On Evolution of Coherent States
as Quantum Counterpart of Classical Dynamics

Lasha Berezhiani and Michael Zantedeschi

Max-Planck-Institut für Physik,
Föhringer Ring 6, 80805 München, Germany
Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,
Theresienstraße 37, 80333 München, Germany

###### Abstract

Quantum dynamics of coherent states is studied within quantum field theory
using two complementary methods: by organizing the evolution as a Taylor
series in elapsed time and by perturbative expansion in coupling within the
interaction-picture formalism. One of the important aspects of our analysis
consists in utilizing the operators and the vacuum of interacting theory in
constructing the states, without invoking asymptotic particles. Focusing on a
coherent state describing a spatially homogeneous field configuration, it is
demonstrated that both adopted methods successfully account for nonlinear
classical dynamics, giving distinguishable quantum effects. In particular,
according to the time-expansion analysis the initial field-acceleration, with
which the field departs from its initial expectation value, is governed by the
tree-level potential with renormalized mass and bare coupling constant. The
interaction-picture computation, instead, can be manipulated to give the
nonlinear dynamics, determined in terms of renormalized coupling and mass.
However, it results in a logarithmic initial-time singularity in the field-
acceleration, reminiscent of the similar behaviour encountered within semi-
classical formalism, for certain choices of the initial state for
fluctuations. Within our coherent-state analysis, the above mentioned
peculiarities are artefacts of an expansion: in the first case over
infinitesimal time, while in the second case in the coupling constant. Despite
this, we show that the evolution obtained within the interaction-picture
analysis is valid for extended period of time. Moreover, on top of the desired
classical dynamics, it serves us with interesting quantum corrections,
previously proposed by Dvali–Gomez–Zell.

##  1 Introduction

As it is well known, the classical field configurations merely serve as an
approximate description of the system. Within the framework of quantum field
theory, the ultimate description of the state is given by a quantum state,
which is an element of the physical Hilbert space. Although the classical
description is extremely accurate for macroscopic systems, there could be
still some cumulative quantum effects that could become relevant in time. In
[1, 2, 3, 4, 5, 6], the quantum corpuscular approach to classical backgrounds
was developed and it was shown that the coherent state description of time-
dependent systems leads to new type of quantum effects, which, in certain
cases, can result in complete breakdown of the classical description. This was
shown to have potentially important ramifications for systems such as black
holes, de Sitter spacetime and cosmic inflation, predicting new deviations
from semi-classical evolution. The importance of these considerations in the
context of the beginning of inflation was discussed in [7]111See also [8],
where the origin of the observed density perturbations was identified, within
the proposal of [5], as the quantum uncertainty in the number of inflaton
constituents (with significantly screened mass-gap, due to collective
gravitational effects) within the Hubble patch..

To be specific, let us begin by recapping the argument of [6]. In a weakly
interacting theory possessing a dimensionless quantum coupling $\alpha\ll 1$,
one may consider a system in a quantum state with large occupation number $N$.
Initially, the dynamics of such a system should be well approximated by its
classical equations of motion. Usually, one may define a classical collective
coupling $\alpha N$, which characterizes the strength of classical
nonlinearities in the equation of motion and consequently sets the classical
time-scale $t_{\rm cl}$, after which nonlinear corrections to the classical
dynamics become important, even if initially the system was prepared in a
state well-described by free waves. Quantum mechanically, the latter
corresponds to the expectation value of the quantum field in a coherent state,
with the mean occupation number determined by the amplitude of the wave in
question. It was argued in [6] that the quantum scattering among the
constituent quanta should lead to the decoherence of the coherent state, which
in turn might cause a quantum departure from the classical evolution, with a
time-scale of significant deviation estimated as

$\displaystyle t_{\rm q}=\frac{t_{\rm cl}}{\alpha}\,;$ (1.1)

referred to as quantum break-time in [6]222However, it was shown in [4] that
the quantum break-time is in general shorter for systems near criticality
(i.e. $\alpha N=1$) and possessing semi-classical Lyapunov exponent.. There,
these concepts were shown to take a particularly simple form, when applied to
the massive scalar field with quartic self-interaction. For this system, a
harmonically oscillating homogeneous field represents an accurate solution to
the full nonlinear equation on timescales

$\displaystyle t\ll t_{\rm cl}\equiv\frac{1}{\omega(\alpha N)}\,;$ (1.2)

where, according to the notation of [6], $\omega$ is the classical frequency
of harmonic oscillations, the quantum coupling $\alpha\equiv\hbar\lambda$ is
connected to the classical parameter $\lambda$ of quartic nonlinearity (with
$\alpha\rightarrow 0$ in the classical limit of vanishing Planck’s constant
$\hbar\rightarrow 0$) and the occupation number (within the Compton volume
$\sim\omega^{-3}$) $N=\frac{A^{2}}{\hbar\omega^{2}}$ is determined in terms of
the amplitude $A$ of the classical wave; implying the independence of $t_{\rm
cl}$ from $\hbar$. Representing the classical background as a coherent state
of zero-momentum particles and estimating the rate of quantum-breaking using
$2\rightarrow 2$ scattering of constituent quanta, it was shown in [6] that
the quantum break-time reduces to the general expression (1.1). Although
$t_{\rm q}\gg t_{\rm cl}$, it was argued in [6] that the gradual quantum
decoherence is a cumulative effect and should persist even after one properly
accounts for the classical nonlinearities, extending the regime of validity
beyond (1.2). Similar results were reproduced for axions in [9]. Recently,
analogous conclusions on quantum breaking were drawn in [10, 11] , based on
numerical analysis within the so-called 2-Particle-Irreducible (2PI) formalism
for $(1+1)$-dimensional self-interacting scalar field theory with a conserved
charge.

The adoption of these arguments to certain physical systems has staggering
ramifications, as we have mentioned at the beginning of this section. Focusing
on cosmic inflation, for concreteness, we reiterate the proposal of [5] and
its consequences for the paradigm. Let us further restrict the discussion to
the so-called “$m^{2}\phi^{2}$-model”, for simplicity. At the end of the slow-
roll phase, the scalar field begins to oscillate and behaves as a non-
relativistic condensate. A quantum state corresponding to such a configuration
consists of large number of zero-momentum $\phi$-particles. The gravitational
background sourced by such a quantum condensate was argued to represent some
sort of a condensate as well, constructed around the Minkowski vacuum by
gravitational degrees of freedom. The motivation for the latter is obvious, if
one ware to drain the scalar condensate the gravitational configuration would
approach the Minkowski space. Rewinding time to the slow-roll stage of
inflation, the similar description was suggested to hold. Although, due to
significant change in the classical dynamics, the degrees of freedom making up
the condensates can go significantly off-shell. In [5], the properties of the
gravitational condensate, i.e. the dispersion relation of constituents, was
determined by demanding the reproduction of the semi-classical properties of
the quasi-de Sitter space. Quantum processes such as particle production
during inflation was identified as the scattering among the condensate
constituents. For example, the microscopic process behind the production of
gravitational waves was identified as the annihilation of the quanta making up
the gravitational background. This led to the conclusion that if these
processes last long enough the quantum depletion of the background should
become significant and invalidate the semiclassical description. In fact, it
was established that if inflaton starts out in the regime of slow-roll eternal
inflation (self-reproduction), the dynamics should become fully quantum before
the scalar field reaches the bottom of the potential and ends inflation.

The goal of this work is to start taking first steps towards establishing the
link between the S-matrix estimates of [5] and the rigorous real-time dynamics
of coherent states for setups such as cosmic inflation. As it will become
clear shortly, the scope of the current work is limited and is by no means
directly applicable to cosmological frameworks. However, our analysis provides
an important foundation for more sophisticated analysis by pinpointing the
possible advantages of designing the states in its entirety.

In general, approaches to computing the corrections to the classical evolution
fall in two main categories. The most commonly used one is the semiclassical
analysis, in which one studies the evolution of perturbations around the
classical background and evaluates the semiclassical back-reaction on the
latter. The expansion, within these methods, can be organized w.r.t. different
parameters (e.g. coupling, $\hbar$ and many others) and a plethora of
different effects can be captured (for instance, 2PI approach capturing
thermalization). See [12, 13, 14, 15, 16, 17, 18, 19, 20] and references
therein for interesting work on the subject. A useful overview of such
methods, along with a complete list of relevant references can be found in
[21]. The goal of this article is not to utilize this methods per se, but
instead we will be questioning some of their underlying assumptions, as it
will become clear. The other category consists of replacing the classical
background by a coherent state (since coherent states represent a proper
quantum counterpart of classical field configurations) of large number of
quanta and studying the evolution as a multi-particle scattering process [1,
2, 3, 4, 5, 6]. In cases when classical evolution is not significantly
affected by interactions, the said state is made of on-shell (asymptotic)
particles, as discussed above333See [22, 23] for the earlier work on
representing classical configurations as coherent states.. In the opposite
situation, the latter approach requires invoking off-shell degrees of freedom
[5, 6], with the off-shell-ness connected with the properties (e.g. dispersion
relation) of the constituent quanta being related to the classical background
and continuously changing in case of nontrivial classical evolution.

The approach adopted in this work belongs to the second category. The
distinguishing feature is the construction of coherent states out of the
vacuum of the interacting theory, canonical field-operator and its conjugate
momentum, without invoking asymptotic approximation for them444Our
construction falls within the category of generalized coherent states; for the
overview of various aspects see [24].. Our work is, in spirit, very similar to
the study of [25], where the quantum mechanical analog case was analysed.
Within our method, the initial construction is exact and valid to all orders
in perturbation theory. Eventually, we are forced to perform a loop expansion
for practical reasons. We show that for the leading order quantum effects it
would have sufficed to construct the coherent state in terms of asymptotic
particles, but our improved construction still helps with some intermediate
technical steps. Furthermore, the adopted construction ensures the consistency
of the perturbative expansion for the state and the dynamics; e.g. for 2-loop
calculation the coupling dependence of the state appears to be relevant.
Moreover, we perform a rigorous computation of nonlinear contributions to the
classical dynamics and its quantum entourage, reproducing the Dvali–Gomez–Zell
formula (1.1) for the quantum break-time.

In this work, we therefore study the quantum evolution of a coherent state in
an interacting quantum field theory. The analysis is performed for a scalar
field with quartic self-interaction, but can be straightforwardly extended to
other theories (to be presented elsewhere). In particular, we study the
Lagrangian555From this point forward, we will be working in units $\hbar=c=1$.

$\displaystyle\mathcal{L}=-\Lambda-\frac{1}{2}Z(\partial_{\mu}\hat{\phi})^{2}-\frac{1}{2}Zm^{2}\hat{\phi}^{2}-\frac{\lambda}{4!}Z^{2}\hat{\phi}^{4}\,;$
(1.3)

where $\Lambda$, $Z$, $m$ and $\lambda$ are the bare vacuum energy, field
normalization, mass and coupling constant respectively; with all of them
expected to be infinite, as usual (keeping in mind that $Z$ does not get a
correction at 1-loop, whithin the theory at hand). The parameters are
considered to be such that there is a single non-degenerate vacuum with
vanishing field-value.

Just like in textbook calculations, we assume that there exists a vacuum state
$|\Omega\rangle$ with unit norm, which is the Hamiltonian eigenstate with
lowest possible eigenvalue. Then, the coherent state corresponding to a
classical field configuration with $\phi_{cl}(x)$ and $\pi_{cl}(x)$ can be
constructed as

$\displaystyle|C\rangle=e^{-i\int
d^{3}x\left(\phi_{cl}(x)\hat{\pi}(x)-\pi_{cl}(x)\hat{\phi}(x)\right)}|\Omega\rangle\,,$
(1.4)

with $\pi$ standing for the conjugate momentum. The convenient property of
this state is that it has unit norm and satisfies the following to all orders
in $\lambda$ at the initial moment of time $t=0$

$\displaystyle\langle C|\hat{\phi}|C\rangle(t=0)=\phi_{cl}(x)\,,$ (1.5)
$\displaystyle\langle C|\hat{\pi}|C\rangle(t=0)=\pi_{cl}(x)\,.$ (1.6)

We adopt two complementary methods for studying the dynamics:

  * (I)

Instead of applying standard field-theoretical methods, we propose a new
approach based on the evaluation of physical quantities in a Taylor series in
time $t$, elapsed since the initial moment. We calculate first several terms
of the expansion, giving it initially in the form that is valid to all orders
in coupling constant. Examining the coherent-state-expectation-value of the
field operator, we find that all manifestly finite contributions are identical
to the iterative solution to the classical equation of motion, at each
calculated order in $t$. Remarkably, these are supplemented with terms
involving vacuum expectation values of singular operators, some of which can
be absorbed via the renormalization of parameters, while the others need to be
resummed together with higher order terms in $t$. We also found that the
initial acceleration of this 1-point-function does not seem to be governed by
the effective potential (a la Coleman-Weinberg [26]), contradicting the
standard intuition. Presumably, this too needs to be resolved through the
resummation. On the flip side, one advantage of the method in question is the
absence of the so-called initial-time singularity, one tends to encounter
within semi-classical techniques for certain initial states for fluctuations;
for the relevant discussion of the latter and the proposed resolutions see
[27, 28, 29, 30, 21].

  * (II)

In the attempt to extend the analysis to finite time-intervals and to
ameliorate the above-mentioned puzzle with leftover divergencies, we have
reanalyzed the evolution of the one-point expectation value in a coherent
state using the interaction-picture approach, by performing a coupling
expansion from the get-go. It is demonstrated that the expectation value of
the field-operator evolves according to the classical equations of motion,
albeit with renormalized parameters, supplemented with finite (yet small)
quantum corrections. The price for this elegant result is the logarithmic
initial-time singularity in the second-time-derivative of the one-point
function in question, along with usual spurious secular divergencies that seem
to be under control for an extended period of time. As we will show, expanding
the result before the final step of the calculation in a Taylor series in
elapsed time, we recover the results of the previous method. Most importantly,
we will discuss the caveats concerning this reproduction. Although these
results are limited by a given order of classical nonlinearities, it should be
straightforward to extend them to higher orders.

Furthermore, we use method (I) to evolve the state itself and compare it to
the classically evolved coherent state, finding the departure after
infinitesimal interval of time with interesting contributions. In (3+1)D,
these corrections appear to be divergent, without any obvious regularization
in sight that could give a physically sound result. We are therefore left with
the only logical conclusion that these infinities need to be resummed with
higher-order corrections in infinitesimal time to give a finite final answer.
Although parametrically the corrections at hand appear to be of interesting
form, assertive claims do not seem possible at this point. However, we show
that the results become finite in (1+1)D, providing us with a tractable
departure from the classical evolution.

The goal of this article, is therefore the following: to build explicitly, at
an operatorial level, the simplest possible coherent state mimicking a
classical configuration and to analyze its dynamics. Our approach is
orthogonal to the commonly used background field methods (e.g. [28]), where
initial conditions for 1-point and 2-point correlation functions are specified
based on semi-classical reasoning and by requiring the absence of the initial
time singularity. Our work helps to elucidate the points that are obscured
within these methods. In fact, our analysis demonstrates that if (non-
squeezed, i.e. simplest) coherent states constructed consistently, without
even invoking asymptotic degrees of freedom and the free vacuum, are to be
physical then the so-called "initial-time singularity" must be an artefact of
the perturbative expansion. Or, one would have to proclaim the inconsistency
of non-squeezed coherent states in four-dimensional quantum field theory in
question.

The article is organized as follows. In Sec. 2, the theoretical framework is
described. In Sec. 3, the main results of the time expansion are shown for a
concrete coherent state, corresponding to the homogeneous classical field
configuration. In particular we evaluate both the overlap between the quantum
and classically evolved coherent states, and the expectation value of some
operators. In Sec. 4, we perform the interaction-picture computation of the
one-point expectation value in the coherent state. Finally, we summarize the
results in Sec. 5 and discuss the outlook. Some of the technical details have
been relegated to appendices.

