%\documentstyle[aps,twocolumn]{revtex}
\documentstyle[aps,prd]{revtex}
\begin{document}

\tighten

\title {Decoherence in Field Theory: General Couplings and
Slow Quenches}

\author{F.\ C.\ Lombardo$^{1}$  \footnote{Electronic address:
lombardo@df.uba.ar}, F.\ D.\ Mazzitelli $^1$ \footnote{Electronic
address: fmazzi@df.uba.ar}, and R.\ J.\ Rivers $^2$
\footnote{Electronic address: r.rivers@ic.ac.uk \\
Permanent address: Imperial College, London SW7 2BZ}}

\address{{\it
$^1$ Departamento de F\'\i sica, Facultad de Ciencias Exactas y Naturales\\
Universidad de Buenos Aires - Ciudad Universitaria,
Pabell\' on I\\
1428 Buenos Aires, Argentina\\
$^2$  Centre of Theoretical Physics, University of Sussex, \\
Brighton, BN1 9QJ, U.K.}}

\maketitle

\begin{abstract}

We study the onset of a classical order parameter after a second-order
phase transition in quantum field theory. We consider a quantum scalar
field theory in which the system-field (long-wavelength modes),
interacts  with its environment,
represented both by a set of scalar fields and by its own
short-wavelength modes. We compute the decoherence times for the
system-field modes and compare them with the other time scales of
the model. We analyze  different couplings between the system and the
environment for both instantaneous and slow quenches.
Within our approximations decoherence is
in general a short time event.
\end{abstract}

\vskip 2cm
PACS numbers: 03.70.+k,  05.70.Fh, 03.65.Yz

%\newpage

\section{Introduction}
%
An understanding of the dynamics of phase transitions is important
in areas of physics as different as particle physics, cosmology
and condensed matter. The first analysis of phase transitions in
relativistic field theory, based on the  picture of a classical
Higgs field (order parameter) rolling down an adiabatic effective
potential \cite{old}, has been shown to be suspect \cite{cormier}.
Alternatively the suggestion by Kibble\cite{Kibble} that, while a
non-adiabatic approach is crucial, causality alone can set
saturated bounds on time and distance scales during a transition,
has been shown to be only partly true, as has its counterpart by
Zurek in condensed matter physics\cite{Zurek}. Numerical
calculations  (using time-dependent Ginzburg-Landau
theory)\cite{laguna} refined the argument for condensed matter
systems, while leaving the original predictions of Zurek largely
unchanged. However, although similar equations have been used
empirically in quantum field theory  to improve upon Kibble's
predictions, only limited progress has been
made\cite{Rivers2,Greg} in justifying the scaling behaviour
proposed in them.

The description of phase transitions from first principles must
take into account both the quantum nature of the order parameter
(usually described by a set of quantum scalar fields) and the
non-equilibrium aspects of the process. This is of course an
extremely complex problem. However, some simple estimates for the
behaviour of a quantum field can be obtained for early times after
the phase transition, before interactions have time to take
effect\cite{Rivers1}: if the quench is fast, the initial stages of
a scalar transition can be described by a free field theory with
inverted mass $M^2<0$. The state of the field is initially
concentrated on a local maximum of the potential, and spreads out
with time. This description is valid for short times, until the
field wave functional explores the ground states of the potential.

The calculations can be improved by
incorporating some backreaction effects in a self-consistent
(Hartree-like) approximation \cite{Greg,Karra,kim,Boya,Ramsey}.
For example, Stephens et
al. \cite{Greg} analyzed the problem by computing numerically the
two-point function for a quantum scalar field in a FRW Universe.
They confirmed Kibble's qualitative critical behaviour.
However, the validity of these non perturbative
approaches in quantum field theory is not well established.

There is an important aspect, the main issue in the present paper,
that has not been completely settled in the previous literature:
the quantum to classical transition of the order parameter. This
transition is crucial to justify several astrophysical
predictions. For example, the very notion of topological defects
(e.g. monopoles) that characterise the domain structure after a
finite-time transition, and whose presence has consequences for
the early universe, is based on the assumption of classical
behaviour for the order parameter\cite{vilen}.

The onset of classical behaviour in {\it closed} quantum systems
has been addressed in Refs.\cite{salman,mottola,devega}. For some
models, classicality was shown to appear as a consequence of huge
particle production, whereby a non-perturbatively large occupation
number of long-wavelength particles produces, on average, a
diagonal density matrix, but this is a {\it late-time} event.
Further, as suggested by quantum mechanical toy models
\cite{diana}, the classical behaviour so obtained could be an
artifact of the Gaussian or $1/N$ approximations. A complete
treatment of the classical limit must include the {\it interaction
of the system with the environment with which inescapably
interacts}. Therefore, one should consider an {\it open} quantum system.

There are several time scales which are relevant for the
description of the onset of a transition\cite{Karra}, apart from
the minimal response time $t_{\rm r}= O(\mu^{-1})$ (particle mass
$\mu$) of the system to the change in temperature. The field
ordering after the transition is due to the growth in amplitude of
its unstable long-wavelength modes, for which the most significant
time is the {\it spinodal} time $t_{\rm sp}$, i.e. the time
interval it takes these unstable field modes to sample the true
minima of the potential. The stable short-wavelength modes of the
field, together with all the other fields with which the $\phi$
interacts, form an environment whose coarse-graining makes the
long-wavelength modes decoherent, as far as the transition is
concerned. As a result, there is an additional time scale
associated with the environment, the {\it decoherence} time
$t_{\rm D}$ \cite{diana}. For $t>t_{\rm D}$ the order parameter
becomes a classical object. It is therefore fundamental to
introduce this time scale into the analysis of the problem.

In a recent paper \cite{lomplb},
we have considered a particular model of an
explicitly {\it open} system, in which classicality is an {\it
early} time event, induced by the environment.
We have shown that under certain approximations the decoherence time is
shorter than the spinodal time. This conclusion is completely
different from that of Refs.\cite{salman,mottola}.
The aim of the present paper is to
extend these results to more general couplings
between the system and the environment, for both
instantaneous and  slow quenches. We
will also provide more details of the calculations.

The paper is organized as follows. In Section \ref{sec:model} we
will introduce our first models. These are theories containing a
system-field $\phi$, coupled to a set of scalar fields $\chi_{\rm
a}$ which constitute an external part of its environment. We
compute the influence functional by integrating out the
environment-fields for different couplings. In Section
\ref{sec:diff} we describe how to obtain the master equation and
we compute the coefficients of the diffusion that enforces
classical behaviour for the simplest case of an instantaneous
quench. Since diffusion coefficients are additive at our level of
approximation, the inclusion of more components of the environment
only serves to shorten the decoherence time.

In Section \ref{sec:tdec} we evaluate upper bounds on the
decoherence times for all these models with this idealised quench,
and we make a detailed comparison with the other relevant temporal
scales. As we will see, the decoherence time depends primarily on
the initial temperature of the environment and its coupling to the
system, and is typically shorter than the spinodal time.

The scaling analysis of Kibble was for quenches implemented with
finite quench rates, as happens in realistic systems. In Section
\ref{sec:slow} we compute the decoherence time evaluation for
slower quenches. In this case, provided quenches are not too slow,
decoherence also happens before the field samples the minimum of
the potential. In Section \ref{sec:final} we summarise our main
conclusions and we write the semiclassical Langevin equation that
describes the stochastic dynamics of the order parameter.

%
\section{The Influence Action for an External  Environment}
\label{sec:model}

For the infinite degree of freedom quantum field undergoing a
continuous transition, the field ordering after the transition
begins is due to the growth in amplitude of its unstable
long-wavelength modes. For these modes the environment consists of
the short-wavelength modes of the field, together with all the
other fields $\chi_{\rm a}$ with which  $\phi$ inevitably
interacts\cite{huzhang,lombmazz,greiner} in the absence of
selection rules. The inclusion of explicit environment fields
$\chi_{\rm a}$ is both a reflection of the fact that a scalar
field in isolation is physically unrealistic, as well as providing
us with a systematic approximation scheme.

Although the system field $\phi$ can never avoid the decohering
environment of its short-wavelength modes, to demonstrate the
effect of an environment we first consider the case in which it is
taken to be composed only of the fields $\chi_{\rm a}$. We are
helped in this by the fact that scalar environments have a
cumulative effect on the onset of classical behaviour. That is,
the inclusion of a further component of the environment {\it
reduces} the time $t_D$ it takes for the system to behave
classically. Thus it makes sense to include the environment one
part after another, since we can derive an {\it upper} bound on
$t_D$ at each step. The short-wavelength modes of the $\phi$ field
will be considered last.

The $\phi$-field describes the scalar order parameter, whose
${\cal Z}_2$ symmetry is broken by a double-well potential.
Specifically, we take the simplest classical action with scalar
environmental fields
%
\begin{equation}
S[\phi , \chi ] = S_{\rm syst}[\phi ] + S_{\rm env}[\chi ] +
S_{\rm int}[\phi ,\chi ],
\label{action0}
\end{equation}
where (with $\mu^2$, $m^2 >0$ )
\begin{equation}
S_{\rm syst}[\phi ] = \int d^4x\left\{ {1\over{2}}\partial_{\mu}
\phi\partial^{\mu} \phi + {1\over{2}}\mu^2 \phi^2 -
{\lambda\over{4!}}\phi^4\right\}, \nonumber
\end{equation}
and
\begin{equation}
S_{\rm env}[\chi_{\rm a} ] = \sum_{\rm a=1}^N\int d^4x\left\{
{1\over{2}}\partial_{\mu}\chi_{\rm a}
\partial^{\mu}
\chi_{\rm a} - {1\over{2}} m_{\rm a}^2 \chi^2_{\rm a}\right\}.
\nonumber
\end{equation}
The most important interactions are
% comprise a sum of
%terms, $S_{\rm int}[\phi ,\chi ]=\sum_{n,m=1,2}S_{(n,m)}[\phi
%,\chi ]$, individually of the form
of the biquadratic form
\begin{equation}
S_{\rm int}[\chi_{\rm a} ] = S_{\rm qu}[\phi ,\chi ] = - \sum_{\rm
a=1}^N\frac{g_{\rm a}}{8} \int d^4x ~ \phi^{\rm 2} (x) \chi^{\rm
2}_{\rm a} (x). \label{Sint}
\end{equation}
Even if there were no external $\chi$ fields with a quadratic
interaction of this kind, the interaction between short and
long-wavelength modes of the $\phi$-field can be recast, in part, as
(\ref{Sint}) (see later), showing that such a term is obligatory.
Later, we shall consider further interactions of the Yukawa form
\begin{equation}
S_{\rm Yuk}[\phi ,\chi ] = - \sum_{\rm a=1}^N\frac{g'_{\rm
a}\mu}{4} \int d^4x ~ \phi^{\rm 2} (x) \chi_{\rm a} (x).
\label{Sint2}
\end{equation}
%
and even linear couplings
\begin{equation}
S_{\rm lin}[\phi ,\chi ] = - \sum_{\rm a=1}^N\frac{g''_{\rm
a}\mu^2}{4} \int d^4x ~ \phi (x) \chi_{\rm a} (x), \label{Sint3}
\end{equation}
since, traditionally, much if not most of the work on decoherence
has been for linear coupling to the environment. Although, in the
context of quantum field theory, such linear couplings signal an
inappropriate field diagonalisation, they are exactly solvable in
some circumstances \cite{kim}. Whether we consider the quadratic,
Yukawa (and linear) interactions as those of the same fields
$\chi_{\rm a}$, or further fields, is immaterial at the level at
which we are working. The generalization to a complex field $\phi$
is straightforward.

