aainstitutetext: Department of Physics & Astronomy, McMaster University, 1280
Main Street West, Hamilton ON, Canada.bbinstitutetext: Perimeter Institute for
Theoretical Physics, 31 Caroline Street North, Waterloo ON,
Canada.ccinstitutetext: Minerva Schools at KGI, 1145 Market Street, San
Francisco, CA 94103, USA.

# Quantum Hotspots: Mean Fields, Open EFTs, Nonlocality and Decoherence Near
Black Holes

C.P. Burgess c R. Holman a,b and G. Kaplanek

(June 2021)

###### Abstract

Effective theories describing black hole exteriors resemble open quantum
systems inasmuch as many unmeasurable degrees of freedom beyond the horizon
interact with those we can see. A solvable Caldeira-Leggett type model of a
quantum field that mixes with many unmeasured thermal degrees of freedom on a
shared surface was proposed in arXiv:2106.09854 to provide a benchmark against
which more complete black hole calculations might be compared. We here use
this model to test two types of field-theoretic approximation schemes that
also lend themselves to describing black hole behaviour: Open EFT techniques
(as applied to the fields themselves, rather than Unruh-DeWitt detectors) and
mean-field methods. Mean-field methods are of interest because the effective
Hamiltonians to which they lead can be nonlocal; a possible source for the
nonlocality that is sometimes entertained as being possible for black holes in
the near-horizon regime. Open EFTs compute the evolution of the field state,
allowing discussion of thermalization and decoherence even when these occur at
such late times that perturbative methods fail (as they often do). Applying
both of these methods to a solvable system identifies their domains of
validity and shows how their predictions relate to more garden-variety
perturbative tools.

##  1 Introduction and Discussion

Quantum fields in black-hole backgrounds have long been known to behave in
surprising ways Bekenstein:1971hc ; Hawking:1974rv , even at energies well
below whatever new physics ultimately describes gravity at its most
foundational level. Although the nature of the approximations being made when
studying these effects was initially puzzling, this quantum-gravitational
interplay has been integrated into the broader framework of theoretical
physics within the formalism of effective field theories (EFTs)
Weinberg:1978kz ; Donoghue:1994dn  (for reviews see Burgess:2003jk ;
Goldberger:2007hy ; Porto:2016pyg ; Donoghue:2017ovt ; EFTBook ).

In the meantime black hole physics grew up, with the discovery of
gravitational waves LIGO  confronting EFT calculations Goldberger:2004jt ;
Goldberger:2005cd ; Porto:2005ac ; Kol:2007bc ; Kol:2007rx ; Gilmore:2008gq ;
Porto:2008jj ; Damour:2009vw ; Emparan:2009at ; Damour:2009wj ; Levi:2015msa
of black hole properties with experimental measurements. This has stimulated
much work and has underlined some of the unique challenges posed when working
with black holes in an EFT framework. One of these challenges asks how the EFT
should handle the large numbers of gapless and dissipative degrees of freedom
Goldberger:2005cd ; Goldberger:2019sya  associated with the black hole’s
entropy.

It would be useful to compare black-hole calculations with similar ones for
well-understood solvable systems that share as many of these features as
possible, and to this end ref. Hotspot  proposed a solvable Caldeira-Leggett
style FeynmanVernon ; CaldeiraLeggett  model in which an ‘external’ massless
quantum field $\phi$ interacts with many unseen gapless thermal fields, but
only on a surface meant as a poor man’s model of the event horizon. Following
Hotspot , in what follows we call such a hot localized source a ‘hotspot’.

In this paper we use this model to explore two types of approximate tools that
both lend themselves to black-hole applications and capture different aspects
of black-hole exceptionalism. (A companion paper Kaplanek:2021xxx  computes
the response an Unruh-DeWitt (qubit) detector that couples to the external
field $\phi$ at a fixed distance from the hotspot.) Applying approximate tools
to a solvable model allows explicit identification of their domain of
validity, which can be useful for applications to more realistic systems for
which a full solution is not known.

The two approximations explored here are late-time Open-EFT methods111Refs.
Kaplanek:2021xxx ; Kaplanek:2019dqu ; Kaplanek:2019vzj ; Kaplanek:2020iay
also explore the use of Open EFT techniques, but do so for the much simpler
case where late-time predictions are only sought for an Unruh-DeWitt qubit
detector Unruh:1976db ; DeWitt:1980hx , rather than for the entire $\phi$
field. (See Decoherence1 ; Decoherence2 ; Decoherence3 ; Decoherence4 ;
Decoherence5 ; Decoherence6 ; Decoherence7 ; Decoherence8 ; Decoherence9 ;
Burgess:2014eoa ; Martin:2018zbe  for related discussions of field decoherence
in cosmology.) and mean-field expansions. Ref. Hotspot  solves the system
dynamics exactly, but does so within the Heisenberg picture and so obscures
how the $\phi$ field state evolves once couplings to the hotspot are switched
on. Open-EFT methods are designed to extract this state evolution, in
principle allowing access to questions such as whether (and how quickly) the
hotspot decoheres the external $\phi$ field. Furthermore it does so with a
domain of validity that allows it to treat phenomena (like thermalization)
that occur at times sufficiently late that naive perturbative methods
generically fail.

Mean-field methods provide a framework within which an effective Hamiltonian
description is possible even while including open-system effects (see for
example EFTBook  for practical examples treated in the same framework used
here). Such a Hamiltonian description need not be guaranteed for generic
systems. Furthermore, the effective mean-field Hamiltonian is often nonlocal
and/or non-Hermitian, making it natural to ask whether mean-field methods
might provide relatively mundane origins for exotic non-Wilsonian behaviour in
the vicinity of black hole horizons; exotic behaviour that is often speculated
to exist near black holes Hawking:1976ra ; Giddings:2006sj ; Skenderis:2008qn
; Almheiri:2012rt ; Almheiri:2013hfa ; Banks:1994ph ; Mathur:2009hf .

Our arguments and results are laid out as follows. §2 starts by briefly
reviewing the hotspot model given in Hotspot  and summarizes the results
computed there that are relevant for later comparisons. The model’s main
variables are a massless field, $\phi$, (meant to represent observable degrees
of freedom on the near side of the horizon) plus a thermal bath (also with
massless fields, $\chi^{a}$) meant to model the dissipative effects of beyond-
the-horizon physics. These fields ‘interact’ locally by mixing only on a
surface meant to represent the horizon itself (though the interaction surface
is not an actual geometrical local horizon).

Once the model is set up §3 formulates the evolution of the external field
$\phi$ after tracing out the thermal degrees of freedom. This section does so
by deriving eq. (3.22); a Nakajima-Zwanzig evolution equation Nak ; Zwan  – a
common open-systems tool – for the reduced density matrix of the field
${\phi}$. We find a limit in which this equation takes a Markovian form –
(3.1.2) – and we solve for the field’s density matrix in this Markovian limit,
doing so perturbatively in the system couplings and working in the field basis
where the solution is a Gaussian state. Using this result we compute the
equal-time correlations $\langle{\phi}(t,{{\bf x}}){\phi}(t,{{\bf y}})\rangle$
– with the result (3.74) – and identify the domain of validity of Markovian
methods by comparing this to the correlator given in §2.

§3 closes by computing a measure of the $\phi$ state’s purity after the
$\chi^{a}$ fields are integrated out, and computes the decoherence rate as a
function of the $\phi$-$\chi$ coupling parameters and hotspot temperature. For
each mode the rate of departure from an initially pure free-vacuum state is
controlled by the time scale $\tilde{g}^{2}/\beta$ where $\beta$ is the
inverse hotspot temperature and $\tilde{g}$ is a measure of the $\phi$-$\chi$
couplings.

Finally, §4 defines what a mean-field Hamiltonian is, and what this definition
means – c.f. eq. (4.15) – for the field $\phi$ in the hotspot model. The
result is in general nonlocal on the interaction surface and in time (with
characteristic nonlocality scales given by the surface radius, $\xi$, and the
inverse temperature, $\beta$), but is typically local in the radial ‘off-
horizon’ direction. This mean-field Hamiltonian is used to compute once again
the system correlation function $\langle{\phi}(t,{{\bf x}}){\phi}(s,{{\bf
y}})\rangle$ for comparison with the results of §2, showing that validity of
mean-field methods requires some couplings (like the $\phi$ self-coupling
$\lambda$) to dominate the temperature-dependent combination
$\tilde{g}^{2}/\beta$.

Taken together, the respective approximate methods described in §3 and §4 hold
in complementary regimes of parameter space, both of which are subsets of the
broader domain of perturbative methods found in Hotspot .

##  2 The hotspot reloaded

This section briefly summarizes the hotspot-model setup as given in Hotspot .
The fields involved consist of an observable sector — a single real massless
scalar field, $\phi$ — and an unmeasured disspative and gapless environment —
$N$ real massless scalar fields, $\chi^{a}$ prepared in a thermal state. These
two systems live in different spatial regions (${\cal R}_{\pm}$) that only
intersect in a small localized domain: a sphere ${\cal S}_{\xi}$ of radius
$\xi$ enclosing the origin that is identified for the two spaces (see Fig. 1).
We choose here not to follow the gravitational back-reaction of these fields,
and so treat ${\cal R}_{\pm}$ as independent flat spatial slices. A surface
${\cal S}_{\xi\pm}$ encircles the origin within each of these spaces, with
${\cal S}_{\xi\pm}$ identified to obtain the interaction surface ${\cal
S}_{\xi}$. The $\chi^{a}$ fields are meant to represent unmeasured degrees of
freedom internal to the black hole with which external fields can interact.

Figure 1: The two spatial branches, ${\cal R}_{+}$ and ${\cal R}_{-}$, in
which the field $\phi$ and the $N$ fields $\chi^{a}$ repsectively live. In
practice the two regions are idealized as flat (though curvature can in
principle also be included) with the spherical interaction region ${\cal
S}_{\xi}$ identified. The two types of fields only couple to one another
within ${\cal S}_{\xi}$ (which effectively becomes the world-line of a point
in the special case that $\xi$ is much smaller than all other scales of
interest). Figure taken from Hotspot .

Interactions on ${\cal S}_{\xi}$ are taken to be bilinear mixings of the
fields, of the form ${\cal L}_{\rm int}=-g_{a}\,\chi^{a}\,\phi$, plus possible
quadratic self-interactions of the fields as required by any renormalization-
group flows.222Such flows arise due to renormalizations of the Coulomb-like
divergences that appear even at the classical level near the interaction
surface Goldberger:2001tn ; deRham:2007mcp ; Burgess:2008yx ; Bayntun:2009im ;
PPEFT . With black holes in mind we consider observables that depend only on
the $\phi$ field and do not directly measure any of the $\chi^{a}$’s.

The action for the model is

$S_{{\scriptscriptstyle U}{\scriptscriptstyle V}}=-\frac{1}{2}\int_{{\cal
R}_{+}^{t}}{\hbox{d}}^{4}x\;\partial_{\mu}\phi\,\partial^{\mu}\phi-\frac{1}{2}\int_{{\cal
R}_{-}^{t}}{\hbox{d}}^{4}x\;\delta_{ab}\,\partial_{\mu}\chi^{a}\partial^{\mu}\chi^{b}-\int_{{\cal
S}_{\xi}^{t}}{\hbox{d}}^{3}x\;\left[G_{a}\,\chi^{a}\phi+\frac{G_{\phi}}{2}\,\phi^{2}\right]\,,$
(2.1)

where ${\cal R}_{\pm}^{t}$ denote spacetime regions whose spatial slices are
${\cal R}_{\pm}$, and the integration for the interaction is over the world-
tube, ${\cal S}_{\xi}^{t}$ swept out by the surface ${\cal S}_{\xi}$ over
time. The couplings $G_{a}$ and $G_{\phi}$ have dimension mass, and $G_{\phi}$
is included because it can be required to renormalize divergences that arise
due to the presence of the $\phi$-$\chi$ mixing $G_{a}$. When performing
calculations we usually specialize to the case where $G_{a}=G$ is
$a$-independent and define the combination

$\widetilde{G}^{2}:=\delta^{ab}G_{a}G_{b}=NG^{2}\,.$ (2.2)

As described in Hotspot  this hotspot model comes in two versions, depending
on whether or not the radius $\xi$ is regarded to be an ultraviolet
scale.333The same distinction also arises for black-hole EFTs, for which the
black hole event horizon can either be regarded as being shrunk to a point –
as in world-line point-particle EFTs Goldberger:2004jt ; Goldberger:2005cd ;
Porto:2005ac ; Kol:2007bc ; Kol:2007rx ; Gilmore:2008gq ; Porto:2008jj ;
Damour:2009vw ; Emparan:2009at ; Damour:2009wj ; Levi:2015msa  – or can be
regarded as being macroscopic Burgess:2018pmm ; Rummel:2019ads . This latter
type of EFT can be used to summarize situations where it is ordinary GR
Price:1986yy ; Thorne:1986iy ; Damour:1978cg ; Parikh:1997ma ; Donnay:2019jiz
or exotic physics Cardoso:2016rao ; Abedi:2016hgu ; Holdom:2016nek ;
Cardoso:2017cqb ; Bueno:2017hyj ; Mark:2017dnq ; Conklin:2017lwb ;
Berti:2018vdi ; Zhou:2016hsh  whose UV near-horizon physics is being
summarized. When $\xi$ is a UV scale the 2-sphere ${\cal S}_{\xi}$ degenerates
to a point and the above description gets replaced by an effective action
organized in powers of $\xi$. The leading $\phi$-dependent interactions in
this effective theory become

$S_{\rm int-
eff}\simeq-\int{\hbox{d}}t\;\left[g_{a}\,\chi^{a}(t,\mathbf{0})\,\phi(t,\mathbf{0})+\frac{\lambda}{2}\,\phi^{2}(t,\mathbf{0})\right]\,,$
(2.3)

where the integration is over the proper time of the interaction point. At
leading order the effective couplings $\lambda$ and $g_{a}$ are related to the
couplings in (2.1) by

$g_{a}=4\pi\xi^{2}G_{a}\quad\hbox{and}\quad\lambda=4\pi\xi^{2}G_{\phi}$ (2.4)

and so have dimensions of length. In this limit it is the quantity

$\tilde{g}^{2}=\delta^{ab}\,g_{a}g_{b}=Ng^{2}$ (2.5)

that plays the same role as did (2.2) when $\xi$ was not small.