##  2 Setup

The Hamiltonian corresponding to the Lagrangian density (1.3) is given by

$\displaystyle\hat{H}=\int
d^{3}x\left(\frac{1}{2}\frac{\hat{\pi}^{2}}{Z}+\frac{1}{2}Z(\partial_{j}\hat{\phi})^{2}+\frac{1}{2}Zm^{2}\hat{\phi}^{2}+\frac{\lambda}{4!}Z^{2}\hat{\phi}^{4}+\Lambda\right)\,,$
(2.1)

where the conjugate momentum is given by $\hat{\pi}=Z\partial_{t}\hat{\phi}$.
We also have the usual equal-time canonical commutation relations

$\displaystyle[\hat{\phi}(x,t),\hat{\pi}(y,t)]=i\delta^{(3)}(x-y)\,,$ (2.2)
$\displaystyle[\hat{\phi}(x,t),\hat{\phi}(y,t)]=[\hat{\pi}(x,t),\hat{\pi}(y,t)]=0\,.$
(2.3)

Let us begin by constraining the parameters of the theory so that the energy
density of the interacting vacuum $|\Omega\rangle$ vanishes, i.e. by requiring

$\displaystyle\langle\Omega|\hat{\mathcal{H}}|\Omega\rangle=0\,,$ (2.4)

with $\mathcal{H}$ denoting the Hamiltonian density operator. For (2.1) we get

$\displaystyle\langle\Omega|\hat{\mathcal{H}}|\Omega\rangle=\Lambda+\frac{1}{2Z}\langle\Omega|\hat{\pi}^{2}|\Omega\rangle+\frac{1}{2}Z\langle\Omega|(\partial_{j}\hat{\phi})^{2}|\Omega\rangle+\frac{1}{2}Zm^{2}\langle\Omega|\hat{\phi}^{2}|\Omega\rangle+\frac{\lambda}{4!}Z^{2}\langle\Omega|\hat{\phi}^{4}|\Omega\rangle\,.$
(2.5)

This can be made to vanish by adjusting the bare vacuum energy $\Lambda$
accordingly.

As already mentioned in the introduction, the coherent states of our main
interest can be parameterised as

$\displaystyle|C\rangle=e^{-i\hat{f}}|\Omega\rangle\,,\quad\text{with}\quad\hat{f}\equiv\int
d^{3}x\left(\phi_{cl}(x)\hat{\pi}(x,0)-\pi_{cl}(x)\hat{\phi}(x,0)\right)\,.$
(2.6)

As the expression speaks for itself, we have presented the state in the
Heisenberg picture; moreover, notice that due to the definition of the
conjugate momentum we have $\pi_{cl}=Z\dot{\phi}_{cl}$.

The Hamiltonian density (2.1) in a coherent state (2.6) can be readily
obtained, after adjusting the vacuum energy to zero, reducing to

$\displaystyle\langle
C|\hat{\mathcal{H}}(x)|C\rangle(t=0)=Z\left[\frac{1}{2}\frac{\pi_{cl}^{2}}{Z^{2}}+\frac{1}{2}\left(\vec{\nabla}\phi_{cl}\right)^{2}+\frac{1}{2}\left(m^{2}+\frac{\lambda
Z}{2}\langle\Omega|\hat{\phi}(x)^{2}|\Omega\rangle\right)\phi_{cl}^{2}+\frac{\lambda}{4!}Z\phi_{cl}^{4}\right]\,.$
(2.7)

Here and throughout this work, we employ the fact that
$\langle\Omega|\hat{\phi}|\Omega\rangle=\langle\Omega|\hat{\phi}^{3}|\Omega\rangle=\langle\Omega|\hat{\pi}|\Omega\rangle=0\,,$
due to $Z_{2}$ symmetry.

This looks encouragingly similar to the classical expression

$\displaystyle\mathcal{H}_{cl}=\frac{1}{2}\pi_{cl}^{2}+\frac{1}{2}\left(\vec{\nabla}\phi_{cl}\right)^{2}+\frac{1}{2}m_{\rm
ph}^{2}\phi_{cl}^{2}+\frac{\lambda_{\rm ph}}{4!}\phi_{cl}^{4}\,.$ (2.8)

However, there are important differences as well. First thing to notice is the
appearance of the divergent equal-point 2-point function. At this point, one
might be inclined to absorb this divergence into the mass. In other words, at
1-loop order (at which, we expect $Z=1$) one could define a physical mass and
coupling as

$\displaystyle m_{\rm
ph}^{2}\stackrel{{\scriptstyle?}}{{\equiv}}\left(m^{2}+\frac{\lambda}{2}\langle\Omega|\hat{\phi}(x)^{2}|\Omega\rangle\right)\,,$
(2.9) $\displaystyle\lambda_{\rm
ph}\stackrel{{\scriptstyle?}}{{\equiv}}\lambda\,;$ (2.10)

which follows from identifying $\mathcal{H}_{cl}$ with the expectation value
of $\hat{\mathcal{H}}$. Although the mass renormalization (2.9) is of the
expected form, the coupling assignment (2.10) seems to be inconsistent with
the standard one for our setup; but we will come to this point later on.

Before specializing on a coherent state describing a particular configuration
of interest, let us make some general remarks that will aid us with technical
calculations.

###  2.1 Heisenberg Picture

Among other quantities, one might be interested in the expectation value of
various operators in a coherent state. In general, it is convenient to discuss
these in Heisenberg picture. Let us consider the following expectation value

$\displaystyle\langle
C|\prod_{i}\mathcal{O}_{i}(\hat{\phi},\hat{\pi};t_{i})|C\rangle\,,$ (2.11)

with $|C\rangle$ being a coherent state (2.6) constructed using operators at
time $t=0$ and $\mathcal{O}_{i}$ standing for a composite operator at a moment
$t_{i}$; in other words, we have equal time products within each
$\mathcal{O}$. Then it is straightforward to show that (2.11) can be expressed
as a vacuum-expectation-value of shifted operators
$\mathcal{O}_{i}(\phi_{cl}+\hat{\phi},\pi_{cl}+\hat{\pi};t_{i})$ evolved from
$t=0$ using the background-field-method Hamiltonian
$\hat{H}[\phi_{cl}+\hat{\phi},\pi_{cl}+\hat{\pi}]$. Namely, we have

$\displaystyle\langle C|\
\prod_{i}\mathcal{O}_{i}(\hat{\phi},\hat{\pi};t_{i})|C\rangle=\langle\Omega|\prod_{i}\left(e^{i\hat{H}[\phi_{cl}+\hat{\phi},\pi_{cl}+\hat{\pi}]t_{i}}\mathcal{O}_{i}(\phi_{cl}+\hat{\phi},\pi_{cl}+\hat{\pi};0)e^{-i\hat{H}[\phi_{cl}+\hat{\phi},\pi_{cl}+\hat{\pi}]t_{i}}\right)|\Omega\rangle\,,~{}~{}$
(2.12)

which is quite intuitive as the exponential operator defining our coherent
state (2.6) is a field displacement operator. For the face value, it appears
as if we have evaluated the expectation value in semi-classical approximation
by quantising perturbations around a fixed classical background and treating
perturbations to be in the vacuum. However, due to the fact that
$|\Omega\rangle$ is the vacuum of the fundamental theory rather than the one
of a semi-classical theory of perturbations, the matters are somewhat more
complicated. As we will see on concrete examples, while the dynamics of
correlators is identical to the one inferred from the background field method,
the initial conditions are quite different. Nevertheless, it is important to
show that (2.12) is exact and holds for any theory. It follows from the
following mathematical identity

$\displaystyle
e^{e^{\hat{X}}\hat{Y}e^{-\hat{X}}}=\sum_{n=0}^{\infty}\frac{1}{n!}(e^{\hat{X}}\hat{Y}e^{-\hat{X}})^{n}=e^{\hat{X}}e^{\hat{Y}}e^{-\hat{X}}\,,$
(2.13)

which holds for any operators $\hat{X}$ and $\hat{Y}$; together with the
observation that for equal time operators we have

$\displaystyle
e^{i\hat{f}}\mathcal{O}(\hat{\phi},\hat{\pi};0)e^{-i\hat{f}}=\mathcal{O}(\phi_{cl}+\hat{\phi},\pi_{cl}+\hat{\pi};0)\,,$
(2.14)

following from the Baker–Campbell–Hausdorff formula.

Regarding the coherent states themselves, we are parameterising them in terms
of two functions of spatial coordinates $\phi_{cl}(x)$ and $\pi_{cl}(x)$, and
a time stamp appearing on the operators invoked in their construction. Using
(2.13), we can write

$\displaystyle|\phi_{cl},\pi_{cl};t\rangle={\rm exp}\left[-i\int
d^{3}x\left(\phi_{cl}(x)\hat{\pi}(x,t)-\pi_{cl}(x)\hat{\phi}(x,t)\right)\right]|\Omega\rangle$
$\displaystyle={\rm exp}\left[-i\int
d^{3}xe^{i\hat{H}t}\left(\phi_{cl}(x)\hat{\pi}(x,0)-\pi_{cl}(x)\hat{\phi}(x,0)\right)e^{-i\hat{H}t}\right]|\Omega\rangle$
$\displaystyle=e^{i\hat{H}t}{\rm exp}\left[-i\int
d^{3}x\left(\phi_{cl}(x)\hat{\pi}(x,0)-\pi_{cl}(x)\hat{\phi}(x,0)\right)\right]e^{-i\hat{H}t}|\Omega\rangle$
$\displaystyle=e^{i\hat{H}t}|\phi_{cl},\pi_{cl};0\rangle\,.$ (2.15)

Notice that this relation is similar to the one we have for field (and
conjugate momentum) eigenstates in Heisenberg picture.

###  2.2 Schrödinger Picture

Besides expectation values, one is usually interested in transition amplitudes
as well, for which it may be convenient to see how the coherent state itself
is evolving.

To see how the coherent state (2.6) evolves in the Schrödinger picture, we use
(2.13) once more to obtain

$\displaystyle|C\rangle(t)=e^{-i\hat{H}t}e^{-i\hat{f}}|\Omega\rangle=e^{-ie^{-i\hat{H}t}\hat{f}e^{i\hat{H}t}}e^{-iE_{\Omega}t}|\Omega\rangle\,;$
(2.16)

with $E_{\Omega}$ denoting the vacuum eigenvalue of the Hamiltonian, which we
have been adjusting to zero (2.4). In other words, the evolved state is the
one created by the same linear combination of $\hat{\phi}$ and $\hat{\pi}$,
but ’evolved’ backwards in time. This seems to correspond to a nontrivial
squeezing of the coherent state. Moreover, comparing to the Heisenberg picture
result, time-evolution simply scans through states (2.15), with fixed
$\phi_{cl}(x)$ and $\pi_{cl}(x)$.

##  3 Concrete Example: Homogeneous background

A state of particular interest is the one corresponding to the homogeneous
field configuration, as being a popular representative of configurations used
in background field calculations. In particular, we focus on a state that
corresponds to a classical homogeneous background with the following initial
conditions

$\displaystyle\phi_{cl}(t=0)=\phi_{0}\,,$ (3.1)
$\displaystyle\pi_{cl}(t=0)=0\,.$ (3.2)

Here we have chosen vanishing initial momentum for simplicity. Classically,
this background will undergo anharmonic oscillations. In order to study the
quantum aspects of this evolution, we need to define the quantum state
corresponding to this classical background.

Following our previous discussion we have (for some rudimentary discussion see
appendix A)

$\displaystyle|C\rangle(t=0)=e^{-i\hat{f}}|\Omega\rangle\,,\qquad\text{with}\qquad\hat{f}\equiv\int
d^{3}x\phi_{0}\hat{\pi}\,;$ (3.3)

In the remainder of this section we will compute the rate of departure of this
state from the classical evolution and the expectation values of different
operators.

###  3.1 Rate of Departure

Applying (2.16) to the homogeneous state at hand, we get

$\displaystyle|C\rangle(t)=e^{-i\phi_{0}\int
d^{3}xe^{-i\hat{H}t}\hat{\pi}(x)e^{i\hat{H}t}}|\Omega\rangle\,.$ (3.4)

Here we choose to work in Schrödinger picture, hence there is no time-label on
$\pi$. At this point, we are interested in analysing the evolution of the
state for an infinitesimal time $t$ and comparing the result to the classical
evolution. In particular, we will be interested in evolving the state up to
$t^{2}$-order, since this is where interesting effects begin to emerge. This
is done, using

$\displaystyle\int d^{3}xe^{-i\hat{H}t}\hat{\pi}(x)e^{i\hat{H}t}=\int
d^{3}x\left[\hat{\pi}(x)+t\left(Zm^{2}\hat{\phi}+\frac{\lambda
Z^{2}}{3!}\hat{\phi}^{3}\right)-\frac{t^{2}}{2}\left(m^{2}\hat{\pi}+\frac{\lambda
Z}{4}\left(\hat{\pi}\hat{\phi}^{2}+\hat{\phi}^{2}\hat{\pi}\right)\right)\right]$
$\displaystyle+\mathcal{O}(t^{3})\,.$ (3.5)

Now, classically we know how the scalar field evolves in infinitesimal time
$t$. Namely, if at $t=0$ we had $\phi=\phi_{0}$ and $\dot{\phi}=0$ then at
time $t$ we would have

$\displaystyle\phi_{cl}(t)=\phi_{0}-\frac{t^{2}}{2}\left(m_{\rm
ph}^{2}\phi_{0}+\frac{\lambda_{\rm
ph}}{3!}\phi_{0}^{3}\right)+\mathcal{O}(t^{3})\,.$ (3.6)

In order to parameterise the deviation of an evolved coherent state from a
coherent state constructed for the latter classical background, we calculate
the overlap between (2.16) and

$\displaystyle|C_{cl}\rangle(t)=e^{-i\int
d^{3}x\left(\phi_{cl}(t)\hat{\pi}-\pi_{cl}(t)\hat{\phi}\right)}|\Omega\rangle\,.$
(3.7)

Here, we will assume $\phi_{cl}(t)$ and $\pi_{cl}(t)=Z\dot{\phi}_{cl}(t)$ to
be given by the above mentioned classical solution, albeit with renormalized
parameters.

By calculating number of commutators, the above expression can be written as

$\displaystyle\langle
C_{cl}|C(t)\rangle=\langle\Omega|e^{\hat{Y}(t)}|\Omega\rangle\,.$ (3.8)

In the limit of infinitesimal time we can straightforwardly evaluate
$\hat{Y}(t)$ up to $t^{2}$ order, arriving at

$\displaystyle\hat{Y}(t)=$ $\displaystyle
it\left[\frac{1}{2}Z\phi_{0}^{2}\left(m_{\rm ph}^{2}-m^{2}+\frac{\lambda_{\rm
ph}-\lambda Z}{3!}\phi_{0}^{2}\right)V+\frac{1}{24}\lambda
Z^{2}\phi_{0}^{4}V\right.$ (3.9)
$\displaystyle~{}~{}~{}~{}+Z\phi_{0}\left(m_{\rm
ph}^{2}-m^{2}+\frac{\lambda_{\rm ph}-\lambda Z}{3!}\phi_{0}^{2}\right)\int
d^{3}x\hat{\phi}(x)$ $\displaystyle~{}~{}~{}~{}\left.-\phi_{0}\int
d^{3}x\left(\frac{\lambda Z^{2}\phi_{0}}{4}\hat{\phi}^{2}+\frac{\lambda
Z^{2}}{3!}\hat{\phi}^{3}\right)\right]$ $\displaystyle-$ $\displaystyle
it^{2}\left[\frac{1}{2}\phi_{0}\left(m_{\rm ph}^{2}-m^{2}+\frac{\lambda_{\rm
ph}-\lambda Z}{3!}\phi_{0}^{2}\right)\int d^{3}x\hat{\pi}\right.$
$\displaystyle~{}~{}~{}~{}~{}\left.-\frac{1}{2}\phi_{0}\int
d^{3}x\left(\frac{\lambda
Z\phi_{0}}{4}(\hat{\pi}\hat{\phi}+\hat{\phi}\hat{\pi})+\frac{\lambda
Z}{4}(\hat{\pi}\phi^{2}+\hat{\phi}^{2}\hat{\pi})\right)\right]+\mathcal{O}(t^{3})\,,$

where $V$ denotes the spatial volume, i.e $V\equiv\int d^{3}x$. Notice that if
quantum and classical evolutions were identical, then we would end up with
$\langle C_{cl}|C(t)\rangle=1$; of course, after taking into account a proper
relation between the physical and bare parameters. Notice that this
calculation is different from the infinitesimal-time transition amplitude
analysis, used in path-integral formalism; the latter will be briefly
discussed in section 3.2.

