the following perturbative quantum calculation, we
indeed fully reproduce (61) in terms of an $n$-photon process (55).

###  IV.2 Quantum calculation

In the present case, all possible Wick contractions lead to non-vanishing
diagrams. However, near the kinematic threshold, one particular diagram
dominates over the others. This is the diagram constructed purely out of the
4-point vertex, as obtained from the digram in fig. 1 upon exchanging $\phi$s
for $\gamma$s and the $\chi$-pair for $s^{+}s^{-}$. Let us denote by
$\delta\Gamma_{4}$ the contribution to the rate from the square of only that
diagram. Isolating such contributions may be useful despite the contribution
of interference terms in the case in which certain diagrams dominate over
others. In terms of the rate (16) in the 2-scalar example, we have

$\delta\Gamma_{4}=2^{-n-1}\Gamma_{n\phi\to 2\chi}\,.$ (62)

Here, the relative factor of $2^{-n}$ comes from the lesser number of Wick
contractions since the complex field $\phi$ replaces the real field $\chi$.
Likewise, there is another relative factor of $1/2$, since the final particles
are not identical. All other diagrams feature photon-insertions via 3-point
vertices. The opposite extreme case of the diagram considered above, arising
from contracting solely 3-point vertices, is depicted in fig. 2.

Figure 2: One of the terms contributing to the amplitude of the process (55).

Let us denote by $\delta\Gamma_{3}$ the contribution based on its square. It
can be obtained in an analogous calculation and the result is

$\frac{\delta\Gamma_{3}}{\delta\Gamma_{4}}=\frac{e^{2n}}{8n}\left(1-\frac{n_{0}^{2}}{n^{2}}\right)^{n}\sim\frac{e^{2n}}{8n}\left(\frac{2\delta}{n_{0}}\right)^{n}\,,$
(63)

where the last relation holds for the dominant process (58). This is strongly
suppressed for sufficiently small $\delta/n_{0}$. The above consideration
indicates that a 3-point vertex results in a suppression factor near the
threshold and that therefore for the process (58) the amplitude is well
approximated by the diagram in fig. 1. We thus have

$\Gamma_{n\gamma\to
s^{+}s^{-}}=\delta\Gamma_{4}\left(1+\mathcal{O}\left(\delta\right)\right)\,.$
(64)

The combinatoric enhancement is the same as in (17). Relating $N$ and $E_{0}$
via (56), we thus have

$\Gamma\sim\frac{1}{8\pi^{3}}Vm^{4}\sqrt{1-\frac{n_{0}^{2}}{n^{2}}}n^{2}\left(\frac{e}{4}\frac{gE_{0}}{m_{e}m}\frac{n_{0}}{n}\right)^{2n}\,,$
(65)

where we have used (17) and the Stirling approximation. For the process (58),
one has $1-(n_{0}/n)^{2}\sim 2\delta/n_{0}$ as well as $(n_{0}/n)^{2n}\approx
e^{-2\delta}$. Using these relations, (65) can be seen to agree completely
with (61), if in the latter one retains only the leading order of

$\text{Erfi}\left(\sqrt{2\delta}\right)=2\sqrt{\frac{2}{\pi}}\sqrt{\delta}\left(1+\frac{2}{3}\delta+\mathcal{O}\left(\delta^{2}\right)\right)\,.$
(66)

The relative error for $\delta\ll 1$ is thus $\sim\delta$ and for $\delta=1$
it is $\approx 0.6$.

##  V Parameter regimes

In the preceding sections, we have considered the rate for a many particle
process of the form $n\to 2$ in the example of three different models. In a
certain regime, we have found a result which scales
$\sim\left(g^{2}N\right)^{n}$. In this section, we are going to look in more
detail at the limits of validity of this leading order approximation. For
definiteness, we are going to base the discussion on the
$\phi^{2}\chi^{2}$-model.

###  V.1 Semiclassical limit

In the limit (1), the following two things happen. First, the backreaction
from the created $\chi$-quanta on the state of $\phi$ vanishes. Secondly, loop
corrections, that are higher order in $g^{2}$, vanish as well.

As a result, any non-perturbative treatment that exploits a semiclassical
approximation and neglects radiative corrections becomes exact. Of such kind
are the semiclassical calculations referenced in the three examples
considered.

Their result can be understood as the resummation of the set of diagrams that
do not vanish in the limit (1). Those diagrams are of the same order in $N$
and in $g^{2}$. At the leading order, $n$, those are the diagrams that we
evaluated in the quantum perturbative calculations. At higher orders, $n+k$,
those are what may be called forward scattering diagrams, with $k$
$\phi$-quanta all scattered into the initial mode.

As we have seen, in the regime (11) (regime (59), respectively), which in
terms of $N$ reads

$n^{2}\gg\frac{g^{2}N}{Vm^{3}}\,,$ (67)

the diagrams beyond the leading order, $k\geq 1$, are negligible. On the other
hand, when approaching the opposite regime, the effect of their resummation is
both to shift the kinematic threshold through an effective contribution to the
particle masses and to alter the scaling of the rate. The power-law growth
with coupling and field strength gets softened before the rate becomes large.
In the case of the Mathieu analysis (in sec.s II and III), that is the
transition between narrow and broad resonance. For the example of pair-
creation in the alternating electric field (sec. IV), the interpolation is
shown in (60) and (61).

Figure 3: Schematic representation of regime boundaries: Above the grey
diagonal line, non-perturbative corrections are negligible (see (67)).
Additional regimes for finite $g^{2}$ (and the example of negligible
$m_{\chi}$): To the left of the blue vertical line, loop corrections to the
potential are negligible (see (69)); Below the blue horizontal line,
contributions of loop induced diagrams are negligible (see (74)). The overlap
of coloured areas is the resulting regime in which the calculation is
perturbative in both $g^{2}N$ and $g^{2}$, i.e., both non-perturbative and
quantum corrections to the leading order approximation are negligible.

Fig. 3 offers a schematic graphical representation of the perturbative and
non-perturbative regime in the parameter plane of $n$ and $g\phi_{0}/m$ as
well as further regimes to be discussed below.

###  V.2 Finite $g^{2}$

Figure 4: ”Symmetrically branching tree” (SBT) diagram: Diagram contributing
to the amplitude of the process $n\phi\to 2\chi$ based on only a single cross
coupling vertex and otherwise only the quartic self-coupling of $\phi$ (with
the most symmetric shape possible).