To keep our calculations tractable, we need the environment to
have a strong impact upon the system-field, but not vice-versa,
whenever possible. The simplest way to implement this is to take a
large number $N\gg 1$ of $\chi_{\rm a}$ fields with comparable
masses $m_{\rm a}\simeq \mu$ weakly coupled to the $\phi$, with
$\lambda$, $g_{\rm a},g'_{\rm a},g''_{\rm a}\ll 1$. Thus, at any
step, there are $N$ weakly coupled environmental fields
influencing the system field, but only one weakly self-coupled
system field to back-react upon the explicit environment.

We shall assume that the initial states of the system and
environment are both thermal, at a temperature $T_{0}>T_{\rm c}$.
We then imagine a change in the global environment (e.g. expansion
in the early universe) that can be characterised by a change in
temperature from $T_{0}$ to $T_{\rm f}<T_{\rm c}$. That is, we do
not attribute the transition to the effects of the
environment-fields.

Given our thermal initial conditions it is not the case that the
full density matrix has $\phi$ and $\chi$ fields uncorrelated
initially, since it is the interactions between them that leads to
the restoration of symmetry at high temperatures. Rather, on
incorporating the hard thermal loop 'tadpole' diagrams of the $\chi$
(and $\phi$) fields in the $\phi$ mass term leads to the effective
action for $\phi$ quasiparticles,
\begin{equation}
S^{\rm eff}_{\rm syst}[\phi ] = \int d^4x\left\{
{1\over{2}}\partial_{\mu} \phi\partial^{\mu} \phi - {1\over{2}}
m_{\phi}^2(T_0) \phi^2 - {\lambda\over{4!}}\phi^4\right\}
\label{Stherm}
\end{equation}
%
where $m_{\phi}^2(T_0)\propto (1-T_0/T_{\rm c})$ for $T\approx
T_c$. As a result, we can take an initial factorised density
matrix at temperature $T_0$ of the form ${\hat\rho}[T_0] =
{\hat\rho}_{\phi}[T_0] \times {\hat\rho}_{\chi}[T_0]$, where
${\hat\rho}_{\phi}[T_0]$ is determined by the quadratic part of
$S^{\rm eff}_{\rm syst}[\phi ]$ and ${\hat\rho}_{\chi}[T_0]$ by
$S_{\rm env}[\chi_{\rm a} ]$. That is, the many $\chi_{\rm a}$
fields have a large effect on $\phi$, but the $\phi$-field has
negligible effect on the $\chi_{\rm a}$.

Provided the change in temperature is not too slow the exponential
instabilities of the $\phi$-field grow so fast that the field has
populated the degenerate vacua well before the temperature has
dropped to $T_{\rm f}\approx 0$. Since the temperature $T_{\rm c}$ has no
particular significance for the environment fields, for these
early times we can keep the temperature of the environment fixed
at $T_{\chi} = T_{0}={\cal O}(T_{\rm c})$ (our calculations are
only at the level of orders of magnitude).  Meanwhile, for
simplicity the $\chi_{\rm a}$ masses are fixed at the common value
$ m\simeq\mu$.

The reduced density matrix $\rho_{{\rm
r}}[\phi^+,\phi^-,t]=\langle\phi^+\vert {\hat\rho}_r (t)\vert
\phi^- \rangle$ describes the evolution of the system under
the influence of the environment, and is defined by
%
\[
\rho_{{\rm r}}[\phi^+,\phi^-,t] = \prod_{{\rm a}=1}^N \int {\cal D}
\chi_{\rm a} ~
\rho[\phi^+,\chi_{\rm a} ,\phi^-,\chi_{\rm a} ,t],
\]
%
where $\rho[\phi^+,\chi^+_{\rm a} ,\phi^-,\chi^-_{\rm a},t]=
\langle\phi^+ \chi^+_{\rm a}\vert {\hat\rho}(t) \vert \phi^-
\chi^-_{\rm a}\rangle$ is the full density matrix. Since we need
to be able to distinguish between different classical system-field
configurations evolving after the transition, we will only be
interested in the field-configuration basis for this reduced
density matrix (this is in analogy with the same choice done in
the usual quantum Brownian motion model, see \cite{zurekTA} as an
example). The environment will have had the effect of making the
system effectively classical once $\rho_{\rm r}(t)$ is,
approximately, diagonal (which is essentially different from the
dephasing effects found in Ref. \cite{salman,mottola}). Quantum
interference can then be ignored and
%the system is said to have
%decohered. At the same time,
we obtain a classical probability distribution from the diagonal
part of $\rho_{\rm r}(t)$, or equivalently, by means of the
reduced Wigner functional, which is positive definite after
decoherence time \cite{diana}. For weak coupling there will be no
'recoherence' at later times in which the sense of classical
probability will be lost \cite{nuno}.

The temporal evolution of $\rho_{\rm r}$ is given by
\[\rho_{\rm r}[\phi_{{\rm
f}}^+,\phi_{{\rm f}}^-,t]= \int d\phi_{{\rm i}}^+ \int d\phi_{{\rm
i}}^- J_{\rm r}[\phi_{{\rm f}}^+,\phi_{{\rm f}}^-,t\vert
\phi_{{\rm i}}^+,\phi_{{\rm i}}^-,t_0] ~~\rho_{\rm r}[\phi_{{\rm
i}}^+ \phi_{{\rm i}}^-,t_0],
\]
where $J_{\rm r}$ is the reduced evolution operator

\begin{equation}
J_{\rm r}[\phi_{{\rm f}}^+,\phi_{{\rm f}}^-,t\vert \phi_{{\rm
i}}^+,\phi_{{\rm i}}^-,t_0] =\int_{\phi_{{\rm i}}^+}^{\phi_{{\rm
f}}^+} {\cal D}\phi^+ \int_{\phi_{{\rm i}}^-}^{\phi_{{\rm f}}^-}
{\cal D}\phi^- ~e^{i\{S[\phi^+] - S[\phi^-]\}} F[\phi^+,\phi^-].
\label{evolred}
\end{equation}
The Feynman-Vernon \cite{feynver} influence functional $F[\phi^+,
\phi^-]$ in (\ref{evolred}) is defined as

\begin{eqnarray}
F[\phi^+,\phi^-] &=& \int d\chi^+_{{\rm a i}} \int d\chi^-_{{\rm a
i}} ~ \rho_{\chi}[\chi_{{\rm a i}}^+,\chi_{{\rm a i}}^-,t_0] \int
d\chi_{{\rm a f}} \int_{\chi^+_{{\rm a i}}}^{\chi_{{\rm a
f}}}{\cal D}\chi^+_{\rm a} \int_{\chi^-_{{\rm  a i}}}^{\chi_{{\rm
a f}}}
{\cal D}\chi^-_{\rm a}\nonumber \\
&\times & \exp{\left(i \{S[\chi^+_{\rm a} ]+S_{{\rm int}}
[\phi^+,\chi^+_{\rm a} ]\}-i\{S[\chi^-_{\rm a}] + S_{{\rm
int}}[\phi^-,\chi^-_{\rm a}]\} \right)}. \nonumber
\end{eqnarray}

Beginning from the initial distribution, peaked around $\phi = 0$,
we follow the evolution of the system under the influence of the
environment fields. From the influence functional we define the
influence action $\delta A[\phi^+,\phi^-]$ and the coarse grained
effective action (CGEA) $A[\phi^+,\phi^-]$ by
%
\begin{equation}
F[\phi^+,\phi^-] = \exp {i
\delta A[\phi^+,\phi^-]},\label{IA}
\end{equation}
%
\begin{equation}A[\phi^+,\phi^-] = S[\phi^+] - S[\phi^-] + \delta
A[\phi^+,\phi^-].
\label{CTPEA}
\end{equation}
%
We will calculate the influence action to lowest non-trivial order
(two vertices) for large $N$ for $S_{\rm int}[\phi ,\chi ] =
S_{\rm qu}[\phi ,\chi ],S_{\rm Yuk}[\phi ,\chi ],S_{\rm lin}[\phi
,\chi ]$, or any linear combination of the three. Thus, we assume
that $N \gg 1$, and weak coupling $\lambda ~,~ g_{\rm a}~,~
g'_{\rm a}~,~ g''_{\rm a}\ll 1$.

As we are considering weak coupling with the environment fields,
we may expand the influence functional $F[\phi^+, \phi^-]$ in
powers of $g,g',g''$ (or alternatively of $\lambda$), up to second
non-trivial order. The general form of the influence action is
then \cite{lombmazz,calhumaz}
%
\begin{eqnarray}
\delta A[\phi^+,\phi^-] = &&\{\langle
S_{\rm int}[\phi^+,\chi^+_{\rm a}]\rangle_0 - \langle
S_{\rm int}[\phi^-,\chi^-_{\rm a}]\rangle_0\}\nonumber \\
&&+{i\over{2}}\{\langle S_{\rm int}^2[\phi^+,\chi^+_{\rm a}]\rangle_0
- \big[\langle
S_{\rm int}[\phi^+,\chi^+_{\rm a}]\rangle_0\big]^2\}\nonumber \\
&&- i\{\langle S_{\rm int}[\phi^+,\chi^+_{\rm a}] S_{\rm int}[\phi^-,\chi^-
_{\rm a}]
\rangle_0 -
\langle S_{\rm int}[\phi^+,\chi^+_{\rm a}]\rangle_0\langle
S_{\rm int}[\phi^-,\chi^-_{\rm a}]\rangle_0\} \label{inflac} \\
&&+{i\over{2}}\{\langle S^2_{\rm int}[\phi^-,\chi^-_{\rm a}]\rangle_0 -
\big[\langle
S_{\rm int}[\phi^-,\chi^-_{\rm a}]\rangle_0\big]^2\},\nonumber
\end{eqnarray}
%
where the quantum average of a functional $B$ of the environmental
fields is defined as
%
\begin{eqnarray}
\langle B[\chi^+_{\rm a},\chi^-_{\rm a}] \rangle_0= \prod_{a=1}^N\int
d\chi^+_{\rm a i}&& \int d\chi^-_{\rm a i}
\rho_{\chi_{\rm a}}[\chi^+_{\rm a i},\chi^-_{\rm a i},t_0]\int
d\chi^+_{\rm a f}\nonumber \\ &&\times
\int_{\chi^+_{\rm a i}}^ {\chi_{\rm a f}}{\cal D}\chi^+_{\rm a}
\int_{\chi^-_{\rm a i}
}^{\chi_{\rm a f}}{\cal
D}\chi^-_{\rm a} \exp {i\{S_0[\chi^+_{\rm a}] - S_0[\chi^-_{\rm a}]\}}
B.
\label{averag}
\end{eqnarray}
%

We define the propagators of the environment field $\chi$ as

\[ G_{\rm ab}(x,y) = \left\{ \matrix{
G_{++}(x,y)=i \langle 0, in\vert T {\hat \chi}^+(x) {\hat
\chi}^+(y)\vert 0, in\rangle, & ~t, t' ~\mbox{both on} ~{\cal C}_+ \cr
G_{--}(x,y)=-i \langle 0, in\vert {\tilde T} {\hat \chi}^-(x)
{\hat \chi}^-(y)\vert 0, in\rangle , & ~t, t' ~\mbox{both on} ~{\cal C}_- \cr
G_{+-} (x,y)=- i \langle 0, in\vert {\hat \chi}^+(x)
{\hat \chi}^-(y)\vert 0, in\rangle, & ~t ~\mbox{on} ~{\cal C}_+, t'
~\mbox{on} ~{\cal C}_- \cr
G_{-+}(x,y)=i \langle 0, in \vert {\hat
\chi}^-(y) {\hat \chi}^+(x)\vert 0, in\rangle, & ~t  ~\mbox{on} ~
  {\cal C}_-, t'~\mbox{on} ~{\cal C}_+ \cr}
\right. \]
where we have introduced the ``closed time path''
\cite{CTP} complex temporal path ${\cal C}=
{\cal C}_+ \cup {\cal C}_-$, going from minus infinity to a final
time $t$ (${\cal C}_+$), and backwards, with a decreasing
(infinitesimal) imaginary part ($\cal C_-$). Time integration over
the contour ${\cal C}$ is defined by $\int_{{\cal C}} dt =\int
_{{\cal C_+}} dt -\int_{{\cal C_-}} dt$. The field operator ${\hat
\chi}$  appearing in the previous equation must be interpreted as
${\hat \chi}(t, x) = {\hat \chi}_{\pm}(t,x)$ if $t \in {\cal
C}_{\pm}$. $G_{++}$ is the time ordered product for  both fields,
while $G_{--}$ is the anti-temporal ordered product, and in this
case $G_{++} = G^*_{--}$. Moreover, the Feynman and Dyson
propagators can be expressed as
%
\begin{equation}
G_{\pm\pm}(x,y)=G_{\pm}(x,y)~\theta(x^0-y^0) + G_{\mp}(x,y)
~\theta(y^0-x^0)
\end{equation}
%
where we are re-denoting $G_{+}=G_{+-}$ and $ G_{-}=G_{-+}$.