Let us proceed to calculating the probability for an evolved state to coincide
with the classically evolved state, resulting in

$\displaystyle|\langle C_{cl}|C(t)\rangle|^{2}=1-t^{2}D+\mathcal{O}(t^{3})\,,$
(3.10)

with

$\displaystyle D=$ $\displaystyle\frac{(\lambda
Z^{2})^{2}\phi_{0}^{4}}{16}\int
d^{3}x_{1}d^{3}x_{2}\left[\langle\hat{\phi}(x_{1})^{2}\hat{\phi}(x_{2})^{2}\rangle-\langle\hat{\phi}(x_{1})^{2}\rangle\langle\hat{\phi}(x_{2})^{2}\rangle\right]$
(3.11) $\displaystyle+\frac{(\lambda Z^{2})^{2}\phi_{0}^{2}}{36}\int
d^{3}x_{1}d^{3}x_{2}\langle\hat{\phi}(x_{1})^{3}\hat{\phi}(x_{2})^{3}\rangle$
$\displaystyle-\frac{\lambda Z^{3}}{3}\phi_{0}^{2}\left(m_{\rm
ph}^{2}-m^{2}+\frac{\lambda_{\rm ph}-\lambda Z}{3!}\phi_{0}^{2}\right)\int
d^{3}x_{1}d^{3}x_{2}\langle\hat{\phi}(x_{1})\hat{\phi}(x_{2})^{3}\rangle$
$\displaystyle+Z^{2}\phi_{0}^{2}\left(m_{\rm ph}^{2}-m^{2}+\frac{\lambda_{\rm
ph}-\lambda Z}{3!}\phi_{0}^{2}\right)^{2}\int
d^{3}x_{1}d^{3}x_{2}\langle\hat{\phi}(x_{1})\hat{\phi}(x_{2})\rangle\,.$

We would like to stress that in this expression, $\langle\ldots\rangle$ stands
for the vacuum expectation value. Simplifying this expression to order
$\lambda^{2}$, we get

$\displaystyle D=$ $\displaystyle Z^{2}\phi_{0}^{2}\left(m_{\rm
ph}^{2}-m^{2}-\frac{\lambda
Z}{2}\langle\hat{\phi}^{2}\rangle+\frac{\lambda_{\rm ph}-\lambda
Z}{3!}\phi_{0}^{2}\right)^{2}\int
d^{3}x_{1}d^{3}x_{2}\langle\hat{\phi}(x_{1})\hat{\phi}(x_{2})\rangle$ (3.12)
$\displaystyle+\frac{(\lambda Z^{2})^{2}\phi_{0}^{4}}{8}\int
d^{3}x_{1}d^{3}x_{2}\langle\hat{\phi}(x_{1})\hat{\phi}(x_{2})\rangle^{2}$
$\displaystyle+\frac{(\lambda Z^{2})^{2}\phi_{0}^{2}}{4}\int
d^{3}x_{1}d^{3}x_{2}\langle\hat{\phi}(x_{1})\hat{\phi}(x_{2})\rangle^{3}+\mathcal{O}(\lambda^{3})\,,$

where we have dropped terms that were obviously higher order than desired.
Provided that $m_{\rm ph}$ and $\lambda_{\rm ph}$ are the renormalized mass
and coupling, there is no possibility for the first line of (3.12) to give
$\lambda^{2}$ contributions.

Therefore, the expression for $D$ simplifies to the last two lines of (3.12),
keeping in mind that, at the order we are working, all the expectation values
need to be evaluated within free theory. Now, these remaining terms look
interesting. In particular, the second line of (3.12) looks like a semi-
classical term, as it goes as $(\lambda N)^{2}$ (with $N\propto\phi_{0}^{2}$
being number of quanta within the Compton volume), while the third line
behaves as $\lambda^{2}N$.

The problem is the evaluation of the integrals. If we compute them by Fourier
transforming the 2-point function and then we exchange the order of momentum
and position integrals, we end up with a divergent result. The IR divergence
can be regulated by taking a finite volume, however the integrals are also UV
divergent. At this point the only way around the latter is to proceed via
analytic continuation. For instance, using the Fourier space expression for
the propagator and exchanging the order of integration we readily obtain

$\displaystyle\int
d^{3}x_{1}d^{3}x_{2}\langle\hat{\phi}(x_{1})\hat{\phi}(x_{2})\rangle^{2}=\frac{V}{4(2\pi)^{3}}\int
d^{3}k\frac{1}{k^{2}+m^{2}}\,.$ (3.13)

To perform the integration we use dimensional regularisation. After
analytically continuing the result back to 3d, we get

$\displaystyle\int
d^{3}x_{1}d^{3}x_{2}\langle\hat{\phi}(x_{1})\hat{\phi}(x_{2})\rangle^{2}=-\frac{V}{16\pi}m\,.$
(3.14)

The problem with the result is the negative sign on the right-hand side of
(3.14) and a negative contribution to $D$ seems to violate unitarity (notice
that for the face value the left-hand side seems manifestly positive
definite). Analogously, the integral appearing in the third line of (3.12) can
be processed in the Fourier space as

$\displaystyle\int
d^{3}x_{1}d^{3}x_{2}\langle\hat{\phi}(x_{1})\hat{\phi}(x_{2})\rangle^{3}=\frac{V}{8(2\pi)^{6}}\int
d^{3}k_{1}d^{3}k_{2}\frac{1}{\sqrt{k_{1}^{2}+m^{2}}\sqrt{k_{2}^{2}+m^{2}}\sqrt{(\vec{k}_{1}+\vec{k}_{2})^{2}+m^{2}}}\,.$
(3.15)

Regretfully, we were unable to evaluate this integral further. Also, it must
be noted that the second line of (3.12) must be made finite independently from
the third line, as they scale differently in $\phi_{0}$.

In the absence of a regularization scheme, that yields a physically sound
finite result, it seems our perturbative expansion in time breaks down here
and the resummation is warranted. In other words, the only conceivable
resolutions to the puzzle at hand seems to be the resummation of this infinity
together with higher order terms in $t$, in order to hopefully achieve a
finite answer.

However, as one can see, the fate of $D$ is dimension sensitive. In fact, in
(1+1)-dimensions both integrals from (3.13) and (3.15) converge. The first can
be easily computed analytically; the second one, on the other hand, requires
numerical integration. Putting results together, the expression reduces to

$\displaystyle|\langle
C_{cl}|C(t)\rangle_{2D}|^{2}=1-t^{2}\cdot\lambda^{2}\phi_{0}^{2}\cdot\frac{L}{64m}\cdot\left(\frac{\alpha}{2}+\phi_{0}^{2}\right)+\mathcal{O}(\lambda^{3},t^{3})\,,$
(3.16)

with $\alpha\approx 3$ and $L$ denoting the spatial system size.

An important point to notice is the lack of a linear term in this expression
of departure of probability from unity, which would have been the rate of
departure. Instead, we have the acceleration of departure. This is connected
to the property of infinitesimal transition amplitudes for coherent states we
are about to discuss in the following section.

Before moving forward to the evolution of a one-point expectation value, where
the treatment of infinities can be discussed in a clearer way, it is worth
pausing to see the fate of these infinities within path-integral formalism for
coherent states. In particular, we will see that the divergencies similar to
those in (3.12) do not pose a problem there.

###  3.2 Path-Integral

Let us now discuss the path-integral formalism for coherent states. This topic
has a long history going back to [31, 32, 33, 34] (see also [24] and
references therein). Here, we simply overview some of the intermediate steps
in the derivation.

The main motivation for this section is to underline the difference between
the philosophy of section 3.1 and the commonly accepted approach to
infinitesimal segments in path-integral. For the coherent states (2.6) the
latter can be calculated as

$\displaystyle\langle\phi^{\prime}_{c},\pi^{\prime}_{c};t+dt|\phi_{c},\pi_{c};t\rangle=\langle\Omega|e^{i\int
d^{3}x\left(\phi^{\prime}_{c}(x)\hat{\pi}(x)-\pi^{\prime}_{c}(x)\hat{\phi}(x)\right)}\cdot
e^{-i\hat{H}dt}\cdot e^{-i\int
d^{3}x\left(\phi_{c}(x)\hat{\pi}(x)-\pi_{c}(x)\hat{\phi}(x)\right)}|\Omega\rangle\,,$
(3.17)

with $dt$ being an infinitesimal time-interval, while
$\\{\phi_{c}(x),\pi_{c}(x)\\}$ and
$\\{\phi^{\prime}_{c}(x),\pi^{\prime}_{c}(x)\\}$ stand for c-number functions
characterising the initial and final states respectively.

We can use the tricks of section 2.1, to show that (3.17) can be rewritten as

$\displaystyle\langle\phi^{\prime}_{c},\pi^{\prime}_{c};t+dt|\phi_{c},\pi_{c};t\rangle$
$\displaystyle=$ $\displaystyle e^{\frac{i}{2}\int
d^{3}z(\delta\phi_{c}(z)\pi_{c}(z)-\phi_{c}(z)\delta\pi_{c}(z))}\times$
$\displaystyle\times\langle\Omega|e^{i\int
d^{3}x\left(\delta\phi_{c}(x)\hat{\pi}(x)-\delta\pi_{c}(x)\hat{\phi}(x)\right)}\cdot
e^{-i\hat{H}[\phi_{c}+\hat{\phi},\pi_{c}+\hat{\pi}]dt}|\Omega\rangle\,,$

without loss of generality; here, we have introduced the notation
$\delta\phi_{c}\equiv\phi^{\prime}_{c}-\phi_{c}$ and
$\delta\pi_{c}\equiv\pi^{\prime}_{c}-\pi_{c}$. Notice that the Hamiltonian is
a functional of shifted fields. At this point, it has not been assumed that as
$dt\rightarrow 0$, field variations should also vanish.

If we were to assume infinitesimal variation of the field and its conjugate
momentum, then the expression would reduce to its familiar form. In other
words, treating $\delta\phi_{c}$ and $\delta\pi_{c}$ to be as infinitesimal as
$dt$ and working to linear order in these quantities, we could easily derive
the following simplified transition amplitude

$\displaystyle\langle\phi^{\prime}_{c},\pi^{\prime}_{c};t+dt|\phi_{c},\pi_{c};t\rangle\Big{|}_{infinitesimal}=e^{i\left(-\bar{H}dt+\frac{1}{2}\int
d^{3}x(\delta\phi_{c}\pi_{c}-\phi_{c}\delta\pi_{c})\right)}\,;$ (3.18)

where the c-number Hamiltonian $\bar{H}$ is defined as

$\displaystyle\bar{H}\equiv\int
d^{3}x\left(\frac{1}{2}\pi_{c}^{2}+\frac{1}{2}(\vec{\nabla}\phi_{c})^{2}+\frac{1}{2}\left(m^{2}+\frac{\lambda}{2}\langle\Omega|\hat{\phi}^{2}|\Omega\rangle\right)\phi_{c}^{2}+\frac{\lambda}{4!}\phi_{c}^{4}\right)\,.$
(3.19)

Notice the appearance of the same mass renormalization as in previous
sections. This should give us a pause, since for the standard path-integral
construction, with field eigenstates, there are no renormalizable infinities
emerging at this stage of the derivation.

However, this assumption of infinitesimal $\delta\phi_{c}$ and $\delta\pi_{c}$
may not be, strictly speaking, warranted for the purposes of path integral
derivation, as for the latter we need intermediate path segments with
arbitrary configurations. A short time interval does not entail that only
paths with small variation contribute. In the standard path-integral
formalism, this complication is avoided due to the orthonormality of the
field-eigenstate basis. Plausibly, one can make the case that most of the
contribution comes from more or less smooth trajectories, but it is an
approximation nonetheless. We need to be extra cautious here, especially
considering that we are after tiny cumulative effects. It must be stressed
that the limit of smooth paths is precisely the approximation that is used in
the literature to construct the path-integral for coherent states [31, 35,
36].

Let us appreciate the fact that for smooth trajectories the transition
amplitude for an infinitesimal segment takes such an elegant form (3.18). In
the standard case, that is if we had dealt with field-eigenstates, there would
have been only a field configuration labelling states and the path-integral
over conjugate momenta on the right-hand-side.

In our case, on the other hand, for an infinitesimal-time-transition-amplitude
that assumes $\delta\phi_{c}$ and $\delta\pi_{c}$ to be $\mathcal{O}(dt)$ or
smaller, we get a pure phase. This is precisely the reason why
$\mathcal{O}(t)$ terms are lacking in (3.10).

Going along with the approximation at hand, we can use (3.18) to construct the
path-integral over a finite period of time by piling up a large number of
infinitesimal segments and integrating over intermediate configurations,
keeping in mind that coherent states provide us with a resolution of identity.
In so many words, we can write

$\displaystyle\langle\phi^{\prime}_{c},\pi^{\prime}_{c};t^{\prime}|\phi_{c},\pi_{c};t\rangle\simeq\int[d\mu(\phi,\pi)]e^{i\int_{t}^{t^{\prime}}dt\left(\int
d^{3}x\dot{\phi}\pi-\bar{H}\right)}\cdot e^{\frac{i}{2}\int
d^{3}x\left[\pi_{c}^{\prime}\phi^{\prime}_{c}-\pi_{c}\phi_{c}\right]}\,,$
(3.20)

with the measure $[d\mu]$ defined using the following resolution of identity

$\displaystyle 1=\int[d\mu(\phi,\pi)]~{}|\phi,\pi\rangle\langle\phi,\pi|\,,$
(3.21)

implying the measure being dimensionless; for the detailed discussion see [24]
and references therein (a brief discussion of Green’s functions in this
formalism is given in appendix B).

Alternatively, one could follow the standard path-integral construction,
utilizing the resolution of identity by means of field and conjugate momentum
eigenstates. In that case, the coherent state appears only in wave-functions
of the initial and final states

$\displaystyle\langle\phi_{c}^{\prime},\pi_{c}^{\prime};t^{\prime}|\phi_{c},\pi_{c};t\rangle=\int\mathcal{D}\phi\,\mathcal{D}\pi~{}{\rm
exp}\left({i\int_{t}^{t^{\prime}}d^{4}x\left(\pi\dot{\phi}-\mathcal{H}(\phi,\pi)\right)}\right)\cdot\langle\Omega|\phi-\phi_{c}^{\prime};t^{\prime}\rangle\cdot\langle\phi-\phi_{c};t|\Omega\rangle$
$\displaystyle\cdot e^{-\frac{i}{2}\int
d^{3}x\left(\phi_{c}\pi_{c}-\phi_{c}^{\prime}\pi_{c}^{\prime}\right)}\cdot
e^{i\int
d^{3}x~{}(\phi(t)\pi_{c}-\phi(t^{\prime})\pi_{c}^{\prime})}\,.~{}~{}~{}$
(3.22)

Here, $|\phi;t\rangle$ appearing on the right-hand-side represents an
eigenstate of $\hat{\phi}(t)$.

Let us finish this section by pointing out important differences between
(3.20) and (3.22). For starters, the Hamiltonians differ by the infinite
contribution that should be absorbed in the renormalization of mass.
Furthermore, we did not have to assume smooth paths in the derivation of
(3.22), making it somewhat superior to (3.20). The price of this elegant
result is the appearance of the wave-function of the vacuum, which would have
to be calculated perturbatively. These wave-functions can be further
simplified by shifting the field-configuration of integration by the classical
function satisfying the appropriate boundary conditions. The last factor from
(3.22) simply imposes the initial and final conditions on the conjugate
momentum.

We will not discuss further the path-integral formalism in this work. It was
mentioned purely for comparative purposes of its infinitesimal segment to the
one computed in the previous subsection.

###  3.3 Expectation Value of the Field

We would like to calculate the evolution of the expectation value of the field
operator in the coherent state (3.3), that is $\langle
C|\hat{\phi}|C\rangle(t)$, and compare it to its classical counterpart.
Applying the general Heisenberg picture formalism we laid out in section 2.1
and expanding the result to order $t^{4}$, we arrive at the following
expression666Some of the relevant commutators can be found in appendix C.