For finite $N$ and $g^{2}$, there are various quantum corrections arising.

Corrections of backreaction-type, such as depletion or evolution of
entanglement, are $1/N$-suppressed [6, 11]. Although such corrections build up
over time, for the first emissions they are negligible in the case of
sufficiently large $N$.

Likewise, corrections due to a single loop or contributions from processes
with a single (non-forward) scattered $\phi$ are negligible due to the extra
power of the coupling $g^{2}$.

In the following we attempt to identify and estimate other loop effects that
may be significant despite $g^{2}\ll 1$. For simplicity, from now on we are
going to focus on the case of a negligible mass $m_{\chi}$.

####  V.2.1 Effective potential

Consider the effective potential at 1-loop [37]. For $\chi=0$, $\phi$ moves in
the potential

$V_{\textit{1Loop}}(\phi,0)=\frac{m^{2}}{2}\phi^{2}+\frac{g^{4}\phi^{4}}{16\pi^{2}}\left(\log\left(\frac{2g^{2}\phi^{2}}{\mu^{2}}\right)-\frac{3}{2}\right)\,,$
(68)

where parameters are defined in $\overline{\text{MS}}$. For $\phi$-values for
which the loop-correction is not negligible, the time evolution $\phi(t)$ as
in (3) as well as the energetics relating $N$ and $\phi_{0}$ as in (9) are
altered and the considerations of the preceding sections do not go through.
Equation (68) implies that quantum corrections of this kind are only
negligible for (with a choice of $\mu\sim m$)

$\frac{\phi_{0}}{m}\ll\frac{1}{g^{2}}\,.$ (69)

This constrains the initial state in terms of the theory parameter $g^{2}$
(see fig. 3).

####  V.2.2 Diagrams with different $n$-scaling

Figure 5: Momentum dependent $4$-$\phi$-vertex induced by a $\chi$-loop with
momenta as occurring in the SBT diagram (fig. 4)

Another kind of quantum correction may come from loop diagrams that compared
to the leading order are suppressed by much higher powers of $g^{2}$, but on
the other hand feature a significantly different scaling with $n$. This may
happen due to different momentum flow through a diagram permitted by different
Wick contractions. In particular, a quartic self-coupling of $\phi$ gives rise
to a well-known case of a diagram with strong $n$-scaling [38, 39]. This is
the ”symmetrically branching tree” (SBT) diagram shown in fig. 4.

The required self-coupling is induced via the cross-coupling through a
$\chi$-loop (see fig. 5) and the corresponding momentum dependent vertex has
the value

$v_{4}\left(q^{2}\right)=-\frac{ig^{4}}{16\pi^{2}}\left(\log\left(\frac{4q^{2}}{\mu^{2}}\right)-2+i\pi\right)+\mathcal{O}\left(g^{6}\right)\,.$
(70)

As is evident from fig.s 4 and 5, we have $m^{2}\leq
q^{2}\leq\frac{1}{9}n^{2}m^{2}$, depending on the position of the vertex in
the diagram. Therefore, there is no significant $n$-scaling implicit in the
momentum dependence of the vertex and $4!|v_{4}|\sim g^{4}$ serves as a good
estimate throughout the diagram.

The rate based on only the square of the SBT diagram can be found in a
calculation analogous to the ones given in [38, 39]. For the present purposes,
the following schematic representation is sufficient:

$\delta\Gamma_{\textit{SBT}}\sim\left(\frac{c}{Vm^{3}}\right)^{n}n!\,,\quad
c\sim g^{4}\,.$ (71)

Due to the presence of the factorial, perturbation theory breaks down for

$n\gtrsim n_{\textit{max}}\equiv\frac{Vm^{3}}{c}\,.$ (72)

Comparing the contribution (71) with the leading order rate (16), we see that
it becomes important for

$n\gtrsim n_{eq}\equiv\sqrt{g^{2}/c}\,,$ (73)

which takes place well before the breakdown. There are of course further
diagrams as well as interference, but we may take (73) as an indication for
the $n$-regime where contributions due to loop-induced couplings are no longer
negligible. The limit of validity of the leading order approximation (16) is
then parametrically given by

$n\ll\frac{1}{g}\,,$ (74)

which is a bound independent of the initial state parameter $\phi_{0}$.

As can be seen in fig. 3, the combination of constraints on $\phi_{0}$ (on
$N$) and $n$ leaves a finite area in which corrections due to effects of
higher order in $g^{2}$ and in $g^{2}N$ are negligible.

##  VI Quantumization and classicalization

It is an outstanding question under what conditions a classical system can
reach a regime in which the classical approximation is no longer applicable
and how rapidly this can happen. In this respect, the first important concept,
introduced within the black hole $N$-portrait [1], was macro-quantumness [5,
40]. This term encoded the previous observations that $1/N$ quantum effects
lead to features that cannot be accounted classically, e.g., emergence of
black hole hair [4].

The question about the speed of the breakdown of classicality was posed in [3]
(and extended in [6, 9]), where this phenomenon was called “quantum breaking”
and the corresponding timescale the “quantum break-time”, $t_{Q}$.

In [3] it was shown that in an $N$-particle system that exhibits a Lyapunov
instability, the quantum break-time can be logarithmically short,

$t_{Q}=\lambda^{-1}\ln(N)\,,$ (75)

where $\lambda$ is the Lyapunov exponent. In particular, this effect was
explicitly demonstrated for the $1+1$-dimensional condensate of bosons on a
ring with attractive interaction [3] (for a more recent discussion of this
model, see [41]).

On the other hand, for a system that exhibits no classical instability, the
situation is different. For such a system the following general bound on the
quantum break-time was imposed [9]:

$t_{Q}\sim\frac{N}{N^{2}\Gamma_{2\to 2}}\,.$ (76)

Here, $\Gamma_{2\to 2}$ is the re-scattering rate of a pair of constituents.
We deliberately kept the $\sim N^{2}$ factor in the denominator, which comes
from the Bose-enhancement of the initial quanta. Basically, it counts the
number of pairs in the condensate, which is $C_{Nn}=N(N-1)/2$. The expression
(76) says that in order for the condensate to quantum break, of order $N$
scattering acts must take place. For the example of a self-coupled scalar of
[9], the expression (76) has been reproduced in [16] via different methods.