###  2.1 Time evolution

The Hamiltonian for this system can be written $H=H_{0}+H_{\rm int}$ where
${H}_{0}:={{\cal H}}_{+}\otimes{\cal I}_{-}+{\cal I}_{+}\otimes{{\cal H}}_{-}$
is the free Hamiltonian, with ${\cal H}_{\pm}$ and ${\cal I}_{\pm}$ the
Hamiltonian and identity operators acting separately within the $\phi$\- and
$\chi$-sectors of the Hilbert space:

${{\cal H}}_{+}:=\frac{1}{2}\int_{{\cal R}_{+}}{\hbox{d}}^{3}{{\bf
x}}\;\Bigl{[}{{\mathfrak{p}}}^{2}+\big{(}\boldsymbol{\nabla}{\phi}\big{)}^{2}\Bigr{]}\quad\hbox{and}\quad{{\cal
H}}_{-}:=\frac{1}{2}\int_{{\cal
R}_{-}}{\hbox{d}}^{3}x\;\Bigl{[}\delta^{ab}{\Pi}_{a}{\Pi}_{b}+\delta_{ab}\boldsymbol{\nabla}{\chi}^{a}\cdot\boldsymbol{\nabla}{\chi}^{b}\Bigr{]}\,,$
(2.6)

and the canonical momenta are defined by ${\mathfrak{p}}:=\partial_{t}{\phi}$
and $\Pi_{a}:=\delta_{ab}\,\partial_{t}{\chi}^{b}$. The interaction
Hamiltonian is similarly

${H}_{\mathrm{int}}=\int_{{\cal
S}_{\xi}}{\hbox{d}}^{2}x\;\left[G_{a}\,{\phi}\otimes{\chi}^{a}+\frac{G_{\phi}}{2}\,\phi^{2}\otimes{\cal
I}_{-}\right]\,,$ (2.7)

which in the point-hotspot limit reduces to

${H}_{\mathrm{int}}(t)\simeq
g_{a}\,{\phi}(t,\boldsymbol{0})\otimes{\chi}^{a}(t,\boldsymbol{0})+\frac{\lambda}{2}\,\phi^{2}(t,\boldsymbol{0})\otimes{\cal
I}_{-}\,.$ (2.8)

Reference Hotspot  solves explicitly for the time-evolution for this model
within the Heisenberg picture, with most of the explicit results given for the
point-hotspot limit (for which $\xi\to 0$). The remainder of this section
quotes a few of the results found for later comparisons.

####  2.1.1 System state

In the Heisenberg picture states do not evolve with time, and correspond to
the initial state of the system in the Schrödinger or interaction pictures.
Correlation functions in Hotspot  are computed assuming the $\phi$ and
$\chi^{a}$ fields are initially uncorrelated, with

$\rho_{0}=\rho_{+}\otimes\rho_{-}\,,$ (2.9)

where the $\phi$ sector is in its standard Minkowski vacuum,
$\ket{\mathrm{vac}}$, and the $\chi^{a}$ sector is in a thermal state,
$\varrho_{\beta}$, with temperature $T=1/\beta$:

$\rho_{+}=\ket{\mathrm{vac}}\bra{\mathrm{vac}}\quad\hbox{and}\quad\rho_{-}=\varrho_{\beta}:=\frac{e^{-\beta{\cal
H}_{-}}}{{\mathrm{Tr}}_{-}[e^{-\beta{\cal H}_{-}}]}\,.$ (2.10)

Here ${\cal H}_{-}$ is the $\chi^{a}$-sector bulk Hamiltonian and the
subscript ‘$-$’ on the trace indicates that it is only taken over the $\chi$
sector.

####  2.1.2 Operator evolution

In Heisenberg picture the entire burden of time evolution falls on the field
operators, which evolve according to

${\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}):={U}^{-1}(t,0)\big{[}{\phi}_{{\scriptscriptstyle S}}({{\bf
x}})\otimes{\cal
I}_{-}\big{]}{U}(t,0)\quad\hbox{and}\quad{\chi}^{a}_{{\scriptscriptstyle
H}}(t,{{\bf x}}):={U}^{-1}(t,0)\big{[}{\cal
I}_{+}\otimes{\chi}^{a}_{{\scriptscriptstyle S}}({{\bf x}})\big{]}{U}(t,0)\,,$
(2.11)

where the full time-evolution operator is ${U}(t,t^{\prime})={\cal
T}\exp\left(-i\int_{t^{\prime}}^{t}{\hbox{d}}s\;{H}(s)\right)$ and ${\cal T}$
denotes time-ordering.

In differential form, a generic Heisenberg-picture operator
$A_{\scriptscriptstyle H}(t)$ satisfies $\partial_{t}A_{{\scriptscriptstyle
H}}(t)=-i\Bigl{[}A_{{\scriptscriptstyle H}}(t),H_{{\scriptscriptstyle
H}}(t)\Bigr{]}$, which in particular implies the fields
$\phi_{\scriptscriptstyle H}$ and $\chi^{a}_{\scriptscriptstyle H}$ satisfy
the equations of motion that follow from the action (2.1):
$\Box\phi=\Box\chi^{a}=0$ for all points exterior to ${\cal S}_{\xi}$. For
points on ${\cal S}_{\xi}$ the presence of the interaction implies the
equations of motion instead impose the boundary condition

$\partial_{r}\phi\Bigr{|}_{r\to\xi}=\Bigl{(}G_{a}\chi^{a}+G_{\phi}\,\phi\Bigr{)}_{r\to\xi}\,,$
(2.12)

(and similarly for $\chi^{a}$). In the $\xi\to 0$ limit of a point source both
the bulk equation and boundary condition are efficiently summarized by the
equations

$(-\partial_{t}^{2}+\nabla^{2})\phi_{{\scriptscriptstyle H}}(t,{{\bf
x}})=\delta^{3}({{\bf x}})\bigg{[}\lambda\phi_{{\scriptscriptstyle
H}}(t,\mathbf{0})+g_{a}\chi_{{\scriptscriptstyle
H}}^{a}(t,\mathbf{0})\bigg{]}$ (2.13)

and

$\delta_{ab}(-\partial_{t}^{2}+\nabla^{2})\chi^{b}_{{\scriptscriptstyle
H}}(t,{{\bf x}})=\delta^{3}({{\bf x}})\;g_{a}\phi_{{\scriptscriptstyle
H}}(t,\mathbf{0})\,.$ (2.14)

It is these last equations that are solved explicitly in Hotspot , with the
result used to compute the time-evolution of the system’s correlation
functions. These integrations are performed assuming the couplings $G_{a}$ and
$G_{\phi}$ (or $g_{a}$ and $\lambda$) turn on suddenly at $t=0$ and remain
time-independent thereafter. These solutions generically diverge as $r\to\xi$
(or ${{\bf r}}\to 0$ in the point-hotspot limit) and because of this these
equation must be regulated, such as by evaluating the boundary conditions at a
small distance $\xi+\epsilon$ from the singular point. The divergences
associated with taking $\epsilon\to 0$ are ultimately absorbed into
renormalizations of couplings like $G_{\phi}$ Goldberger:2001tn ;
deRham:2007mcp ; Burgess:2008yx ; Bayntun:2009im ; PPEFT

###  2.2 Correlation functions

The main result of Hotspot  is the calculation of the correlation functions
for the fields, whose results we now quote for future use.

####  2.2.1 $\phi$ correlation functions

The $\phi$-field Wightman function is defined by

$W_{\beta}(t,{{\bf x}};t^{\prime},{{\bf
x}}^{\prime}):=\mathrm{Tr}\Bigl{[}\phi_{{\scriptscriptstyle H}}(t,{{\bf
x}})\phi_{{\scriptscriptstyle H}}(t^{\prime},{{\bf
x}}^{\prime})\rho_{0}\Bigr{]}\,,$ (2.15)

where $\rho_{0}$ is the state given in (2.9), with explicit formulae given in
the small-hotspot limit ($\xi\to 0$).

The result computed to leading nontrivial order in $\tilde{g}^{2}$ and
$\lambda$ turns out to be given by

$\displaystyle W_{\beta}(t,{{\bf x}};t^{\prime},{{\bf
x}}^{\prime})\simeq\frac{1}{4\pi^{2}\big{[}-(t-t^{\prime}-i\delta)^{2}+|{{\bf
x}}-{{\bf x}}^{\prime}|^{2}\big{]}}$
$\displaystyle\qquad+\frac{\lambda}{16\pi^{3}}\bigg{(}\frac{\Theta(t-|{{\bf
x}}|)}{|{{\bf x}}|}\frac{1}{(t-t^{\prime}-|{{\bf x}}|-i\delta)^{2}-|{{\bf
x}}^{\prime}|^{2}}+\frac{\Theta(t^{\prime}-|{{\bf x}}^{\prime}|)}{|{{\bf
x}}^{\prime}|}\frac{1}{(t-t^{\prime}+|{{\bf x}}^{\prime}|-i\delta)^{2}-|{{\bf
x}}|^{2}}\bigg{)}$ $\displaystyle\qquad-\frac{\tilde{g}^{2}\Theta(t-|{{\bf
x}}|)\Theta(t^{\prime}-|{{\bf x}}^{\prime}|)}{64\pi^{2}\beta^{2}|{{\bf
x}}||{{\bf x}}^{\prime}|\sinh^{2}\left[\frac{\pi}{\beta}(t-|{{\bf
x}}|-t^{\prime}+|{{\bf x}}^{\prime}|-i\delta)\right]}$
$\displaystyle\qquad+\frac{\tilde{g}^{2}}{32\pi^{4}}\bigg{(}-\frac{\Theta(t-|{{\bf
x}}|)}{|{{\bf x}}|}\frac{t-t^{\prime}-|{{\bf x}}|}{\big{[}(t-t^{\prime}-|{{\bf
x}}|-i\delta)^{2}-|{{\bf
x}}^{\prime}|^{2}\big{]}^{2}}+\frac{\Theta(t^{\prime}-|{{\bf
x}}^{\prime}|)}{|{{\bf x}}^{\prime}|}\frac{t-t^{\prime}+|{{\bf
x}}^{\prime}|}{\big{[}(t-t^{\prime}+|{{\bf x}}^{\prime}|-i\delta)^{2}-|{{\bf
x}}|^{2}\big{]}^{2}}\bigg{)}$
$\displaystyle\qquad+\frac{\tilde{g}^{2}}{64\pi^{4}}\bigg{(}\frac{\delta(t-|{{\bf
x}}|)}{|{{\bf x}}|\big{[}-(t^{\prime}+i\delta)^{2}-|{{\bf
x}}^{\prime}|^{2}\big{]}}+\frac{\delta(t^{\prime}-|{{\bf
x}}^{\prime}|)}{|{{\bf x}}^{\prime}|\big{[}-(t-i\delta)^{2}-|{{\bf
x}}|^{2}\big{]}}\bigg{)}\qquad\hbox{(perturbative)}\,,$

where the delta functions and step functions describe the transients due to
the switch-on of couplings at $t=|{{\bf x}}|=0$. The infinitesimal $\delta\to
0^{+}$ is taken to zero at the end of the calculation.

Of most interest here is the form for the correlation function inside the
future light-cone of this switch-on (i.e. for $t>|{{\bf x}}|$ and
$t^{\prime}>|{{\bf x}}^{\prime}|$), for which this perturbative expression
becomes

$\displaystyle W_{\beta}(t,{{\bf x}};t^{\prime},{{\bf x}}^{\prime})$
$\displaystyle\simeq$
$\displaystyle\frac{1}{4\pi^{2}\big{[}-(t-t^{\prime}-i\delta)^{2}+|{{\bf
x}}-{{\bf x}}^{\prime}|^{2}\big{]}}+\frac{\lambda}{16\pi^{3}|{{\bf x}}||{{\bf
x}}^{\prime}|}\bigg{[}\frac{|{{\bf x}}|+|{{\bf
x}}^{\prime}|}{(t-t^{\prime}-i\delta)^{2}-(|{{\bf x}}+|{{\bf
x}}^{\prime}|)^{2}}\bigg{]}$
$\displaystyle\qquad-\frac{\tilde{g}^{2}}{64\pi^{2}\beta^{2}|{{\bf x}}||{{\bf
x}}^{\prime}|\sinh^{2}\left[\frac{\pi}{\beta}(t-|{{\bf x}}|-t^{\prime}+|{{\bf
x}}^{\prime}|-i\delta)\right]}$
$\displaystyle\qquad+\frac{\tilde{g}^{2}}{32\pi^{4}}\bigg{(}-\frac{1}{|{{\bf
x}}|}\frac{t-t^{\prime}-|{{\bf x}}|}{\big{[}(t-t^{\prime}-|{{\bf
x}}|-i\delta)^{2}-|{{\bf x}}^{\prime}|^{2}\big{]}^{2}}+\frac{1}{|{{\bf
x}}^{\prime}|}\frac{t-t^{\prime}+|{{\bf
x}}^{\prime}|}{\big{[}(t-t^{\prime}+|{{\bf x}}^{\prime}|-i\delta)^{2}-|{{\bf
x}}|^{2}\big{]}^{2}}\bigg{)}$ (inside light cone, perturbative)

The exact correlator is also computed in Hotspot  and the result is compared
there to this perturbative limit, verifying that the full dependence on
$\lambda$ agrees with RG-improved resummations using point-particle EFT
boundary-condition based methods PPEFT . Because the full result is not needed
in what follows it is not repeated here, beyond observing that the
perturbative result emerges from the full one once it is expanded in powers of

$\frac{\tilde{g}^{2}}{16\pi^{2}\epsilon\tau}\left(1+\frac{\lambda}{4\pi\epsilon}\right)^{-1}\ll
1\quad\hbox{and}\quad\frac{\tilde{g}^{2}}{16\pi^{2}\epsilon\beta}\left(1+\frac{\lambda}{4\pi\epsilon}\right)^{-1}\ll
1\,.$ (2.18)

For later comparisons it is useful to focus on the equal-time special case of
these formulae, for which $t^{\prime}=t$. Of particular interest when
comparing with other approximation schemes is the equal-time limit of the
perturbative result (2.2.1), which is

$\displaystyle W_{\beta}(t,{{\bf x}};t,{{\bf x}}^{\prime})$
$\displaystyle\simeq$ $\displaystyle\frac{1}{4\pi^{2}|{{\bf x}}-{{\bf
x}}^{\prime}|^{2}}-\frac{\lambda}{16\pi^{3}|{{\bf x}}||{{\bf
x}}^{\prime}|(|{{\bf x}}|+|{{\bf x}}^{\prime}|)}$
$\displaystyle\qquad\qquad-\frac{\tilde{g}^{2}}{64\pi^{2}\beta^{2}|{{\bf
x}}||{{\bf x}}^{\prime}|\sinh^{2}\left[\frac{\pi}{\beta}(|{{\bf x}}|-|{{\bf
x}}^{\prime}|)\right]}+\frac{\tilde{g}^{2}}{16\pi^{4}(|{{\bf x}}|^{2}-|{{\bf
x}}^{\prime}|^{2})^{2}}\ .$ (inside light cone, perturbative)

####  2.2.2 $\chi^{a}$ correlation functions

Reference Hotspot  also gives the explicit form for the $\chi^{a}$ free-field
thermal correlator. Evaluated at spacetime points $x=(t,\mathbf{x})$ and
$x^{\prime}=(t^{\prime},\mathbf{x}^{\prime})$ the result (at large $N$) is

$\displaystyle\langle\chi^{a}(t,{{\bf x}})\chi^{b}(t^{\prime},{{\bf
x}}^{\prime})\rangle_{\beta}$ $\displaystyle=$
$\displaystyle\frac{\delta^{ab}}{8\pi\beta|\mathbf{x}-\mathbf{x}^{\prime}|}\left\\{\coth\left[\dfrac{\pi}{\beta}\left(t-t^{\prime}+|\mathbf{x}-\mathbf{x}^{\prime}|-i\delta\right)\right]\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.-\coth\left[\dfrac{\pi}{\beta}\left(t-t^{\prime}-|\mathbf{x}-\mathbf{x}^{\prime}|-i\delta\right)\right]\right\\}\,,$

in agreement with standard formulae Thermal . In this expression the limit
$\delta\to 0^{+}$ is to be taken at the end of the calculation.