$\displaystyle\langle
C|\hat{\phi}(x)|C\rangle(t)=\phi_{0}-\frac{t^{2}}{2}\left[\phi_{0}\left(m^{2}+\frac{\lambda
Z}{2}\langle\Omega|\hat{\phi}(x)^{2}|\Omega\rangle\right)+\frac{\lambda
Z}{3!}\phi_{0}^{3}\right]-\frac{t^{3}}{3!}\left[\frac{\lambda}{2}\phi_{0}\langle\Omega|\hat{\pi}\hat{\phi}+\hat{\phi}\hat{\pi}|\Omega\rangle\right]$
$\displaystyle+\frac{t^{4}}{4!}\left[\lambda
Z\phi_{0}\langle\Omega|(\partial_{\mu}\hat{\phi})^{2}+m^{2}\hat{\phi}^{2}|\Omega\rangle+\phi_{0}\left(m^{4}+\lambda
Zm^{2}\langle\Omega|\hat{\phi}^{2}|\Omega\rangle+\frac{5}{2\cdot 3!}(\lambda
Z)^{2}\langle\Omega|\hat{\phi}^{4}|\Omega\rangle\right)\right.$
$\displaystyle\left.+\frac{\lambda Z}{3!}\phi_{0}^{3}\left(4m^{2}+5\lambda
Z\langle\Omega|\hat{\phi}^{2}|\Omega\rangle\right)+\frac{(\lambda
Z)^{2}}{2\cdot 3!}\phi_{0}^{5}\right]+\mathcal{O}(t^{5})\,.~{}~{}~{}$ (3.23)

Important point to notice is that $t^{3}$ term vanishes because it can be
written as $\partial_{t}\langle\Omega|\hat{\phi}^{2}|\Omega\rangle$, which
must vanish for the Poincare invariant vacuum.

Let us try renormalizing parameters at hand, in such a way that makes (3.23)
finite at 1-loop order. Curiously enough, the finite part of (3.23) matches
the classical evolution exactly at each order in $t$; we have checked this
statement explicitly up to $t^{7}$ corrections. In order to avoid the
accidental mismanagement of classical nonlinearities, we perform the loop
expansion by keeping $\lambda$ finite but treating the loop factor
$\frac{\lambda}{16\pi^{2}}$ to be small. As a result, in dimensional
regularisation, (3.23) reduces to

$\displaystyle\langle C|\hat{\phi}(x)|C\rangle(t)\Big{|}_{\rm 1-loop}=$
$\displaystyle\phi_{0}-\frac{t^{2}}{2}\left[\phi_{0}m_{\rm
ph}^{2}+\frac{\lambda}{3!}\phi_{0}^{3}\right]$ (3.24)
$\displaystyle+\frac{t^{4}}{4!}\left[\phi_{0}m_{\rm
ph}^{4}+\frac{2\lambda}{3}\phi_{0}^{3}m_{\rm
ph}^{2}+\lambda\phi_{0}^{3}\Sigma+\frac{\lambda^{2}}{2\cdot
3!}\phi_{0}^{5}\right]\,.$

where $m_{\rm ph}^{2}\equiv m^{2}+\Sigma$, we have taken into account that
$\langle\Omega|(\partial_{\mu}\hat{\phi})^{2}+m^{2}\hat{\phi}^{2}|\Omega\rangle$=0,
in dimensional regularization and have set $Z=1$, since it acquires
corrections only starting at 2-loop in $\phi^{4}$ theory777Notice that
$\lambda$ and $Z$ always appear in $\lambda Z$ combination here.. Also, in
this expression $\Sigma$ denotes the usual 1-loop bubble integral

$\displaystyle\Sigma\equiv\frac{\lambda}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{-i}{k_{\mu}^{2}+m^{2}}\,.$
(3.25)

In other words, 1-loop computation entails keeping terms linear in $\Sigma$
only.

The key observation we would like to stress on, concerns the initial field
acceleration. Being given by the $t^{2}$-term of (3.23), it seems to be
determined in terms of the renormalized mass and the bare coupling constant.
The latter being infinite in the standard renormalization prescription, our
result seems quite strange. One would expect the renormalized coupling to be
the one driving the acceleration from the initial value of the field. In order
to make the point even clearer, let us simply take Hamilton’s operator
equation (in the Heisenberg picture) and find its expectation value in the
coherent state (3.3), at the initial time,

$\displaystyle\Big{[}\frac{\partial^{2}}{\partial t^{2}}\langle
C|\hat{\phi}|C\rangle-\Delta\langle C|\hat{\phi}|C\rangle+m^{2}\langle
C|\hat{\phi}|C\rangle+\frac{\lambda Z}{3!}\langle
C|\hat{\phi}^{3}|C\rangle\Big{]}_{t=0}=0\,.$ (3.26)

The second term vanishes due to the homogeneity of the coherent state and the
vanishing of the one-point vacuum expectation value. The last two terms also
simplify to give a result identical to the one deduced from (3.23)

$\displaystyle\frac{\partial^{2}}{\partial t^{2}}\langle
C|\hat{\phi}|C\rangle\Big{|}_{t=0}=-\phi_{0}\left(m^{2}+\frac{\lambda
Z}{2}\langle\Omega|\hat{\phi}(x)^{2}|\Omega\rangle\right)-\frac{\lambda
Z}{3!}\phi_{0}^{3}\,.$ (3.27)

It must be emphasized that this expression is exact up to this point. Even if
one could question the validity of the time-series expansion adopted earlier,
Hamilton’s operator equation serves as a core of any perturbative calculation
in quantum field theory.

The surprising feature of (3.27) lies in the fact that the initial
acceleration of the field value is driven by the bare coupling constant,
rather than the one running as a function of $\phi_{0}$. In other words, the
effective potential felt by the field at $t=0$ lacks the logarithmic
correction encountered in the celebrated Coleman-Weinberg’s work [26]. This
may not sound too surprising as the latter deals with constant backgrounds,
while here we are dealing with an oscillating one (although at the initial
moment $\dot{\phi}=0$). In general, the 1-loop effective action can be
organised in a derivative expansion

$\displaystyle\Gamma[\phi]=\int
d^{4}x\Big{(}-V(\phi^{2})-F(\phi^{2})(\partial_{\mu}\phi)^{2}+P((\partial_{\mu}\phi)^{2})+\ldots\Big{)}\,,$
(3.28)

with ellipsis standing for terms with more than one derivative per field. For
constant and homogeneous field configurations one legitimately assumes tree-
level values for $F$ and $P$, while including loop effects in the potential.
For time-dependent backgrounds, on the other hand, one needs to be more
cautious. For instance, it is straightforward to show that up to
$\sim\lambda^{2}$ terms, at 1-loop, one gets

$\displaystyle F\sim
1+\lambda^{2}\frac{\phi^{2}}{m^{2}}\,\qquad\text{and}\qquad
P\sim\frac{\lambda^{2}}{m^{4}}(\partial_{\mu}\phi)^{4}\,;$ (3.29)

see [37, 38] for the relevant discussion. Schematic equations for the
homogeneous field that follow, take the following form

$\displaystyle\left(1+\frac{\lambda^{2}\phi^{2}}{m^{2}}+\frac{\lambda^{2}\dot{\phi}^{2}}{m^{4}}\right)\ddot{\phi}=-V^{\prime}(\phi^{2})\phi-\lambda^{2}\frac{\phi}{m^{2}}\dot{\phi}^{2}\,.$
(3.30)

Although the effective potential depends on renormalized running-coupling,
bringing the expression in parentheses to the right-hand-side may
significantly affect it. In other words, some of the derivative terms of the
effective action modify the background field dynamics at the same level as the
radiatively generated potential terms888Let us stress that this statement
holds only in cases when tree-level potential gives a leading contribution to
the dynamics and one is interested in sub-leading quantum effects. There are
supersymmetric inflationary scenarios in which the inflaton field rolls solely
due to radiative corrections to the potential [39, 40]. In those cases the
derivative corrections to the effective action give a negligible contribution
compared to the quantum potential terms. We would like to thank Gia Dvali for
the discussion on this matter.. In a similar fashion, if we were to consider a
configuration with initial $\dot{\phi}$, the last term on the right hand side
of (3.30) could give vanishing contribution to $\ddot{\phi}$, together with
the combination of mass-term and the third term in parentheses.

It must be mentioned that the full expression for $F$ obtained in [37, 38]
does not seem to lead to the reconciliation with our result for coherent
states. In fact, being of semi-classical nature, there is no reason to expect
the effective field-acceleration from (3.28) to reproduce the fully quantum
result. It must simply serve as a lesson, not to draw conclusions for the
dynamical problem based on the static-background analysis. More detailed
discussion of these points from the effective action perspective is beyond
this work and will be presented elsewhere.

Here, we can shed some light on the possibility of the above-mentioned idea of
potential cancelations among derivative terms within the coherent state
formalism by introducing non-vanishing initial momentum for the field. For
this, let us consider the following initial state

$\displaystyle|C\rangle=e^{-i\hat{f}}|\Omega\rangle\,,\quad\text{with}\quad\hat{f}\equiv\int
d^{3}x\left(\phi_{0}\hat{\pi}(x,0)-\pi_{0}\hat{\phi}(x,0)\right)\,.$ (3.31)

Assuming $Z=1$ for notational simplicity and keeping in mind that it could be
easily reintroduced, we get the following extension of (3.23)

$\displaystyle\langle C|\hat{\phi}(x)|C\rangle(t)=\phi_{0}+\pi_{0}\cdot
t-\frac{t^{2}}{2}\left[\phi_{0}\left(m^{2}+\frac{\lambda}{2}\langle\Omega|\hat{\phi}(x)^{2}|\Omega\rangle\right)+\frac{\lambda}{3!}\phi_{0}^{3}\right]$
$\displaystyle-\frac{t^{3}}{3!}\left[\pi_{0}\left(m^{2}+\frac{\lambda}{2}\langle\Omega|\hat{\phi}(x)^{2}|\Omega\rangle\right)+\frac{\lambda}{2}\phi_{0}^{2}\pi_{0}\right]+\mathcal{O}(t^{4})\,.$
(3.32)

The interesting property of this relation is that the initial momentum does
not enter at $t^{2}$ order, which seems to reassure the statements made around
(3.30).

Closing the bracket on this somewhat speculative attempt for the
interpretation of our field-acceleration result, let us go back to (3.23) and
think about an alternative way around the puzzle of having the bare coupling
constant being in control of the initial evolution. Splitting the infinite
bare coupling into the physical one and a counter-term, as per usual
$\lambda=\lambda_{\rm ph}+\delta\lambda$, one can easily convince oneself that
the only possibility of getting rid of the infinite leftover term
$\left[-\frac{t^{2}}{2}\frac{\delta\lambda}{3!}\phi_{0}^{3}\right]$ is in
combination with higher order terms in $t$. For example, there is an infinite
term of the form $\left[\frac{t^{4}}{4!}\frac{\delta\lambda^{2}}{2\cdot
3!}\phi_{0}^{5}\right]$ descending from the last $t^{4}$-term of (3.23), along
with other similar ones. The question is if they can be combined into a
function of the combination $[\delta\lambda\phi_{0}^{2}t^{2}]$, vanishing in
the infinite limit for the latter. We will tackle this question of time-series
resummation within the interaction-picture formalism in section 4.

###  3.4 Curious Case of Two-Point Function

Finally we come to the 2-point correlation function calculated in the coherent
state

$\displaystyle\langle
C(t=0)|\hat{\phi}(x,t)\hat{\phi}(y,0)|C(t=0)\rangle=\phi_{0}^{2}+\langle\Omega|\hat{\phi}(x,0)\hat{\phi}(y,0)|\Omega\rangle+t\langle\Omega|\hat{\pi}(x,0)\hat{\phi}(y,0)|\Omega\rangle$
$\displaystyle-\frac{t^{2}}{2}\left[\phi_{0}^{2}\left(m^{2}+\frac{\lambda}{2}\langle\Omega|\hat{\phi}^{2}|\Omega\rangle+\frac{\lambda}{3!}\phi_{0}^{2}\right)+\langle\Omega|\hat{\phi}(y,0)\left(-\Delta+m^{2}+\frac{\lambda}{2}\phi_{0}^{2}+\frac{\lambda}{3!}\hat{\phi}^{2}(x)\right)\hat{\phi}(x,0)|\Omega\rangle\right]$
$\displaystyle+\mathcal{O}(t^{3})\,.~{}~{}$ (3.33)

Here, the notation $|C(t=0)\rangle$ has been invoked to underline the fact
that the coherent state has been constructed using operators at $t=0$, as we
are using the Heisenberg picture.

Few interesting comments are in order here:

  * •

The first two terms of (3.33) should be compared to the result of the
background field method. In our case, the second term is a vacuum expectation
value, while in the background field method the latter gets replaced with a
two-point function for perturbations; the dispersion relation of which depends
on the background configuration.

  * •

On the second line of (3.33), the first term looks encouraging as it is
precisely the combination we have dealt with in (3.23). In other words, the
renormalization prescription that makes $t^{2}$-term of (3.23) finite takes
care of that combination here as well.

  * •

The second term of the second line of (3.33) is also quite appealing, as the
operator there resembles a background dependent operator appearing in the
equation of motion for perturbations. It must be stressed that this result is
reminiscent of the general results regarding Heisenberg picture operators; see
section 2.1.

Furthermore, we can repackage (3.33) in a simpler form by means of Hamilton’s
equation as

$\displaystyle\langle
C(t=0)|\hat{\phi}(x,t)\hat{\phi}(y,0)|C(t=0)\rangle=\phi_{0}^{2}+\langle\Omega|\hat{\phi}(x,t)\hat{\phi}(y,0)|\Omega\rangle$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{t^{2}}{2}\left[\phi_{0}^{2}\left(m^{2}+\frac{\lambda}{2}\langle\Omega|\hat{\phi}^{2}|\Omega\rangle+\frac{\lambda}{3!}\phi_{0}^{2}\right)+\frac{\lambda}{2}\phi_{0}^{2}\langle\Omega|\hat{\phi}(x,0)\hat{\phi}(y,0)|\Omega\rangle\right]+\mathcal{O}(t^{3})\,.$
(3.34)

The first two terms are self-explanatory, radiative corrections must be taken
care of as usual in order to make the second term finite. Namely, at 1-loop
the mass renormalization must be the standard one. As a result, examining the
second line of (3.34) and comparing it with the 1-point function case, we
realize that the divergent part of the bare coupling, appearing on the second
line of (3.34), needs to be resummed with higher order $t$-terms in a similar
fashion (see the argument at the end of section 3.3).

##  4 Interaction Picture

In previous sections we have tackled the problem of evolution from a
relatively unconventional side. That is, we have chosen the time interval as a
primary parameter for organising the expansion, instead of the coupling. One
of the advantages of that approach is that it is quite straightforward to
account for classical nonlinearities through the underlying quantum process.

In this section we would like to pursue a more standard avenue, involving the
expansion in coupling from the get-go but keeping the time-interval of
evolution arbitrary. As we are about to show, this corresponds to the
resummation of the time-expansion of section 3.3 at a given order in the
coupling constant.