The explicit expressions for the propagators are
%
\begin{equation}
G_{\rm ab}(x,y) = \int {d^4k\over{(2\pi)^4}} e^{ik(x-y)}
\left[G_{{\rm ab}}^{{\rm T}=0}(k) +G_{{\rm ab}}^{{\rm T}}(k)\right]
,\end{equation}
%
with
%
\begin{eqnarray}
G_{++}^{{\rm T}=0}(k)&=& {1\over{k^2 - m^2 +
i\epsilon}},\nonumber \\
G_{{\rm ab}}^{{\rm T}}(k) &=& (-2\pi i) \delta (k^2 - m^2) n(k),
\nonumber \\
  n(k) &=& [e^{-{\vert k_0\vert\over{k_B T}}} - 1]^{-1},
  \nonumber
  \end{eqnarray}
  %
and
%
\begin{equation}
G_{\pm}^{{\rm T}=0}(k)= \mp 2\pi i \delta (k^2 - m^2) \theta
  (\pm k^0).
  \label{tcero}
  \end{equation}
  %

For weak couplings it is relatively easy to compute the upper
bound on the decoherence time due to the interactions $S_{\rm
int}[\phi ,\chi ]$ of (\ref{Sint}), (\ref{Sint2}) and
(\ref{Sint3}). We shall find that, in our particular model, it is
in general shorter than the spinodal time $t_{\rm sp}$, defined as the
time for which
\begin{equation}
\langle \phi^2\rangle_t \sim \eta^2=
6\mu^2/\lambda\,\,\, .\label{deftsp}
\end{equation}
It is for this reason that, with the qualifications below, we can
use perturbation theory. In consequence, by the time that the
field is ordered it can be taken to be classical. This has
implications\cite{lomplb2} for the formation of the defects that
are a necessary byproduct of transitions.

\subsection{Quadratic coupling to the environment}
%
To see how this happens we first consider the case in which the
environment is taken to be composed of the $N$ fields $\chi_{\rm
a}$, interacting through the biquadratic interaction (\ref{Sint})
alone.

Before we examine the model in  detail, there are some general
observations to be made about initial conditions, and the way in
which the transition is implemented. The model has a continuous
transition at a temperature $T_{\rm c}$. The environmental fields
$\chi_{\rm a}$ reduce $T_{\rm c}$ and, in order that $T_{\rm
c}^2=\frac {\mu^2}{\lambda + \sum g_{\rm a}}\gg \mu^2$, we must
take $\lambda + \sum g_{\rm a} \ll 1$. For one-loop consistency at
second order in our subsequent calculation of the dissipation
coefficient that enforces classicality, we assume that $N\gg 1$.
For order of magnitude estimations it is sufficient to take
identical $g_{\rm a} = g/\sqrt{N}$, whereby $1\gg 1/\sqrt N\gg
g\simeq \lambda$. Further, with this choice the hard thermal loop
contribution of the $\phi$-field to the $\chi_{\rm a}$ thermal
masses is, relatively, $O(1/\sqrt{N})$ and will be ignored (the
$\chi_{\rm a}$ 'tadpole' diagrams completely overwhelm the $\phi$
self-interaction tadpole diagram in generating the $\phi$ thermal
mass).

With $\eta = \sqrt{6\mu^2/\lambda}$ determining the position of
the minima of the potential and the final value of the order
parameter, this choice of coupling and environments gives the
hierarchy of scales
\[
\mu^2\ll T_{\rm c}^2 =
O\bigg(\frac{\eta^2}{\sqrt{N}}\bigg)\ll\eta^2,
\]
important in establishing a reliable approximation scheme.
Further, with this choice the dominant hard loop contribution of
the $\phi$-field to the $\chi_{\rm a}$ thermal masses is
\[
\delta m^2_T = O (g T^2_{\rm c}/\sqrt{ N}) = O(\mu^2/N)\ll \mu^2.
\]
Similarly, the two-loop (setting sun) diagram which is the first
to contribute to the discontinuity of the $\chi$-field propagator
is of magnitude
\[
g^2 T_{\rm c}^2/N  = O(g\mu^2/N^{3/2})\ll\delta m^2_T,
\]
in turn. That is, the effect of the thermal bath on the
propagation of the environmental $\chi$-fields is ignorable.

This was our intention in model-making; to construct an
environment that reacted on the system field, but was not reacted
upon by it to any significant extent. We stress that this is not a
Hartree or large-N approximation of the type that, to date, has
been the standard way to proceed for a {\it
closed} system. In particular, the infinite $N$ limit does not
exist. However, the goal is the same of establishing a hierarchy
of diagrams, of which we retain only the most dominant, {\it
independent} of $N$, except implicitly through $T_{\rm c}$, for initial
temperatures $T_0 = O(T_{\rm c})$.

With $S_{\rm qu}[\phi ,\chi ]$ rewritten as
\begin{equation}
S_{\rm qu}[\phi ,\chi ] = -{g\over{8\sqrt N}} \sum_{{\rm a}=1}^N
\int d^4x ~ \phi^2 (x) \chi^2_{\rm a} (x),
\end{equation}
we can evaluate the influence action from Eqs.(\ref{inflac}) to 
(\ref{tcero}). We find
%
\begin{eqnarray}
\delta A_{\rm qu}[\phi^+, \phi^-] &=& {g^2\over{64}}\int d^4x \int
d^4y \left[\phi^{+2}(x) i G_{++}^{2}(x-y) \phi^{+2}(y) -
\phi^{+2}(x)i G_{+}^{2}(x-y) \phi^{-2}(y) \right. \nonumber \\
&-&\left. \phi^{-2}(x)i G_{-}^{2}(x-y) \phi^{+2}(y) + \phi^{-2}(x)
i G_{--}^{2} (x-y)\phi^{-2}(y)\right]. \label{inf}
\end{eqnarray}
%