##  3 Open EFT late-time field evolution

The solution to the hotspot problem given in Hotspot  is provided in the
Heisenberg picture, and a drawback of this picture is that it obscures how the
system’s state evolves. This is unfortunate because it makes it difficult to
compare with much of the literature on open quantum systems, which is often
phrased in terms of the system’s reduced density matrix (see for example
EFTBook ).

In this section we aim to make this comparison more transparent, by solving
the hotspot problem using the Schrödinger-picture (in practice we use the
interaction picture once we resort to perturbative methods), computing in
particular the reduced density matrix $\sigma_{{\scriptscriptstyle S}}(t)$

$\sigma_{{\scriptscriptstyle
S}}(t):={\mathrm{Tr}}_{-}[\rho_{{\scriptscriptstyle S}}(t)]\,,$ (3.1)

for the $\phi$-field (defined as the partial trace of the full Schrödinger-
picture density matrix $\rho_{{\scriptscriptstyle S}}(t)$ over the $\chi^{a}$
sector).

The cumbersome nature of the Schrödinger picture for field theories prevented
us from solving exactly for $\sigma_{{\scriptscriptstyle S}}(t)$ even within
the hotspot model (though we have no reason to believe that this cannot be
done). So we instead compute this evolution perturbatively in the hotspot
couplings, also restricting for simplicity to the case of a point-like hotspot
($\xi\to 0$). A major drawback of using perturbation theory, however, is that
perturbative methods intrinsically break down at late times, seeming to put
beyond reach a reliable calculation of phenomena like decoherence or
thermalization. We therefore also adapt Open EFT techniques EFTBook ;
Burgess:2014eoa ; Agon:2014uxa ; Burgess:2015ajz ; Braaten:2016sja ;
Kaplanek:2019dqu ; Kaplanek:2019vzj ; Kaplanek:2020iay  to verify that they
allow the perturbative result to be resummed to extend the perturbative domain
of validity to very late times.444Open EFT techniques were also applied to the
hotspot in ref. Kaplanek:2021xxx , but only to obtain the late-time
thermalization behaviour of an Unruh-DeWitt detector Unruh:1976db ;
DeWitt:1980hx  that sits at rest displaced from the hotspot. Doing so also
allows the testing of these tools in this relatively unfamiliar quantum-field
setting.

###  3.1 Open EFT evolution

In principle the time evolution of the system’s state is given within the
Interaction picture by the Liouville equation,

$\frac{\partial\rho}{\partial t}\ =\ -i\big{[}H_{\rm int}(t),\rho(t)\big{]}$
(3.2)

where $H_{\mathrm{int}}(t)$ is given by eq. (2.7), and where $\rho(t)$ is the
interaction-picture state

$\rho(t)=e^{+iH_{0}t}\rho_{{\scriptscriptstyle S}}(t)\,e^{-iH_{0}t}$ (3.3)

where $H_{0}={\cal H}_{+}\otimes{\cal I}_{-}+{\cal I}_{+}\otimes{\cal H}_{-}$
is the free Hamiltonian for the combined system. It is easy to see from the
above that the Schrödinger-picture reduced density matrix
$\sigma_{{\scriptscriptstyle S}}(t)$ is related to the interaction-picture
reduced density matrix $\sigma(t)$ through the relation

$\sigma(t)=e^{+i{\cal H}_{+}t}\sigma_{{\scriptscriptstyle S}}(t)\,e^{-i{\cal
H}_{+}t}\ .$ (3.4)

When solving these equations we assume the uncorrelated initial state
$\rho(0)=\rho_{0}=\rho_{+}\otimes\rho_{-}$ given in (2.9) where
$\rho_{+}=\ket{\mathrm{vac}}\bra{\mathrm{vac}}$ and $\rho_{-}=\varrho_{\beta}$
is the thermal configuration for the $\chi$ sector.

####  3.1.1 Nakajima-Zwanzig equation

In principle the evolution of $\sigma(t)$ is given by taking the trace of
(3.2) over all unmeasured degrees of freedom (in this case the fields
$\chi^{a}$). In perturbation theory one usually first formally solves (3.2)
and then takes the trace of the result, leading to

$\sigma(t)=\sigma(0)-i\int_{0}^{t}{\hbox{d}}s\;\mathrm{Tr}_{-}\Bigl{[}H_{\rm
int}(s)\,,\rho_{0}\Bigr{]}+(-i)^{2}\int_{0}^{t}{\hbox{d}}s_{1}\int_{0}^{s_{1}}{\hbox{d}}s_{2}\;\mathrm{Tr}_{-}\Bigl{[}H_{\rm
int}(s_{1})\,,\Bigl{[}H_{\rm int}(s_{2})\,,\rho_{0}\Bigr{]}\Bigr{]}+\cdots$
(3.5)

where $\mathrm{Tr}_{-}$ denotes the partial trace only over the $\chi^{a}$
sector. The drawback of this expression is the relatively complicated
dependence of its right-hand side on the full system’s state. Because the
right-hand side refers explicitly to the initial state $\rho_{0}$ successive
terms in the series generically grow without bound for large $t$, causing the
perturbative approximation to fail at late times and precluding accessing
issues like thermalization and late-time decoherence.

The better route for late-time purposes is to take the trace of the
differential relation (3.2) and to eliminate from this the dependence of the
right-hand side on any unmeasured degrees of freedom (for a review of the
steps given below see for example EFTBook ). The good news is that because of
the linearity of (3.2) this can be done in great generality, with the
resulting evolution equation for $\sigma(t)$ known as the Nakajima-Zwanzig
equation Nak ; Zwan . Although this is a textbook derivation, we now describe
it in some detail since it is not often applied to quantum fields (as we do
here) in the relativity literature.

The logic of the derivation proceeds as follows. One first defines the super-
operator $\mathscr{P}$ acting on operators in the Hilbert space by

$\mathscr{P}({\cal O})\ =\ {\mathrm{Tr}}_{-}\left[{\cal
O}\right]\otimes\rho_{-}\ ,$ (3.6)

where $\rho_{-}=\varrho_{\beta}$ is the time-independent initial thermal
density matrix for the fields $\chi^{a}$. This is a projection super-operator
because it satisfies $\mathscr{P}^{2}=\mathscr{P}$, as therefore must its
complement $\mathscr{R}=1-\mathscr{P}$. The definition (3.6) is defined so
that it projects the full density matrix $\rho(t)$ onto the reduced density
matrix $\sigma(t)$,

$\mathscr{P}\big{[}\rho(t)\big{]}=\sigma(t)\otimes\rho_{-}\ ,$ (3.7)

and so $\mathscr{R}[\rho(t)]$ can be regarded as describing all of the
unmeasured parts of the full density matrix.

Our goal is therefore to rewrite the Liouville equation as a coupled set of
evolution equations for the mutually exclusive quantities
$\mathscr{P}\big{[}\rho(t)\big{]}$ and $\mathscr{R}\big{[}\rho(t)\big{]}$, and
then eliminate $\mathscr{R}\big{[}\rho(t)\big{]}$ from these by solving its
equation as a function of $\mathscr{P}\big{[}\rho(t)\big{]}$. To this end
write the interaction-picture Liouville equation (3.2) in terms of a linear
Liouville super-operator,

$\partial_{t}\rho=\mathscr{L}_{t}(\rho)\quad\quad\quad\mathrm{where}\quad\quad\quad\mathscr{L}_{t}(\rho)=-i[H_{\rm
int}(t),\rho\,]\ ,$ (3.8)

and project the result using the operators $\mathscr{P}$ and $\mathscr{R}$.
Since $\mathscr{P}+\mathscr{R}=1$ this leads to

$\mathscr{P}(\partial_{t}\rho)=\mathscr{P}\mathscr{L}_{t}\mathscr{P}(\rho)+\mathscr{P}\mathscr{L}_{t}\mathscr{R}(\rho)$
(3.9)

and

$\mathscr{R}(\partial_{t}\rho)=\mathscr{R}\mathscr{L}_{t}\mathscr{P}(\rho)+\mathscr{R}\mathscr{L}_{t}\mathscr{R}(\rho)\,.$
(3.10)

The unmeasured degrees of freedom are eliminated by formally solving eq.
(3.10):

$\mathscr{R}[\rho(t)]={\cal
G}(t,0)\mathscr{R}[\rho(0)]+\int_{0}^{t}{\hbox{d}}s\
\mathcal{G}(t,s)\mathscr{R}\mathscr{L}_{t}\mathscr{P}[\rho(s)]$ (3.11)

with

$\mathcal{G}(t,s)=1+\sum_{n=1}^{\infty}\int_{s}^{t}{\hbox{d}}s_{1}\
\cdots\int_{s}^{s_{n-1}}{\hbox{d}}s_{n}\
\mathscr{R}\mathscr{L}_{s_{1}}\cdots\mathscr{R}\mathscr{L}_{s_{n}}\,,$ (3.12)

and inserting the result into (3.9). This yields the desired self-contained
evolution equation for $\mathscr{P}[\rho(t)]$:

$\mathscr{P}[\partial_{t}\rho(t)]=\mathscr{P}\mathscr{L}_{t}\mathscr{P}[\rho(t)]+\mathscr{P}\mathscr{L}_{t}{\cal
G}(t,0)\mathscr{R}[\rho(0)]+\int_{0}^{t}{\hbox{d}}s\
\mathcal{K}(t,s)[\rho(s)]\quad$ (3.13)

with kernel

$\mathcal{K}(t,s)=\mathscr{P}\mathscr{L}_{t}{\cal
G}(t,s)\mathscr{R}\mathscr{L}_{s}\mathscr{P}\,.$ (3.14)

For uncorrelated initial states of the form $\rho(0)=\rho_{+}\otimes\rho_{-}$
the second term on the right-hand side of (3.13) vanishes because
$\mathscr{P}\big{[}\rho(0)\big{]}=\rho(0)$ and so
$\mathscr{R}\big{[}\rho(0)\big{]}=0$.

In what follows we wish to use (3.13) but work only to second order in $H_{\rm
int}(t)$, which means expanding out the kernel $\mathcal{K}(t,s)$ to second
order in $\mathscr{L}_{t}$. At this order we can therefore take ${\cal
G}(t,s)\simeq 1$ in $\mathcal{K}(t,s)$, which becomes

$\mathcal{K}(t,s)\simeq\mathscr{P}\mathscr{L}_{t}\mathscr{R}\mathscr{L}_{s}\mathscr{P}\,,$
(3.15)

and so (3.13) simplifies to

$\displaystyle\mathscr{P}[\partial_{t}\rho(t)]$ $\displaystyle\simeq$
$\displaystyle\mathscr{P}\mathscr{L}_{t}\mathscr{P}[\rho(t)]+\int_{0}^{t}{\hbox{d}}s\
\mathscr{P}\mathscr{L}_{t}\mathscr{R}\mathscr{L}_{s}\mathscr{P}[\rho(s)]\,.$
(3.16)

Writing this out explicitly using the definitions of $\mathscr{P}$,
$\mathscr{R}$ and $\mathscr{L}_{t}$ then gives the more explicit form

$\displaystyle\frac{\partial\sigma}{\partial t}$ $\displaystyle\simeq$
$\displaystyle-i\;{\mathrm{Tr}}_{-}\Bigl{\\{}\Bigl{[}H_{\rm
int}(t),\sigma(t)\otimes\rho_{-}\Bigr{]}\Bigr{\\}}$
$\displaystyle\quad-\int_{0}^{t}{\hbox{d}}s\
{\mathrm{Tr}}_{-}\left\\{\bigg{[}H_{\rm int}(t)\,,\,\Bigl{[}H_{\rm
int}(s)\,,\,\sigma(s)\otimes\rho_{-}\Bigr{]}-{\mathrm{Tr}}_{-}\Bigl{(}\Bigl{[}H_{\rm
int}(s),\sigma(s)\otimes\rho_{-}\Bigr{]}\Bigr{)}\otimes\rho_{-}\bigg{]}\right\\}\,.$

To apply this expression to the hotspot fields expand $H_{\rm int}(t)$ in
terms of a basis of operators with the factorized form

$H_{\rm int}(t)={\cal A}^{\scriptscriptstyle A}(t)\otimes{\cal
B}_{\scriptscriptstyle A}(t)\,,$ (3.18)

where ${\cal A}^{\scriptscriptstyle A}$ acts only in the $\phi$ sector and
${\cal B}_{\scriptscriptstyle A}$ acts only in the $\chi^{a}$ sector. With
this choice (3.16) simplifies to become

$\displaystyle\frac{\partial\sigma}{\partial t}\simeq-i\;\Bigl{[}{\cal
A}_{a}(t)\,,\,\sigma(t)\Bigr{]}{{\langle\langle}\,{\cal
B}^{a}(t)\,{\rangle\rangle}}$ (3.19)
$\displaystyle\qquad\quad+\int_{0}^{t}{\hbox{d}}s\ \bigg{(}\Bigl{[}{\cal
A}^{a}(s)\,\sigma(s)\,,{\cal A}^{b}(t)\Bigr{]}{{\langle\langle}\,{\cal
B}_{b}(t){\cal B}_{a}(s)\,{\rangle\rangle}}+\Bigl{[}{\cal
A}^{b}(t)\,,\sigma(s){\cal A}^{a}(s)\Bigr{]}{{\langle\langle}\,{\cal
B}_{a}(s)\,{\cal B}_{b}(t)\,{\rangle\rangle}}\bigg{)}$
$\displaystyle\qquad\qquad-\int_{0}^{t}{\hbox{d}}s\ \bigg{(}\Bigl{[}{\cal
A}^{a}(s)\sigma(s)\,,{\cal A}^{b}(t)\Bigr{]}+\Bigl{[}{\cal
A}^{b}(t)\,,\sigma(s){\cal
A}^{a}(s)\Bigr{]}\Bigr{]}\bigg{)}{{\langle\langle}\,{\cal
B}_{b}(t)\,{\rangle\rangle}}{{\langle\langle}\,{\cal
B}_{a}(s)\,{\rangle\rangle}}\,.$

where ${{\langle\langle}\,{\cal
O}\,{\rangle\rangle}}:={\mathrm{Tr}}_{-}\,[\rho_{-}\,{\cal
O}]={\mathrm{Tr}}_{-}\,[\varrho_{\beta}{\cal O}]$ is the thermal trace for
operators acting purely in the $\chi$ sector.