In other words, the main engine for quantum breaking is a gradual loss of
coherence due to scattering of small number of constituents into the external
quanta, as opposed to non-gradual, i.e., single-process transitions with the
participation of many constituents. For a stable classical system, single-
process transitions into a quantum state are suppressed.

To this type of processes, in the present paper, we give the special name of
quantumization. Although the term can be defined in a broader sense, the
processes that we have studied here, specifically amount to transitions from
an initially classical state into a quantum state with a small number of
constituents. Such processes can be viewed as the transitions from macro to
micro systems.

The opposite process, a single-process transition from a few-particle quantum
state into a classical one, is known as classicalization [22, 23, 25, 24, 26,
27].

Our analysis contributes into understanding of physics of both processes and
in their very different manifestations.

In particular while unsuppressed classicalization is possible in certain
systems, quantumization is universally suppressed. At first glance, this may
sound rather surprising, since both processes originate from the same basic
phenomenon: transition from few to many, or vice versa. For definiteness,
under “few” let us think of two-particle state $\left|\,2\right>$ and under
“many” the state $\left|\,n\right>$ of $n\gg 1$ quanta. We shall assume that
both are approximated as valid asymptotic $S$-matrix states.

The basic ingredient controlling both transition processes is the square of
the $S$-matrix element $|\left<2\,\right|\hat{S}\left|\,n\right>|^{2}$. This
element is always suppressed. In fact using very general arguments based on
the effective Hamiltonian and locality of the Hilbert space, one can argue
that at weak coupling and large $n$ this element is bounded from above (cf.
[27], which refines [26]),

$|\left<2\,\right|\hat{S}\left|\,n\right>|^{2}\leqslant e^{-n}\,.$ (77)

However, in the estimate of the total probability of the transition, the
matrix element is summed over the degeneracy of the final states [1],

$\Gamma_{i\to
f}\propto\sum_{f}\,|\left<2\,\right|\hat{S}\left|\,n\right>|^{2}\,.$ (78)

Depending on whether the out state is $\left<2\,\right|$ or
$\left<n\,\right|$, the degeneracy factors can be dramatically different. In
particular, the degeneracy of two-particle states can never be exponentially
large, without compromising the validity of the theory, whereas the degeneracy
of $n$-particle states can [27]. This is the core reason for why
quantumization and classicalization are realised in nature very differently.

We shall discuss the two phenomena separately and compare them. For
definiteness, in the following, we are going to refer to the
$\phi^{2}\chi^{2}$-example. The discussion equally applies to the other
examples.

###  VI.1 Quantumization

The bound (76) provides an explanation of the fact that quantumization
phenomena are commonly not observed. For example, telecommunication does not
suffer from complications due to transitions of electromagnetic waves to
mostly quantum states upon encountering a charged particle. Neither is
quantumization expected for objects that are not directly observed. To repeat
the example given in the introduction, a black hole is not expected to decay
into a pair of high energetic photons although no conservation law would bar
such a transition.

The systems studied above allow us to observe the reason for suppression
explicitly. In the considered systems, quantumization would correspond to the
process $n\rightarrow 2$ with $n\sim N$. In this case, an order one fraction
of the energy stored in the initial coherent state is transferred to the
created particle pair.

Since the basic $S$-matrix element is suppressed, the rate requires
enhancement due to combinatorics associated with either initial or the final
states or both.

One possibility is the enhancement due to large number of choices of $n$
quanta from the total set of initial particles.

However, we must note that the increase of the initial occupation number of
quanta up to a level $N\sim g^{-4}$, will invalidate the weak coupling
treatment. Firstly, in such a regime, the collective interaction from the
rest, substantially modifies the dispersion relation of each particle.
Secondly, for such a high occupation number, the condensate develops Lyapunov
instabilities and a coherently oscillating field (3) no longer represents a
valid classical background. This is also clear from the one-loop effective
potential (68), which corrects the free oscillations for amplitudes exceeding
$m/g^{2}$. Correspondingly, the validity of our treatment is limited by (69),
which in terms of $N$ reads

$N\ll\frac{Vm^{3}}{g^{4}}\,.$ (79)

To make the above explicit in the example of a condensate decaying into a
particle pair, let us compare the predicted decay time to the timescale of an
oscillation, $\sim m^{-1}$, or to the one required for a complete gradual
decay,

$\tau_{\textit{gradual}}\lesssim N\,\Gamma_{2\to 2}^{-1}\sim
m^{-1}N\frac{Vm^{3}}{g^{4}}\,.$ (80)

This upper bound is neglecting the Bose-enhancement due to the $N$ quanta in
the initial state (which is taken into account in (76)) as well as the
accumulating Bose-enhancement due to the created $\chi$-particles
(corresponding to the occurrence of parametric resonance in the Mathieu-
equation (25)). We shall see that nevertheless non-gradual decay is negligible
during timescales on the order of (80).

The coherent superposition is sharply peaked around the mean occupation $N$,
but it has non-zero support for other occupations, including ones higher than
$N$. In order to parametrize the range of $n$ corresponding to $n\sim N$ we
may therefore introduce a parameter $r$ and sum the rates for processes with
$N/r\leq n\leq r\,N$ (where appropriate rounding is implicit and values
$r\gtrsim 10$ are sensible). This range of $n$ is in the perturbative regime
(67) as long as

$r^{2}\ll NVm^{3}/g^{2}\,.$ (81)

Depending on the number state in the coherent superposition, for a given $n$,
there is a number $n^{\prime}$ of unscattered initial quanta. The sum over
final states therefore includes a sum over $n^{\prime}$. To see what effective
combinatoric enhancement that sum amounts to for a given $n$, we note that

$\displaystyle\sum_{n^{\prime}=0}^{\infty}\left|\langle
n^{\prime}|\hat{a}^{n}|c\rangle\right|^{2}=\sum_{n^{\prime}=0}^{\infty}\begin{pmatrix}n+n^{\prime}\\\
n\end{pmatrix}\left|\langle 0|\hat{a}^{n}|n\rangle\langle
n+n^{\prime}|c\rangle\right|^{2}=$ $\displaystyle=$
$\displaystyle\frac{N^{n}}{n!}\left|\langle
0|\hat{a}^{n}|n\rangle\right|^{2}\,,$ (82)

where the occupations of the number states such as $|n\rangle$ and of the
coherent state $|c\rangle$ are all referring to the same mode and
$\hat{a}|c\rangle=\sqrt{N}|c\rangle$. We therefore have for the rate of non-
gradual decay

$\Gamma_{n\sim N}\sim\sum_{n=N/r}^{r\,N}\frac{N^{n}}{n!}\Gamma_{n\to 2}\,.$
(83)

In the regime where for the quantity $\Gamma_{n\to 2}$ the result (16) is
valid, we see that the timescale $\Gamma^{-1}_{n\sim N}$ is much larger than
(80), as long as (81) is fulfilled. If instead the scaling (71) is valid, the
summation range of $n$ is within the perturbative unitarity bound (72) if
$N\leq n_{\textit{max}}/f$, which coincides with (79). In that case we see
that $\Gamma^{-1}_{n\sim N}$ is still much larger than (80).