Focusing on the homogeneous coherent state (3.3) studied earlier, we are after
the evolution of one-point expectation value

$\displaystyle\langle C|\hat{\phi}(t,x)|C\rangle\,.$ (4.1)

Using the standard method we can convert this into the following out-of-time-
ordered vacuum expectation value in the interaction picture

$\displaystyle\langle
C|\hat{\phi}(t,x)|C\rangle=\lim_{T\rightarrow\infty}\frac{\langle
0|U(T,0)e^{i\phi_{0}\int
d^{3}x^{\prime}\hat{\pi}_{I}(0,x^{\prime})}U(0,t)\hat{\phi}_{I}(t,x)U(t,0)e^{-i\phi_{0}\int
d^{3}x^{\prime\prime}\hat{\pi}_{I}(0,x^{\prime\prime})}U(0,-T)|0\rangle}{\langle
0|U(T,-T)|0\rangle}\,.$

Here the subscript ’$I$’ denotes the corresponding operator to be in the
interaction picture, possessing the standard free-particle expansion in ladder
operators, ’$|0\rangle$’ stands for the free-theory vacuum and $U$ is the
usual time-ordered exponential of the interaction Hamiltonian, i.e.
$U(t,t_{0})\equiv T\\{{\rm exp}(\int_{t_{0}}^{t}dt~{}H_{I})\\}$; with
$H_{I}\equiv\lambda/4!\int d^{3}z\hat{\phi}_{I}^{4}(t,z)$. As it stands, the
expectation value in a coherent state has been recast as a vacuum expectation
value for asymptotic states with some operator insertions; with the
information about the state being imprinted in the exponential field-
displacement operators. The latter have a simple effect on enclosed operators,
that follows from the Baker–Campbell–Hausdorff identity,

$\displaystyle e^{i\phi_{0}\int
d^{3}x^{\prime}\hat{\pi}_{I}(0,x^{\prime})}\hat{\phi}_{I}(t,x)e^{-i\phi_{0}\int
d^{3}x^{\prime\prime}\hat{\pi}_{I}(0,x^{\prime\prime})}=\phi_{0}{\rm
cos}(mt)+\hat{\phi}_{I}(t,x)\,;$ (4.2)

where the first term represents a homogeneous solution to free equations of
motion, with desired initial conditions.

We evaluate the above expression for the one-point function explicitly up to
$\lambda^{2}$ order in appendix D. Since in this work we are primarily
interested in 1-loop results, let us drop higher order contributions and
present a clean result valid up to quadratic order in classical nonlinearities
and to the first order in the loop factor (i.e. in $\hbar$, if we were to
reinstate it)

$\displaystyle\langle C|\hat{\phi}(t,x)|C\rangle_{1-loop}=$
$\displaystyle\phi_{0}{\rm cos}(m_{\rm
ph}t)-\frac{\lambda\phi_{0}^{3}}{3!}\frac{{\rm sin}(m_{\rm ph}t)(6m_{\rm
ph}t+{\rm sin}(2m_{\rm ph}t))}{16m_{\rm ph}^{2}}$ (4.3)
$\displaystyle+\frac{\lambda^{2}\phi_{0}^{5}}{4!^{2}2^{6}m^{4}}\Big{(}(23-72m^{2}t^{2}){\rm
cos}(mt)-24~{}{\rm cos}(3mt)+{\rm cos}(5mt)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+12~{}mt(8~{}{\rm
sin}(mt)-3~{}{\rm sin}(3mt))\Big{)}$
$\displaystyle+\frac{\lambda^{2}}{2}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{{\rm
sin}(2E_{p}(t_{1}-t_{2}))}{(2E_{p})^{2}}\frac{{\rm
sin}(m(t-t_{1}))}{m}\Phi(t_{1})\Phi(t_{2})^{2}$
$\displaystyle+\mathcal{O}(\lambda^{3})\,,$

where, $\Phi(t)\equiv\phi_{0}{\rm cos}(mt)$ and $m_{\rm ph}$ is defined in the
usual manner as

$\displaystyle m_{\rm ph}^{2}\equiv
m^{2}+\frac{\lambda}{2}\langle\hat{\phi}_{I}^{2}\rangle\,.$ (4.4)

Here the second term is the expectation value in a free theory vacuum, a.k.a.
’Bubble Diagram’. Notice that one can use $m$ and $m_{\rm ph}$ interchangeably
in (4.3) everywhere except for the first line, due to the order of
approximation.

Interestingly, and expectantly, our calculation recovers the classical
nonlinear evolution through quantum dynamics, supplemented with quantum
corrections. Namely, the first three terms are precisely what one would obtain
by solving the classical homogeneous equation perturbatively in $\lambda$. The
last term, on the other hand, is a 1-loop divergent contribution. It is
therefore natural to expect this divergence to renormalize the coupling
constant, as the mass renormalization has been already taken care of.
Neglecting this prejudice, the fate of the divergence should be determined by
its time dependence, which we are about to check.

Naively, the momentum integral in the last term of (4.3) appears to be
linearly divergent, which is not the behaviour we would expect for the one-
loop diagram responsible for the coupling constant renormalization. However,
such assessment fails due to the appearance of the oscillatory function of
energy. After swapping momentum and time integrals and explicitly performing
the latter, it becomes evident that the divergence is logarithmic.

In order to verify that the computation of the current section is up to this
point compatible with the time-expansion analysis of previous sections, we may
blatantly expand (4.3) in Taylor series to $t^{4}$ order (preceded by explicit
integration over $t_{1,2}$). It is a matter of straightforward computation to
show that this leads to the result identical to (3.24). In other words, the
last term of (4.3) is $\mathcal{O}(t^{4})$, in time-expansion, and thus does
not contribute to the initial field-acceleration. For the latter we once again
arrive at

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}^{2}\langle
C|\hat{\phi}(t,x)|C\rangle_{1-loop}=-m_{\rm
ph}^{2}\phi_{0}-\frac{\lambda}{3!}\phi_{0}^{3}\,.$ (4.5)

For the physical discussion and potential interpretation of this result, the
reader is redirected to section 3.3.

At this point one may still wonder, whether in the current interaction-picture
computation, the time-expansion of the integrand in the last term of (4.3) is
legitimate (due to the fact that every single term in this expansion seems to
diverge). Instead, let us proceed by isolating the divergent part and then
compute the leftover finite contribution without time-expansion. For this we
use the trick that has been used before in a similar context in [21]. In
particular, in order to rewrite the last term of (4.3) in a manifestly
logarithmically divergent form, we rewrite the momentum-dependent oscillatory
term as

$\displaystyle\sin{\left(2E_{p}(t_{1}-t_{2})\right)}=\frac{1}{2E_{p}}\frac{d}{dt_{2}}\cos{\left(2E_{p}(t_{1}-t_{2})\right)}\,.$
(4.6)

Integration of this by parts in the integrand of (4.3) leads to the emergence
of three terms: two boundary terms and a term with derivative acting on
$\Phi(t_{2})$. The upper boundary term, after further integration over $t_{1}$
and remembering that at this order in our expansion we can replace $m$ with
$m_{\rm ph}$, gives the following contribution

$\displaystyle\langle
C|\hat{\phi}(t,x)|C\rangle\supset\frac{\lambda^{2}\phi_{0}^{3}}{2}\frac{\sin\left(m_{\text{ph}}t\right)\left(6m_{\text{ph}}t+\sin\left(2m_{\text{ph}}t\right)\right)}{16\,m_{\text{ph}}^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}\,\frac{1}{(2E_{p})^{3}}\,.$
(4.7)

Interestingly, this term exhibits the time-dependence identical to the second
term of (4.3), with the momentum integral corresponding to the 1-loop diagram
responsible for the coupling constant renormalization in scattering
computations. Therefore, the contribution in question can be absorbed by the
term linear in $\lambda$ as a coupling renormalization

$\displaystyle\lambda_{\rm
ph}=\lambda-3\lambda^{2}\int\frac{d^{3}p}{(2\pi)^{3}}\,\frac{1}{(2E_{p})^{3}}\,.$
(4.8)

The remarkable outcome of the adopted trick lies in the fact that the other
two terms resulting from integration-by-parts are finite. The lower boundary
term (henceforth denoted as $F_{\rm L}$) descending from the last line of
(4.3), gets expressed through Meijer G-functions. The numerical behaviour of
this contribution has been depicted on Fig. 1. Away from the initial moment,
it seems to undergo steady oscillations. However, it exhibits peculiar
behaviour near $t=0$, namely

$\displaystyle\lim_{t\rightarrow 0}F_{\rm
L}(t)=\frac{\lambda^{2}\phi_{0}^{3}}{64\pi^{2}}\left(-\frac{1}{2}+\gamma+{\rm
ln}\left(mt\right)\right)t^{2}+\mathcal{O}(t^{4})\,;$ (4.9)

which implies that the contribution to the field-acceleration, i.e.
$\partial_{t}^{2}\langle C|\hat{\phi}(t,x)|C\rangle$, has a logarithmic
initial time singularity. This is believed to be an artefact of the weak
coupling expansion; see [21] and references therein. Moreover, the unusual
behaviour we have encountered in the time-expansion analysis is related to
$F_{\rm L}(t)$-contribution as well. Namely, if we were to expand the
integrand of $F_{\rm L}(t)$ in a Taylor series in $t$, before momentum
integration, we would get a logarithmically UV-diverging $t^{2}$-term (which
would undo the coupling renormalization) and quadratically UV-diverging
$t^{4}$-term. In other words, the peculiar initial time behaviour exhibited by
(4.9) seems to be the price for recovering the standard renormalization
prescription.

Figure 1: Numerical comparison of finite quantum corrections descendent from
the last term of (4.3). The blue curve denotes $F_{L}$, the orange one
corresponds to $F_{n}$ and the $green$ one represents $F_{a}$.

The remaining term resulting from integrating (4.6) in the last term of (4.3),
consist of two parts. The part that can be analytically calculated (which we
denote as $F_{a}(t)$) and the one requiring numerical integration (denoted by
$F_{n}(t)$). Together with $F_{\rm L}$, they are depicted in Fig. 1,
demonstrating that eventually $F_{a}$ dominates over the other contributions.
The analytic expression for $F_{a}(t)$ takes the following form

$\displaystyle F_{a}(t)=\frac{\lambda^{2}\phi_{0}^{3}}{32\pi^{2}}\frac{{\rm
sin}(mt)(2mt+{\rm sin}(2mt))}{16m^{2}}\,.$ (4.10)

Putting everything together, (4.3) can be written as

$\displaystyle\langle C|\hat{\phi}(t,x)|C\rangle_{1-loop}=$
$\displaystyle\phi_{0}{\rm cos}(m_{\rm ph}t)-\frac{\lambda_{\rm
ph}\phi_{0}^{3}}{3!}\frac{{\rm sin}(m_{\rm ph}t)(6m_{\rm ph}t+{\rm
sin}(2m_{\rm ph}t))}{16m_{\rm ph}^{2}}$ (4.11)
$\displaystyle+\frac{\lambda_{\rm ph}^{2}\phi_{0}^{5}}{4!^{2}2^{6}m_{\rm
ph}^{4}}\Big{(}(23-72m_{\rm ph}^{2}t^{2}){\rm cos}(m_{\rm ph}t)-24~{}{\rm
cos}(3m_{\rm ph}t)+{\rm cos}(5m_{\rm ph}t)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+12~{}m_{\rm ph}t(8~{}{\rm
sin}(m_{\rm ph}t)-3~{}{\rm sin}(3m_{\rm ph}t))\Big{)}$
$\displaystyle+\frac{\lambda_{\rm ph}^{2}\phi_{0}^{3}}{32\pi^{2}}\frac{{\rm
sin}(m_{\rm ph}t)(2m_{\rm ph}t+{\rm sin}(2m_{\rm ph}t))}{16m_{\rm
ph}^{2}}+F_{n}(t)+F_{\rm L}(t)$ $\displaystyle+\mathcal{O}(\lambda^{3})\,,$

Figure 2: The blue curve denotes the second term of (4.11), while the green
one depicts $F_{a}$ (rescaled by the loop factor for the visibility). The plot
is displayed in units of $\lambda_{\rm ph}\phi_{0}^{3}/m_{\rm ph}^{2}$.

As one can easily notice, besides the renormalization of parameters, there are
actual quantum corrections to the classical dynamics. Moreover, the dominant
quantum effect can be evaluated analytically and it exhibits significant
similarity with the linear classical correction to the dynamics; (ignoring
numerical coefficients of order unity) the former is suppressed by the loop
factor. Moreover, it is easy to see that these contributions are always out of
phase, as depicted in Fig. 2 for convenience. Due to the linear growth of the
correction in question, one might expect it to eventually dominate over the
first term of (4.11). However, taking into account the secular instability
present in classical corrections, which we know for a fact to be an artefact
of the approximation, one may wonder whether the observed quantum instability
is equally spurious. In fact, at the level of the carried-out computation, the
classical evolution breaks down (departs from full nonlinear evolution) around
the timescale

$\displaystyle t_{\rm cl}=\left[\frac{\lambda_{\rm ph}\phi_{0}^{2}}{16m_{\rm
ph}}\right]^{-1}\,.$ (4.12)

In a similar fashion, if we were to estimate the time it would take for the
quantum contributions to become of order $\phi_{0}$, we would obtain

$\displaystyle t_{\rm q}=t_{\rm cl}\left(\frac{\lambda_{\rm
ph}}{16\pi^{2}}\right)^{-1}\,.$ (4.13)

It must be stressed that this timescale matches the expression for the
quantum-break-time given in [6], according to which, assuming re-scattering of
the background constituents to be the underlying process responsible for
quantum breaking, and assuming $2\rightarrow 2$ scattering to be the simplest
contributing process (treating particles to be collectively off-shell, thus
ignoring the kinematic prohibition of the process) one would get the rate of
quantum depletion to be given by $N/t_{\rm q}$ where $N$ is the number of
particles per Compton-volume.

Regretfully, more definite statements require going to higher orders in
$\lambda$. In particular, we know for a fact that there are processes that
change the total number of particles, the most elementary one being
$4\rightarrow 2$, with rate proportional to $\lambda^{4}$; see the discussion
in [6]. In other words, we expect the quantum part of (4.11) to have higher
order corrections in the classical coupling $\lambda$. Therefore, it remains
an open question whether the observed quantum secular instability survives in
the resummed theory.

##  5 Summary and Outlook

In this work, we have studied the real-time quantum dynamics of coherent
states in the case of an interacting quantum field theory. One of the vital
details about the setup is the build-up over the vacuum of the interacting
theory, rather than of the free Hamiltonian. In this way we manage to bypass
some of the unnecessary approximations, which in turn helps us to avoid some
of the complications that could arise in case of inconsistent perturbative
treatment of the dynamics and the state itself.

In order to establish the gradual departure of physical observables from their
classical trajectories, we have begun the analysis by organizing the dynamics
as a Taylor series in time elapsed since the initial moment. This method
seemed particularly promising for computing the initial values of certain
physical quantities. Below we recap some of our central results in this
regard.

Considering a coherent state (2.6) corresponding to the configuration with
homogeneous field $\phi_{0}$ and its momentum $\pi_{0}$, the initial
expectation values of the Hamiltonian density and the field-acceleration take
the following form at one-loop999Notice that the exact results given by (2.7)
and (3.27) are equally elegant. However, we refrain from making statements
beyond 1-loop here, since we have mainly focused our discussion on the latter.

$\displaystyle\langle C|\hat{\mathcal{H}}(x)|C\rangle(t=0)$
$\displaystyle=\frac{1}{2}\pi_{0}^{2}+\frac{1}{2}m_{\rm
ph}\phi_{0}^{2}+\frac{\lambda}{4!}\phi_{0}^{4}\,,$ (5.1)
$\displaystyle\partial_{t}^{2}\langle C|\hat{\phi}(x)|C\rangle(t=0)$
$\displaystyle=-m_{\rm ph}^{2}\phi_{0}-\frac{\lambda}{3!}\phi_{0}^{3}\,,$
(5.2)

where $m_{\rm ph}$ stands for the renormalized mass. The equation (5.2) simply
follows from the expectation value of Hamilton’s operator equation, evaluated
at the initial moment. The unexpected property of these results lies in the
fact that these quantities seem to be governed by the tree-level potential
(i.e. no evidence for coupling renormalization at this level). In fact, as we
have argued qualitatively in section 3.3, the standard expectation regarding
the field-acceleration needs to be reconsidered by including radiatively
induced derivative interactions in the effective action, as they seem to
contribute to the dynamics as significantly as the potential terms
(irrespective of the rate of the dynamics). Unfortunately, after all is said
and done the manifestly infinite contributions remain (through the bare
coupling) and need to be disposed off via resummation.

Another interesting observation we would like to bring attention to, concerns
the initial-time two-point function. In particular, the following result holds
to all orders in perturbation theory101010The only property of the theory used
in deriving this expression was that the vacuum expectation value of odd
number of fields vanishes.

$\displaystyle\langle
C|\hat{\phi}(0,x)\hat{\phi}(0,y)|C\rangle=\phi_{0}^{2}+\langle\Omega|\hat{\phi}(0,x)\hat{\phi}(0,y)|\Omega\rangle\,.$
(5.3)

The striking similarity with the background field technique is obvious here.
Note however that the second term is a background independent vacuum two-point
function, while in the semiclassical treatment it comes out to be the
correlator for fluctuations with background-dependent dispersion relation.