For the point-hotspot system the interaction Hamiltonian given in (2.8) has
the form
$H_{\mathrm{int}}(t)=g_{a}\phi(t,\mathbf{0})\otimes\chi^{a}(t,\mathbf{0})+\frac{1}{2}\lambda\phi^{2}(t,\mathbf{0})\otimes{\cal
I}_{-}$ (in the Interaction picture) and so using
${{\langle\langle}\,\chi^{a}\,{\rangle\rangle}}=0$ the second-order Nakajima-
Zwanzig equation reduces to

$\displaystyle\frac{\partial\sigma}{\partial
t}\simeq-\frac{i\lambda}{2}\Bigl{[}\phi^{2}(t,\mathbf{0}),\sigma(t)\Bigr{]}+g_{a}g_{b}\int_{0}^{t}{\hbox{d}}s\
\bigg{(}{{\langle\langle}\,\chi^{b}(t,\mathbf{0})\chi^{a}(s,\mathbf{0})\,{\rangle\rangle}}\Bigl{[}\phi(s,\mathbf{0})\,\sigma(s)\,,\phi(t,\mathbf{0})\Bigr{]}$
(3.20)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+{{\langle\langle}\,\chi^{a}(s,\mathbf{0})\,\chi^{b}(t,\mathbf{0})\,{\rangle\rangle}}\Bigl{[}\phi(t,\mathbf{0})\,,\sigma(s)\phi(s,\mathbf{0})\Bigr{]}\bigg{)}$

We write the thermal correlation function for two $\chi$ fields as

$g_{a}g_{b}{{\langle\langle}\,\chi^{b}(t,\mathbf{0})\chi^{a}(s,\mathbf{0})\,{\rangle\rangle}}=:\tilde{g}^{2}\mathscr{W}(t-s)=-\frac{\tilde{g}^{2}}{4\beta^{2}}\;{\rm
csch}^{2}\left[\frac{\pi}{\beta}(t-s-i\delta)\right]$ (3.21)

where $\delta=0^{+}$ goes to zero at the end of any calculation and the first
equality defines the function $\mathscr{W}(t)$ with
$\tilde{g}^{2}:=\delta^{ab}\,g_{a}g_{b}$ as given in (2.5), while the second
equality uses the explicit form for the ${{\bf x}}^{\prime}\to{{\bf x}}$ limit
of the correlation function given in (2.2.2).

Finally, after a change of integration variables $s\to t-s$ we arrive at the
form for the Nakajima-Zwanzig equation whose properties are explored below:

$\displaystyle\frac{\partial\sigma}{\partial
t}\simeq-\frac{i\lambda}{2}\Bigl{[}\phi^{2}(t,\mathbf{0}),\sigma(t)\Bigr{]}$
(3.22) $\displaystyle\qquad\qquad\qquad+g^{2}\int_{0}^{t}{\hbox{d}}s\
\bigg{(}\mathscr{W}(s)\Bigl{[}\phi(t-s,\mathbf{0})\,\sigma(t-s)\,,\phi(t,\mathbf{0})\Bigr{]}+\mathscr{W}^{\ast}(s)\Bigl{[}\phi(t,\mathbf{0})\,,\sigma(t-s)\phi(t-s,\mathbf{0})\Bigr{]}\bigg{)}\
.$

####  3.1.2 Markovian limit

Since the correlation function $\mathscr{W}(s)$ is sharply peaked about $s=0$
and falls off exponentially fast like $\mathscr{W}(s)\propto e^{-{2\pi
s}/{\beta}}$ for $s\gg\beta$, the integral simplifies if the rest of the
integrand varies more slowly in the region where $\mathscr{W}$ varies quickly.
In such a case the integral is well-approximated by expanding the rest of the
integrand in powers of $s$, using

$\displaystyle\phi(t-s)\sigma(t-s)\simeq\phi(t,\mathbf{0})\,\sigma(t)-s\big{[}\partial_{t}\phi(t,\mathbf{0})\,\sigma(t)+\phi(t,\mathbf{0})\partial_{t}\sigma(t)\big{]}+\ldots$
(3.23)

beneath the integral sign in (3.22). Notice that this assumes both $\phi$ and
$\sigma$ vary slowly, and so its justification requires both that $\sigma(t)$
should be slowly varying and that we work in an effective description that
keeps only those modes of $\phi$ whose energies satisfy $E\ll 1/\beta$. This
becomes relevant when choosing momentum cutoffs for later integrals, since
this Markovian derivation requires $\Lambda\ll 1/\beta$. Part of the
discussion to follow aims to identify more precisely the region of parameter
space for which this Markovian approximation is valid (which we find by asking
when the subleading terms in the series (3.23) are small).

Keeping only the leading-order term in the Taylor-series (3.23) yields the
approximate equation of motion

$\frac{\partial\sigma}{\partial
t}\simeq-\frac{i\lambda}{2}\Bigl{[}\phi^{2}(t,\mathbf{0}),\sigma(t)\Bigr{]}+\tilde{g}^{2}\mathscr{C}\Bigl{[}\phi(t,\mathbf{0})\,\sigma(t)\,,\phi(t,\mathbf{0})\Bigr{]}+\tilde{g}^{2}\mathscr{C}^{*}\Bigl{[}\phi(t,\mathbf{0})\,,\sigma(t)\phi(t,\mathbf{0})\Bigr{]}\,.$
(3.24)

where the coefficient is given by

$\mathscr{C}(t):=\int_{0}^{t}{\hbox{d}}s\,\mathscr{W}(s)$ (3.25)

and the approximate equality requires $t\gg\beta$ so that the integration
includes the strong peaking of $\mathscr{W}(s)$ (with exponential fall-off)
noted above. Although the coefficient $\mathscr{C}(t)$ is in principle a
function of $t$, in practice the narrowly peaked form of $\mathscr{W}(t)$
ensures it approaches a constant exponentially quickly once $t\gg\beta$. The
value of this constant can be evaluated explicitly by taking the upper
integration limit to infinity and evaluating the resulting integral using
expression (3.21) for $\mathscr{W}(t)$:

$\mathscr{C}(t)\simeq\mathscr{C}_{\infty}=\int_{0}^{\infty}{\hbox{d}}s\,\mathscr{W}(s)=-\frac{1}{4\beta^{2}}\int_{0}^{\infty}\frac{{\hbox{d}}s}{\sinh^{2}\big{[}\tfrac{\pi}{\beta}(s-i\delta)\big{]}}\simeq\frac{1}{4\pi\beta}-\frac{i}{4\pi^{2}\beta}\left[\frac{\beta}{\delta}+\mathcal{O}\left(\text{\scalebox{0.85}{$\frac{\delta}{\beta}$}}\right)\right]\,,$
(3.26)

for $t\gg\beta$. The divergence as $\delta\to 0$ is a reflection of the
divergence of the integrand as $s\to 0$.

It is sometimes useful to convert (3.24) to the Schrödinger picture, as is
done by noting that

$e^{-i{\cal H}_{+}t}\frac{\partial\sigma}{\partial t}e^{+i{\cal
H}_{+}t}=\frac{\partial\sigma_{{\scriptscriptstyle S}}}{\partial t}+i[{\cal
H}_{+},\sigma_{{\scriptscriptstyle S}}(t)]\ .$ (3.27)

which follows from the relation (3.4), giving

$\frac{\partial\sigma_{{\scriptscriptstyle S}}}{\partial
t}\simeq-i\Bigl{[}{\cal
H}_{+}+\text{\scalebox{0.85}{$\frac{\lambda}{2}$}}\phi_{{\scriptscriptstyle
S}}^{2}(\mathbf{0}),\sigma_{{\scriptscriptstyle
S}}(t)\Bigr{]}+\tilde{g}^{2}\mathscr{C}\Bigl{[}\phi_{{\scriptscriptstyle
S}}(\mathbf{0})\,\sigma_{{\scriptscriptstyle S}}(t),\phi_{{\scriptscriptstyle
S}}(\mathbf{0})\Bigr{]}+\tilde{g}^{2}\mathscr{C}^{*}\Bigl{[}\phi_{{\scriptscriptstyle
S}}(\mathbf{0})\,,\sigma_{{\scriptscriptstyle S}}(t)\phi_{{\scriptscriptstyle
S}}(\mathbf{0})\Bigr{]}\,.$ (3.28)

Using this in (3.28) finally gives

$\displaystyle\frac{\partial\sigma_{{\scriptscriptstyle S}}}{\partial t}$
$\displaystyle\simeq$ $\displaystyle-i\Bigl{[}{\cal
H}_{+}+\text{\scalebox{0.85}{$\frac{\lambda}{2}$}}\phi_{{\scriptscriptstyle
S}}^{2}(\mathbf{0}),\sigma_{{\scriptscriptstyle S}}(t)\Bigr{]}$
$\displaystyle\
\quad+\frac{\tilde{g}^{2}}{4\pi}\left(\frac{1}{\beta}-\frac{i}{\pi\delta}\right)\Bigl{[}\phi_{{\scriptscriptstyle
S}}(\mathbf{0})\,\sigma_{{\scriptscriptstyle
S}}(t)\,,\phi_{{\scriptscriptstyle
S}}(\mathbf{0})\Bigr{]}+\frac{\tilde{g}^{2}}{4\pi}\left(\frac{1}{\beta}+\frac{i}{\pi\delta}\right)\Bigl{[}\phi_{{\scriptscriptstyle
S}}(\mathbf{0})\,,\sigma_{{\scriptscriptstyle
S}}(t)\,\phi_{{\scriptscriptstyle S}}(\mathbf{0})\Bigr{]}$ $\displaystyle=$
$\displaystyle-i\Bigl{[}{\cal H}_{+}+\text{\scalebox{0.85}{$\frac{\lambda_{\rm
ren}}{2}$}}\phi_{{\scriptscriptstyle
S}}^{2}(\mathbf{0}),\sigma_{{\scriptscriptstyle
S}}(t)\Bigr{]}+\frac{\tilde{g}^{2}}{4\pi\beta}\Bigl{(}\Bigl{[}\phi_{{\scriptscriptstyle
S}}(\mathbf{0})\,\sigma_{{\scriptscriptstyle
S}}(t)\,,\phi_{{\scriptscriptstyle
S}}(\mathbf{0})\Bigr{]}+\Bigl{[}\phi_{{\scriptscriptstyle
S}}(\mathbf{0})\,,\sigma_{{\scriptscriptstyle
S}}(t)\,\phi_{{\scriptscriptstyle S}}(\mathbf{0})\Bigr{]}\Bigr{)}\,,$

which shows that the divergence can be absorbed into the renormalization

$\lambda_{\mathrm{ren}}\ :=\ \lambda-\frac{\tilde{g}^{2}}{2\pi^{2}\delta}\,.$
(3.30)

####  3.1.3 Evolution equation in a field basis

It is easiest to solve an equation like (3.1.2) in a basis that diagonalizes
the interaction Hamiltonian, and in this instance this suggests using a basis
of field eigenstates defined as the basis that diagonalizes the Schrödinger-
picture field operator $\phi(0,{{\bf x}})=\phi_{{\scriptscriptstyle S}}({{\bf
x}})$:

$\phi_{{\scriptscriptstyle S}}({{\bf
x}})\ket{\varphi{\scriptstyle(\cdot)}}=\varphi({{\bf
x}})\ket{\varphi{\scriptstyle(\cdot)}}$ (3.31)

where the eigenvalue $\varphi({{\bf x}})$ is a real-valued function of
position.

The wave-functional of the free vacuum $\ket{\hbox{vac}}$ in this
reprsentation is given by a gaussian Weinberg:1995mt

$\displaystyle\bra{\varphi{\scriptstyle(\cdot)}}\hbox{vac}{\rangle}$
$\displaystyle=$
$\displaystyle\sqrt{\mathcal{N}_{0}}\;\exp\left[-\frac{1}{2}\int{\hbox{d}}^{3}{{\bf
x}}\int d^{3}{{\bf y}}\ \mathcal{E}({{\bf x}},{{\bf y}})\varphi({{\bf
x}})\varphi({{\bf y}})\right]$ $\displaystyle=$
$\displaystyle\sqrt{\mathcal{N}_{0}}\;\exp\left[-\frac{1}{2}\int\frac{{\hbox{d}}^{3}p}{(2\pi)^{3}}\;E_{p}\;\varphi_{{\bf
p}}\varphi_{-{{\bf p}}}\right]$

where the kernel $\mathcal{E}({{\bf x}},{{\bf y}})$ is given by

$\mathcal{E}({{\bf x}},{{\bf y}})=\int\frac{d^{3}{{\bf p}}}{(2\pi)^{3}}\
E_{p}\ e^{i{{\bf p}}\cdot({{\bf x}}-{{\bf y}})}\,,$ (3.33)

where $E_{p}:=|{{\bf p}}|$ and the normalization factor $\mathcal{N}_{0}$ is
determined by using the normalization condition
$\langle\hbox{vac}|\hbox{vac}\rangle=1$.