After defining $$\Delta_2 ={1\over{2}}(\phi^{+2} - \phi^{-2})
~~~;~~~ \Sigma_2 ={1\over{2}}(\phi^{+2} + \phi^{-2}),$$ and using
simple and well known identities between propagators, the real and
imaginary parts of the influence action can be written as
%
\begin{equation}
{\rm Re} \delta A_{\rm qu} = {g^2\over{8}} \int d^4 x\int d^4y ~
\Delta_2 (x) K_{\rm q}(x-y) \Sigma_2 (y),\label{realpartIA}
\end{equation}
%
\begin{equation}
{\rm Im} \delta A_{\rm qu} = - {g^2\over{16}} \int d^4x\int d^4y ~
\Delta_2 (x) N_{\rm q} (x,y) \Delta_2 (y),\label{imaginarypartIA}
\end{equation}
%
where
%
\begin{equation}
K_{\rm q} (x-y) = {\rm Im} G_{++}^2(x,y) \theta (y^0-x^0)
\end{equation}
%
is the dissipation kernel, and
%
\begin{equation}
  N_{\rm q} (x-y) = {\rm Re} G_{++}^2(x,y)
\end{equation}
%
is the noise (diffusion) kernel. Non-leading terms are smaller by a factor
${\cal O}(N^{1/2})$.

Only the real part of the influence functional in
Eq.(\ref{realpartIA}) contains divergences and must be
renormalised. For example, the term ${\rm Im} G_{++}^2\Delta_2 (x)
\Delta_2 (y)$ renormalises the coupling constant. It is important
to note that the imaginary part of the influence functional Eq.
(\ref{imaginarypartIA}) does not contain divergences. This
imaginary part of the influence action (the kernel $N_{\rm q}$)
contains the information about the decoherence effects, that we
will consider in the next Section.

\subsection{Yukawa and linear couplings to the environment}
%
Another possibility that needs to be considered is that of a
Yukawa coupling of the $\phi$-field to the environment, since this
is a  common coupling for gauge and fermionic theories. For
simplicity we take Yukawa couplings equal, as $g_{\rm a}'=g'/\sqrt{N}$,
and
%
\begin{equation}
S_{\rm Yuk}[\phi ,\chi ] = - {g'\mu\over 4\sqrt N}\sum_{{\rm
a}=1}^{\rm N} \int d^4x ~ \phi^{\rm 2}(x) \chi_{\rm a} (x).
\end{equation}
%
The Yukawa interaction preserves the $\phi\rightarrow -\phi$
symmetry of $ S_{\rm syst}[\phi ]$. For the moment we treat this
as an {\it additional} set of interactions to the biquadratic
interactions, whereby $T_{\rm c}$ is qualitatively unchanged. Had there
been no biquadratic interaction the dominant contributions to the
thermal mass term $m_{\phi}^2 (T_0)$ are $O(g'^2 T_0)$, swamped by
the $O(\lambda T_0^2)$ from the short-wavelength modes of the
$\phi$-field itself. As a result, we would have had
$T_{\rm c}^2=O(\mu^2/\lambda) = O(\eta^2)$, rather than
$O(\eta^2/\sqrt{N})$. We will consider this possibility later.

The influence action is still obtained following Eqs.(\ref{inflac}) to
(\ref{tcero}). Thus, the influence functional can be written like
\begin{eqnarray}
\delta A_{\rm Yuk}[\phi^+, \phi^-] &=& {g'^{2}\mu^2\over{32}}\int
d^4x \int d^4y \left[\phi^{+{\rm 2}}(x) G_{++}(x-y) \phi^{+{\rm
2}}(y) + \phi^{+{\rm 2}}(x) G_{+}(x-y) \phi^{-{\rm 2}}(y) \right.
\nonumber
\\ &+&\left. \phi^{-{\rm 2}}(x) G_{-}(x-y) \phi^{+{\rm 2}}(y) -
\phi^{-{\rm 2}}(x) G_{--}(x-y)\phi^{-{\rm 2}}(y)\right].
\label{inf2}
\end{eqnarray}
With $\Delta_2$ and $\Sigma_2$ as before, we are able to write the
real and imaginary parts of the influence functional as

\begin{equation}
{\rm Re} \delta A_{\rm Yuk} =  {g'^{2}\mu^2\over{8}} \int d^4
x\int d^4y ~ \Delta_2 (x) K_{\rm y}(x-y) \Sigma_2 (y),
\end{equation}
and
\begin{equation}
{\rm Im} \delta A_{\rm Yuk} = -  {g'^{2}\mu^2\over{16}} \int
d^4x\int d^4y ~ \Delta_2 (x) N_{\rm y}(x-y) \Delta_2
(y),\end{equation} where 

\begin{equation}K_{\rm y} = {\rm Re}G_{++}(x,y)\theta
(y^{0}-x^{0}),\end{equation} 

and
\begin{equation}N_{\rm y} = {\rm Im}G_{++}(x,y).\end{equation}

It is worth noting that, as we are considering models with
spontaneous symmetry breaking, linear couplings with external
fields are not a natural choice since they break the vacuum
degeneracy. Nonetheless, we shall find the decoherence time to be
so short that the quantum field never has time to reach the vacua
of the theory before it becomes classic, and we include it for
completeness. Again, choosing couplings equal, for the sake of
argument as $g''_{\rm a}=g''/\sqrt{N}$, we take the linear interaction
as

\begin{equation}
S_{\rm lin}[\phi ,\chi ] = - {g''\mu^2\over 4\sqrt N}\sum_{{\rm
a}=1}^{\rm N} \int d^4x ~ \phi (x) \chi_{\rm a} (x).
\end{equation}
%

Using again simple
relationship between Green functions, and defining $\Delta_{\rm 1}
= (\phi^{+} - \phi^{-})/2$ and $\Sigma_{\rm 1}= (\phi^{+} +
\phi^{-})/2$, we are able to write the real and imaginary parts of
the influence functional as

\begin{equation}
{\rm Re} \delta A_{\rm lin} =  {g''^{2}\mu^4\over{8}} \int d^4
x\int d^4y ~ \Delta_{\rm 1} (x) K_{\rm y}(x-y) \Sigma_{\rm 1} (y),
\end{equation}
and
\begin{equation}
{\rm Im} \delta A_{\rm lin} = -  {g''^{2}\mu^4\over{16}} \int
d^4x\int d^4y ~ \Delta_{\rm 1}(x) N_{\rm y}(x-y) \Delta_{\rm
1}(y).\end{equation} Note that as the environmental fields
enter linearly in both couplings
(linear and Yukawa), the kernels appearing into the influence functional
are the same.

\subsection{Including the $\phi$-field short-wavelength modes}
\label{sec:weak}

In our present model the environment fields $\chi_{\rm a}$ are not
the only decohering agents. The environment is also constituted by
the short-wavelength modes of the self-interacting field $\phi$.
Therefore, we split the field as $\phi = \phi_< + \phi_>$, and
define the system by
\begin{equation}\phi_<(\vec x, t) = \int_{\vert \vec k\vert < \Lambda}
{d^3\vec k\over{(2 \pi)^3}} ~ \phi(\vec k, t) \exp{i \vec k . \vec
x},\label{sys1}\end{equation} and the environment by
\begin{equation}\phi_>(\vec x, t) = \int_{\vert \vec k\vert > \Lambda}
{d^3\vec k\over{(2 \pi)^3}} ~ \phi(\vec k, t) \exp{i \vec k . \vec
x}.\label{env2}\end{equation} The system-field contains the modes
with wavelengths longer than the critical value $\Lambda^{-1}$,
while the bath or environment-field contains wavelengths shorter
than $\Lambda^{-1}$. In order to consider only the unstable modes
inside our system, we will set this critical scale $\Lambda$ of
the order of $\mu$. Doing that, we may ensure that every mode
$k_0$ which satisfies $k_0 < \mu$ is in the unstable region and
therefore is included in our system as a long-wavelength mode.
In practice, whether the
separation is made at $k = \mu$ exactly or at $k\approx\mu$ is
immaterial\cite{Karra} by time $t_D$, when the power of the
$\phi$-field fluctuations is peaked at $k_0\ll\mu$. The effect is
to give a separation of system from environment through the
decomposition of $S[\phi , \chi ]$ of (\ref{action0}) as
\begin{equation}
S[\phi , \chi ] = S_{\rm syst}[\phi_< ] + S_{\rm env}[\chi
,\phi_>] + S_{\rm couple}[\phi_> ,\chi ], \label{action}
\end{equation}
where 
\begin{eqnarray}
S_{\rm env}[\chi , \phi_>] &=& S_{\rm env}[\chi ] + \int d^4x\left\{
{1\over{2}}\partial_{\mu} \phi_{>}\partial^{\mu} \phi_{>} -
{1\over{2}}\mu^2 \phi_{>}^2\right\}\nonumber
\\
S_{\rm couple}[\phi_{<} ,\chi ] &=& S_{\rm int}[\phi_{<} ,\chi
]-{\lambda\over{4}}\int d^4x \,(\phi_{<}(x) \phi_{>} (x))^2.
\nonumber
\end{eqnarray}
All terms omitted in the expansion are not relevant\cite{lombmazz,greiner}
for the one-loop calculations for the long-wavelength modes that
we shall now consider.\footnote{Strictly speaking, for $\mu/3<k_0<\mu$
one should include an additional term
proportional to $\phi_<^3\phi_>$ in the interaction Lagrangian.
See Ref. \cite{lombmazz} for details.}

This gives an additional one-loop contribution $\delta A_>$ to
$\delta A$ as in (\ref{inf}), but with
             $G_{\pm\pm}$ constructed form the
             short-wavelength modes of the $\phi$-field as it evolves from the
top of the potential hill. Without the additional powers of
$N^{-1}$ to order contributions the one-loop calculation is
unreliable. However, it is
             not necessary to calculate $\delta A_>$ in order to
             get a good estimate of $t_D$ since it can only shorten the
decoherence time
             $t_D$. In fact, we would not expect the inclusion of the
             $\phi$-field to give a qualitative change.
              The effect is that the short-wavelength
             modes in the one-loop diagrams from which they are calculated have
             been kept at the initial temperature $T_0$, on the grounds that
             passing through the transition quickly has no effect on them.  The
             quench mimics a slower evolution of temperature in which only the
             long-wavelength modes show instabilities that the transition
             induces. That is, with $g\simeq\lambda$ and no $1/N$ factor,
             the short-wavelength $\phi$
             modes give a comparable contribution,
             qualitatively, as {\it all} the explicit
             environmental fields put together. However, at an order of
             magnitude level there is no change, since the effect is to
             replace $g^2$ by $g^2 + O(\lambda^2) = O(g^2)$.

\section{Master Equations and the Diffusion Coefficients}
\label{sec:diff}

In this Section we will obtain the evolution equation for the
reduced density matrix (master equation), paying particular
attention to the diffusion term, which is responsible for
decoherence. Before doing this, we briefly review the case of
the quantum Brownian motion \cite{qbm}.
Denote by $x$ the coordinate of the
Brownian particle, and by $q_{\rm i}$ the coordinates of the oscillators
in the environment. For a linear coupling $x q_{\rm i}$, the  master
equation for the reduced density matrix $\rho_{\rm r}(x,x',t)$ is
of the form \cite{qbm,unruh}
\begin{eqnarray}
i\partial_t \rho_{\rm r}(x,x',t)&=&\langle x\vert[H,\rho_{\rm r}]\vert x'
\rangle
-i\gamma(t)(x-x')(\partial_x - \partial_{x'})\rho_{\rm r}(x,x',t)\nonumber\\
&&+ f(t)(x-x')(\partial_x + \partial_{x'})
\rho_{\rm r}(x,x',t) - iD(t)(x-x')^2\rho_{\rm r}(x,x',t),
\label{meqbm}
\end{eqnarray}
where the coefficients $\gamma (t), D(t)$ and $f(t)$ depend on the
properties of the environment.
For non-linear couplings like $x^{\rm n}q_{\rm i}^{\rm m}$, one expects
the master equation to contain
terms of the form
$iD^{({\rm n},{\rm m})}(t)(x^{\rm n}-x'^{\rm n})^2\rho_{\rm r}$.

The first term on the RHS of Eq.(\ref{meqbm}) gives the usual Liouville-like
evolution; the term
proportional to $\gamma$ produces dissipation ($\gamma$ is the
relaxation coefficient). The term
proportional to the diffusion coefficient $D(t)$, which is proportional to
$(x-x')^2$ and positive definite, gives the main