Likewise, the rate of quantumization cannot be enhanced significantly through
degeneracy of the final $2$-particle states. The reason is that the
exponential increase of $2$-particle degeneracy would make the theory strongly
coupled, thereby invalidating the degrees of freedom.

A simple way for illustrating this is to notice that the degeneracy of final
states can be increased by endowing the $\chi$-particles with an internal
“flavor” quantum number $j=1,2,...,N_{f}$. For example, $\chi_{j}$ can form an
$N_{f}$-dimensional representation of the $SO(N_{f})$ symmetry group. This of
course will increase the rate (16) of the transition $n\phi\to 2\chi$ by a
factor $N_{f}$. However, $N_{f}$ cannot be arbitrarily large. The requirement
that $\phi$ and $\chi$-particles be valid (weakly interacting) degrees of
freedom, puts the following bound on the collective coupling:

$N_{f}g^{4}\,\lesssim\,1\,.$ (84)

Violation of this bound changes the regime of the theory. In particular this
is signalled by the breakdown of the loop expansion.

The above bound makes the enhancement of the quantumization process negligible
as compared to the exponential suppression of the matrix element (77).

Notice that the suppression of a fast transition from the classical state of
$n$ $\phi$s into a quantum state of two $\chi$s also illustrates the
difficulty of rapid generation of quantum entanglement from the classical
state. Indeed, the $SO(N_{f})$-invariant $2$-particle state,

$\left|\,2\right>=\frac{1}{\sqrt{N_{f}}}\sum_{j=1}^{N_{f}}\left|\,\chi_{j}\right>\times\left|\,\chi_{j}\right>\,,$
(85)

is entangled with respect to the flavor quantum number. However, its
production rate is highly suppressed. Instead in a classically stable system
the entanglement can only be generated gradually and requires the time of
order (76) [6, 7, 12].

###  VI.2 Classicalization

The story is very different for classicalization processes [22]. Such a
process comes from the inverse transition $2\to n$ [23, 25, 26, 27].

Of course, for a fixed pair of initial and final states, the absolute values
of $S$-matrix elements for transitions in both directions, $|S_{2\to n}|$ and
$|S_{n\to 2}|$, are exactly equal. However, the degeneracy of an $n$-particle
final state is potentially much higher. If the degeneracy is sufficiently
high, it can overcome the suppression. This is the essence of classicalization
[26, 27].

For example, endowing $\phi_{j}$ with the $SO(N_{f})$ index, exponentially
increases the degeneracy of the available states with $n$ $\phi$-bosons [27].
Of course, by symmetry, the transition elements to each member of the same
irreducible representation are equal. The end-result is enhanced by the
dimensionality of the representation. For example, a symmetric state of $n$
quanta has degeneracy $C_{N_{f}+n,n}=(N_{f}+n)!/n!N_{f}!$, which scales as
$e^{n}$ for $N_{f}\sim n$. Of course, we do not claim that a simple
$\phi^{2}\chi^{2}$-theory exhibits classicalization at order-one rate, but the
tendency of enhancement of classicalizing transition $2\to n$ versus
quantumization process $n\to 2$ is clear. 111Notice that one could try to
enhance the rate of an $n\to 2$ quantumization process by taking the initial
$n$-particle state as the normalized invariant integral over the entire
representation space. This would enhance the amplitude by the dimensionality
of the representation. However, such an initial state represents a
superposition of all basis state vectors and cannot be regarded as classical.
Therefore, such a transition does not fall in our category of quantumization
processes. However, such transitions can potentially play an important role in
the evolution of a macroscopic system, such as a black hole, after its quantum
break-time, since by then the system can evolve into an entangled
superposition of many microstates (see the discussion in VII).

Correspondingly, the $n$-particle states of high entropy, the so-called
saturons, can in principle be produced with unsuppressed rate in a collision
of two energetic quanta [27]. Nature provides an example of a saturon in the
form of black holes.

It has been believed for a long time [42, 43, *AmatiStar, 45, *GrossStar] that
a black hole can form in a collision of two particles with unsuppressed
probability. This view is supported by the semiclassical intuition that a
black hole must form whenever energy is localized within its gravitational
radius. In particular, this feature served as the basis for the idea of self-
completion of gravity [47].

However, as usual, without a microscopic theory of a black hole, it is
impossible to verify and understand the underlying quantum mechanism for its
unsuppressed formation. This is provided by the black hole $N$-portrait [1],
where a black hole is described as a condensate of $N$ soft gravitons, with
the microstate entropy $\sim N$.

In this theory, the creation of a black hole from colliding quanta represents
an instance of classicalization, $2\to N$, in which a highly degenerate
$N$-graviton state is formed [1]. Due to the high multiplicity involved in the
process, the probability of transition to each microstate is suppressed by
$e^{-N}$ [25, 28]. Despite this, the total probability of black hole formation
is of order one. The reason for this is found in the non-perturbative
enhancement of the transition rate due to degeneracy of the $N$-graviton
state, which accounts for the black hole entropy [1, 25].

The high multiplicity nature of a black hole likewise serves as an explanation
of stability against instantaneous decay. Indeed, such a decay, $N\to 2$, is
controlled by the same exponentially suppressed $S$-matrix element as the
formation process, $N\to 2$ [1, 25]. However, the crucial difference from the
formation case is the lack of the entropy-enhancement. This is due to an
insufficient degeneracy of the final $2$-particle state. This explains why a
classical black hole cannot explode into two (or few) particles.

Of course, the term “explosion into few particles” in the case of a black hole
must be taken with a grain of salt, since the decay products gravitate.
Correspondingly, if the particles materialize at separations not much larger
than the initial gravitational radius, the accompanying classical
gravitational field is too strong to be neglected. Calculationwise, this
merely amounts to a dressing of the outgoing energetic quanta with a classical
gravitational field in the form of a coherent state of secondary “softer”
gravitons.