In the field basis the reduced density matrix is a time-dependent functional
of the basis field configurations $\varphi_{1}({{\bf x}})$ and
$\varphi_{2}({{\bf x}})$, with components

$\sigma_{{\scriptscriptstyle
S}}[\varphi_{1},\varphi_{2};t]:={\langle}\varphi_{1}{\scriptstyle(\cdot)}|\sigma_{{\scriptscriptstyle
S}}(t)|\varphi_{2}{\scriptstyle(\cdot)}{\rangle}$ (3.34)

which the Markovian equation (3.1.2) implies must satisfy

$\displaystyle\frac{\partial\sigma_{{\scriptscriptstyle
S}}[\varphi_{1},\varphi_{2};t]}{\partial t}$ $\displaystyle\simeq$
$\displaystyle-i\bra{\varphi_{1}{\scriptstyle(\cdot)}}\Bigl{[}{\cal
H}_{+},\sigma_{{\scriptscriptstyle
S}}(t)\Bigr{]}\ket{\varphi_{2}{\scriptstyle(\cdot)}}$
$\displaystyle\qquad-\bigg{[}\frac{\tilde{g}^{2}}{4\pi\beta}\big{(}\varphi_{1}(\mathbf{0})-\varphi_{2}(\mathbf{0})\big{)}^{2}+\frac{i\lambda_{\rm
ren}}{2}\big{(}\varphi_{1}^{2}(\mathbf{0})-\varphi^{2}_{2}(\mathbf{0})\big{)}\bigg{]}\sigma_{{\scriptscriptstyle
S}}[\varphi_{1},\varphi_{2};t]\ .$

We henceforth drop the subscript ‘ren’ on the renormalized coupling parameter
$\lambda$. Evaluating the commutator term using

$\bra{\varphi{\scriptstyle(\cdot)}}{\cal H}_{+}\ket{\Psi}\ =\
\frac{1}{2}\int{\hbox{d}}^{3}{{\bf
x}}\bigg{[}-\frac{\delta^{2}}{\delta\varphi({{\bf
x}})^{2}}+\big{|}\boldsymbol{\nabla}\varphi({{\bf
x}})\big{|}^{2}\bigg{]}{\langle}\varphi{\scriptstyle(\cdot)}|\Psi{\rangle}\ ,$
(3.36)

for any state $|\Psi\rangle$, the equation of motion for
$\sigma_{\scriptscriptstyle S}$ finally becomes

$\displaystyle\frac{\partial\sigma_{{\scriptscriptstyle
S}}[\varphi_{1},\varphi_{2};t]}{\partial
t}\simeq-\frac{i}{2}\int{\hbox{d}}^{3}{{\bf
x}}\bigg{[}-\frac{\delta^{2}}{\delta\varphi_{1}({{\bf
x}})^{2}}+\big{|}\boldsymbol{\nabla}\varphi_{1}({{\bf
x}})\big{|}^{2}+\frac{\delta^{2}}{\delta\varphi_{2}({{\bf
x}})^{2}}-\big{|}\boldsymbol{\nabla}\varphi_{2}({{\bf
x}})\big{|}^{2}\bigg{]}\sigma_{{\scriptscriptstyle
S}}[\varphi_{1},\varphi_{2};t]$
$\displaystyle\qquad\qquad\qquad\qquad-\frac{\tilde{g}^{2}}{4\pi\beta}\big{(}\varphi_{1}(\mathbf{0})-\varphi_{2}(\mathbf{0})\big{)}^{2}\sigma_{{\scriptscriptstyle
S}}[\varphi_{1},\varphi_{2};t]-\frac{i\lambda}{2}\big{(}\varphi_{1}^{2}(\mathbf{0})-\varphi^{2}_{2}(\mathbf{0})\big{)}\sigma_{{\scriptscriptstyle
S}}[\varphi_{1},\varphi_{2};t]\ .$

As is easily verified, when $\lambda=\tilde{g}=0$ this equation has as a
solution

$\displaystyle\sigma_{\scriptscriptstyle S}(t,\varphi_{1},\varphi_{2})$
$\displaystyle=$
$\displaystyle\langle\varphi_{1}{\scriptstyle(\cdot)}|\hbox{vac}\rangle\,\langle\hbox{vac}|\varphi_{2}{\scriptstyle(\cdot)}\rangle$
(3.38) $\displaystyle=$ $\displaystyle{\cal
N}_{0}\exp\left(-\frac{1}{2}\int{\hbox{d}}^{3}{{\bf x}}\int{\hbox{d}}^{3}{{\bf
y}}\;\mathcal{E}({{\bf x}}-{{\bf y}})\big{[}\varphi_{1}({{\bf
x}})\varphi_{1}({{\bf y}})+\varphi_{2}({{\bf x}})\varphi_{2}({{\bf
y}})\big{]}\right)$

where the second equality uses (3.1.3) and the kernel $\mathcal{E}({{\bf
x}}-{{\bf y}})$ is as defined in equation (3.33).

###  3.2 Solutions for the reduced density matrix

We next solve eq. (3.1.3) for the $\phi$-sector density matrix in the presence
of the hotspot interactions.

####  3.2.1 Gaussian ansatz

Keeping in mind that the hotspot ‘interactions’ are all bilinear in the fields
we seek solutions to (3.1.3) subject to the more general Gaussian ansatz

$\displaystyle\sigma_{{\scriptscriptstyle S}}[\varphi_{1},\varphi_{2};t]$
$\displaystyle=$ $\displaystyle{\cal
N}(t)\exp\bigg{(}-\frac{1}{2}\int{\hbox{d}}^{3}{{\bf
x}}\int{\hbox{d}}^{3}{{\bf y}}\;\bigg{\\{}\mathcal{A}_{1}({{\bf x}},{{\bf
y}};t)\,\varphi_{1}({{\bf x}})\varphi_{1}({{\bf y}})+\mathcal{A}_{2}({{\bf
x}},{{\bf y}};t)\,\varphi_{2}({{\bf x}})\varphi_{2}({{\bf y}})$ (3.39)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\quad\quad+2{\cal
B}({{\bf x}},{{\bf y}};t)\,\varphi_{1}({{\bf x}})\varphi_{2}({{\bf
y}})\bigg{\\}}\bigg{)}\,,$

with the kernels ${\cal A}_{1}$, ${\cal A}_{2}$ and ${\cal B}$ to be
determined. Note that we can without loss of generality assume the symmetry

${\cal A}_{j}({{\bf x}},{{\bf y}};t)={\cal A}_{j}({{\bf y}},{{\bf x}};t)\ ,$
(3.40)

and that hermiticity of the reduced density matrix –
$\sigma_{{\scriptscriptstyle
S}}^{\ast}[\varphi_{1},\varphi_{2};t]=\sigma_{{\scriptscriptstyle
S}}[\varphi_{2},\varphi_{1};t]$ – implies ${\cal N}^{\ast}(t)={\cal N}(t)$ and

${\cal A}_{1}^{\ast}({{\bf x}},{{\bf y}};t)={\cal A}_{2}({{\bf x}},{{\bf
y}};t)\quad\hbox{and}\quad{\cal B}^{\ast}({{\bf x}},{{\bf y}};t)={\cal
B}({{\bf y}},{{\bf x}};t)\,.$ (3.41)

Notice that $\sigma_{\scriptscriptstyle S}=|\Psi\rangle\,\langle\Psi|$ for a
gaussian pure state
$\langle\varphi{\scriptstyle(\cdot)}|\Psi\rangle\propto\exp\left[-\frac{1}{2}\,K({{\bf
x}},{{\bf y}})\,\varphi({{\bf x}})\varphi({{\bf y}})\right]$ only if

${\cal A}_{1}({{\bf x}},{{\bf y}})={\cal A}^{*}_{2}({{\bf x}},{{\bf
y}})=K({{\bf x}},{{\bf y}})\quad\hbox{and}\quad{\cal B}({{\bf x}},{{\bf
y}})=0\,.$ (3.42)

with the free vacuum (3.38) corresponding to the choice $K({{\bf x}},{{\bf
y}})={\cal E}({{\bf x}}-{{\bf y}})$. Since the first of these is an automatic
consequence of (3.41), this shows that ${\cal A}_{1}$ and ${\cal A}_{2}$ can
be regarded as the deformations of the ground state away from the free result
due to the interactions, while having ${\cal B}\neq 0$ corresponds to the
interaction causing the initially pure state to become mixed.

The kernels are obtained by plugging the ansatz (3.39) into (3.1.3) and
equating the coefficients of the different functional forms on both sides of
the equation. The details are worked out in Appendix A.1, with the results
simply quoted here. Equating the coefficients of terms independent of
$\varphi_{i}$ implies

$\frac{1}{{\cal N}}\frac{\partial{\cal N}}{\partial t}\ =\
-\frac{i}{2}\int{\hbox{d}}^{3}{{\bf x}}\Bigl{[}{\cal A}_{1}({{\bf x}},{{\bf
x}};t)-{\cal A}_{2}({{\bf x}},{{\bf x}};t)\Bigr{]}\,.$ (3.43)

This expression can also be derived from the condition that
${\mathrm{Tr}}_{+}\sigma_{\scriptscriptstyle S}(t)=1$ for all times. The
coefficient of $\varphi_{1}({{\bf x}})\varphi_{1}({{\bf y}})$ similarly gives

$\displaystyle\frac{\partial{\cal A}_{1}({{\bf x}},{{\bf y}};t)}{\partial t}$
$\displaystyle=$ $\displaystyle-i\nabla_{{\bf x}}^{2}\delta^{3}({{\bf
x}}-{{\bf
y}})+\left(\frac{\tilde{g}^{2}}{2\pi\beta}+i\lambda\right)\delta^{3}({{\bf
x}})\delta^{3}({{\bf y}})$
$\displaystyle\quad\quad\quad\quad\quad\quad+\int{\hbox{d}}^{3}{{\bf
z}}\;\Bigl{[}-i{\cal A}_{1}({{\bf z}},{{\bf x}};t){\cal A}_{1}({{\bf z}},{{\bf
y}};t)+i{\cal B}({{\bf x}},{{\bf z}};t){\cal B}({{\bf y}},{{\bf
z}};t)\Bigr{]}\,,$

while the coefficient of $\varphi_{2}({{\bf x}})\varphi_{2}({{\bf y}})$ leads
to

$\displaystyle\frac{\partial{\cal A}_{2}({{\bf x}},{{\bf y}};t)}{\partial t}$
$\displaystyle=$ $\displaystyle i\nabla_{{\bf x}}^{2}\delta^{3}({{\bf
x}}-{{\bf
y}})+\left(\frac{\tilde{g}^{2}}{2\pi\beta}-i\lambda\right)\delta^{3}({{\bf
x}})\delta^{3}({{\bf y}})$
$\displaystyle\quad\quad\quad\quad\quad\quad+\int{\hbox{d}}^{3}{{\bf
z}}\;\Bigl{[}i{\cal A}_{2}({{\bf x}},{{\bf z}};t){\cal A}_{2}({{\bf y}},{{\bf
z}};t)-i{\cal B}({{\bf z}},{{\bf x}};t){\cal B}({{\bf z}},{{\bf
y}};t)\Bigr{]}\,.$

Finally, the coefficient of $\varphi_{1}({{\bf x}})\varphi_{2}({{\bf y}})$
gives

$\displaystyle\frac{\partial{\cal B}({{\bf x}},{{\bf y}};t)}{\partial t}$
$\displaystyle=$
$\displaystyle-\frac{\tilde{g}^{2}}{2\pi\beta}\delta^{3}({{\bf
x}})\delta^{3}({{\bf y}})$
$\displaystyle\quad\quad\quad\quad\quad\quad+\int{\hbox{d}}^{3}{{\bf
z}}\;\Bigl{[}-i{\cal A}_{1}({{\bf z}},{{\bf x}};t){\cal B}({{\bf z}},{{\bf
y}};t)+i{\cal B}({{\bf x}},{{\bf z}};t){\cal A}_{2}({{\bf z}},{{\bf
y}};t)\Bigr{]}\,.$

The implications of these equations are simpler to see in momentum space, so
we define

$\displaystyle{\cal A}_{j}({{\bf x}},{{\bf y}};t)$ $\displaystyle=$
$\displaystyle\int\frac{{\hbox{d}}^{3}{{\bf
k}}}{(2\pi)^{3}}\int\frac{{\hbox{d}}^{3}{{\bf q}}}{(2\pi)^{3}}\;A_{j}({{\bf
k}},{{\bf q}};t)\;e^{+i{{\bf k}}\cdot{{\bf x}}}\,e^{-i{{\bf q}}\cdot{{\bf
y}}}$ (3.47) $\displaystyle\hbox{and}\quad{\cal B}({{\bf x}},{{\bf y}};t)$
$\displaystyle=$ $\displaystyle\int\frac{{\hbox{d}}^{3}{{\bf
k}}}{(2\pi)^{3}}\int\frac{{\hbox{d}}^{3}{{\bf q}}}{(2\pi)^{3}}\;B({{\bf
k}},{{\bf q}};t)\;e^{+i{{\bf k}}\cdot{{\bf x}}}\,e^{-i{{\bf q}}\cdot{{\bf
y}}}\,,$

for which the symmetry ${\cal A}_{j}({{\bf x}},{{\bf y}};t)={\cal A}_{j}({{\bf
y}},{{\bf x}};t)$ of (3.40) implies

$\displaystyle A_{j}({{\bf k}},{{\bf q}};t)\ =\ A_{j}(-{{\bf q}},-{{\bf
k}};t)\ .$ (3.48)

In terms of these equation (3.43) becomes

$\displaystyle\frac{1}{{\cal N}}\frac{\partial{\cal N}}{\partial t}\ =\
-\frac{i}{2}\int\frac{{\hbox{d}}^{3}{{\bf
k}}}{(2\pi)^{3}}\;\Bigl{[}A_{1}({{\bf k}},{{\bf k}};t)-A_{2}({{\bf k}},{{\bf
k}};t)\Bigr{]}$ (3.49)

while equations (3.2.1) through (3.2.1) become

$\displaystyle\frac{\partial A_{1}({{\bf k}},{{\bf q}};t)}{\partial t}$
$\displaystyle=$ $\displaystyle i(2\pi)^{3}|{{\bf k}}|^{2}\delta^{3}({{\bf
k}}-{{\bf q}})+\left(\frac{\tilde{g}^{2}}{2\pi\beta}+i\lambda\right)$
$\displaystyle\quad\quad\quad+\int\frac{{\hbox{d}}^{3}{{\bf
p}}}{(2\pi)^{3}}\Bigl{[}-iA_{1}({{\bf k}},{{\bf p}};t)A_{1}({{\bf p}},{{\bf
q}};t)+iB({{\bf k}},{{\bf p}};t)B(-{{\bf q}},-{{\bf p}};t)\Bigr{]}\,,$
$\displaystyle\frac{\partial A_{2}({{\bf k}},{{\bf q}};t)}{\partial t}$
$\displaystyle=$ $\displaystyle-i(2\pi)^{3}|{{\bf k}}|^{2}\delta^{3}({{\bf
k}}-{{\bf q}})+\left(\frac{\tilde{g}^{2}}{2\pi\beta}-i\lambda\right)$
$\displaystyle\quad\quad\quad+\int\frac{{\hbox{d}}^{3}{{\bf
p}}}{(2\pi)^{3}}\Bigl{[}iA_{2}({{\bf k}},{{\bf p}};t)A_{2}({{\bf p}},{{\bf
q}};t)-iB(-{{\bf p}},-{{\bf k}};t)B({{\bf p}},{{\bf q}};t)\Bigr{]}$

and

$\frac{\partial B({{\bf p}},{{\bf q}};t)}{\partial
t}=-\frac{\tilde{g}^{2}}{2\pi\beta}+\int\frac{{\hbox{d}}^{3}{{\bf
k}}}{(2\pi)^{3}}\Bigl{[}-iA_{1}({{\bf p}},{{\bf k}};t)B({{\bf k}},{{\bf
q}};t)+iB({{\bf p}},{{\bf k}};t)A_{2}({{\bf k}},{{\bf q}};t)\Bigr{]}\,.$
(3.52)

These are the equations we solve in the next few sections. Notice in
particular that (3.2.1) or (3.52) implies $\tilde{g}^{2}/\beta\neq 0$ is an
obstruction to $B({{\bf q}},{{\bf x}},t)=0$ being a solution.