contribution to the decoherence since it produces a diagonalisation of
the reduced density matrix \cite{qbm}.
Indeed, let us we write
\begin{equation}
\rho_{\rm r}[x,x';t] = G[x,x',t] \exp{\Big[ - (x-x')^2
\int_0^t D(s)~ ds\Big]}
.\label{qbmdecay0}\end{equation}
Inserting this expression into the master equation it is easy to
to see that the differential equation for $G[x,x',t]$
contains the usual Liouville-term plus additional
contributions proportional to $D,\gamma$, and $f$. However,
none of these additional terms
is imaginary with the right sign.
An approximate solution
of Eq.(\ref{meqbm}) is therefore \cite{jpp}
\begin{equation}
\rho_{\rm r}[x,x';t] \approx \rho^{\rm u}_{\rm r}[x,x',t]
\exp{\Big[-(x-x')^2 \int_0^t D(s) ~ ds\Big]}
,\label{qbmdecay}\end{equation} where $\rho^{\rm u}_{\rm r}$ takes
into account the unitary evolution (i.e. without environment). It
is obvious from this (and also setting $x=x'$ in
Eq.(\ref{meqbm})), that the diagonal density matrix just evolves
like the unitary matrix (the environment has almost no-effect on
the diagonal par of $\rho_{\rm r}$).

\subsection{Instantaneous Quench}

Let us now turn to quantum field theory. Since it is the
system-field $\phi$ field whose behaviour changes dramatically on
taking $T_{\phi}$ through $T_{\rm c}$, we will first adopt an {\it
instantaneous} quench  for $T_{\phi}$ from $T_0$ to $T_{\rm f}=0$
at time $t=0$, in which $m^{2}_{\phi}(T)$ changes sign and
magnitude instantly, from $m_{\phi}^2(T_0) = O(\mu^2)$ to
$m_{\phi}^2(t)=-\mu^2, t>0$. Slower quenches in which
$m^{2}_{\phi}(T)$ varies linearly in time  will be considered in
Section \ref{sec:slow}.  We will follow closely
Ref.\cite{lombmazz}. The first step in the evaluation of the
master equation is the calculation of the density matrix
propagator $J_{\rm r}$ from Eq.(\ref{evolred}). In order to solve
the functional integration which defines the reduced propagator,
we perform a saddle point approximation
%
\begin{equation}
J_{\rm r}[\phi^+_{\rm f},\phi^-_{\rm f},t\vert\phi^+_{\rm i},\phi^-_{\rm i},
t_0] \approx \exp{ i A[\phi^+_{\rm cl},\phi^-_{\rm cl}]},
\label{prosadle}
\end{equation}
%
where $\phi^\pm_{\rm cl}$ is the solution of the equation of
motion ${\delta Re A\over\delta\phi^+}\vert_{\phi^+=\phi^-}=0$
with boundary conditions $\phi^\pm_{\rm cl}(t_0)=\phi^\pm_{\rm i}$
and $\phi^\pm_{\rm cl}(t)=\phi^\pm_{\rm f}$. It is very difficult
to solve this equation analytically. For simplicity, we assume
that the system-field contains only one Fourier mode with $\vec k
= \vec k_0$. We are motivated in this by the observation that the
ordering of the field is due to the exponential growth of the
long-wavelength unstable modes. Long-wavelengths, for which $\vert
k_0\vert^2 < \mu^2$, start growing exponentially as soon as the
quench is performed. They increasingly bunch  about a wavenumber
$k_0 < \mu$  which diminishes with time initially as $k_0^2 =
{\cal O}(\mu/t)$. Modes with $\vert k_0\vert^2
 > \mu^2$ will oscillate.

 The classical solution for $\vec k = \vec k_0$ is of the form
\[
\phi_{\rm cl}(\vec x, s) =  f(s,t)\cos(\vec k_0 . \vec x),
\]
where $f(s,t)$ satisfies the boundary conditions $f(0,t)=
\phi_{\rm i}$ and $f(t,t) = \phi_{\rm f}$. Qualitatively, since
the instantaneous quench corresponds to sitting the field at the
top ($\phi = 0$) of the hill, $f(s,t)$ grows exponentially with
$s$ for $t\lesssim t_{\rm sp}$, and oscillates for $ t_{\rm
sp}<s<t$ when $t\gtrsim t_{\rm sp}$. We shall therefore
approximate its time dependence for $t\lesssim t_{\rm sp}$ as
\[
f(s,t) = \phi_{\rm i} u_1(s,t) + \phi_{\rm f} u_2(s,t),\,\,
\]
where
             $u_1(0,t) = 1$, $u_1(t,t) = 0$ and
             $u_2(0,t) = 0$, $u_2(t,t) = 1$, with solution
             \[
             u_1(s,t) =  {\sinh[\omega_0 (t - s)]
             \over{\sinh(\omega_0 t)}},\,\,u_2(s,t)=  {\sinh(\omega_0 s) \over
             {\sinh(\omega_0 t)}}, \,\,\,
             \]
with $\omega_0^2 = \mu^2 - k_0^2$.

In order to obtain the master equation we must
compute the final time derivative of the propagator $J_{\rm r}$. After
that, all the dependence on the initial field configurations
$\phi^\pm_{\rm i}$ (coming from
the classical solutions $\phi^\pm_{\rm cl}$) must be eliminated.
The free propagator, defined as
%
\begin{equation}
J_0[\phi^+_{\rm f}, \phi^-_{\rm f}, t\vert \phi^+_{\rm i},
\phi^-_{\rm i}, 0] =
\int_{\phi^+_{\rm i}}^{\phi^+_{\rm f}}{\cal D}\phi^+ \int_{\phi^-_{\rm i}}
^{\phi^-_{\rm f}}
{\cal D}\phi^- \exp\{i [ S_0(\phi^+) -
S_0(\phi^-)]\};
\label{propdeJ0}
\end{equation}
%
satisfies the general identities \cite{lombmazz,qbm}
%
\begin{equation}
\phi^+_{\rm cl}(s) J_0 =
\Big[ \phi^+_{\rm f} [u_2(s,t) - \frac{{\dot u}_2(t,t)}{{\dot
             u}_1(t,t)}u_1(s,t)] -  i {u_1(s,t)\over{{\dot u}_1(t,t)}}
\partial_{\phi^+_{\rm f}}\Big]J_0,
\label{rel1}
\end{equation}
%
and
%
\begin{equation}
\phi^-_{\rm cl}(s) J_0 =
\Big[ \phi^-_{\rm f} [u_2(s,t) - \frac{{\dot u}_2(t,t)}{{\dot
             u}_1(t,t)}u_1(s,t)] + i {u_1(s,t)\over{{\dot u}_1(t,t)}}
\partial_{\phi^-_{\rm f}}\Big]J_0.
\label{rel11}
\end{equation}
%
These identities allow us to remove the initial field
configurations  $\phi^\pm_{\rm i}$, by expressing them in terms of the
final fields  $\phi^\pm_{\rm f}$ and the derivatives
$\partial_{\phi^\pm_{\rm f}}$, and
obtain the master equation.

The full equation is very complicated and, as in the quantum
Brownian motion case, it depends on the system-environment
coupling. In what follows we will compute the diffusion
coefficients for the different couplings described in the previous
section.

\subsection{Quadratic coupling to $\chi$ fields}

As we are solely interested in decoherence, it is sufficient to
calculate the correction to the usual unitary evolution coming
from the noise kernel (imaginary part of the influence action).
Taking the temporal derivative of $J_{\rm r}$ in
Eq.(\ref{prosadle}) we obtain
\begin{eqnarray}i\partial_t J_{\rm r}[\phi^+_{\rm f},\phi^-_{\rm f},t\vert
\phi^+_{\rm i},\phi^-_{\rm i},0] &=&
\bigg\{h_{\rm ren}[\phi^+] - h_{\rm ren}[\phi^-] \nonumber \\
&-& i {g^2\over{64}} \Delta_{2,\rm f} V \int_0^t ds
\Delta_{\rm 2,cl}(s)\left[{\rm Re} G_{++}^{2}(2k_0;t-s)\right. \nonumber \\
&+& \left. 2 {\rm Re} G_{++}^{2} (0;t-s)\right] + ...
\bigg\}J_{\rm r}[\phi^+_{\rm f},\phi^-_{\rm f},t\vert \phi^+_{\rm
i},\phi^-_{\rm i},0],\label{timeder} \end{eqnarray} where $h_{\rm
ren}$ is the (renormalised) system-field Hamiltonian,
$G^2_{++}(k_0; t-s)$ is the Fourier transform of the square of the
$\chi$ propagator at finite temperature, and $\Delta_{2,\rm f} =
(\phi^{+2}_{\rm f} - \phi_{\rm f}^{-2})/2$. The ellipsis denotes
other terms coming from the time derivative that do not contribute
to the diffusive effects (we are ignoring all terms not
proportional to $\Delta_{\rm f}^2$ in the final equation). Thus,
using properties (\ref{rel1}) and (\ref{rel11}) we can remove the
initial boundary condition dependence and obtain the master
equation, as

\begin{equation}
i {\dot \rho}_{\rm r} = \langle \phi^+_{\rm f}\vert [H,\rho_{\rm
r}] \vert \phi^-_{\rm f}\rangle - i {g^2\over{16}} V \Delta_{\rm
f}^2 D_{\rm qu}(\omega_0, t) \rho_{\rm r}+ ... \label{master}
\end{equation}
%
The volume factor $V$ that appears in the master equation is due
to the fact we are considering a density matrix which is a
functional of two different field configurations, $\phi^\pm(\vec
x) = \phi^\pm \cos \vec k_0 . \vec x$, which are spread over all
space. The time dependent diffusion coefficient $D_{\rm qu
}(\omega_0, t)$ is given by
%
\begin{equation}
D_{\rm qu}(\omega_0, t) = \int_0^t ds ~ u(s) \left[{\rm Re}
G_{++}^2(2k_0; t-s) + 2 {\rm Re}G_{++}^2(0; t-s)\right],
\label{diff}
\end{equation}
%
where $u(s) =\bigg[u_2(s,t) - \frac{{\dot u}_2(t,t)}{{\dot
             u}_1(t,t)}u_1(s,t)\bigg]^{2} = \cosh^2[\omega_0 ( t - s)]$ when
$t \lesssim t_{\rm sp}$, and it is an oscillatory function of time
when $t \gtrsim t_{\rm sp}$.

In the high temperature limit ($k_{\rm B}T \gg \mu$), the explicit expression
for the diffusion coefficient
is
%
\begin{eqnarray}
D_{\rm qu}(\omega_0, t)={(k_{\rm B}T)^2\over{64\pi^2}} \int_0^tds~
\cosh^2[\omega (t - s)]&& \left\{{1\over{k_0}} \int_0^\infty
dp~{p\over{p^2 + \mu^2}}\int_{\sqrt{\vert p - 2 k_0\vert^2 +
\mu^2}}^{\sqrt
{\vert p + 2 k_0\vert^2 + \mu^2}}{du\over{u}}\right. \nonumber \\
&& \times \left.~\cos[(\sqrt{p^2 + \mu^2}+ u)
(t-s)]\right. \nonumber \\
&+& \left. \int_0^\infty dp~{p^2\over{(p^2 + \mu^2)^2}} \cos[2\sqrt{p^2 +
\mu^2}(t-s)]\right\},
\label{D}
\end{eqnarray}
%
where we have set $m^2 = \mu^2$.

We have performed the integrations in Eq.(\ref{D}) in part
analytically, and in part numerically. After doing the
temporal integration, we obtain one term
which is an oscillatory function of time, and another one
which is exponential in time.
The outcome is that, for the unstable modes in the high
temperature limit, modes such that $k_0 < \mu$ (although not very
close to $\mu$), we have found the following diffusion coefficient
for  short-times $\mu t \ll 1$:
%
\begin{equation}
D_{\rm qu}(k_0, t) \approx {(k_{\rm B} T)^2\over{\mu}} t.
\label{unstD}
\end{equation}
For $k_0\approx 0$ this is approximately independent of
wavenumber.

For longer times $D_{\rm qu}(k_0, t)$ shows the exponential growth
\begin{equation}
D_{\rm qu}(k_0, t) \approx {(k_{\rm B} T)^2\over{\mu^3}}
\sqrt{\mu^2 - k_0^2}~ \exp [2\sqrt{\mu^2 - k_0^2}t],
\label{unstD2}
\end{equation}
%
associated with the instability of the $k_0$ mode. For $t > t_{\rm
sp}$ the diffusion coefficient stops growing, and oscillates
around its value at $t = t_{\rm sp}$ ($D_{\rm qu}(k_0, t= t_{\rm
sp})$). The approximate equalities allow for ${\cal O}(1)$
prefactors.

%
%\begin{equation}
%D(k_0, t) \approx {(k_{\rm B} T)^2\over{\mu^2}}
%~ \times ~ \{\mbox{ oscillating function of time}\}.
%\end{equation}
%

\subsection{Yukawa coupling}

For coupling to a single Yukawa field we have the same formal
structure of the master equation. The temporal diffusion
coefficient is now given by
\begin{equation}
D_{\rm Yuk}(k_0, t) = \int_0^t ds \cosh^2[\omega_0(t - s)]
\left[{\rm Im} G_{++}(2k_0; t-s) + 2 {\rm Im} G_{++}(0;
t-s)\right], \label{diff2}
\end{equation}
where
\begin{equation}
{\rm Im} G_{++}(2k_0; s) = \frac{k_B
T}{4}\frac{\cos[\sqrt{4k_{0}^{2} + \mu^2 }s]}{4k_{0}^{2} + \mu^2},
\end{equation}
whereby, for the unstable long-wavelengths we have
\begin{equation}
D_{\rm Yuk}(k_0, t) = {k_{\rm B}T\over{4}}\int_0^t ds\,\cosh^{2}