It has been suggested [48] that another candidate for the saturated entropy
state is a Color Glass Condensate of gluons in QCD [49]. The formation of this
state in collision of protons can then be viewed as a process of
classicalization. It has also been proposed [27] that confinement in QCD with
large number of colors can be interpreted as the process of formation of a
saturated state of gluons due to excessive color degeneracy.

The formation of saturon bound states in a $2\to N$ process within the Gross-
Neveu model [50] has been studied in [51]. This analysis clearly indicates the
entropy enhancement of an otherwise exponentially suppressed transition rate.

##  VII Summary and outlook

In the present paper we have resolved a class of non-perturbatuve
semiclassical phenomena in terms of perturbative quantum processes involving
many quanta. This continues the line previously applied to objects such as
black holes [1] and de Sitter [1, 6] (see section I for a more complete list
of references). This resolution makes certain properties very transparent. One
very important physical implication is the universal suppression of
quantumization, defined as a single-process transition from a classical object
to a few-particle quantum state.

Classically stable objects, such as coherently oscillating fields, black
holes, or de Sitter space do not “explode” via creation of few highly
energetic quanta. Rather they loose coherence gradually, via quantum processes
that involve scattering of a small number of constituents. Likewise,
entanglement cannot be generated via a single-process transition but requires
a gradual development.

The fundamental effect unifying all these seemingly distinct systems is the
suppression of many-particle $S$-matrix elements describing $N\to 2$ (or
$N\to{\rm few}$) transitions between basic number eigenstates. Previously,
this has been argued to be the case in $2\to N$ graviton scattering [1] and
explicitly verified in [25]. The suppression of $2\to N$ matrix-elements has
also been argued on very general grounds in [27].

The present study reduces the phenomenon to its bare essentials and explicitly
traces its origin within the most elementary quantum field theories.
Specifically, we have explicitly demonstrated the effect within the simplest
interacting scalar and gauge theories.

Our analysis is directly extendable to the case of particle-creation in an
oscillating background spin-2 field, which, in particular, has been used as a
resolution of de Sitter space [6].

We have shown that in the large-$N$ double-scaling limit (1), the perturbative
quantum results fully match the semiclassical computations, which usually are
considered to be non-perturbative in nature.

At the same time, our analysis helps to visualize why the phenomenon of
classicalization, based on the inverse transition ${\rm few}\to N$, may be
unsuppressed if the $N$-particle state has a sufficient microstate degeneracy
[27].

In particular, this is the case for black holes [1]. The microstate degeneracy
does not assist the process of quantumization if the initial state of the
object is described by one particular microstate. Such absence of
superposition is a necessary condition of classicality.

It is important that the classicality of a state is distinguished from its
macroscopic nature. After $t_{Q}$ the system, e.g., a black hole, continues to
be macroscopic but it is no longer classical. Due to this, past the half-decay
time all bets are off.

For example, after $t_{Q}$, dramatic modifications in black hole evolution can
take place. In particular, as shown in [7], the backreaction from the stored
quantum information (the so-called “memory burden” effect [52]) becomes
overwhelming and the Hawking-approximation breaks down fully.

Past this point, the following two pathways were outlined in [7]. One, as
indicated by simulations of prototype systems, is the slowdown of the
initially dominant gradual decay process. This, however, does not exclude the
development of a new classical instability and correspondingly the appearance
of a Lyapunov exponent leading to the disintegration of the black hole [7].
Then, in the light of [3] and of the present discussion, this can serve as a
trigger of unsuppressed quantum processes. That is, at the level of present
understanding, past its half decay, we cannot exclude a further explosive
evolution of a black hole, neither quantum nor classical.

###### Acknowledgements.

It is a pleasure to thank Max Warkentin for conversations.
The figures of diagrams have been created using TikZ-Feynman [53].
This work was supported in part by the Humboldt Foundation under Humboldt
Professorship Award, by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - EXC-2111 -
390814868, and Germany’s Excellence Strategy under Excellence Cluster Origins.