Moving forward, we have computed first several terms in the time-series of
$\langle C|\hat{\phi}(t,x)|C\rangle(t)$, illustrated in (3.23). It is
straightforward to see that the effect of classical nonlinearities is
correctly captured by every computed term. In fact, all finite terms obtained
in this expansion seem to correspond to classical contributions; this
statement has been verified explicitly up to $t^{7}$ order. We would like to
emphasize that the reproduction of classical contributions at each order in
$t$ is performed in full generality without resorting to the weak coupling
expansion. In this approach, quantum effects seem to appear in the form of
vacuum expectation values of singular operators, some of which can be absorbed
by the parameters of the theory, while the others need to be resummed with
higher order terms in $t$.

Due to the above-mentioned need for the resummation of the time-series, we
have performed the calculation of $\langle C|\hat{\phi}(t,x)|C\rangle(t)$
within the interaction-picture framework. Even though the starting point of
every single calculation invokes the infinitesimal time limit of the
Hamiltonian flow, the method in question manages the resummation of the time-
series at a given order in perturbation theory (in coupling constant),
yielding a result valid for extended periods of time. In particular, we have
evaluated the quantity in question up to $\lambda^{2}$ order, further
simplifying the result by dropping 2-loop contributions, leading to (4.11),
which includes nonlinear classical effects up to order $\lambda^{2}$ as well
as 1-loop quantum corrections; with parameters being renormalized as per the
standard prescription. The appearance of the standard coupling
renormalization, unlike (5.2), shows potential in terms of recovering the
effects of Coleman-Weinberg effective potential to some extent, but for that
one needs to extend our 1-loop calculation to higher orders in $\lambda$
(classical coupling) and perform the resummation. The outcome of the latter is
especially interesting, due to the appearance of spurious secular divergencies
in (4.11) that are obviously an artefact of the perturbative calculation. The
obtained quantum corrections to the dynamics have interesting properties, with
the most notable ones being:

  * •

Logarithmic initial-time singularity

$\lim_{t\rightarrow 0}\partial_{t}^{2}\langle
C|\hat{\phi}(t,x)|C\rangle\supset\frac{\lambda^{2}\phi_{0}^{3}}{32\pi^{2}}{\rm
ln}(mt).$

Similar pathologies have been previously encountered in the literature.
Namely, in the background field methods one sometimes considers the initial
state of perturbations to be given by the free-theory vacuum for fluctuations
leading to initial-time singularities, which is believed to be an artefact of
the adopted, strictly speaking ill-defined, treatment of the initial state
(see the discussion in [21] and references therein). In our coherent state
calculation, on the other hand, we have not invoked any free-theory states or
operators in the construction. However, we have shown that the perturbative
expansion of the interaction picture and subsequent termination of the series
at 1-loop effectively reduced our expression to the one we would obtain if the
initial coherent state were built from a free theory vacuum.

  * •

Growing quantum correction to the classical dynamics. It has a form similar to
the classical secular (spurious) instability, albeit with a loop-factor
suppression. Therefore, it is incapable of giving a significant correction to
the dynamics; the perturbative expansion breaks down before quantum terms have
time to grow sufficiently. However, if taken for the face value and compared
to the initial field displacement $\phi_{0}$, it leads to the estimate of [6]
for the quantum break-time (i.e. the time it takes for the quantum evolution
to depart significantly from the classical dynamics), which was obtained using
the simplest $2\rightarrow 2$ re-scattering of the constituents as a source of
quantum breaking.

We have also studied the quantum evolution of a coherent state for
infinitesimal time $t$, and its subsequent overlap with a coherent state
constructed out of the classically evolved field configuration. We have
computed the probability for a quantum state at time $t$ to coincide with the
aforementioned classical counterpart. The deviation is schematically given as

$\displaystyle|\langle C_{cl}|C(t)\rangle|^{2}=1-t^{2}D+\mathcal{O}(t^{3})\,.$
(5.4)

The notable fact about this expression is that the probability departs from
unity quadratically in $t$, which is due to the fact that the infinitesimal
transition amplitude between coherent states is a pure phase at leading order
in $t$ (see the discussion of sections 3.1 and 3.2). Regretfully, the
coefficient $D$ is divergent in (3+1)-dimensions and a resummation of higher
order terms in $t$ seems to be required. However, in (1+1)-dimensions one gets
an elegant result given by (3.16), which shows that the characteristic
timescale of departure from the classical path behaves as $\lambda^{-1}$.
This, on the other hand, corresponds to a parametrically shorter time than
(4.13).

It is quite possible for the computed departure probability to capture more of
the microscopic dynamics than the one-point function. However, the above-
mentioned peculiar result may be an artefact of the infinitesimal-time
approximation. In particular, examining the quantum contribution to (4.11) it
is easy to see that, while over timescales longer than the inverse-mass the
correction is linear in time (giving $t_{\rm q}\sim\lambda^{-2}$), for
infinitesimal time-interval the corrections become quadratic in time thus
giving $t_{\rm q}\sim\lambda^{-1}$. In other words, departure rates observed
for infinitesimal time may be part of oscillatory behaviour, similar to
(4.11).

We would like to conclude by briefly mentioning future directions. Our
calculation can be straightforwardly extended to include higher order
corrections in coupling, which may clarify whether the growing quantum effects
(depicted in (4.11)) are spurious or not. Moreover, as we have already pointed
out, going to higher orders in coupling may open up another channel of
depletion connected to the particle number changing processes. Alternatively
one could introduce another light particle coupled to $\phi$, unlocking the
two-particle annihilation channel. It would be interesting to see how the
particle production in the adopted coherent state formalism compares to the
standard semi-classical analysis which studies the production of particles in
the time-dependent classical background. These and other related questions
will be discussed elsewhere.

## Acknowledgements

We would like to thank Gia Dvali for many illuminating discussions and
comments on the manuscript. We also thank Giordano Cintia, Cesar Gomez,
Georgios Karananas, Otari Sakhelashvili, Dimitris Skliros, Tanmay Vachaspati
and Sebastian Zell for valuable discussions.

## Appendix A: Coherent State: Normal Ordering

In this appendix, we would like to discuss a point that could, in principle,
be used to question the legitimacy of the adopted construction. We have
considered the following coherent state

$\displaystyle|C\rangle=e^{-i\hat{f}}|\Omega\rangle\,,\qquad\text{with}\qquad\hat{f}\equiv\int
d^{3}x\phi_{0}\hat{\pi}\,;$ (A-I)

focusing on a homogeneous one for simplicity. This may appear to be
problematic, because of the same point singularity. Even though, the state
defined this way is properly normalised, i.e. $\langle C|C\rangle=1$, one
should keep in mind that the exponential operator is defined as its Taylor
series

$\displaystyle|C\rangle=\sum_{n=0}^{\infty}\frac{(-i\phi_{0})^{n}}{n!}\int
d^{3}x_{1}\ldots
d^{3}x_{n}\hat{\pi}(x_{1})\ldots\hat{\pi}(x_{n})|\Omega\rangle\,.$ (A-II)

Almost every term in this expansion possesses a singularity, for instance the
operator

$\displaystyle\int d^{3}x_{1}d^{3}x_{2}\hat{\pi}(x_{1})\hat{\pi}(x_{2})$
(A-III)

is singular. Easiest way to convince oneself in this statement is to calculate
the vacuum expectation value

$\displaystyle\int
d^{3}x_{1}d^{3}x_{2}\langle\Omega|\hat{\pi}(x_{1})\hat{\pi}(x_{2})|\Omega\rangle\,,$
(A-IV)

which diverges. It is straightforward to show that to the leading order in
perturbation theory it possesses an infrared divergence (i.e. is proportional
to the volume of the system). The ultra-violate singularity from the
overlapping points, on the other hand, can be regulated. At this point we may
wonder whether singularities must be brushed off term-by-term from (A-II),
using normal ordering. This is a legitimate point, since there would be no
objection against the state constructed using non-singular operators acting on
the vacuum. Indeed, let us see what happens when we go down this road. For
illustrative purposes, we will perform analysis up to certain order in
$\phi_{0}$, in particular we begin by improving our coherent state as

$\displaystyle|C\rangle=\left(1-i\phi_{0}\int
d^{3}x_{1}\hat{\pi}(x_{1})+\frac{(-i\phi_{0})^{2}}{2}\int
d^{3}x_{1}d^{3}x_{2}:\hat{\pi}(x_{1})\hat{\pi}(x_{2}):\right.$
$\displaystyle\left.+\frac{(-i\phi_{0})^{3}}{3!}\int
d^{3}x_{1}d^{3}x_{2}d^{3}x_{3}:\hat{\pi}(x_{1})\hat{\pi}(x_{2})\hat{\pi}(x_{3}):\right.$
$\displaystyle\left.+\frac{(-i\phi_{0})^{4}}{4!}\int
d^{3}x_{1}d^{3}x_{2}d^{3}x_{3}d^{3}x_{4}:\hat{\pi}(x_{1})\hat{\pi}(x_{2})\hat{\pi}(x_{3})\hat{\pi}(x_{4}):+\ldots\right)|\Omega\rangle\,.$
(A-V)

As one can see, we have simply replaced operators with their normal ordered
versions. See [41, 42] for the discussion on complete normal ordering. For
convenience, we give the explicit expressions that will be of use

$\displaystyle:\hat{\pi}(x_{1})\hat{\pi}(x_{2}):=\hat{\pi}(x_{1})\hat{\pi}(x_{2})-\langle\Omega|\hat{\pi}(x_{1})\hat{\pi}(x_{2})|\Omega\rangle\,,$
(A-VI)
$\displaystyle:\hat{\pi}(x_{1})\hat{\pi}(x_{2})\hat{\pi}(x_{3}):=\hat{\pi}(x_{1})\hat{\pi}(x_{2})\hat{\pi}(x_{3})-3\hat{\pi}(x_{1})\langle\Omega|\hat{\pi}(x_{2})\hat{\pi}(x_{3})|\Omega\rangle\,,$
(A-VII)
$\displaystyle:\hat{\pi}(x_{1})\hat{\pi}(x_{2})\hat{\pi}(x_{3})\hat{\pi}(x_{4}):=\hat{\pi}(x_{1})\hat{\pi}(x_{2})\hat{\pi}(x_{3})\hat{\pi}(x_{4})-6:\hat{\pi}(x_{1})\hat{\pi}(x_{2}):\langle\Omega|\hat{\pi}(x_{3})\hat{\pi}(x_{4})|\Omega\rangle$
$\displaystyle\hskip
135.0pt-\langle\Omega|\hat{\pi}(x_{1})\hat{\pi}(x_{2})\hat{\pi}(x_{3})\hat{\pi}(x_{4})|\Omega\rangle\,.$
(A-VIII)

Here, we have implicitly assumed that both sides are integrated over
coordinates. Although we have gotten rid of singularities, we need to check
the normalization. It is easy to see that (A-V) does not have a unit norm.
Upon normalization we get

$\displaystyle|C\rangle=$ $\displaystyle\Big{(}1-\frac{\phi_{0}^{2}}{2}\int
d^{3}y_{1}d^{3}y_{2}\langle\Omega|\hat{\pi}(y_{1})\hat{\pi}(y_{2})|\Omega\rangle$
(A-IX) $\displaystyle+\frac{\phi_{0}^{4}}{24}\int
d^{3}y_{1}d^{3}y_{2}d^{3}y_{3}d^{3}y_{4}\langle\Omega|\hat{\pi}(y_{1})\hat{\pi}(y_{2})\hat{\pi}(y_{3})\hat{\pi}(y_{4})|\Omega\rangle+\ldots\Big{)}$
$\displaystyle\times\left(1-i\phi_{0}\int
d^{3}x_{1}\hat{\pi}(x_{1})+\frac{(-i\phi_{0})^{2}}{2}\int
d^{3}x_{1}d^{3}x_{2}:\hat{\pi}(x_{1})\hat{\pi}(x_{2}):\right.$
$\displaystyle\left.+\frac{(-i\phi_{0})^{3}}{3!}\int
d^{3}x_{1}d^{3}x_{2}d^{3}x_{3}:\hat{\pi}(x_{1})\hat{\pi}(x_{2})\hat{\pi}(x_{3}):\right.$
$\displaystyle\left.+\frac{(-i\phi_{0})^{4}}{4!}\int
d^{3}x_{1}d^{3}x_{2}d^{3}x_{3}d^{3}x_{4}:\hat{\pi}(x_{1})\hat{\pi}(x_{2})\hat{\pi}(x_{3})\hat{\pi}(x_{4}):+\ldots\right)|\Omega\rangle\,.$

The first line represents the normalization factor. It is straightforward to
show that upon expansion this normalization factor simply removes the normal
ordering and reduces (A-IX) to (A-II). In conclusion we are back to square
one.

For non-interacting case, one can easily re-sum the normalization factor. In
fact, the normalized normal ordered state in that case reduces to

$\displaystyle|C\rangle=e^{-\frac{\phi_{0}^{2}}{2}\int
d^{3}x_{1}d^{3}x_{2}\langle\hat{\pi}(x_{1})\hat{\pi}(x_{2})\rangle}:e^{-i\phi_{0}\int
d^{3}x\hat{\pi}(x)}:|\Omega\rangle\,.$ (A-X)

This is precisely what one gets in quantum mechanics. Expanding in terms
creation-annihilation operators we would arrive at the familiar expression.

In conclusion, the imposition of normal ordering brings the norm away from
$1$, which can be fixed by introducing a constant normalization factor,
amounting to the removal of the normal ordering.

## Appendix B: Path-Integral: Green’s Functions

Here, we would like to discuss a generating functional for Green’s functions
within the path-integral formalism overviewed in section 3.2. In other words,
we are interested in a path-integral representation of the transition
amplitude with field insertions

$\displaystyle\langle\phi_{out},\pi_{out};t_{out}|\prod_{k}\hat{\phi}(x_{k},t_{k})|\phi_{in},\pi_{in};t_{in}\rangle\,,\quad\text{with}\quad
t_{k}>t_{k+1},~{}~{}\forall~{}k\,.$ (B-I)

As usual, we will dissect the time interval into infinitesimal ones by
inserting the resolution of identity, as a result some of the segments will
add up with a field operator sandwiched inside. Having already discussed
segments without insertions in section 3.2, let us focus on the one with a
field insertion

$\displaystyle\langle\phi^{\prime}_{c},\pi^{\prime}_{c};t+dt|\hat{\phi}(x,\tau)|\phi_{c},\pi_{c};t\rangle\quad\text{with}\quad\tau\in[t,t+dt]\,.$
(B-II)

This element can be straightforwardly evaluated to the linear order in $dt$
and field variations as

$\displaystyle\lim_{dt\rightarrow
0}\langle\phi^{\prime}_{c},\pi^{\prime}_{c};t+dt|\hat{\phi}(x,\tau)|\phi_{c},\pi_{c};t\rangle=\left[1-idt\bar{H}[\phi_{c},\pi_{c}]\right]\phi_{c}(x,t)-idt\int
d^{3}z\Big{(}\dot{\pi}_{c}(z)\langle\hat{\phi}(z)\hat{\phi}(x)\rangle$
$\displaystyle+\partial_{\vec{z}}\phi_{c}(z)\langle\partial_{\vec{z}}\hat{\phi}(z)\hat{\phi}(x)\rangle+m^{2}\phi_{c}(z)\langle\hat{\phi}(z)\hat{\phi}(x)\rangle+\frac{\lambda}{3!}\phi_{c}^{3}(z)\langle\hat{\phi}(z)\hat{\phi}(x)\rangle+\frac{\lambda}{3!}\phi_{c}(z)\langle\hat{\phi}^{3}(z)\hat{\phi}(x)\rangle\Big{)}\,;$
(B-III)

where, $\langle\ldots\rangle$ stands for the vacuum expectation value. The
second line can be simplified further using Heisenberg’s operator equation
equation of motion, but it’s unnecessary. Notice that we will, once again,
pile-up intervals of this kind together. As a result, the infinitesimal term
$dt\,\bar{H}$ will turn into an exponentiated time-integral. The infinitesimal
term involving expectation values, on the other hand, receives only
contributions from intervals with field insertions and thus vanishes in the
limit $dt\rightarrow 0$.