[\sqrt{\mu^2 - k_{0}^2}(t-s)] \left\{{\cos [\sqrt{4k_{0}^2 +
\mu^2}(t-s)]\over{(4k_0^2 + \mu^2)}} + {\cos[\mu
(t-s)]\over{\mu^2}}\right\}.
\end{equation}
On performing an early-time expansion
  we obtain
\begin{equation}D_{\rm Yuk}(k_0, t)\approx {k_{\rm B} T\over{\mu^2}}~t.
\label{Dyu1}\end{equation} For larger times $D_{\rm Yuk}(k_0, t)$
again shows the exponential growth
\begin{equation}
D_{\rm Yuk}(k_0, t) \approx \frac{k_B T}{\mu^4}\sqrt{\mu^2 -
k_0^2} \exp [2\sqrt{\mu^2 - k_0^2}t], \label{Dyu2}
\end{equation}
%
of (\ref{unstD2}).

\subsection{Linear coupling}

For linear coupling the  diffusion coefficient is
\begin{equation}
D_{\rm lin}(k_0, t) = {k_{\rm B}T\over{4}}\int_0^t ds\,\cosh
[\sqrt{\mu^2 - k_{0}^2}(t-s)]{\cos [\sqrt{k_{0}^2 +
\mu^2}(t-s)]\over{k_0^2 + \mu^2}}. \label{Dlin}
\end{equation}
In the same approximation as before, we find the small-time
result
\begin{equation}D_{\rm lin}(k_0, t)\approx {k_{\rm B} T\over{\mu^2}} ~t.
\end{equation}

For larger times the exponential growth is now

\begin{equation}
D_{\rm lin}(k_0, t) \approx \frac{k_B T}{\mu^4}\sqrt{\mu^2 -
k_0^2}\exp [\sqrt{\mu^2 - k_0^2}t].  \label{DL}
\end{equation}
%
\subsection{Including the $\phi$-field short-wavelength modes}
\label{sec:weak2}

The effect of the separation of the long-wavelength modes of the
system from the short-wavelength modes of the environment through the
interaction term
\begin{equation}
-{\lambda\over{4}}\int d^4x \,\phi_{<}(x)^2 \phi_{>} (x)^2
\label{qq}
\end{equation}
of (\ref{action}) gives an additional one-loop contribution
$D_{\phi}(k_0,t)$ to the diffusion function with the same $u(s,t)$
but a $G_{++}$ constructed form the short-wavelength modes of 
the $\phi$-field, as we have mentioned 
at the end of Section II. However, since the contribution of
$D_{\phi}(k_0,t)$ to the overall diffusion function is positive we
can derive an {\it upper} bound on the decoherence time $t_D$ from
the reliable diffusion functions $D_{\chi}(k_0,t)$, $D_{\rm
Yuk}(k_0,t)$ (and $D_{\rm lin}(k_0,t)$).


\section{Decoherence Times for the Instantaneous Quench}
\label{sec:tdec}
In the quantum field scenario, the effect of the diffusion
coefficient on the decoherence process can be seen considering
the following approximate solution to the master equation
%
\begin{equation} \rho_{\rm r}[\phi^+, \phi^-; t] \approx
\rho^{\rm u}_{\rm r}[\phi^+, \phi^-; t] ~ \exp \left[-V\sum_j
\Gamma_{\rm j} \int_0^t ds ~D_{\rm j}(k_0, s) \right],
\end{equation}
where $\rho^{\rm u}_{\rm r}$ is the solution of the unitary part
of the master equation (i.e. without environment), and $j$ labels
the environments (in this case the quadratic, Yukawa, or linear
coupling with the environment-fields respectively). In terms of
the dimensionless fields $\bar\phi = (\phi^+ + \phi^-)/2\mu,$ and
$ \delta = (\phi^+ - \phi^-)/2\mu$ we have $\Gamma_{\rm qu}
=(1/16)g^2\mu^4 \bar\phi^2 \delta^2$ for the quadratic coupling,
$\Gamma_{\rm Yuk} =(1/16)g'^2\mu^4 \bar\phi^2 \delta^2$ for the
Yukawa coupling, and $\Gamma_{\rm lin} =(1/16)g''^2\mu^2 \delta^2$
for linear coupling.

The system will decohere when the non-diagonal elements of the
reduced density matrix are much smaller than the diagonal ones.
We therefore look at the ratio
\begin{equation}
\left\vert \frac {\rho_{\rm r}[\bar\phi+\delta,\bar\phi-\delta;t]}
{\rho_{\rm r}[\bar\phi,\bar\phi;t]} \right\vert \approx \left\vert
\frac {\rho_{\rm r}^{\rm u}[\bar\phi+\delta, \bar\phi-\delta;t]}
{\rho_{\rm r}^{\rm u}[\bar\phi,\bar\phi;t]} \right\vert ~ \exp
\bigg[-V \sum_j\Gamma_{\rm j}\int_0^t ds ~D_{\rm j}(k_0, s) \bigg
]\, . \label{ratio}\end{equation}

It is not possible to obtain an analytic expression for the ratio
of unitary density matrices that appears in Eq.(\ref{ratio}). The
simplest approximation is to neglect the self-coupling of the
system field \cite{kim,guthpi}. In this case the unitary density
matrix remains Gaussian at all times as
\begin{equation}\left\vert \frac {\rho_{\rm r}^{\rm u}[\bar\phi+\delta,
\bar\phi-\delta;t]}
{\rho_{\rm r}^{\rm u}[\bar\phi,\bar\phi;t]}
\right\vert = \exp [-{T_{\rm c}\over{\mu}}\delta^2 p^{-1}(t)]
,
\label{uratio}
\end{equation}
where $p^{-1}(t)$, essentially $\mu^2
\langle\phi^2\rangle_t^{-1}$, decreases exponentially with time to
a value ${\cal O}(\lambda)$. This approximation can be improved by
means of a Hartree-like approximation \cite{kim,devega,diana}. In this case
the ratio is still given by Eq.(\ref{uratio}), but now $p^{-1}(t)$
decreases more slowly as $t$ approaches $t_{\rm sp}$.

In both approximations, the unitary density matrix itself is a
Gaussian with an increasing width. In consequence, one can easily
show that the associate Wigner function is positive definite, and
that it becomes sharply peaked around the classical trajectories
in phase space \footnote{It can be shown from its very definition
that when the width of the Gaussian density matrix increases like
$f(t)$ the width of the Wigner function decreases as $f(t)^{-1}$}.
As long as only initial Gaussian states are considered, and within
an approximation that does not break Gaussianity, one would
conclude that the system has classical behaviour for long times,
i.e. by the time when the Wigner function is sufficiently peaked
around the classical trajectories. Note that in this case the
system becomes "classical"  after a period of time that depends on
the width of the initial state and on the shape of the potential.
However, this argument is only true for the upside-down harmonic
oscillator \cite{guthpi}, where the Gaussianity of the density
matrix and the positivity of the Wigner function are exactly
preserved by the dynamical evolution. When the self-coupling is
taken into account, a full numerical calculation \cite{diana}
shows that $\rho_{\rm r}^{\rm u}$ becomes a non-diagonal and
non-Gaussian function. Consequently the associated Wigner function
becomes non-positive in some regions of phase space. Only when the
full reduced density matrix becomes diagonal, the positivity of
the Wigner function is restored everywhere. Therefore, in order to
obtain classical behaviour, the relevant part of the reduced
density matrix is the term proportional to the diffusion
coefficient in Eq.(\ref{ratio}), since it is this that enforces
its diagonalisation.

The decoherence time  $t_{D}$ sets the scale after which we
have a classical system-field configuration, and depends strongly
on the properties of the environment. It is constrained by

\begin{equation}
1 \gtrsim V \sum_j\Gamma_{\rm j} \int_{0}^{t_{\rm D}} ds ~D_{\rm
j}(k_0,s). \label{Dsum}
\end{equation}
and corresponds to the time after which we are able to distinguish
between two different field amplitudes (inside a given volume
$V$).

Eq.(\ref{Dsum}) will be satisfied because one of its terms will
grow faster than the others, rather than because many terms will
each give a small fraction of unity. Specifically, we have (up to
numerical factors $O(1)$)
\begin{eqnarray}
&V& \Gamma_{\rm qu} \int_{0}^{t_{D}} ds ~D_{\rm qu}(k_0,s): V
\Gamma_{\rm Yuk} \int_{0}^{t_{D}} ds ~D_{\rm Yuk}(k_0,s): V
\Gamma_{\rm lin} \int_{0}^{t_{D}} ds ~D_{\rm lin}(k_0,s)
\nonumber
\\
&\sim& g\,\,:\,\,
g'\bigg(\frac{\mu}{T_{\rm c}}\bigg)\,\,:\,\,
g''\frac{1}{\delta^2}\bigg(\frac{\mu}
{T_{\rm c}}\bigg)\,e^{-\mu t_D }.
\end{eqnarray}
Since $T_{\rm c}\gg\mu$ and $\delta\gg 1$ then, for $\mu t_D\gg 1$, we
have
\begin{equation}
V \Gamma_{\rm qu} \int_{0}^{t_{D}} ds ~D_{\rm qu}(k_0,s)\,\,
\gg\,\, V \Gamma_{\rm Yuk} \int_{0}^{t_{D}} ds
~D_{\rm Yuk}(k_0,s)\,\,\gg\,\, V \Gamma_{\rm lin} \int_{0}^{t_{D}} ds ~
D_{\rm lin}(k_0,s).
\end{equation}
if $g,g'$ and $g''$ are comparable. Thus, assuming that the
biquadratic term is present, it is sufficient to evaluate the
constraint on the time $t_{D}$ from this biquadratic
interaction alone.

Suppose we reduce the couplings $g \sim \lambda$ of the system
$\phi$-field to its environment. Since, as a one-loop construct,
$\Gamma\propto g^2 \sim  \lambda^2$ our first guess would be that
as $g$, $\lambda$ decrease, then $t_D$ increases and the system
takes longer to become classical. Although this is the usual
result for Brownian motion, say, it is not the case for quantum
field theory phase transitions. The reason is twofold. Firstly,
there is the effect that $\Gamma\propto T_0^2$, and $T_0^2
\propto\lambda^{-1}$ is non-perturbatively large for a phase
transition.  Secondly, because of the non-linear coupling to the
environment, obligatory for quantum field theory,
$\Gamma\propto{\bar\phi}^2$. The completion of the transition
finds ${\bar\phi}^2\simeq\eta^2\propto\lambda^{-1}$ also
non-perturbatively large. This suggests that $\Gamma$, and hence
$t_D$, can be independent of $\lambda$.  This would not be the
case for a linear coupling to the environment, or a cold initial
state in which $\phi$ is peaked about $\phi = 0$. In fact, the
situation is a little more complicated, but the end result that
$t_D$ does not increase (relative to $t_{\rm sp}$) as the couplings
become weaker remains true. However, if the interactions became
stronger, the Wigner functional would cease to remain positive
after the transition and we would lose the classical probability
interpretation.

In order to quantify the decoherence time we have to fix
the values of $V$, $\delta$, and $\bar\phi$. $V$ is understood as
the minimal volume inside which we there are no coherent
superpositions of macroscopically distinguishable states for the
field. Thus, our choice is that this volume factor is ${\cal
O}(\mu^{-3})$  since $\mu^{-1}$ (the Compton wavelength) is the
smallest scale at which we need to look. Inside this volume, we do
not discriminate between field amplitudes which differ by $ {\cal
O}(\mu) $, and therefore we take $\delta \sim {\cal O}(1)$.  For
$\bar\phi$ we set $\bar\phi^2\sim {\cal O}(\alpha /\lambda)$,
  where $\lambda\leq\alpha\leq 1$ is to be determined self-consistently.
We evaluate now the decoherence times for the simplest
possibilities. The order parameter itself is the $k_0=0$ mode of
the field. The domain structure at $t_{\rm sp}$, when the transition
is first implemented, is determined by wavenumbers $k_0\approx 0$.
Since there is no singular behaviour at $k_0=0$ it is sufficient,
for our purposes, to consider the case $k_0\approx 0$, when $t_D$
is independent of $k_0$. We shall comment briefly on the case
$k_0\neq 0$ at the end of this Section.

\subsection{Quadratic coupling}

From the short-time expression for the diffusion coefficient