## References

  * Dvali and Gomez [2013a] G. Dvali and C. Gomez, Black Hole’s Quantum N-Portrait, Fortsch. Phys. 61, 742 (2013a), arXiv:1112.3359 [hep-th] .
  * Dvali and Gomez [2014a] G. Dvali and C. Gomez, Black Holes as Critical Point of Quantum Phase Transition, Eur. Phys. J. C74, 2752 (2014a), arXiv:1207.4059 [hep-th] .
  * Dvali  _et al._ [2013] G. Dvali, D. Flassig, C. Gomez, A. Pritzel, and N. Wintergerst, Scrambling in the Black Hole Portrait, Phys. Rev. D88, 124041 (2013), arXiv:1307.3458 [hep-th] .
  * Dvali and Gomez [2013b] G. Dvali and C. Gomez, Black Hole’s 1/N Hair, Phys. Lett. B719, 419 (2013b), arXiv:1203.6575 [hep-th] .
  * Dvali and Gomez [2012] G. Dvali and C. Gomez, Black Hole Macro-Quantumness, (2012), arXiv:1212.0765 [hep-th] .
  * Dvali and Gomez [2014b] G. Dvali and C. Gomez, Quantum Compositeness of Gravity: Black Holes, AdS and Inflation, JCAP 1401, 023, arXiv:1312.4795 [hep-th] .
  * Dvali  _et al._ [2020] G. Dvali, L. Eisemann, M. Michel, and S. Zell, Black hole metamorphosis and stabilization by memory burden, Phys. Rev. D 102, 103523 (2020), arXiv:2006.00011 [hep-th] .
  * Dvali and Gomez [2016] G. Dvali and C. Gomez, Quantum Exclusion of Positive Cosmological Constant?, Annalen Phys. 528, 68 (2016), arXiv:1412.8077 [hep-th] .
  * Dvali  _et al._ [2017] G. Dvali, C. Gomez, and S. Zell, Quantum Break-Time of de Sitter, JCAP 1706, 028, arXiv:1701.08776 [hep-th] .
  * Dvali  _et al._ [2019] G. Dvali, L. Eisemann, M. Michel, and S. Zell, Universe’s Primordial Quantum Memories, JCAP 1903, 010, arXiv:1812.08749 [hep-th] .
  * Dvali [2020] G. Dvali, $S$-Matrix and Anomaly of de Sitter, Symmetry 13, 3 (2020), arXiv:2012.02133 [hep-th] .
  * Dvali [2021a] G. Dvali, Quantum gravity in species regime, (2021a), arXiv:2103.15668 [hep-th] .
  * Berezhiani  _et al._ [2022a] L. Berezhiani, G. Dvali, and O. Sakhelashvili, de Sitter space as a BRST invariant coherent state of gravitons, Phys. Rev. D 105, 025022 (2022a), arXiv:2111.12022 [hep-th] .
  * Gibbons and Hawking [1977] G. W. Gibbons and S. W. Hawking, Cosmological Event Horizons, Thermodynamics, and Particle Creation, Phys. Rev. D15, 2738 (1977).
  * Dvali and Zell [2018] G. Dvali and S. Zell, Classicality and Quantum Break-Time for Cosmic Axions, JCAP 07, 064, arXiv:1710.00835 [hep-ph] .
  * Berezhiani and Zantedeschi [2021] L. Berezhiani and M. Zantedeschi, Evolution of coherent states as quantum counterpart of classical dynamics, Phys. Rev. D 104, 085007 (2021), arXiv:2011.11229 [hep-th] .
  * Berezhiani  _et al._ [2022b] L. Berezhiani, G. Cintia, and M. Zantedeschi, Background-field method and initial-time singularity for coherent states, Phys. Rev. D 105, 045003 (2022b), arXiv:2108.13235 [hep-th] .
  * Kühnel [2015] F. Kühnel, Thoughts on the Vacuum Energy in the Quantum N-Portrait, Mod. Phys. Lett. A 30, 1550197 (2015), arXiv:1408.5897 [gr-qc] .
  * Kuhnel and Sandstad [2015] F. Kuhnel and M. Sandstad, Corpuscular Consideration of Eternal Inflation, Eur. Phys. J. C 75, 505 (2015), arXiv:1504.02377 [gr-qc] .
  * Berezhiani [2017] L. Berezhiani, On Corpuscular Theory of Inflation, Eur. Phys. J. C 77, 106 (2017), arXiv:1610.08433 [hep-th] .
  * Kofman  _et al._ [1997] L. Kofman, A. D. Linde, and A. A. Starobinsky, Towards the theory of reheating after inflation, Phys. Rev. D 56, 3258 (1997), arXiv:hep-ph/9704452 .
  * Dvali  _et al._ [2011a] G. Dvali, G. F. Giudice, C. Gomez, and A. Kehagias, UV-Completion by Classicalization, JHEP 08, 108, arXiv:1010.1415 [hep-ph] .
  * Dvali  _et al._ [2011b] G. Dvali, C. Gomez, and A. Kehagias, Classicalization of Gravitons and Goldstones, JHEP 11, 070, arXiv:1103.5963 [hep-th] .
  * Dvali [2017] G. Dvali, Strong Coupling and Classicalization, Subnucl. Ser. 53, 189 (2017), arXiv:1607.07422 [hep-th] .
  * Dvali  _et al._ [2015] G. Dvali, C. Gomez, R. S. Isermann, D. Lüst, and S. Stieberger, Black hole formation and classicalization in ultra-Planckian 2→N scattering, Nucl. Phys. B 893, 187 (2015), arXiv:1409.7405 [hep-th] .