Therefore, it seems one can make the case for the familiar path-integral
representation for Green’s functions (as stated in [36])

$\displaystyle\langle\phi_{out},\pi_{out};t_{out}|\prod_{k}\hat{\phi}(x_{k},t_{k})|\phi_{in},\pi_{in};t_{in}\rangle\simeq\int[d\mu(\phi,\pi)]\prod_{k}\phi(x_{k},t_{k})e^{i\int_{t_{in}}^{t_{out}}dt\left(\int
d^{3}x\dot{\phi}\pi-\bar{H}\right)}$ $\displaystyle\times e^{i\int
d^{3}x\left[\pi_{out}\phi_{out}-\pi_{in}\phi_{in}\right]}\,.$ (B-IV)

From this, one can define the generating functional and quantum effective
action as usual, through the Legendre transform.

Next, we could derive Coleman-Weinberg potential using the usual background
field method; underlining the fact that all the complications we have
encountered with infinites in (3.12) are hidden in ’$\simeq$’ sign appearing
in (B-IV).

## Appendix C: Expectation Value of the Field Operator

Here we give expressions for the relevant commutation relation used in section
3.3 to evolve the expectation value of the field operator using the relations
given in sections 2.1 and the Baker–Campbell–Hausdorff formula. In particular

$\displaystyle[\hat{H},\hat{\phi}(x)]=-\frac{i}{Z}\hat{\pi}\,,$ (C-I)
$\displaystyle[\hat{H},[\hat{H},\hat{\phi}(x)]]=-\Delta\hat{\phi}(x)+m^{2}\hat{\phi}+\frac{\lambda
Z}{3!}\hat{\phi}^{3}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\int
d^{3}z\vec{\nabla}^{(z)}\left(\delta^{(3)}(x-z)\vec{\nabla}^{(z)}\hat{\phi}(z)\right)\,,$
(C-II)

(The last term is a boundary term, the likes of which we will be dropping; by
assuming that the observable will be evaluated away from the boundary.)

$\displaystyle[\hat{H},[\hat{H},[\hat{H},\hat{\phi}(x)]]]=-\frac{i}{Z}\left[(-\Delta+m^{2})\hat{\pi}+\frac{\lambda
Z}{4}\left(\hat{\pi}\hat{\phi}^{2}+\hat{\phi}^{2}\hat{\pi}\right)\right]\,,$
(C-III)
$\displaystyle[\hat{H},[\hat{H},[\hat{H},[\hat{H},\hat{\phi}(x)]]]]=(-\Delta+m^{2})^{2}\hat{\phi}+\frac{\lambda
Z}{3!}(-\Delta+m^{2})\hat{\phi}^{3}+\frac{\lambda
Z}{2}\hat{\phi}^{2}(-\Delta+m^{2})\hat{\phi}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{\lambda}{4Z}\left(\hat{\pi}^{2}\hat{\phi}+2\hat{\pi}\hat{\phi}\hat{\pi}+\hat{\phi}\hat{\pi}^{2}\right)+\frac{(\lambda
Z)^{2}}{2\cdot 3!}\hat{\phi}^{5}\,.$ (C-IV)

We have used these expressions, together with the fact that the commutators of
shifted operators (appearing in (2.12)) obey similar relations, in deriving
(3.23).

## Appendix D: Interaction-Picture Computation

Here, we would like to give some of the intermediate steps used in arriving at
the results of section 4. In a nutshell, this appendix represents the recount
of the standard interaction-picture steps applied to the evaluation of (4).

The operators $\hat{\phi}_{I}$ and $\hat{\pi}_{I}$, appearing in section 4,
are the usual interaction picture operators

$\displaystyle\hat{\phi}_{I}(t,x)=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{p}}}\left(\hat{a}_{p}e^{ip_{\mu}x^{\mu}}+\hat{a}^{\dagger}_{p}e^{-ip_{\mu}x^{\mu}}\right)\Big{|}_{x^{0}=t-t_{0}}\,,$
(D-I)
$\displaystyle\hat{\pi}_{I}(t,x)=\int\frac{d^{3}p}{(2\pi)^{3}}(-i)\sqrt{\frac{E_{p}}{2}}\left(\hat{a}_{p}e^{ip_{\mu}x^{\mu}}-\hat{a}^{\dagger}_{p}e^{-ip_{\mu}x^{\mu}}\right)\Big{|}_{x^{0}=t-t_{0}}\,,$
(D-II)

with $p^{0}=E_{p}\equiv\sqrt{m^{2}+\vec{p}^{2}}$ and $t_{0}$ being a fiducial
moment of time at which the ladder operators have been defined (not to be
confused with moment of construction of our coherent state $t=0$); as usual
$t_{0}$ drops out from observables.

The time evolution operator $U$, connecting the interaction-picture operators
to $\hat{\phi}$ and $\hat{\pi}$, is the usual one of interaction picture

$\displaystyle U(t,t_{0})$
$\displaystyle=T\left\\{e^{-i\int_{t_{0}}^{t}dt^{\prime}H_{I}(t^{\prime})}\right\\}$
(D-IV)
$\displaystyle=1+(-i)\int_{t_{0}}^{t}dt_{1}H_{I}(t_{1})+(-i)^{2}\int_{t_{0}}^{t}dt_{1}\int_{t_{0}}^{t_{1}}dt_{2}H_{I}(t_{1})H_{I}(t_{2})+\ldots\,.$

The last equality follows from the definition of the time-ordered product. We
have chosen to expand it in terms of non-ordered products because the
expectation value we are calculating is also non-ordered.

We evaluate the one-point expectation value (4) to order $\lambda^{2}$ piece-
by-piece, starting with the inner part

$\displaystyle
U(0,t)\hat{\phi}_{I}(t,x)U(t,0)=\hat{\phi}_{I}(t,x)+i\frac{\lambda}{3!}\int_{0}^{t}dt_{1}\int
d^{3}z~{}\hat{\phi}_{I}^{3}(t_{1},z)D(t_{1}-t,z-x)$
$\displaystyle+\frac{\lambda^{2}}{3!4!}\int
d^{3}z_{1}d^{3}z_{2}\Big{\\{}-\int_{0}^{t}dt_{1}\int_{t_{1}}^{t}dt_{2}~{}\hat{\phi}_{I}^{4}(t_{1},z_{1})\hat{\phi}_{I}^{3}(t_{2},z_{2})D(t_{2}-t,z_{2}-x)$
$\displaystyle+\int_{0}^{t}dt_{1}\int^{t_{1}}_{0}dt_{2}~{}\hat{\phi}_{I}^{3}(t_{1},z_{1})\hat{\phi}_{I}^{4}(t_{2},z_{2})D(t_{1}-t,z_{1}-x)\Big{\\}}$
$\displaystyle+\mathcal{O}(\lambda^{3})\,.$ (D-V)

Here, the function $D$ stands for the following commutator

$\displaystyle
D(t_{1}-t,z-x)\equiv[\hat{\phi}_{I}(t_{1},z),\hat{\phi}_{I}(t,x)]=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{2E_{p}}\Big{(}e^{ip_{\mu}(z-x)^{\mu}}-e^{-ip_{\mu}(z-x)^{\mu}}\Big{)}\Big{|}_{p^{0}=E_{p}}\,.$
(D-VI)

Moving forward, it is straightforward to show that

$\displaystyle e^{i\phi_{0}\int
d^{3}x^{\prime}\hat{\pi}_{I}(0,x^{\prime})}\hat{\phi}_{I}(t,x)e^{-i\phi_{0}\int
d^{3}x^{\prime\prime}\hat{\pi}_{I}(0,x^{\prime\prime})}=\Phi(t)+\hat{\phi}_{I}(t,x)\,,\quad\text{with}\quad\Phi(t)\equiv\phi_{0}{\rm
cos}(mt)\,.$ (D-VII)

Notice that the exponential operator still acts as a displacement operator
but, unlike earlier calculations, instead of displacing the field by
$\phi_{0}$ it does so by the time-dependent classical configuration of free
theory. This is easy to understand, as field and momentum operators are at
different times here, while in the earlier discussion they were at equal
times; also, operators are the ones of the interaction picture, hence the
appearance of the harmonic classical background. (Remember that earlier we saw
that even in full theory one gets a full anharmonic classical background in
similar calculation; the observation that was made perturbatively in time.)

Combining everything we have the following

$\displaystyle A\equiv e^{i\phi_{0}\int
d^{3}x^{\prime}\hat{\pi}_{I}(0,x^{\prime})}U(0,t)\hat{\phi}_{I}(t,x)U(t,0)e^{-i\phi_{0}\int
d^{3}x^{\prime\prime}\hat{\pi}_{I}(0,x^{\prime\prime})}-\Phi(t)=\hat{\phi}_{I}(t,x)$
$\displaystyle+i\frac{\lambda}{3!}\int_{0}^{t}dt_{1}\int
d^{3}z~{}(\Phi(t_{1})+\hat{\phi}_{I}(t_{1},z))^{3}D(t_{1}-t,z-x)$
$\displaystyle+\frac{\lambda^{2}}{3!4!}\int
d^{3}z_{1}d^{3}z_{2}\Big{\\{}-\int_{0}^{t}dt_{1}\int_{t_{1}}^{t}dt_{2}~{}(\Phi(t_{1})+\hat{\phi}_{I}(t_{1},z_{1}))^{4}(\Phi(t_{2})+\hat{\phi}_{I}(t_{2},z_{2}))^{3}D(t_{2}-t,z_{2}-x)$
$\displaystyle+\int_{0}^{t}dt_{1}\int^{t_{1}}_{0}dt_{2}~{}(\Phi(t_{1})+\hat{\phi}_{I}(t_{1},z_{1}))^{3}(\Phi(t_{2})+\hat{\phi}_{I}(t_{2},z_{2}))^{4}D(t_{1}-t,z_{1}-x)\Big{\\}}$
$\displaystyle+\mathcal{O}(\lambda^{3})\,.~{}~{}~{}~{}~{}~{}~{}~{}$ (D-VIII)

Another quantity we need to calculate is $U(T,0)\hat{O}U(0,-T)$, with the
inserted operator being the entire previous expression. Since we are
interested in order $\lambda^{2}$ corrections and that the leading order term
in the above expression is linear in the field (keeping in mind that the
vacuum expectation value of the odd number of fields vanishes), we only
require the following to the linear order in coupling

$\displaystyle
U(T,0)\hat{O}U(0,-T)=\hat{O}+(-i)\int_{0}^{T}d\tau~{}H_{I}(\tau)\hat{O}+(-i)\hat{O}\int_{-T}^{0}d\tau~{}H_{I}(\tau)+\mathcal{O}(\lambda^{2})\,.$
(D-IX)

Therefore, the entire one point function takes the following form

$\displaystyle\langle
C|\hat{\phi}(t,x)|C\rangle=\lim_{T\rightarrow\infty}\frac{1}{\langle
0|U(T,-T)|0\rangle}\Big{\\{}\Phi(t)\langle 0|U(T,-T)|0\rangle+\langle
0|A|0\rangle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle+\frac{\lambda^{2}}{3!4!}\int_{0}^{t}dt_{1}\int
d^{3}zd^{3}y~{}D(t_{1}-t,z-x)\Big{(}\int_{-T}^{T}d\tau\langle
0|\hat{\phi}_{I}^{4}(\tau,y)|0\rangle\Phi^{3}(t_{1})$
$\displaystyle+3\int_{0}^{T}d\tau\Phi(t_{1})\langle
0|\hat{\phi}_{I}^{4}(\tau,y)\hat{\phi}_{I}^{2}(t_{1},z)|0\rangle+3\int_{-T}^{0}d\tau\Phi(t_{1})\langle
0|\hat{\phi}_{I}^{2}(t_{1},z)\hat{\phi}_{I}^{4}(\tau,y)|0\rangle\Big{)}\Big{\\}}\,;$
(D-X)

with $A$ given by (D-VIII). Notice that some of the disconnected contributions
cancel trivially. One can easily recognize that the term with the integral
over $\tau\in(-T,T)$ should cancels with the combination of normalisation pre-
factor and one of the terms from $A$, just like disconnected diagrams cancel
in the standard calculations.

Let us begin evaluating $\langle 0|A|0\rangle$ order by order in $\lambda$. To
the zeroth order we obviously have simply $\Phi(t)$, which is the classical
background of a free theory. The first order correction is given by

$\displaystyle\langle 0|A|0\rangle=i\lambda\int_{0}^{t}dt_{1}\int
d^{3}z\left(\frac{1}{3!}\Phi^{3}(t_{1})+\frac{1}{2}\Phi(t_{1})\langle\hat{\phi}_{I}^{2}(t_{1},z)\rangle\right)D(t_{1}-t,z-x)+\mathcal{O}(\lambda^{2})\,.~{}~{}~{}$
(D-XI)

From this expression it immediately follows that, after substituting it into
(D-X), the order $\lambda$ part of $\langle 0|U(T,-T)|0\rangle$ in combination
with the $\Phi^{3}$-term of (D-XI) cancels with the first $\lambda^{2}$ term.
Notice that $\langle\hat{\phi}_{I}^{2}(t_{1},z)\rangle$ is an equal time and
point expectation value and as such independent of the space-time coordinates.
Henceforth, it will be denoted simply as $\langle\hat{\phi}_{I}^{2}\rangle$.
After using the explicit expression for $D$, the result simplifies to

$\displaystyle\langle
0|A|0\rangle=-\frac{\lambda\langle\hat{\phi}_{I}^{2}\rangle}{2}\cdot\phi_{0}\frac{t~{}{\rm
sin}(mt)}{2m}+\frac{\lambda}{3!}\int_{0}^{t}dt_{1}\Phi^{3}(t_{1})\frac{1}{m}{\rm
sin}\left(m(t_{1}-t)\right)+\mathcal{O}(\lambda^{2})\,.$ (D-XII)

This result makes perfect physical sense. In fact, the first term together
with $\Phi(t)$ in (D-X), can be repackaged into $\phi_{0}\,{\rm
cos}\left(\sqrt{m^{2}+\frac{\lambda}{2}\langle\hat{\phi}_{I}^{2}\rangle}~{}t\right)$,
up to $\lambda^{2}$ corrections, therefore corresponding to the mass
renormalization. The third term, on the other hand, is simply the leading
order classical correction due to nonlinearities, which can be easily
integrated to give

$\displaystyle\Phi^{(1)}(t)=-\frac{\lambda\phi_{0}^{3}}{3!}\frac{{\rm
sin}(mt)(6mt+{\rm sin(2mt)})}{16m^{2}}\,.$ (D-XIII)

Here, label ’$(1)$’ marks the fact that this correction is linear in coupling.