Eq.(\ref{unstD}) is very easy to show that decoherence does not
occur while $\mu t \ll 1$ since, assuming that it were the case,
we get the inconsistency

\begin{equation}
t_{D} = \left[{1\over{V g^2 \mu^3 {\bar\phi}^2\delta^2
(k_{\rm B}T)^2}}\right]^{1\over{2}}\sim {1\over{\mu}}
\left[{(\lambda^2 + \lambda \sqrt{N}g)\over{\alpha g^2}}
\right]^{1\over{2}} \sim {N^{1\over{4}}\over{\alpha \mu}}~{\cal O}(1) \gg
{1\over{\mu}}.
\end{equation}

In this, and all subsequent calculations, we take $\lambda\sim g$.
Thus, in order to evaluate the upper bound on the decoherence
time, we have to use Eq.(\ref{unstD2}) for longer times. Therefore
we obtain

\begin{equation}
\exp [2\mu t_{D}]\lesssim {1\over{V\mu g^2 {\bar\phi}^2
\delta^2 (k_{\rm B}T)^2}} \sim {\lambda \sqrt{N} g\over{\alpha
g^2}} = {\cal O}({\sqrt{N}\over{\alpha}}),\end{equation}
from
which the decoherence time is bounded by

\begin{equation}
\mu t_D\lesssim {1\over{4}}\ln N -{1\over{2}}\ln\alpha\simeq \ln
(\frac{\eta}{T_{\rm c}\sqrt{\alpha}})\label{tdecalpha}.
\end{equation}
The value of $\alpha$ is determined as $\alpha \simeq
\sqrt{\mu/T_{\rm c}}$ from the condition that, at time $t_D$, $\langle
\phi^2\rangle\sim\alpha\eta^2$; thus $t_D$ is
\[
\mu t_D\lesssim {1\over{4}}\ln {N T_{\rm c}\over{\mu}}.
\]
For comparison, from Eq.(\ref{deftsp}) we find
\begin{equation}
\mu t_{\rm sp} \sim \frac{1}{4}\ln (\frac{\sqrt N}{\lambda})\sim
\ln (\frac{\eta}{\sqrt{\mu T_{\rm c}}}).
  \label{tsp}
\end{equation}
As a result, $1 < \mu t_D \leq \mu t_{\rm sp}$, with
\begin{equation}
\mu t_{\rm sp} -\mu t_D\gtrsim\frac{1}{4}\ln (\frac{T_{\rm c}}{\mu})>1,
\label{dt}
\end{equation}
for weak enough coupling, or high enough initial temperatures (we
have taken $T_0 \sim T_{\rm c}$ throughout).
%
This is our main result, that for the physically relevant modes
(with small $k_0$) classical behaviour has been established before
the spinodal time, when the ground states have became populated.
This result goes in the direction of justifying the use of
classical numerical simulations for the analysis of the dynamics
of the long-wavelengths modes after the quench.

We can take this further, in that $N$ only occurs implicitly
through $T_{\rm c}$. If there were no quadratically coupled $\chi$
fields, but only the quadratic coupling (\ref{qq}) due to the
interactions between the long and short-wavelength modes, the
one-loop diagram from them alone would give a bound on the
decoherence time $t_D$ of the {\it identical} form (\ref{dt}). The
proviso is that $T_{\rm c}$ is now $O(\eta)$, since the effect of Yukawa
couplings on the critical temperature is, relatively,
$O(T_{\rm c}^{-1})$. The inclusion of both effects (internal and
external environments) only makes the inequality (\ref{dt})
stronger, with the original $T_{\rm c}$.

We can say even more in that, for an
instantaneous quench, nonlinear behaviour only becomes important
in an interval $\Delta t$, $\mu\Delta t = O(1)$, before the
spinodal time\cite{Karra}. When (\ref{dt}) is valid, we see that
$\rho_{\rm r}$ becomes diagonal before non-linear terms could be
relevant. In this sense, classical behaviour has been achieved
before quantum effects could destroy the positivity of the Wigner
function $W_{\rm r}$. Really, our $t_{\rm D}$ sets the time after
which we have a classical probability distribution (positive
definite) even for times $t > t_{\rm sp}$. The existence of the
environment is crucial in doing this. Of course, for non-Gaussian
or delocalised (in the field space) initial states, it is clear
that $W_{\rm r}$ will be non-positive definite even in the linear
regime, and therefore $t_{\rm D}$ should be smaller than the one
we evaluated here. In the present work, $t_{\rm D}$ is the {\it
classicalisation time}-bound.

\subsection{Yukawa couplings}

We have seen before  that, even in the absence of external $\chi$
fields coupling biquadratically with the $\phi$ field, we have the
same biquadratic behaviour from its short-wavelength modes. Thus,
we never have  stand-alone Yukawa interactions. One power of
$T/\mu$ down, for comparable couplings they will have no
qualitative effect on $t_D$, reducing it only by a small fraction.
However, we note that, even if we took just the contribution from
the Yukawa couplings in (\ref{Dsum}), the bound that it imposes on
$t_D$ is $O(t_{\rm sp})$.

For long-times, this is

\begin{equation}\exp [2\mu t_{\rm D}] \approx {1\over{V g'^2 {\bar\phi}^2
\delta^2 (k_{\rm B}T)}} \sim {\lambda\over{ \alpha g
'^2}}.\end{equation}
That is,
\begin{equation} \mu t_{\rm D} \sim {1\over{2}} \ln {1\over{\alpha \lambda}}
\simeq {5\over{4}}\ln {T_{\rm c}\over{\mu}}.
\end{equation}
In the absence of biquadratic $\chi$ fields $T_{\rm c} = O(\eta)$,
whereby $t_D\approx 5t_{\rm sp}/2$, larger than $t_{\rm sp}$, but not
introducing any new timescales.

\subsection{Linear couplings}

As we have already observed, early studies of decoherence were
confined largely to quantum mechanical systems, for which the
environment was typically a collection of harmonic oscillators, to
which the system coupled linearly. Such systems have the virtue of
exact solvability (or closed equations) and have been very
instructive. However, in the context of quantum field theory
linear terms are usually a signal of an inappropriate choice of
field basis. Nonetheless, as an aside we note that, even in the
absence of short-wavelength modes (as applies to finite degree of
freedom quantum mechanics) and biquadratic and Yukawa
interactions, the decoherence time $t_D$ is $O(t_{\rm sp})$, yet
again, for comparable coupling strengths. The reason is as for
Yukawa couplings: since the diffusion term grows exponentially,
because of long wavelength instabilities, the bound on $t_D$ has
only a logarithmic dependence on parameters, almost guaranteeing
no new scales.

In this case, for long time-scales we have the bound

\begin{equation}
\exp [\mu t_{\rm D}] \approx {\mu^2\over{V g''^2 \delta^2 (k_{\rm
B}T)}} \sim {\sqrt{\lambda}\over{g''^2}},\end{equation} from where
we can show $\mu t_{\rm D} \simeq 3 \ln T_{\rm c}/\mu$. Again,
this bound on $t_D$ is $O(t_{\rm sp})$, but larger than $t_{\rm sp}$. This
shows how adopting linear coupling to an environment in mimicry of
quantum mechanics can be misleading.

\subsection{The case $k_0\neq 0$}

Our earlier discussion was for modes for which $k_0\approx 0$,
including the important order-parameter itself ($k_0 = 0$).
However, if we are looking for structure in the field we need to
get away from $k_0 = 0$. It is apparent that the decoherence time
increases rapidly as we approach $k_0 = \mu$. Provided $k_0$ is
not too close to $\mu$ it is sufficient to replace $\mu t_D$ by
$t_D(k_0)\sqrt{\mu^2-k_0^2}$ in the calculations above to
determine the bound on the decoherence time for the mode $k_0$. An
immediate question is to ask what is the maximum value $k_{\rm
max}$ for which $t_{\rm D}(k_{\rm max})= t_{\rm sp}$. After this
replacement in
Eqs. (\ref{tdecalpha}) and (\ref{tsp}) we obtain ($\alpha = 1$)
\begin{equation}
\frac{k^2_{\rm max}}{\mu^2}\approx \frac{\ln (T_{\rm c}/\mu)}{\ln (\eta
/\sqrt{\mu T_{\rm c}})}. \label{kmax}
\end{equation}

Thus, provided $k < k_{\rm max}$, we have
\begin{equation} 1 < \mu t_{\rm D} < \mu t_{\rm sp}.\end{equation}

Finally, there is a time $t({\tilde k})$ when the mode ${\tilde
k}$ would become the dominant
wavenumber. This ``quantum time'' is determined by the instabilities
of the long-wavelength modes, which increasingly bunch about the wavenumber
$\tilde k^2(t)\sim \mu / t({\tilde k})$\cite{Rivers2,Karra}. At 
time $t_D(k_{\rm max}) = t_{\rm sp}$
\begin{equation}
\frac{\tilde k^2(t_{\rm sp})}{\mu^2} \sim \frac{1}{\mu t_D}\sim
\frac{1}{\ln (\eta /\sqrt{\mu T_{\rm c}})}< \frac{k^2_{\rm max}}{\mu^2}
\end{equation}
from (\ref{kmax}) above. That is, the dominant modes at time
$t_{\rm sp}$ have already decohered by this time.

In all of the above, coefficients $O(1)$ in the arguments of the
logarithms have been neglected. Thus, although the results are
true for couplings that are weak enough, a specific choice of
parameters would require a more detailed analysis.

\section{Slow Quenches}
\label{sec:slow}

More realistically, any temperature quench occurs in a finite
time. The  question is, how this slow quench affect our main
result about the decoherence time? To address this issue, we
assume that the quench begins at $t=0$ and ends at time $t =
\tau_{\rm q}$, with $\tau_{\rm q}\gg t_{\rm r}\sim \mu^{-1}$. At
the qualitative level at which we are working it is sufficient to
take $m_{\phi}^2(T_0) = \mu^2$ exactly. Most simply, we consider a
quench linear in time, with temperature $T(t)$, for which the mass
function is of the following form \cite{bowick}
\[ m^2(t) = m_{\phi}^2(T(t)) = \left\{ \matrix{
\mu^2&\mbox{for} ~ t \le 0\cr \mu^2 - {2t\mu^2\over{\tau_{\rm
q}}}& ~~~~~~~~\mbox{for}  ~ 0 < t \le \tau_{\rm q}\cr -
\mu^2&~\mbox{for} ~ t \ge \tau_{\rm q}\cr} \right.
\]

In order to find  the master equation (or more strictly the
diffusion coefficient) we follow the same procedure as in Section
III, assuming a dominant wavenumber $k_0$. That is, we look for
classical solutions of the form

\[
\phi_{\rm cl}(\vec x, s) =  f(s,t)\cos(\vec k_0 . \vec x),
\]
where $f(s,t)$ satisfies
$f(0,t)= \phi_{\rm i}$ and $f(t,t) = \phi_{\rm f}$. We write it as
\[
f(s,t) = \phi_{\rm i} u_1(s,t) + \phi_{\rm f} u_2(s,t),\,\,
\]
where  $u_i(s,t)$ are solutions of the equation

\begin{equation}\left[{d^2\over{ds^2}} + k^2 + \mu^2 - 
{2\mu^2 t\over{\tau_{\rm q}}}\right]
u_i(s,t) = 0,\end{equation}
with boundary conditions $u_1(0,t) = 1$, $u_1(t,t) = 0$ and
$u_2(0,t) = 0$, $u_2(t,t) = 1$.

The solution is given by
             \[
             u_1(s,t) =  {-Ai[{\Delta t\over{t_{\rm k}}}] Bi[{\Delta s
\over{t_{\rm k}}}] + Ai[{\Delta s\over{t_{\rm k}}}] Bi[{\Delta t
\over{t_{\rm k}}}]\over{Ai[-\mu^2 t^2_{\rm k}] Bi[{\Delta t
\over{t_{\rm k}}}] - Ai[{\Delta t\over{t_{\rm k}}}] Bi[-\mu^2 t^2_{\rm k}]}},
\]

\[u_2(s,t)= {-Ai[{\Delta s\over{t_{\rm k}}}] Bi[-\mu^2 t^2_{\rm k}]
+ Ai[-\mu^2 t^2_{\rm k}] Bi[{\Delta s
\over{t_{\rm k}}}]\over{Ai[-\mu^2 t^2_{\rm k}] Bi[{\Delta t
\over{t_{\rm k}}}] - Ai[{\Delta t\over{t_{\rm k}}}] Bi[-\mu^2 t^2_{\rm k}]}} .
             \]
with $Ai[s]$, $Bi[s]$ the Airy functions; $\Delta s = s - \mu^2
t^3_{\rm k} $, $\Delta t = t - \mu^2 t^3_{\rm k} $, and $t^3_{\rm
k} = \tau_{\rm q}/\mu^2$. In the causal analysis of
Kibble\cite{Kibble} $t_{\rm k}$ is the earliest time in which domains