  * Dvali [2018a] G. Dvali, Classicalization Clearly: Quantum Transition into States of Maximal Memory Storage Capacity, (2018a), arXiv:1804.06154 [hep-th] .
  * Dvali [2021b] G. Dvali, Entropy Bound and Unitarity of Scattering Amplitudes, JHEP 03, 126, arXiv:2003.05546 [hep-th] .
  * Addazi  _et al._ [2017] A. Addazi, M. Bianchi, and G. Veneziano, Glimpses of black hole formation/evaporation in highly inelastic, ultra-planckian string collisions, JHEP 02, 111, arXiv:1611.03643 [hep-th] .
  * Kibble [1965] T. W. B. Kibble, Frequency Shift in High-Intensity Compton Scattering, Phys. Rev. 138, B740 (1965).
  * Birrell and Davies [1984] N. D. Birrell and P. C. W. Davies,  _Quantum Fields in Curved Space_, Cambridge Monographs on Mathematical Physics (Cambridge Univ. Press, Cambridge, UK, 1984).
  * Mukhanov and Winitzki [2007] V. Mukhanov and S. Winitzki,  _Introduction to quantum effects in gravity_ (Cambridge University Press, 2007).
  * Landau and Lifshitz [1976] L. Landau and S. Lifshitz, E.,  _Mechanics, Vol.1, pp. 80-84_ (Butterworth-Heinemann, 1976).
  * Marinov and Popov [1977] M. S. Marinov and V. S. Popov, Electron-Positron Pair Creation from Vacuum Induced by Variable Electric Field, Fortsch. Phys. 25, 373 (1977).
  * Brezin and Itzykson [1970] E. Brezin and C. Itzykson, Pair production in vacuum by an alternating field, Phys. Rev. D 2, 1191 (1970).
  * Ringwald [2001] A. Ringwald, Pair production from vacuum at the focus of an X-ray free electron laser, Phys. Lett. B 510, 107 (2001), arXiv:hep-ph/0103185 .
  * Schwinger [1951] J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951).
  * Coleman and Weinberg [1973] S. R. Coleman and E. J. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev. D 7, 1888 (1973).
  * Goldberg [1990] H. Goldberg, Breakdown of perturbation theory at tree level in theories with scalars, Phys. Lett. B 246, 445 (1990).
  * Cornwall [1990] J. M. Cornwall, On the High-energy Behavior of Weakly Coupled Gauge Theories, Phys. Lett. B 243, 271 (1990).
  * Flassig  _et al._ [2013] D. Flassig, A. Pritzel, and N. Wintergerst, Black holes and quantumness on macroscopic scales, Phys. Rev. D 87, 084007 (2013), arXiv:1212.3344 [hep-th] .
  * Kovtun and Zantedeschi [2020] A. Kovtun and M. Zantedeschi, Breaking BEC, JHEP 07, 212, arXiv:2003.10283 [hep-th] .
  * ’t Hooft [1987] G. ’t Hooft, Graviton Dominance in Ultrahigh-Energy Scattering, Phys. Lett. B 198, 61 (1987).
  * Amati  _et al._ [1987] D. Amati, M. Ciafaloni, and G. Veneziano, Superstring Collisions at Planckian Energies, Phys. Lett. B 197, 81 (1987).
  * Amati  _et al._ [1988] D. Amati, M. Ciafaloni, and G. Veneziano, Classical and Quantum Gravity Effects from Planckian Energy Superstring Collisions, Int. J. Mod. Phys. A 3, 1615 (1988).
  * Gross and Mende [1988] D. J. Gross and P. F. Mende, String Theory Beyond the Planck Scale, Nucl. Phys. B 303, 407 (1988).
  * Gross and Mende [1987] D. J. Gross and P. F. Mende, The High-Energy Behavior of String Scattering Amplitudes, Phys. Lett. B 197, 129 (1987).
  * Dvali and Gomez [2010] G. Dvali and C. Gomez, Self-Completeness of Einstein Gravity, (2010), arXiv:1005.3497 [hep-th] .
  * Dvali and Venugopalan [2022] G. Dvali and R. Venugopalan, Classicalization and unitarization of wee partons in QCD and gravity: The CGC-black hole correspondence, Phys. Rev. D 105, 056026 (2022), arXiv:2106.11989 [hep-th] .
  * Gelis  _et al._ [2010] F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, The Color Glass Condensate, Ann. Rev. Nucl. Part. Sci. 60, 463 (2010), arXiv:1002.0333 [hep-ph] .
  * Gross and Neveu [1974] D. J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field Theories, Phys. Rev. D 10, 3235 (1974).
  * Dvali and Sakhelashvili [2022] G. Dvali and O. Sakhelashvili, Black-hole-like saturons in Gross-Neveu, Phys. Rev. D 105, 065014 (2022), arXiv:2111.03620 [hep-th] .
  * Dvali [2018b] G. Dvali, A Microscopic Model of Holography: Survival by the Burden of Memory, (2018b), arXiv:1810.02336 [hep-th] .
  * Ellis [2017] J. Ellis, TikZ-Feynman: Feynman diagrams with TikZ, Comput. Phys. Commun. 210, 103 (2017), arXiv:1601.05437 [hep-ph] .