####  3.2.2 Perturbative solution

As is easily verified, for $\lambda=\tilde{g}=0$ these above equations are
solved by

$B({{\bf p}},{{\bf q}},t)=0\quad\hbox{and}\quad A_{j}({{\bf k}},{{\bf q}})\ =\
(2\pi)^{3}|{{\bf k}}|\;\delta^{3}({{\bf k}}-{{\bf q}})\,,$ (3.53)

corresponding to the vacuum solution of (3.38). We next seek solutions that
are perturbatively close to this vacuum solution, as should be possible for
small $\tilde{g}$ and $\lambda$.

To this end we write

$A_{j}({{\bf k}},{{\bf q}};t)=(2\pi)^{3}|{{\bf k}}|\;\delta^{3}({{\bf
k}}-{{\bf q}})+{\mathfrak{a}}_{j}({{\bf k}},{{\bf
q}};t)+\ldots\quad\hbox{and}\quad B({{\bf k}},{{\bf
q}};t)={\mathfrak{b}}({{\bf k}},{{\bf q}};t)+\ldots\,,$ (3.54)

and linearize eqs. (3.49) through (3.52) in the perturbations
${\mathfrak{a}}_{j}$ and ${\mathfrak{b}}$. The resulting evolution equations
decouple, to become

$\frac{\partial{\mathfrak{a}}_{1}({{\bf k}},{{\bf q}};t)}{\partial
t}=\frac{\tilde{g}^{2}}{2\pi\beta}+i\lambda-i(|{{\bf k}}|+|{{\bf
q}}|)\,{\mathfrak{a}}_{1}({{\bf k}},{{\bf q}};t)\,,$ (3.55)
$\frac{\partial{\mathfrak{a}}_{2}({{\bf k}},{{\bf q}};t)}{\partial
t}=\frac{\tilde{g}^{2}}{2\pi\beta}-i\lambda+i(|{{\bf k}}|+|{{\bf
q}}|)\,{\mathfrak{a}}_{2}({{\bf k}},{{\bf q}};t)$ (3.56)

and

$\frac{\partial{\mathfrak{b}}({{\bf k}},{{\bf q}};t)}{\partial
t}=-\frac{\tilde{g}^{2}}{2\pi\beta}-i\big{(}|{{\bf k}}|-|{{\bf
q}}|\big{)}\,{\mathfrak{b}}({{\bf k}},{{\bf q}};t)\,.$ (3.57)

These are to be solved subject to the initial conditions

${\mathfrak{a}}_{1}({{\bf k}},{{\bf q}};0)={\mathfrak{a}}_{2}({{\bf k}},{{\bf
q}};0)={\mathfrak{b}}({{\bf k}},{{\bf q}};0)=0\,,$ (3.58)

since the scalar $\phi$ is starts off in its vacuum state.

The solutions to these initial-value problems are given by

${\mathfrak{a}}_{1}({{\bf k}},{{\bf
q}};t)=\left(\lambda-\frac{i\tilde{g}^{2}}{2\pi\beta}\right)\frac{1-e^{-i\big{(}|{{\bf
k}}|+|{{\bf q}}|\big{)}t}}{|{{\bf k}}|+|{{\bf q}}|}\,,$ (3.59)
${\mathfrak{a}}_{2}({{\bf k}},{{\bf
q}};t)=\left(\lambda+\frac{i\tilde{g}^{2}}{2\pi\beta}\right)\frac{1-e^{+i\big{(}|{{\bf
k}}|+|{{\bf q}}|\big{)}t}}{|{{\bf k}}|+|{{\bf q}}|}$ (3.60)

and

${\mathfrak{b}}({{\bf k}},{{\bf
q}};t)=\left(\frac{i\tilde{g}^{2}}{2\pi\beta}\right)\frac{1-e^{-i(|{{\bf
k}}|-|{{\bf q}}|)t}}{|{{\bf k}}|-|{{\bf q}}|}\,.$ (3.61)

In the limit ${{\bf k}}={{\bf q}}$ (or in the limit of small $t$) this last
solution simplifies to

${\mathfrak{b}}({{\bf k}},{{\bf
q}};t)\to-\frac{\tilde{g}^{2}t}{2\pi\beta}\qquad\hbox{when }\quad{{\bf
k}}\to{{\bf q}}\,.$ (3.62)

As remarked earlier — and explored in more detail in §3.4 — nonzero
${\mathfrak{b}}$ is a signature of $\sigma_{\scriptscriptstyle S}$ becoming a
mixed state, and so (3.61) shows that this only happens for nonzero
$\tilde{g}^{2}/\beta$. Furthermore when $\tilde{g}^{2}/\beta$ is nonzero there
can be no static solution with $\partial_{t}{\mathfrak{b}}=0$, (and in
particular no solution with ${\mathfrak{b}}=0$) and $|{\mathfrak{b}}({{\bf
k}},{{\bf k}};t)|$ monotonically increases. The coupling $\lambda$, by
contrast, just deforms the ground state but leaves it pure.

The normalization ${\cal N}(t)$ is found in a similar way. Using the above
solution in (3.49) allows it to be written

$\displaystyle\frac{\partial_{t}{\cal N}}{{\cal N}}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\int\frac{{\hbox{d}}^{3}{{\bf
k}}}{(2\pi)^{3}}\Bigl{[}{\mathfrak{a}}_{1}({{\bf k}},{{\bf
k}};t)-{\mathfrak{a}}_{2}({{\bf k}},{{\bf k}};t)\Bigr{]}$ (3.63)
$\displaystyle=$ $\displaystyle\int\frac{{\hbox{d}}^{3}{{\bf
k}}}{(2\pi)^{3}|{{\bf k}}|}\bigg{[}\lambda\cos(|{{\bf k}}|t)\sin(|{{\bf
k}}|t)-\frac{\tilde{g}^{2}}{2\pi\beta}\sin^{2}(|{{\bf k}}|t)\bigg{]}\,.$

The integral on the right-hand side diverges in the ultraviolet, which we
regulate using a momentum cutoff $|{{\bf k}}|<\Lambda$, leading to the result

${\cal
N}(t)=\exp\left(C_{0}-\frac{\tilde{g}^{2}\Lambda^{2}t}{16\pi^{3}\beta}-\frac{\lambda}{16\pi^{2}t}\sin(2\Lambda
t)+\frac{\tilde{g}^{2}}{16\pi^{3}\beta t}\sin^{2}(\Lambda t)\right)\,,$ (3.64)

where $C_{0}$ is the integration constant. In terms of the initial condition
${\cal N}(0)={\cal N}_{0}$, where ${\cal N}_{0}$ is the normalization of the
free-vacuum state, we have

${\cal N}_{0}=\exp\left(C_{0}-\frac{\lambda\Lambda}{8\pi^{2}}\right)\,,$
(3.65)

and so

${\cal N}(t)={\cal
N}_{0}\;\exp\bigg{\\{}\frac{\Lambda}{8\pi^{2}}\bigg{[}-\frac{\tilde{g}^{2}\Lambda
t}{2\pi\beta}\left(1-\frac{\sin^{2}(\Lambda t)}{(\Lambda
t)^{2}}\right)+\lambda\left(1-\frac{\sin\left(2\Lambda t\right)}{2\Lambda
t}\right)\bigg{]}\bigg{\\}}\ .$ (3.66)

The significance of the divergences in the time-dependence of ${\cal N}(t)$ is
discussed further in §3.4.

###  3.3 Equal-time $\phi$-correlator

This section uses the reduced density matrix for $\phi$ computed in the
previous section to calculate the $\langle\phi\,\phi\rangle$ two-point
function. Comparison of the result with the Wightman function given in §2
provides a check on the domain of validity of the Nakajima-Zwanzig late-time
evolution.

The correlator evaluated at $t=t^{\prime}$ is convenient because it has a
simple representation in terms of the reduced density matrix. This is because
the only evolution operators that appear are the ones that convert between
Heisenberg, interaction and Schrödinger pictures:

$\mathrm{Tr}\Bigl{[}{\phi}_{\scriptscriptstyle H}(t,{{\bf
x}})\,{\phi}_{\scriptscriptstyle H}(t,{{\bf
x}}^{\prime})\,\rho_{0}\Bigr{]}={\mathrm{Tr}}_{+}\Bigl{[}{\phi}(t,{{\bf
x}})\,{\phi}(t,{{\bf
x}}^{\prime})\,\sigma(t)\Bigr{]}={\mathrm{Tr}}_{+}\Bigl{[}{\phi}_{{\scriptscriptstyle
S}}({{\bf x}})\,{\phi}_{{\scriptscriptstyle S}}({{\bf
x}}^{\prime})\,\sigma_{{\scriptscriptstyle S}}(t)\Bigr{]}\,,$ (3.67)

where ${\mathrm{Tr}}_{+}$ in the second two terms denotes a trace only over
the $\phi$ sector of the Hilbert space.

The calculation based on the field-representation of the Schrödinger-picture
reduced density matrix evaluates the trace using a partition of unity written
as a functional integral over the field eigenstates,

$\displaystyle{\mathrm{Tr}}_{+}\Bigl{[}\phi_{{\scriptscriptstyle
S}}(\mathbf{x})\phi_{{\scriptscriptstyle
S}}(\mathbf{x}^{\prime})\sigma_{{\scriptscriptstyle S}}(t)\Bigr{]}$
$\displaystyle=$ $\displaystyle\int{\cal
D}\varphi\;\bra{\varphi{\scriptstyle(\cdot)}}\phi_{{\scriptscriptstyle
S}}(\mathbf{x})\,\phi_{{\scriptscriptstyle
S}}(\mathbf{x}^{\prime})\,\sigma_{{\scriptscriptstyle
S}}(t)|\varphi{\scriptstyle(\cdot)}\rangle$ $\displaystyle=$
$\displaystyle\int{\cal
D}\varphi\;\varphi(\mathbf{x})\,\varphi(\mathbf{x}^{\prime})\,\sigma_{{\scriptscriptstyle
S}}[\varphi,\varphi;t]\,,$

with $\sigma_{\scriptscriptstyle S}[\varphi_{1},\varphi_{2};t]$ given in terms
of the kernels ${\cal A}_{j}$ and ${\cal B}$ as in (3.39). Our focus here is
in particular on the $\lambda$ and $\tilde{g}$ dependent parts of the density
matrix, since Appendix A.3 verifies that the above functional integral
correctly reproduces the free-field Wightman function inasmuch as it shows
that in the limit $\lambda=\tilde{g}=0$ eq. (3.3) reproduces the usual
expression

$\bra{\mathrm{vac}}\phi(t,{{\bf x}})\phi(t,{{\bf
x}}^{\prime})\ket{\mathrm{vac}}=\frac{1}{4\pi^{2}|{{\bf x}}-{{\bf
x}}^{\prime}|^{2}}\,.$ (3.69)

Eq. (3.3) shows that only the diagonal part of the reduced density matrix is
required to compute the equal-time Wightman function. Explicitly, this is
given by

$\sigma_{{\scriptscriptstyle S}}[\varphi,\varphi;t]={\cal
N}(t)\exp\bigg{(}-\frac{1}{2}\int{\hbox{d}}^{3}{{\bf
x}}\int{\hbox{d}}^{3}{{\bf x}}^{\prime}\;{\cal M}({{\bf x}},{{\bf
x}}^{\prime};t)\varphi({{\bf x}})\varphi({{\bf x}}^{\prime})\bigg{)}\,,$
(3.70)

where

${\cal M}({{\bf x}},{{\bf
x}}^{\prime};t):=2\mathrm{Re}\Bigl{[}\mathcal{A}_{1}({{\bf x}},{{\bf
x}}^{\prime};t)+{\cal B}({{\bf x}},{{\bf y}};t)\Bigr{]}\,,$ (3.71)

and we use the symmetry (3.41). Evaluating the gaussian integrals then implies

$\displaystyle{\mathrm{Tr}}_{+}[\phi_{{\scriptscriptstyle
S}}(\mathbf{x})\phi_{{\scriptscriptstyle
S}}(\mathbf{x}^{\prime})\sigma_{{\scriptscriptstyle S}}(t)]$ $\displaystyle=$
$\displaystyle{\cal N}(t)\int{\cal
D}\varphi\;\varphi(\mathbf{x})\varphi(\mathbf{x}^{\prime})\;e^{-\frac{1}{2}\int\mathrm{d}^{3}{{\bf
z}}_{1}\int\rm{d}^{3}{{\bf z}}_{2}\;{\cal M}({{\bf
z}}_{1},\mathbf{z}_{2};t)\varphi({{\bf z}}_{1})\varphi({{\bf z}}_{2})}$ (3.72)
$\displaystyle=$ $\displaystyle{\cal M}^{-1}({{\bf x}},{{\bf
x}}^{\prime};t)\,,$

where ${\cal M}^{-1}({{\bf x}},{{\bf x}}^{\prime};t)$ is the inverse of ${\cal
M}({{\bf x}},{{\bf x}}^{\prime};t)$, in the sense that
$\int{\hbox{d}}^{3}{{\bf z}}\;{\cal M}^{-1}({{\bf x}},{{\bf z}};t){\cal
M}({{\bf z}},{{\bf x}}^{\prime};t)=\delta^{3}({{\bf x}}-{{\bf x}}^{\prime})$.
The components ${\cal M}^{-1}({{\bf x}},{{\bf x}}^{\prime})$ are computed
explicitly in Appendix A.2 using (3.71) and the perturbative solutions for
${\cal A}_{j}$ and ${\cal B}$ given earlier. Using the result in (3.72) then
gives

$\displaystyle{\mathrm{Tr}}_{+}[\phi_{{\scriptscriptstyle
S}}(\mathbf{x})\phi_{{\scriptscriptstyle
S}}(\mathbf{x}^{\prime})\sigma_{{\scriptscriptstyle S}}(t)]$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}|{{\bf x}}-{{\bf
x}}^{\prime}|^{2}}-\frac{\lambda}{16\pi^{3}(|{{\bf x}}|^{2}-|{{\bf
x}}^{\prime}|^{2})}\bigg{[}\frac{1}{|{{\bf x}}^{\prime}|}\,\Theta(t-|{{\bf
x}}^{\prime}|)-\frac{1}{|{{\bf x}}|}\,\Theta(t-|{{\bf x}}|)\bigg{]}$ (3.73)
$\displaystyle\qquad\qquad\qquad\qquad+\frac{\tilde{g}^{2}}{32\pi^{3}\beta|{{\bf
x}}||{{\bf x}}^{\prime}|}\;\Theta(t-|{{\bf x}}|)\,\delta(|{{\bf x}}|-|{{\bf
x}}^{\prime}|)\,.$