could have formed. Our analysis suggests that this time is not
$t_{\rm k}$, but $t_{\rm sp}$. Calculation shows that
\begin{equation}\langle\phi^2\rangle_t \sim {T_{\rm c}\over{\mu t_{\rm k}^2}}
\exp{\left\{ {4\over{3}}\left({t\over{t_{\rm
k}}}\right)^{3\over{2}} \right\}}, \label{phi2}
\end{equation}
whereby
\begin{equation}
\exp{\left\{ {4\over{3}}\left({t_{\rm sp}\over{t_{\rm
k}}}\right)^{3\over{2}} \right\}} \sim  {\mu \eta^2\over{T_{\rm
c}}} t_{\rm k}^2. \label{sp}
\end{equation}
Although $t_{\rm sp}$ and $t_{\rm k}$ are different in principle, and
their ratio can be treated as large in asymptotic expansions where
it occurs in exponents, in practice logarithms are small and
$t_{\rm sp} = O(t_{\rm k})$.

We can formally find the master equation in the same way that
we did in Section III. For the case of quadratic coupling,  it is
given again by Eq.(\ref{master}), with a diffusion coefficient  of
the form
\begin{equation}
D(k_0,t) = \int_0^t ~ ds ~ u(s,t)~ F(s,t),
\end{equation}
with
$$u(s,t) =\bigg[u_2(s,t) - \frac{{\dot u}_2(t,t)}{{\dot
             u}_1(t,t)}u_1(s,t)\bigg]^{2},$$
and
$$
F(s,t) = {\rm Re}
G_{++}^2(2k_0; t-s) + 2 {\rm Re}G_{++}^2(0; t-s).$$

It is important to note that $D$ is dominated by the $s \approx 0$
contribution into the integral. Indeed, it is easy to prove that
$u(0,t) = ({{\dot u}_2(t,t)\over{{\dot u}_1(t,t)}})^2 \gg 1$,
while $u(t,t) = 1$. Therefore, we can approximate the diffusion
coefficient by
\begin{equation}D(t) = F(0,t) ~ u(0,t) ~
\int_0^{\mu^{-1}} ds ~ {u(s,t)\over{u(0,t)}}
\label{D0}\end{equation}
where we used the fact that $F(s,t)$ is bounded at $s=0$.

Assuming large $\Delta t$ (and $\Delta s$), which means $t, s >> t_{\rm k}$,
we can use the asymptotic expansions of the Airy functions
and their derivatives for the evaluation of $u_i(s,t)$. In particular
we obtain
\begin{eqnarray}
u(0,t)&\simeq&{1\over 4 \mu t_{\rm k}}\sqrt{t\over t_{\rm k}}
\exp{\left\{ {4\over{3}}\left({t\over{t_{\rm k}}}\right)^{3\over{2}}
\right\}}\nonumber\\
\dot u(0,t)&\simeq&{1\over 2 \mu^2t_{\rm k}^3}\sqrt{t\over t_{\rm k}}
\exp{\left\{ {4\over{3}}\left({t\over{t_{\rm k}}}\right)^{3\over{2}}
\right\}}.
\end{eqnarray}
Therefore,  it
is straightforward  to check that in these cases the integral in
Eq.(\ref{D0}) is given by

\[
\mu ~\int_0^{\mu^{-1}}ds \left[1 + s {{\dot u}(0,t)\over{u(0,t)}} + ...~
\right] \approx \mu ~\int_0^{\mu^{-1}}ds \left[ 1 + s {2\over{\mu t^2_{\rm k}}}
+ ... \right] ,
\]
and it is ${\cal O}(1)$, due to the fact that $\mu t_{\rm k} \gg 1$.

\subsection{Decoherence times}

We can estimate the decoherence time as in previous Sections from
\begin{equation}
1 \approx V\Gamma\int_0^{t_{D}} dt ~ D(t),
\end{equation}
where $V$ is the volume, $\Gamma = \lambda^2 \delta^2 {\bar\phi}^2
\mu^4$, and $D(t) \sim (T_{\rm c}^2/\mu^2) u(0,t)$. As before, we
take $\delta^2 \sim {\cal O}(1)$. ${\bar\phi}^2 = \alpha /\lambda$
where $\alpha$ must be evaluated self-consistently, but now taking
into account that the quench is slow. As in the case of the
instantaneous quench, we only need  to consider the bound on $t_D$
due to quadratic interactions. Again it is sufficient to consider
only modes with $k_0\approx 0$.

 For quadratic interactions alone the decoherence time
reads
\begin{equation}
\exp{\left\{ {4\over{3}}\left({t_D\over{t_{\rm k}}}\right)^{3\over{2}}
\right\}}
\sim {\mu^2\over{\lambda T_{\rm c}^2 \alpha}}.
\end{equation}
Using $\langle\phi^2\rangle$ of (\ref{phi2}) we obtain $\alpha^2 =
{1\over{t_{\rm k}^2\mu T_{\rm c}}}$. Thus the decoherence time
satisfies
\begin{equation}
\exp{\left\{ {4\over{3}}\left({t_D\over{t_{\rm k}}}\right)^{3\over{2}}
\right\}}
\sim {\eta^2\over{T_{\rm c}^2}} t_{\rm k} \sqrt{\mu T_{\rm c}}.
\end{equation}

If we compare it with the spinodal one of (\ref{sp}), we get
\begin{equation}
t_{\rm sp}^{{3\over{2}}} - t_{D}^{{3\over{2}}} \sim
{3\over{4}}~ t_{\rm k}^{{3\over{2}}}~ \ln t_{\rm k}\sqrt{\mu T_{\rm c}},
\end{equation}
or, more transparently,
\begin{equation}
(\mu t_{\rm sp})^{{3\over{2}}} - (\mu t_{D})^{{3\over{2}}} \gtrsim
{1\over{4}}~ \sqrt{\mu \tau_{\rm q}}~ \ln (\mu \tau_{\rm q}),
\end{equation}
which shows that $t_D < t_{\rm sp}$, as in the instantaneous case.
The inclusion of further interactions, including the
self-interaction with short-wavelength ($k > \mu$) modes can only
reduce $t_D$ further.

Our approximation scheme that we have just
outlined depends, as for the instantaneous quench, on the peaking
of the power in the field fluctuations at long wavelength
$k_0\ll\mu$ by time $t_{\rm sp}$. This motivated our adopting a
dominant wavelength approximation, necessary to make the
calculation possible. Whereas peaking is inevitable for the
instantaneous quench for weak enough coupling that is not the case
for very slow quenches. In such cases our approximations break
down and a different analysis is required. The details are rather
messy, but a sufficient condition for our approximation to be
valid is that $\mu\tau_{\rm q} \lesssim\eta/\mu$\cite{Karra}.
Tighter, but less transparent bounds can be given\cite{Karra}.
When these bounds are satisfied the minimum wavelength for which
the modes decohere by time $t_{\rm sp}$ can be shown\cite{lomplb2} to
be shorter than that which characterises domain size at that time.

\section{Final remarks}
\label{sec:final}
%

We first summarize the results contained in this paper, in which
we have shown how the environment leads to the decoherence of the
order parameter after a transition. After the integration over the
scalar environment-fields $\chi_{\rm a}$ in Section II, we have obtained
the coarse-grained effective action (CGEA) for the system (field
$\phi$). From the imaginary part of the CGEA we obtained in
Section III the diffusion coefficient of the master equation, at
1-loop and in the high temperature environment limit. Terms
omitted are relatively $O(N^{-1/2})$ for $N$ weakly coupled
environmental fields. Subsequently, we evaluated the decoherence
time for the long-wavelength modes of the system-field (for each
of the different couplings with the environment considered) both
for an instantaneous and for slow quenches. This decoherence time
depends on the coupling between system and bath, the self-coupling
of the system (through the environment temperature $T$), and the
mass. Some previous works about classicality in field theory
\cite{salman,mottola} did not consider the influence of the
environment. For such closed systems the classical behaviour only
emerges a long time after the quench. In our model, we have shown
that the decoherence time is in general  smaller than the spinodal
time. This  result justifies in part the use of phenomenological
stochastic equations to describe the dynamical evolution of the
system field, as we will now discuss.

As it is well known \cite{lombmazz,greiner}, for $\phi^2\chi^m$
interactions one can regard the imaginary part of $\delta A$ as
coming from noise source $\xi_{\rm m}(x)$, with a Gaussian
functional probability distribution. Taking the biquadratic
coupling to the external environment $\chi$-field first leads to a
noise, termed $\xi_2$, say, with distribution
\begin{equation}
P[\xi_{2}(x)]= N_{\xi_{2}} \exp\bigg\{-{1\over{2}}\int d^4x\int
d^4y ~\xi_{2} (x) \Big[ g^2 N_{\rm q}\Big]^{-1}\xi_{2} (y)\bigg\},
\end{equation}
%
where $N_{\xi_{\rm m}}$ is a normalization factor.  This enables
us to write the imaginary part of the influence action as a
functional integral over the Gaussian field $\xi_{2} (x)$
%
\begin{eqnarray}
\int {\cal D}\xi_{2} (x) P[\xi_{2} ]&&\exp{\left[ -i
\bigg\{\Delta_{2} (x) \xi_{2} (x) \bigg\}\right]}\nonumber
\\ &=& \exp{\bigg\{-i\int d^4x\int d^4y \ \Big[\Delta_{2}(x)
~g^2 N_{\rm q}(x,y)~ \Delta_{2}(y)\Big] \bigg\}}.\end{eqnarray}
%
In consequence, the CGEA can be rewritten as
%
\begin{equation}A[\phi^+,\phi^-]=-{1\over{i}} \ln  \int {\cal D} \xi_{2}
  P[\xi_{2}]
\exp\bigg\{i S_{\rm eff}[\phi^+,\phi^-, \xi_{2}]\bigg\},
\end{equation}
%
where
%
\begin{equation}
S_{\rm eff}[\phi^+,\phi^-,\xi_{2}]= {\rm Re} A[\phi^+,\phi^-]-
\int d^4x\Big[\Delta_{2} (x) \xi_{2}(x) \Big].
\end{equation}
%

Therefore, taking the functional variation
%
\begin{equation}
\left.{\delta S_{\rm eff}[\phi^+,\phi^-, \xi_{2}]\over{\delta
\phi^+}}\right\vert_{\phi^+=\phi^-}=0,
\end{equation}
%
we are able to obtain the ``semiclassical-Langevin'' equation for the
system-field \cite{lombmazz,greiner}

\begin{eqnarray}
\Box \phi (x) - {\tilde\mu}^2 \phi +
{{\tilde\lambda}\over{6}}\phi^3(x) &+& g^2 \phi (x) \int d^4y ~
K_{\rm q}(x-y)~ \phi^2(y) = \phi (x) \xi_{2}(x)\label{lange2}
,\end{eqnarray}
where  ${\tilde\mu}$ and
$ {\tilde\lambda}$ are constants ``renormalised'' because of
the coupling with the environment.

Each part of the environment that we include leads to a further
'dissipative' term on the left hand side of (\ref{lange2}) with a
countervailing noise term on the right hand side. Once we include
the interactions between $\phi_<$ and $\phi_>$ at one-loop level
we get a similar
equation to (\ref{lange2}) from the term Eq.(\ref{qq}) alone.
However, although the
$\phi_<\phi^3_>$ and $\phi^3_<\phi_>$ terms were ignorable in the
bounding of $t_D$, in the Langevin equations they give further
terms, with quadratic $\phi^2\xi_3$ noise and linear noise $\xi_1$
respectively.

The inclusion of Yukawa interactions leads to the inclusion of
further terms in (\ref{lange2}) of the same form. Only for the
linear interaction with the environment (to the exclusion of
self-interaction) we do recover the additive noise that has been
the basis for stochastic equations in relativistic field theory
that confirm the scaling behaviour of Kibble's and Zurek's
analysis \cite{laguna}.

For times later than $t_{\rm sp}$, neither perturbation
theory nor more general non-Gaussian methods are valid.
It is
difficult to imagine an {\it ab initio} derivation of the dissipative and
noise terms  from the full quantum field theory. In this sense,
a reasonable alternative
is to analyze numerically  phenomenological stochastic
equations
and check the robustness of the predictions
against different choices of the dissipative kernels and of the
type of noise.

\acknowledgments
F.C.L. and F.D.M. were supported by Universidad
de Buenos Aires, CONICET, Fundaci\'on Antorchas and ANPCyT. R.J.R.
was supported, in part, by the COSLAB programme of the European
Science Foundation.

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\end{document}