In order to compute quadratic corrections, we will need the correlation
function

$\displaystyle\langle
0|\hat{\phi}_{I}(t_{1},z_{1})\hat{\phi}_{I}(t_{2},z_{2})|0\rangle=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{2E_{p}}e^{i\vec{p}\cdot(\vec{z_{1}}-\vec{z_{2}})}\cdot
e^{-iE_{p}(t_{1}-t_{2})}\,;$ (D-XIV)

other correlators that make appearance in $A$-term and follows trivially from
Wick’s theorem are

$\displaystyle\langle
0|\hat{\phi}_{I}(t_{1},z_{1})\hat{\phi}_{I}^{3}(t_{2},z_{2})|0\rangle=$
$\displaystyle 3\langle
0|\hat{\phi}_{I}(t_{1},z_{1})\hat{\phi}_{I}(t_{2},z_{2})|0\rangle\cdot\langle\hat{\phi}_{I}^{2}\rangle\,,$
(D-XV) $\displaystyle\langle
0|\hat{\phi}_{I}^{3}(t_{1},z_{1})\hat{\phi}_{I}^{3}(t_{2},z_{2})|0\rangle=$
$\displaystyle 6\langle
0|\hat{\phi}_{I}(t_{1},z_{1})\hat{\phi}_{I}(t_{2},z_{2})|0\rangle^{3}$ (D-XVI)
$\displaystyle+3\langle
0|\hat{\phi}_{I}(t_{1},z_{1})\hat{\phi}_{I}(t_{2},z_{2})|0\rangle\cdot\langle\hat{\phi}_{I}^{2}\rangle^{2},$
$\displaystyle\langle
0|\hat{\phi}_{I}^{4}(t_{1},z_{1})\hat{\phi}_{I}^{2}(t_{2},z_{2})|0\rangle=$
$\displaystyle 12\langle
0|\hat{\phi}_{I}(t_{1},z_{1})\hat{\phi}_{I}(t_{2},z_{2})|0\rangle^{2}\langle\hat{\phi}_{I}^{2}\rangle$
(D-XVII) $\displaystyle+3\langle\hat{\phi}_{I}^{2}\rangle^{3}\,.$

Next we evaluate terms descending from the $\lambda^{2}$ part of (D-VIII), it
is fairly straightforward to show that only terms with nontrivial power of
$\langle 0|\hat{\phi}_{I}(t_{1},z_{1})\hat{\phi}_{I}(t_{2},z_{2})|0\rangle$
contribute,

$\displaystyle\langle
0|A|0\rangle\Big{|}_{\lambda^{2}}=\Phi^{(2)}(t)+F_{1}(t)\langle\hat{\phi}_{I}^{2}\rangle+F_{2}(t)\langle\hat{\phi}_{I}^{2}\rangle^{2}+L_{coupling}+L_{sunrise}.$
(D-XVIII)

Here $\Phi^{(2)}(t)$ stands for the quadratic (in $\lambda$) corrections to
the classical background and is given by

$\displaystyle\Phi^{(2)}(t)=\frac{\lambda^{2}\phi_{0}^{5}}{4!^{2}2^{6}m^{4}}\Big{(}(23-72m^{2}t^{2}){\rm
cos}(mt)-24~{}{\rm cos}(3mt)+{\rm cos}(5mt)$ $\displaystyle+12~{}mt(8~{}{\rm
sin}(mt)-3~{}{\rm sin}(3mt))\Big{)}\,.$ (D-XIX)

The functions $F_{1,2}$ are given as follows

$\displaystyle
F_{1}(t)=-\frac{\lambda^{2}\phi_{0}^{3}}{4!2^{5}m^{4}}\Big{(}2(-1+6m^{2}t^{2}){\rm
cos}(mt)+2~{}{\rm cos}(3mt)+mt(-13~{}{\rm sin}(mt)+3~{}{\rm
sin}(3mt))\Big{)}\,,~{}$ $\displaystyle
F_{2}(t)=\frac{\lambda^{2}\phi_{0}}{32m^{3}}t\Big{(}-mt~{}{\rm cos}(mt)+{\rm
sin}(mt)\Big{)}\,.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

It is straightforward to check that (D-XIX) coincides with the
$\lambda^{2}$-contribution to classical dynamics. Moreover, the second term of
(D-XVIII) corresponds to 1-loop mass renormalization in $\Phi^{(1)}(t)$. The
third term of (D-XVIII), on the other hand, contributes to 2-loop mass
renormalization $\Phi(t)$. Unlike our previous results, here we get nontrivial
loop diagrams as well. In particular, $L_{coupling}$ involves the loop-
integral corresponding to the renormalization of coupling, $L_{sunrise}$
corresponds to the so called sunrise 2-loop diagram. To be more concrete, for
the former we have

$\displaystyle
L_{coupling}=\frac{\lambda^{2}}{2}\int\frac{d^{3}p}{(2\pi)^{3}}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}~{}\frac{{\rm
sin}(E_{p}(t_{1}-t_{2}))}{(2E_{p})^{2}}\frac{{\rm
sin}\left(m(t-t_{1})\right)}{m}$
$\displaystyle\times\left(\Phi(t_{1})\Phi^{2}(t_{2})+\Phi(t_{1})\langle\hat{\phi}_{I}^{2}\rangle\right)\,.$
(D-XX)

(Notice that terms containing $\langle\hat{\phi}_{I}^{2}\rangle$ correspond to
two-loop diagram, in which one of the propagators in the loop has an attached
bubble.) It is easy to check, that if we were to expand 1-loop part of
$L_{coupling}$ in Taylor series in $t$, we would find that it is
$\mathcal{O}(t^{4})$. This seems to be consistent with the result of previous
sections.

For the sunrise integral, on the other hand, we have

$\displaystyle
L_{sunrise}=\frac{\lambda^{2}}{3}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{1}{2E_{p}\cdot
2E_{p^{\prime}}\cdot 2E_{p+p^{\prime}}}\times$
$\displaystyle\times\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}~{}{\rm
sin}\Big{(}(E_{p}+E_{p^{\prime}}+E_{p+p^{\prime}})(t_{1}-t_{2})\Big{)}\frac{{\rm
sin}\left(m(t-t_{1})\right)}{m}\Phi(t_{2})\,.$ (D-XXI)

Combining everything we have gotten so far and renormalising the mass we
arrive at

$\displaystyle\Phi(t)+\langle 0|A|0\rangle=$ $\displaystyle\phi_{0}{\rm
cos}(m_{\rm ph}t)-\frac{\lambda\phi_{0}^{3}}{3!}\frac{{\rm sin}(m_{\rm
ph}t)(6m_{\rm ph}t+{\rm sin(2m_{\rm ph}t)})}{16m_{\rm ph}^{2}}$ (D-XXII)
$\displaystyle+\frac{\lambda^{2}\phi_{0}^{5}}{4!^{2}2^{6}m^{4}}\Big{(}(23-72m^{2}t^{2}){\rm
cos}(mt)-24~{}{\rm cos}(3mt)+{\rm cos}(5mt)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+12~{}mt(8~{}{\rm
sin}(mt)-3~{}{\rm sin}(3mt))\Big{)}$
$\displaystyle+L_{coupling}+L_{sunrise}+\mathcal{O}(\lambda^{3})\,,$

This already looks very promising, but before concluding let us take care of
terms from (D-X) with $T$ as a limit of integration.

As a result of straightforward computation, all of the $T$-dependent terms
cancel, except one; from which $T$ drops out in $T\rightarrow\infty$ limit.
Putting everything together we have the following

$\displaystyle\langle C|\hat{\phi}(t,x)|C\rangle=\Phi(t)+\langle
0|A|0\rangle-\frac{\lambda^{2}\langle\hat{\phi}_{I}^{2}\rangle}{2m}\int_{0}^{t}dt_{1}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{(2E_{p})^{3}}\Phi(t_{1}){\rm
sin}(m(t_{1}-t)){\rm cos}(2E_{p}t_{1})$
$\displaystyle+\mathcal{O}(\lambda^{3})\,.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(D-XXIII)

Obviously, the second term in this expression corresponds to a 2-loop
correction, similar to $L_{sunrise}$, with 1-loop result nicely summarized in
(4.3).

## References

  * [1] G. Dvali and C. Gomez, “Black Hole’s Quantum N-Portrait,” Fortsch. Phys. 61, 742-767 (2013) doi:10.1002/prop.201300001 [arXiv:1112.3359 [hep-th]].
  * [2] G. Dvali and C. Gomez, “Black Holes as Critical Point of Quantum Phase Transition,” Eur. Phys. J. C 74, 2752 (2014) doi:10.1140/epjc/s10052-014-2752-3 [arXiv:1207.4059 [hep-th]].
  * [3] G. Dvali and C. Gomez, “Black Hole Macro-Quantumness,” [arXiv:1212.0765 [hep-th]].
  * [4] G. Dvali, D. Flassig, C. Gomez, A. Pritzel and N. Wintergerst, “Scrambling in the Black Hole Portrait,” Phys. Rev. D 88, no. 12, 124041 (2013) doi:10.1103/PhysRevD.88.124041 [arXiv:1307.3458 [hep-th]].
  * [5] G. Dvali and C. Gomez, “Quantum Compositeness of Gravity: Black Holes, AdS and Inflation,” JCAP 1401, 023 (2014) doi:10.1088/1475-7516/2014/01/023 [arXiv:1312.4795 [hep-th]].
  * [6] G. Dvali, C. Gomez and S. Zell, “Quantum Break-Time of de Sitter,” JCAP 06, 028 (2017) doi:10.1088/1475-7516/2017/06/028 [arXiv:1701.08776 [hep-th]].
  * [7] L. Berezhiani and M. Trodden, “How Likely are Constituent Quanta to Initiate Inflation?,” Phys. Lett. B 749, 425-430 (2015) doi:10.1016/j.physletb.2015.08.007 [arXiv:1504.01730 [hep-th]].
  * [8] L. Berezhiani, “On Corpuscular Theory of Inflation,” Eur. Phys. J. C 77, no.2, 106 (2017) doi:10.1140/epjc/s10052-017-4672-5 [arXiv:1610.08433 [hep-th]].
  * [9] G. Dvali and S. Zell, “Classicality and Quantum Break-Time for Cosmic Axions,” JCAP 07, 064 (2018) doi:10.1088/1475-7516/2018/07/064 [arXiv:1710.00835 [hep-ph]].
  * [10] A. Kovtun and M. Zantedeschi, “Breaking BEC,” [arXiv:2003.10283 [hep-th]].
  * [11] A. Kovtun and M. Zantedeschi, [arXiv:2008.02187 [hep-th]].
  * [12] F. Cooper, S. Habib, Y. Kluger, E. Mottola, J. P. Paz and P. R. Anderson, Phys. Rev. D 50, 2848-2869 (1994) doi:10.1103/PhysRevD.50.2848 [arXiv:hep-ph/9405352 [hep-ph]].
  * [13] S. Habib, Y. Kluger, E. Mottola and J. P. Paz, Phys. Rev. Lett. 76, 4660-4663 (1996) doi:10.1103/PhysRevLett.76.4660 [arXiv:hep-ph/9509413 [hep-ph]].
  * [14] F. Cooper, S. Habib, Y. Kluger and E. Mottola, Phys. Rev. D 55, 6471-6503 (1997) doi:10.1103/PhysRevD.55.6471 [arXiv:hep-ph/9610345 [hep-ph]].
  * [15] G. Aarts, G. F. Bonini and C. Wetterich, Phys. Rev. D 63, 025012 (2001) doi:10.1103/PhysRevD.63.025012 [arXiv:hep-ph/0007357 [hep-ph]].
  * [16] B. Mihaila, F. Cooper and J. F. Dawson, Phys. Rev. D 63, 096003 (2001) doi:10.1103/PhysRevD.63.096003 [arXiv:hep-ph/0006254 [hep-ph]].
  * [17] J. Berges, Nucl. Phys. A 699, 847-886 (2002) doi:10.1016/S0375-9474(01)01295-7 [arXiv:hep-ph/0105311 [hep-ph]].
  * [18] G. Aarts, D. Ahrensmeier, R. Baier, J. Berges and J. Serreau, Phys. Rev. D 66, 045008 (2002) doi:10.1103/PhysRevD.66.045008 [arXiv:hep-ph/0201308 [hep-ph]].
  * [19] J. Berges, “Introduction to nonequilibrium quantum field theory,” AIP Conf. Proc. 739, no.1, 3-62 (2004) doi:10.1063/1.1843591 [arXiv:hep-ph/0409233 [hep-ph]].
  * [20] S. Borsanyi and U. Reinosa, Phys. Rev. D 80, 125029 (2009) doi:10.1103/PhysRevD.80.125029 [arXiv:0809.0496 [hep-th]].
  * [21] E. A. Calzetta and B.-L. B. Hu, “Nonequilibrium Quantum Field Theory”
  * [22] R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766-2788 (1963) doi:10.1103/PhysRev.131.2766
  * [23] T. Kibble, “Frequency Shift in High-Intensity Compton Scattering,” Phys. Rev. 138, B740-B753 (1965) doi:10.1103/PhysRev.138.B740
  * [24] W. Zhang, D. Feng and R. Gilmore, “Coherent States: Theory and Some Applications,” Rev. Mod. Phys. 62, 867-927 (1990) doi:10.1103/RevModPhys.62.867
  * [25] T. Vachaspati, Phys. Rev. D 95, no.12, 125002 (2017) doi:10.1103/PhysRevD.95.125002 [arXiv:1704.06235 [hep-th]].
  * [26] S. R. Coleman and E. J. Weinberg, “Radiative Corrections as the Origin of Spontaneous Symmetry Breaking,” Phys. Rev. D 7, 1888-1910 (1973) doi:10.1103/PhysRevD.7.1888
  * [27] F. Cooper and E. Mottola, “Initial Value Problems in Quantum Field Theory in the Large N Approximation,” Phys. Rev. D 36, 3114 (1987) doi:10.1103/PhysRevD.36.3114
  * [28] J. Baacke, K. Heitmann and C. Patzold, “On the choice of initial states in nonequilibrium dynamics,” Phys. Rev. D 57, 6398-6405 (1998) doi:10.1103/PhysRevD.57.6398 [arXiv:hep-th/9711144 [hep-th]].
  * [29] D. Boyanovsky, H. de Vega, R. Holman and M. Simionato, “Dynamical renormalization group resummation of finite temperature infrared divergences,” Phys. Rev. D 60, 065003 (1999) doi:10.1103/PhysRevD.60.065003 [arXiv:hep-ph/9809346 [hep-ph]].
  * [30] J. Baacke, D. Boyanovsky and H. de Vega, “Initial time singularities in nonequilibrium evolution of condensates and their resolution in the linearized approximation,” Phys. Rev. D 63, 045023 (2001) doi:10.1103/PhysRevD.63.045023 [arXiv:hep-ph/9907337 [hep-ph]].
  * [31] J. R. Klauder, “The Action option and a Feynman quantization of spinor fields in terms of ordinary C numbers,” Annals Phys. 11, 123-168 (1960) doi:10.1016/0003-4916(60)90131-7
  * [32] S. S. Schweber, “On Feynman Quantization”, Journal of Mathematical Physics 3, 5, 831-842 (1962) doi:10.1063/1.1724296
  * [33] F. Berezin, “Non-wiener continual integrals,” Teor. Mat. Fiz. 6, 194-212 (1971) doi:10.1007/BF01036576
  * [34] J. R. Klauder, “Path Integrals and Stationary Phase Approximations,” Phys. Rev. D 19, 2349 (1979) doi:10.1103/PhysRevD.19.2349
  * [35] H. Kuratsuji and T. Suzuki, “Path Integral Approach to Many-Body Systems and Classical Quantization of Time-Dependent Mean Field”, Progress of Theoretical Physics Supplement, Nos. 74-75, pp. 209-220 (1983) doi:10.1143/PTPS.74.209
  * [36] W. Zhang, “Coherent states in field theory,” [arXiv:hep-th/9908117 [hep-th]].
  * [37] J. Iliopoulos, C. Itzykson and A. Martin, Rev. Mod. Phys. 47, 165 (1975) doi:10.1103/RevModPhys.47.165
  * [38] C. M. Fraser, “Calculation of Higher Derivative Terms in the One Loop Effective Lagrangian,” Z. Phys. C 28, 101 (1985) doi:10.1007/BF01550255
  * [39] G. R. Dvali, Q. Shafi and R. K. Schaefer, “Large scale structure and supersymmetric inflation without fine tuning,” Phys. Rev. Lett. 73, 1886-1889 (1994) doi:10.1103/PhysRevLett.73.1886 [arXiv:hep-ph/9406319 [hep-ph]].
  * [40] S. Dimopoulos, G. R. Dvali and R. Rattazzi, “Dynamical inflation and unification scale on quantum moduli spaces,” Phys. Lett. B 410, 119-124 (1997) doi:10.1016/S0370-2693(97)00970-2 [arXiv:hep-ph/9705348 [hep-ph]].
  * [41] D. P. Skliros, “Tadpoles, Cephalopods, and ’Complete Normal Ordering’,” PoS PLANCK2015 (2015), 123 doi:10.22323/1.258.0123 [arXiv:1510.02549 [hep-th]].
  * [42] J. Ellis, N. E. Mavromatos and D. P. Skliros, “Complete Normal Ordering 1: Foundations,” Nucl. Phys. B 909 (2016), 840-879 doi:10.1016/j.nuclphysb.2016.05.024 [arXiv:1512.02604 [hep-th]].