Specializing to the forward light cone of the switch-on of couplings — i.e. to
$t>|{{\bf x}}|$ and $t>|{{\bf x}}^{\prime}|$ — the above becomes

${\mathrm{Tr}}_{+}[\phi_{{\scriptscriptstyle
S}}(\mathbf{x})\phi_{{\scriptscriptstyle S}}({{\bf
x}}^{\prime})\sigma_{{\scriptscriptstyle S}}(t)]\simeq\frac{1}{4\pi^{2}|{{\bf
x}}-{{\bf x}}^{\prime}|^{2}}-\frac{\lambda}{16\pi^{3}|{{\bf x}}|\;|{{\bf
x}}^{\prime}|(|{{\bf x}}|+|{{\bf
x}}^{\prime}|)}+\frac{\tilde{g}^{2}}{32\pi^{3}\beta|{{\bf x}}||{{\bf
x}}^{\prime}|}\;\delta(|{{\bf x}}|-|{{\bf x}}^{\prime}|)\ .$ (3.74)

#### Comparison with Heisenberg picture

Expression (3.74) is to be compared with the Wightman function computed in
Heisenberg picture, whose expansion at linear order in $\lambda$ and
$\tilde{g}^{2}$ is quoted in (2.2.1) and whose equal-time limit is given in
(2.2.1), reproduced for convenience here:

$\displaystyle W_{\beta}(t,{{\bf x}};t,{{\bf x}}^{\prime})$
$\displaystyle\simeq$ $\displaystyle\frac{1}{4\pi^{2}|{{\bf x}}-{{\bf
x}}^{\prime}|^{2}}-\frac{\lambda}{16\pi^{3}|{{\bf x}}||{{\bf
x}}^{\prime}|(|{{\bf x}}|+|{{\bf x}}^{\prime}|)}$
$\displaystyle\qquad\qquad-\frac{\tilde{g}^{2}}{64\pi^{2}\beta^{2}|{{\bf
x}}||{{\bf x}}^{\prime}|\sinh^{2}\left[\frac{\pi}{\beta}(|{{\bf x}}|-|{{\bf
x}}^{\prime}|)\right]}+\frac{\tilde{g}^{2}}{16\pi^{4}(|{{\bf x}}|^{2}-|{{\bf
x}}^{\prime}|^{2})^{2}}\,.$

Although the free and the $\lambda$-dependent terms here agree with those in
(3.74), the same is in general not true for the $\tilde{g}^{2}$-dependent
terms. This need not be a problem since they should only be expected to agree
within the domain of validity of both, and (3.74) is derived under more
restrictive assumptions.

Recall in particular that the derivation within the Schrödinger picture
approximated the Nakajima-Zwanzig equation (3.22) using a Markovian limit in
which $\sigma(t-s)\phi(t-s,\mathbf{0})\simeq\sigma(t)\phi(t,\mathbf{0})$ is
used under the integral sign. Doing this assumes both
$\sigma_{\scriptscriptstyle S}$ and $\phi$ vary very slowly on the time-scale
$\beta$ for which the thermal $\chi^{a}$ correlator was sharply peaked (see
the discussion surrounding eq. (3.23)). This is only valid if the UV cutoff,
$\Lambda$, for the $\phi$-field modes satisfies $\beta\Lambda\ll 1$, since
only in this EFT is the field $\phi$ sufficiently slowly varying.

Within this regime the term involving the hyperbolic function has a
microscopic width and so approaches a delta function. To see this explicitly
it is easier to work in momentum space, in which case (3.3) is given by the
following mode sum

$\displaystyle W_{\beta}(t,{{\bf x}};t,{{\bf x}}^{\prime})$
$\displaystyle\simeq$ $\displaystyle\frac{1}{4\pi^{2}|{{\bf x}}-{{\bf
x}}^{\prime}|}\int_{0}^{\infty}{\hbox{d}}p\;\sin(p|{{\bf x}}-{{\bf
x}}^{\prime}|)-\frac{\lambda}{16\pi^{3}|{{\bf x}}||{{\bf
x}}^{\prime}|}\int_{0}^{\infty}{\hbox{d}}p\;e^{-ip|{{\bf
x}}^{\prime}|}\sin\big{[}p(|{{\bf x}}|+|{{\bf x}}^{\prime}|)\big{]}$ (3.76)
$\displaystyle\qquad+\frac{\tilde{g}^{2}}{64\pi^{4}|{{\bf x}}||{{\bf
x}}^{\prime}|}\int_{0}^{\infty}{\hbox{d}}p\;p\left\\{\;\cos\big{[}p(|{{\bf
x}}|+|{{\bf x}}^{\prime}|)\big{]}+\frac{2\cos\big{[}p(|{{\bf x}}|-|{{\bf
x}}^{\prime}|)\big{]}}{e^{\beta p}-1}\;\right\\}\,.$

Because $p<\Lambda\ll\beta^{-1}$ we can expand the temperature-dependent last
term using $\beta p\ll 1$,

$\frac{\tilde{g}^{2}}{64\pi^{4}|{{\bf x}}||{{\bf
x}}^{\prime}|}\int_{0}^{\infty}{\hbox{d}}p\;p\left\\{\cos\big{[}p(|{{\bf
x}}|+|{{\bf x}}^{\prime}|)\big{]}+\bigg{[}\frac{2}{\beta p}-1+\frac{\beta
p}{6}+{\cal O}(p^{3}\beta^{3})\bigg{]}\cos\big{[}p(|{{\bf x}}|-|{{\bf
x}}^{\prime}|)\big{]}\;\right\\}\,,$ (3.77)

and perform the momentum integrals term-by-term, giving

$\frac{\tilde{g}^{2}}{64\pi^{4}\beta|{{\bf x}}||{{\bf
x}}^{\prime}|}\left\\{-\frac{1}{(|{{\bf x}}|+|{{\bf
x}}^{\prime}|)^{2}}+\bigg{[}\frac{2\pi}{\beta}\delta(|{{\bf x}}|-|{{\bf
x}}^{\prime}|)+\frac{1}{(|{{\bf x}}|-|{{\bf
x}}^{\prime}|)^{2}}-\frac{\beta}{6}\delta^{\prime\prime}(|{{\bf x}}|-|{{\bf
x}}^{\prime}|)+\ldots\bigg{]}\right\\}\,.$ (3.78)

Using this in the Wightman function gives

$\displaystyle W_{\beta}(t,{{\bf x}};t,{{\bf x}}^{\prime})$
$\displaystyle\simeq$ $\displaystyle\frac{1}{4\pi^{2}|{{\bf x}}-{{\bf
x}}^{\prime}|^{2}}-\frac{\lambda}{16\pi^{3}|{{\bf x}}||{{\bf
x}}^{\prime}|\big{(}|{{\bf x}}|+|{{\bf x}}^{\prime}|\big{)}}$
$\displaystyle\qquad\qquad\quad+\frac{\tilde{g}^{2}}{32\pi^{3}\beta|{{\bf
x}}||{{\bf x}}^{\prime}|}\,\delta\big{(}|{{\bf x}}|-|{{\bf
x}}^{\prime}|\big{)}+\frac{\tilde{g}^{2}}{16\pi^{4}\big{(}|{{\bf
x}}|^{2}-|{{\bf x}}^{\prime}|^{2}\big{)}^{2}}+{\cal O}(\beta)\,.$

Notice that the leading term in this expansion indeed matches the Schrodinger-
picture calculation. This comparison reveals more explicitly the long-
wavelength domain of validity inherent in the Markovian limit,555At least in
the way it is derived here. A Markovian limit with a broader domain of
validity might also be possible, such as if only $\sigma_{\scriptscriptstyle
S}(t-s)$ is expanded in powers of $s$ without also expanding $\phi(t-s)$.
Examples along these lines are seen in some simpler examples involving qubits
interacting with fields Kaplanek:2019vzj . revealing it to depend on both an
expansion in powers of $\tilde{g}^{2}$ and on taking long wavelengths compared
to the thermal length scale $\beta$.

###  3.4 Decoherence

Earlier sections explore the behaviour of the $\phi$-field state by computing
its reduced density matrix once the hotspot fields $\chi^{a}$ are all
integrated out. It was there argued that the kernel ${\cal B}({{\bf x}},{{\bf
x}}^{\prime};t)$ — or equivalently ${\mathfrak{b}}({{\bf k}},{{\bf q}};t)$ —
provides a diagnostic of whether the state is pure or mixed (because pure
states seem to require ${\cal B}={\mathfrak{b}}=0$.)

That is what makes eqs. (3.57) and (3.61) so interesting; they quantify the
rate with which decoherence accumulates when hotspot couplings are turned on
with $\phi$ prepared in its vacuum state. These equations show in particular
the growth of decoherence at early times has a momentum-independent universal
rate, set by $\tilde{g}^{2}/(2\pi\beta)$. Since all momenta start to decohere
with the same rate, the accumulated decoherence should be dominated by the
highest momenta, for which there is the most available phase space. It is for
this reason that measures of integrated decoherence – such as the purity
diagnostic computed explicitly below – tend to diverge in the ultraviolet, at
least so long as the cutoff remains below the characteristic hotspot
temperature, as assumed for the Markovian approximation above.

####  3.4.1 Purity

To pin down the decoherence process more precisely it is useful to have a
practical diagnostic for state purity. A standard choice for this is often
${\mathrm{Tr}}_{+}[\sigma_{\scriptscriptstyle S}^{2}]$ since this is unity if
and only if the state is pure (in which case $\sigma_{\scriptscriptstyle
S}^{2}=\sigma_{\scriptscriptstyle S}$).

This trace can be computed as a function of the state kernels ${\cal A}_{j}$
and ${\cal B}$, as follows:

$\displaystyle\mathrm{Tr}[\sigma_{{\scriptscriptstyle S}}^{2}]$
$\displaystyle=$ $\displaystyle\int{\cal D}\varphi_{1}\int{\cal
D}\varphi_{2}\;\bra{\varphi_{1}}\sigma_{{\scriptscriptstyle
S}}(t)\ket{\varphi_{2}}\bra{\varphi_{2}}\sigma_{{\scriptscriptstyle
S}}(t)\ket{\varphi_{1}}$ $\displaystyle=$ $\displaystyle{\cal
N}(t)^{2}\int{\cal D}\varphi_{1}\int{\cal
D}\varphi_{2}\;\exp\left\\{-\frac{1}{2}\int\mathrm{d}^{3}{{\bf
x}}\int\mathrm{d}^{3}{{\bf y}}\;\bigg{[}2\,\mathrm{Re}\Bigl{[}{\cal
A}_{1}({{\bf x}},{{\bf y}};t)\Bigr{]}\Bigl{(}\varphi_{1}({{\bf
x}})\varphi_{1}({{\bf y}})+\varphi_{2}({{\bf x}})\varphi_{2}({{\bf
y}})\Bigr{)}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+4\,\mathrm{Re}\Bigl{[}{\cal
B}({{\bf x}},{{\bf y}};t)\Bigr{]}\varphi_{1}({{\bf x}})\varphi_{2}({{\bf
y}})\bigg{]}\right\\}$ $\displaystyle=$ $\displaystyle{\cal
N}(t)^{2}\left\\{\det\left[\frac{1}{2\pi}\left(\begin{matrix}2\,\mathrm{Re}[{\cal
A}_{1}(t)]&2\,\mathrm{Re}[{\cal B}(t)]\\\ 2\,\mathrm{Re}[{\cal
B}(t)]&2\,\mathrm{Re}[{\cal
A}_{1}(t)]\end{matrix}\right)\right]\right\\}^{-1/2}\,.$

In the last line we write quantities ${\cal B}({{\bf x}},{{\bf y}})$ as
matrices, with rows and columns labelled by position. In this notation the
matrix $\mathrm{Re}[{\cal B}(t)]$ is symmetric, since the identity ${\cal
B}^{\ast}({{\bf x}},{{\bf y}};t)={\cal B}({{\bf y}},{{\bf x}};t)$ means

$\mathrm{Re}[{\cal B}]({{\bf x}},{{\bf y}};t)=\frac{{\cal B}({{\bf x}},{{\bf
y}};t)+{\cal B}^{\ast}({{\bf x}},{{\bf y}};t)}{2}=\frac{{\cal B}({{\bf
x}},{{\bf y}};t)+{\cal B}({{\bf y}},{{\bf x}};t)}{2}\,,$ (3.81)

a fact that has been used in writing the last line of (3.4.1).

This result can be further simplified using the following identity666To see
this, take the determinant of both sides of $\left[\begin{smallmatrix}I&I\\\
0&I\end{smallmatrix}\right]\left[\begin{smallmatrix}X&Y\\\
Y&X\end{smallmatrix}\right]\left[\begin{smallmatrix}I&-I\\\
0&I\end{smallmatrix}\right]=\left[\begin{smallmatrix}X+Y&0\\\
Y&X-Y\end{smallmatrix}\right]$. that applies for any two square matrices $X$
and $Y$:

$\det\left(\begin{matrix}X&Y\\\ Y&X\end{matrix}\right)=\det(X-Y)\det(X+Y)\,.$
(3.82)

This leads to

$\displaystyle\mathrm{Tr}[\sigma_{{\scriptscriptstyle S}}^{2}]$
$\displaystyle=$ $\displaystyle{\cal
N}(t)^{2}\left\\{\det\left[\frac{1}{\pi}\Bigl{(}\mathrm{Re}[{\cal
A}_{1}(t)]-\mathrm{Re}[{\cal
B}(t)]\Bigr{)}\right]\det\left[\frac{1}{\pi}\Bigl{(}\mathrm{Re}[{\cal
A}_{1}(t)]+\mathrm{Re}[{\cal B}(t)]\Bigr{)}\right]\right\\}^{-1/2}$ (3.83)
$\displaystyle=$
$\displaystyle\frac{\det\left[\frac{1}{\pi}\Bigl{(}\mathrm{Re}[{\cal
A}_{1}]+{\cal
B}(t)\Bigr{)}\right]}{\sqrt{\det\left[\frac{1}{\pi}\Bigl{(}\mathrm{Re}[{\cal
A}_{1}(t)]-\mathrm{Re}[{\cal
B}(t)]\Bigr{)}\right]\det\left[\frac{1}{\pi}\Bigl{(}\mathrm{Re}[{\cal
A}_{1}(t)]+\mathrm{Re}[{\cal B}(t)]\Bigr{)}\right]}}$ (3.84)

which uses

${\cal N}(t)^{2}=\det\left[\frac{1}{\pi}\Bigl{(}\mathrm{Re}[{\cal
A}_{1}]+{\cal B}(t)\Bigr{)}\right]=\det{\cal M}(t)\,.$ (3.85)

Notice that this last relation verifies that ${\cal B}({{\bf x}},{{\bf y}})=0$
implies $\mathrm{Tr}[\sigma_{{\scriptscriptstyle S}}^{2}]=1$, so ${\cal B}=0$
is sufficient to ensure that $\sigma_{{\scriptscriptstyle S}}$ describes a
pure state.