## References

  * Dvali and Gomez [2013a] G. Dvali and C. Gomez, Black Hole’s Quantum N-Portrait, Fortsch. Phys. 61, 742 (2013a), arXiv:1112.3359 [hep-th] .
  * Dvali and Gomez [2014a] G. Dvali and C. Gomez, Black Holes as Critical Point of Quantum Phase Transition, Eur. Phys. J. C74, 2752 (2014a), arXiv:1207.4059 [hep-th] .
  * Dvali  _et al._ [2013] G. Dvali, D. Flassig, C. Gomez, A. Pritzel, and N. Wintergerst, Scrambling in the Black Hole Portrait, Phys. Rev. D88, 124041 (2013), arXiv:1307.3458 [hep-th] .
  * Dvali and Gomez [2013b] G. Dvali and C. Gomez, Black Hole’s 1/N Hair, Phys. Lett. B719, 419 (2013b), arXiv:1203.6575 [hep-th] .
  * Dvali and Gomez [2012] G. Dvali and C. Gomez, Black Hole Macro-Quantumness, (2012), arXiv:1212.0765 [hep-th] .
  * Dvali and Gomez [2014b] G. Dvali and C. Gomez, Quantum Compositeness of Gravity: Black Holes, AdS and Inflation, JCAP 1401, 023, arXiv:1312.4795 [hep-th] .
  * Dvali  _et al._ [2020] G. Dvali, L. Eisemann, M. Michel, and S. Zell, Black hole metamorphosis and stabilization by memory burden, Phys. Rev. D 102, 103523 (2020), arXiv:2006.00011 [hep-th] .
  * Dvali and Gomez [2016] G. Dvali and C. Gomez, Quantum Exclusion of Positive Cosmological Constant?, Annalen Phys. 528, 68 (2016), arXiv:1412.8077 [hep-th] .
  * Dvali  _et al._ [2017] G. Dvali, C. Gomez, and S. Zell, Quantum Break-Time of de Sitter, JCAP 1706, 028, arXiv:1701.08776 [hep-th] .
  * Dvali  _et al._ [2019] G. Dvali, L. Eisemann, M. Michel, and S. Zell, Universe’s Primordial Quantum Memories, JCAP 1903, 010, arXiv:1812.08749 [hep-th] .
  * Dvali [2020] G. Dvali, $S$-Matrix and Anomaly of de Sitter, Symmetry 13, 3 (2020), arXiv:2012.02133 [hep-th] .
  * Dvali [2021a] G. Dvali, Quantum gravity in species regime, (2021a), arXiv:2103.15668 [hep-th] .
  * Berezhiani  _et al._ [2022a] L. Berezhiani, G. Dvali, and O. Sakhelashvili, de Sitter space as a BRST invariant coherent state of gravitons, Phys. Rev. D 105, 025022 (2022a), arXiv:2111.12022 [hep-th] .
  * Gibbons and Hawking [1977] G. W. Gibbons and S. W. Hawking, Cosmological Event Horizons, Thermodynamics, and Particle Creation, Phys. Rev. D15, 2738 (1977).
  * Dvali and Zell [2018] G. Dvali and S. Zell, Classicality and Quantum Break-Time for Cosmic Axions, JCAP 07, 064, arXiv:1710.00835 [hep-ph] .
  * Berezhiani and Zantedeschi [2021] L. Berezhiani and M. Zantedeschi, Evolution of coherent states as quantum counterpart of classical dynamics, Phys. Rev. D 104, 085007 (2021), arXiv:2011.11229 [hep-th] .
  * Berezhiani  _et al._ [2022b] L. Berezhiani, G. Cintia, and M. Zantedeschi, Background-field method and initial-time singularity for coherent states, Phys. Rev. D 105, 045003 (2022b), arXiv:2108.13235 [hep-th] .
  * Kühnel [2015] F. Kühnel, Thoughts on the Vacuum Energy in the Quantum N-Portrait, Mod. Phys. Lett. A 30, 1550197 (2015), arXiv:1408.5897 [gr-qc] .
  * Kuhnel and Sandstad [2015] F. Kuhnel and M. Sandstad, Corpuscular Consideration of Eternal Inflation, Eur. Phys. J. C 75, 505 (2015), arXiv:1504.02377 [gr-qc] .
  * Berezhiani [2017] L. Berezhiani, On Corpuscular Theory of Inflation, Eur. Phys. J. C 77, 106 (2017), arXiv:1610.08433 [hep-th] .
  * Kofman  _et al._ [1997] L. Kofman, A. D. Linde, and A. A. Starobinsky, Towards the theory of reheating after inflation, Phys. Rev. D 56, 3258 (1997), arXiv:hep-ph/9704452 .
  * Dvali  _et al._ [2011a] G. Dvali, G. F. Giudice, C. Gomez, and A. Kehagias, UV-Completion by Classicalization, JHEP 08, 108, arXiv:1010.1415 [hep-ph] .
  * Dvali  _et al._ [2011b] G. Dvali, C. Gomez, and A. Kehagias, Classicalization of Gravitons and Goldstones, JHEP 11, 070, arXiv:1103.5963 [hep-th] .
  * Dvali [2017] G. Dvali, Strong Coupling and Classicalization, Subnucl. Ser. 53, 189 (2017), arXiv:1607.07422 [hep-th] .
  * Dvali  _et al._ [2015] G. Dvali, C. Gomez, R. S. Isermann, D. Lüst, and S. Stieberger, Black hole formation and classicalization in ultra-Planckian 2→N scattering, Nucl. Phys. B 893, 187 (2015), arXiv:1409.7405 [hep-th] .
  * Dvali [2018a] G. Dvali, Classicalization Clearly: Quantum Transition into States of Maximal Memory Storage Capacity, (2018a), arXiv:1804.06154 [hep-th] .
  * Dvali [2021b] G. Dvali, Entropy Bound and Unitarity of Scattering Amplitudes, JHEP 03, 126, arXiv:2003.05546 [hep-th] .
  * Addazi  _et al._ [2017] A. Addazi, M. Bianchi, and G. Veneziano, Glimpses of black hole formation/evaporation in highly inelastic, ultra-planckian string collisions, JHEP 02, 111, arXiv:1611.03643 [hep-th] .
  * Kibble [1965] T. W. B. Kibble, Frequency Shift in High-Intensity Compton Scattering, Phys. Rev. 138, B740 (1965).
  * Birrell and Davies [1984] N. D. Birrell and P. C. W. Davies,  _Quantum Fields in Curved Space_, Cambridge Monographs on Mathematical Physics (Cambridge Univ. Press, Cambridge, UK, 1984).
  * Mukhanov and Winitzki [2007] V. Mukhanov and S. Winitzki,  _Introduction to quantum effects in gravity_ (Cambridge University Press, 2007).
  * Landau and Lifshitz [1976] L. Landau and S. Lifshitz, E.,  _Mechanics, Vol.1, pp. 80-84_ (Butterworth-Heinemann, 1976).
  * Marinov and Popov [1977] M. S. Marinov and V. S. Popov, Electron-Positron Pair Creation from Vacuum Induced by Variable Electric Field, Fortsch. Phys. 25, 373 (1977).
  * Brezin and Itzykson [1970] E. Brezin and C. Itzykson, Pair production in vacuum by an alternating field, Phys. Rev. D 2, 1191 (1970).
  * Ringwald [2001] A. Ringwald, Pair production from vacuum at the focus of an X-ray free electron laser, Phys. Lett. B 510, 107 (2001), arXiv:hep-ph/0103185 .
  * Schwinger [1951] J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951).
  * Coleman and Weinberg [1973] S. R. Coleman and E. J. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev. D 7, 1888 (1973).
  * Goldberg [1990] H. Goldberg, Breakdown of perturbation theory at tree level in theories with scalars, Phys. Lett. B 246, 445 (1990).
  * Cornwall [1990] J. M. Cornwall, On the High-energy Behavior of Weakly Coupled Gauge Theories, Phys. Lett. B 243, 271 (1990).
  * Flassig  _et al._ [2013] D. Flassig, A. Pritzel, and N. Wintergerst, Black holes and quantumness on macroscopic scales, Phys. Rev. D 87, 084007 (2013), arXiv:1212.3344 [hep-th] .
  * Kovtun and Zantedeschi [2020] A. Kovtun and M. Zantedeschi, Breaking BEC, JHEP 07, 212, arXiv:2003.10283 [hep-th] .
  * ’t Hooft [1987] G. ’t Hooft, Graviton Dominance in Ultrahigh-Energy Scattering, Phys. Lett. B 198, 61 (1987).
  * Amati  _et al._ [1987] D. Amati, M. Ciafaloni, and G. Veneziano, Superstring Collisions at Planckian Energies, Phys. Lett. B 197, 81 (1987).
  * Amati  _et al._ [1988] D. Amati, M. Ciafaloni, and G. Veneziano, Classical and Quantum Gravity Effects from Planckian Energy Superstring Collisions, Int. J. Mod. Phys. A 3, 1615 (1988).
  * Gross and Mende [1988] D. J. Gross and P. F. Mende, String Theory Beyond the Planck Scale, Nucl. Phys. B 303, 407 (1988).
  * Gross and Mende [1987] D. J. Gross and P. F. Mende, The High-Energy Behavior of String Scattering Amplitudes, Phys. Lett. B 197, 129 (1987).
  * Dvali and Gomez [2010] G. Dvali and C. Gomez, Self-Completeness of Einstein Gravity, (2010), arXiv:1005.3497 [hep-th] .
  * Dvali and Venugopalan [2022] G. Dvali and R. Venugopalan, Classicalization and unitarization of wee partons in QCD and gravity: The CGC-black hole correspondence, Phys. Rev. D 105, 056026 (2022), arXiv:2106.11989 [hep-th] .
  * Gelis  _et al._ [2010] F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, The Color Glass Condensate, Ann. Rev. Nucl. Part. Sci. 60, 463 (2010), arXiv:1002.0333 [hep-ph] .
  * Gross and Neveu [1974] D. J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field Theories, Phys. Rev. D 10, 3235 (1974).
  * Dvali and Sakhelashvili [2022] G. Dvali and O. Sakhelashvili, Black-hole-like saturons in Gross-Neveu, Phys. Rev. D 105, 065014 (2022), arXiv:2111.03620 [hep-th] .
  * Dvali [2018b] G. Dvali, A Microscopic Model of Holography: Survival by the Burden of Memory, (2018b), arXiv:1810.02336 [hep-th] .
  * Ellis [2017] J. Ellis, TikZ-Feynman: Feynman diagrams with TikZ, Comput. Phys. Commun. 210, 103 (2017), arXiv:1601.05437 [hep-ph] .