These expressions can be made even more explicit by switching to momentum
space and evaluating them using the above perturbative solution near the free
vacuum. In this case repeated use of the identity $\det=\exp\,\mathrm{Tr}\log$
leads to

$\displaystyle\mathrm{Tr}[\sigma_{{\scriptscriptstyle S}}(t)^{2}]$
$\displaystyle=$ $\displaystyle\exp\left[\int{\hbox{d}}^{3}{{\bf
k}}\;\frac{\pi}{|{{\bf k}}|}\cdot\frac{2\tilde{g}^{2}{\mathfrak{b}}({{\bf
k}},{{\bf k}};t)}{(2\pi)^{4}}+{\cal O}(\tilde{g}^{4})\right]$ $\displaystyle=$
$\displaystyle\exp\left[-\frac{\tilde{g}^{2}t}{16\pi^{4}\beta}\int\frac{{\hbox{d}}^{3}{{\bf
k}}}{|{{\bf k}}|}+{\cal O}(\tilde{g}^{4})\right]$ $\displaystyle=$
$\displaystyle\exp\left[-\frac{\tilde{g}^{2}\Lambda^{2}t}{8\pi^{3}\beta}+{\cal
O}(\tilde{g}^{4})\right]\,,$

revealing the divergence described above. These ultimately arise because the
Markovian derivation required $\Lambda\ll 1/\beta$. Presumably the same
calculation would not have diverged if $\Lambda$ could have been taken larger
than the temperature, though we have not succeeded yet in capturing this
evolution in a fuller calculation.

Notice that the purity starts out at $1$ since the initial vacuum state is
pure, and then drops monotonically as time passes. A strictly perturbative
calculation would only have been able to capture the leading contribution in
powers of $\tilde{g}^{2}$, and so would have given

$\mathrm{Tr}[\sigma_{{\scriptscriptstyle S}}(t)^{2}]\simeq
1-\frac{\tilde{g}^{2}\Lambda^{2}t}{8\pi^{3}\beta}+{\cal
O}(\tilde{g}^{4})\qquad\hbox{(perturbative)}\,,$ (3.87)

with the linear secular growth with $t$ eventually causing the perturbative
calculation to break down. It is the deviation of Nakajima-Zwanzig equation
from straight-up perturbation theory that allows the resummation of the
secularly growing terms to all orders in $\tilde{g}^{2}t$ into the exponential
form visible in (3.4.1), along the lines also seen in simpler examples (such
as in Kaplanek:2019dqu ; Kaplanek:2019vzj ; Kaplanek:2020iay ; Burgess:2015ajz
). This allows us to see that it approaches $0$ for late times, corresponding
to a maximally mixed state (in $D$-dimensional quantum mechanics, a maximally
mixed state has purity $1/D$ and in the present instance $D\to\infty$).

##  4 Mean-field methods

One of our goals is to explore the nature of nonlocality in open systems, in
hopes that this can shed light on whether nonlocality can also arise near
horizons for black holes. Nonlocality in this context traditionally means the
extent to which the effective action (or Hamiltonian) is not simply the
integral of a Hamiltonian density that depends only on fields and their
derivatives at a single point (a property normally held by Wilsonian actions
as a consequence of cluster decomposition and microcausality Weinberg:1995mt
).

Locality is a question for which the language of open quantum systems used
above is less well adapted because it frames its predictions in terms of the
evolution of the reduced density matrix that is obtained after tracing out any
dependence on unmeasured sectors of the Hilbert space, as given by the
Nakajima-Zwanzig equation Nak ; Zwan . Although entanglement and information
exchange ensure this equation naturally involves nonlocality (particularly in
time), the absence of an effective Hamiltonian or action in this approach
makes its lessons relatively obscure.

To help understand the issues that are at play this section sets up the mean-
field approximation for the hotspot system, since the mean-field limit
provides a natural definition of an effective hamiltonian (and effective
action) that can provide a good approximation to the system’s evolution in
some circumstances even for open systems EFTBook . Part of the purpose is to
identify the kinds of nonlocality that can emerge for the mean-field
Hamiltonian, but at the same time also to identify precisely when the mean-
field description is a good approximation to the full dynamics. The hotspot
provides a relatively simple test laboratory for exploring these ideas.

###  4.1 Definitions

The essence of the mean-field approximation is that averages in the unmeasured
(or ‘environment’) part of the open system dominate the fluctuations in this
sector, allowing an informative expansion in powers of small deviations from
the mean. There is some freedom in how to set up this expansion when working
beyond the leading order, which we first summarize before computing how things
look specifically in the hotspot example.

We use, as before, the ‘double-bracket’ notation

${{\langle\langle}\,{\cal
O}\,{\rangle\rangle}}:={\mathrm{Tr}}_{-}\Bigl{[}\rho_{\beta}\,{\cal
O}\Bigr{]}\quad\hbox{where}\quad\rho_{\beta}:={\cal
I}_{+}\otimes\varrho_{\beta}$ (4.1)

with $\varrho_{\beta}$ being the thermal state appearing in (2.10) and the
partial trace running only over the environmental sector of the Hilbert space
(which in the hotspot example is the sector spanned at the initial time by the
$\chi^{a}$ fields). For example, for ${\cal O}=\sum_{n}{\cal
A}_{n}\otimes{\cal B}_{n}$ expanded in terms of a basis of operators acting in
the $\phi$ and $\chi^{a}$ sectors,

${{\langle\langle}\,{\cal O}\,{\rangle\rangle}}=\sum_{n}{\cal
A}_{n}\;{\mathrm{Tr}}_{-}\Bigl{[}\varrho_{\beta}\,{\cal
B}_{n}\Bigr{]}=\sum_{n}{\cal A}_{n}\,\bigl{\langle}{\cal
B}_{n}\bigr{\rangle}_{\beta}\,.$ (4.2)

In particular, ${{\langle\langle}\,{\cal O}\,{\rangle\rangle}}$ is an operator
that acts only in the measured sector (i.e. the $\phi$ sector of the hotspot
example).

####  4.1.1 Mean-field evolution

For a perturbative analysis it is worth specializing to the interaction
picture, for which expectation values for observables evolve according to

${\cal A}(t):=\mathrm{Tr}\Bigl{[}\rho_{\scriptscriptstyle S}(t)\,{\cal
O}_{\scriptscriptstyle S}\Bigr{]}=\mathrm{Tr}\Bigl{[}\rho_{\scriptscriptstyle
I}(t){\cal O}_{\scriptscriptstyle
I}(t)\Bigr{]}=\mathrm{Tr}\Bigl{[}\rho_{\scriptscriptstyle
I}(0)V^{\star}(t){\cal O}_{\scriptscriptstyle I}(t)V(t)\Bigr{]}\,,$ (4.3)

where $V(t)=U_{0}^{\star}(t)U(t)$ is the interaction-picture evolution
operator for the state $\rho_{\scriptscriptstyle I}(t)$. In what follows we
drop the subscript ‘$I$’ for interaction-picture quantities.

For the observables that only measure the $\phi$ sector the Schrödinger
picture operator has the factorized form ${\cal O}_{\scriptscriptstyle
S}={\cal O}_{{\scriptscriptstyle S}+}\otimes{\cal I}_{-}$. This factorization
remains true in the interaction picture provided that the free part of the
Hamiltonian does not couple the two sectors to one another. That is, if
$H=H_{0}+H_{{\rm int}}$ where $H_{0}={\cal H}_{+}\otimes{\cal I}_{-}+{\cal
I}_{+}\otimes{\cal H}_{-}$, then

$U_{0}(t):=\exp\Bigl{[}-iH_{0}t\Bigr{]}=U_{+}(t)\otimes U_{-}(t)\,,$ (4.4)

and so in the interaction picture

${\cal O}(t):=U_{0}^{\star}(t)\,{\cal O}_{\scriptscriptstyle
S}\,U_{0}(t)={\cal O}_{+}(t)\otimes{\cal I}_{-}$ (4.5)

with ${\cal O}_{+}(t)=U_{+}^{\star}(t)\,{\cal O}_{{\scriptscriptstyle
S}+}\,U_{+}(t)$.

Now comes the main point: in the mean-field approximation the interaction-
picture evolution operator is well-approximated within the observable sector
by the operator

$\displaystyle\overline{V}(t)$ $\displaystyle:=$
$\displaystyle{{\langle\langle}\,V(t)\,{\rangle\rangle}}$ $\displaystyle=$
$\displaystyle I-i\int_{0}^{t}{\hbox{d}}\tau\,{{\langle\langle}\,H_{\rm
int}(\tau)\,{\rangle\rangle}}+\frac{1}{2}(-i)^{2}\int_{0}^{t}{\hbox{d}}\tau_{1}\int_{0}^{\tau_{1}}{\hbox{d}}\tau_{2}\,{{\langle\langle}\,H_{\rm
int}(\tau_{1})\,H_{\rm int}(\tau_{2})\,{\rangle\rangle}}+\cdots\,,$

where $H_{\rm int}(t)$ denotes the usual interaction-picture interaction
Hamiltonian $H_{\rm int}(t)=U^{\star}_{0}(t)H_{\rm int}U_{0}(t)$. Writing the
full evolution operator as a mean-field part plus the rest (with the rest here
called the ‘diffuse’ evolution, in analogy to optics)

$V(t)=:\overline{V}(t)\otimes{\cal I}_{-}+{\cal V}(t)\,,$ (4.7)

the mean-field approximation is a good one when contributions of the diffuse
evolution operator, ${\cal V}(t)$ are parametrically small.

The point of defining the mean-field evolution using $\overline{V}(t)$ (as
opposed to simply averaging the Hamiltonian, say) is that this ensures that
for any observable of the form (4.5) there is no cross-interference – to all
orders in perturbation theory – between the mean field evolution and diffuse
evolution. That is, computing the expectation of ${\cal O}(t)$ using (4.7)
gives

$\mathrm{Tr}\Bigl{[}\rho(t)\,{\cal
O}(t)\Bigr{]}=\mathrm{Tr}\Bigl{[}V(t)\rho(0)V^{\star}(t)\,{\cal
O}(t)\Bigr{]}=\mathrm{Tr}_{+}\Bigl{[}\overline{V}(t)\rho_{+}\overline{V}^{\star}(t)\,{\cal
O}_{+}(t)\Bigr{]}+\mathrm{Tr}\Bigl{[}{\cal V}(t)\rho(0)\,{\cal
V}^{\star}(t){\cal O}(t)\Bigr{]}\,,$ (4.8)

with the first trace being only over the measured ($\phi$) sector, and we
assume the uncorrelated initial conditions (2.9) and (2.10). Notice the
absence of any cross terms involving both $\overline{V}$ and ${\cal V}$, which
vanish because (4.1.1) and (4.7) together imply ${{\langle\langle}\,{\cal
V}(t)\,{\rangle\rangle}}=0$.

####  4.1.2 Mean-field Hamiltonian

Given the mean-field evolution, the mean-field interaction Hamiltonian,
$\overline{H}_{\rm int}(t)$, is defined as the operator that generates
$\overline{V}(t)$, or equivalently is related to $\overline{V}(t)$ by the
usual iterative expression

$\overline{V}(t)=I-i\int_{0}^{t}{\hbox{d}}\tau\,\overline{H}_{\rm
int}(\tau)+\frac{1}{2}(-i)^{2}\int_{0}^{t}{\hbox{d}}\tau_{1}\int_{0}^{\tau_{1}}{\hbox{d}}\tau_{2}\,\overline{H}_{\rm
int}(\tau_{1})\,\overline{H}_{\rm int}(\tau_{2})+\cdots\,.$ (4.9)

Comparing this with (4.1.1) we read off

$\overline{H}_{\rm int}(t)={{\langle\langle}\,H_{\rm
int}(t)\,{\rangle\rangle}}-i\int_{0}^{t}{\hbox{d}}\tau\;{{\langle\langle}\,\delta
H_{\rm int}(t)\,\delta H_{\rm int}(\tau)\,{\rangle\rangle}}+\cdots\,,$ (4.10)

where the ellipses represent terms at least third order in $\delta H_{\rm
int}$ and

$\delta H_{\rm int}(t):=H_{\rm int}(t)-{{\langle\langle}\,H_{\rm
int}(t)\,{\rangle\rangle}}\,.$ (4.11)

Notice that starting at second order in $\delta H_{\rm int}$ the mean-field
Hamiltonian need not be hermitian, since $\overline{H}_{\rm
int}(t)={\mathfrak{H}}(t)+i\,{\mathfrak{I}}(t)$ with

$\displaystyle{\mathfrak{H}}(t)$ $\displaystyle=$
$\displaystyle{{\langle\langle}\,H_{\rm
int}(t)\,{\rangle\rangle}}-\frac{i}{2}\int_{0}^{t}{\hbox{d}}\tau\;{{\langle\langle}\,\bigl{[}\delta
H_{\rm int}(t)\,,\delta H_{\rm int}(\tau)\bigr{]}\,{\rangle\rangle}}+\cdots$
$\displaystyle\hbox{and}\quad{\mathfrak{I}}(r)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{0}^{t}{\hbox{d}}\tau\;{{\langle\langle}\,\bigl{\\{}\delta
H_{\rm int}(t)\,,\delta H_{\rm
int}(\tau)\bigr{\\}}\,{\rangle\rangle}}+\cdots\,.$ (4.12)

The failure of unitarity for $\overline{V}(t)$ that arises in this way merely
reflects the relatively artificial nature of the mean-field/diffuse split,
since ‘diffuse’ interactions with fluctuations in the environment can deplete
probability from the mean-field description.

These diffuse interactions are themselves described perturbatively by

${\cal V}(t)\simeq-i\int_{0}^{t}{\hbox{d}}\tau\;\delta H_{\rm
int}(\tau)+\cdots\,,$ (4.13)

where ellipses now represent terms second-order in $H_{\rm int}$. (Recall that
knowing ${\cal V}$ at linear order suffices to compute observables to second
order because expressions like (4.8) depend quadratically on ${\cal V}$.) It
is only the total evolution (with mean-field and diffuse contributions
combined) that must be unitary, and this can be expressed as a generalization
of the optical theorem, relating the imaginary part of $\overline{H}_{\rm
int}$ to the rate of diffuse scattering EFTBook . This makes the relative size
of the imaginary and real parts of $\overline{H}_{\rm int}$ a proxy for the
relative importance of diffuse and mean-field evolution.

####  4.1.3 Domain of validity

Broadly speaking, mean-field descriptions arise as good approximations for
real systems in two common ways, depending on whether or not
${{\langle\langle}\,H_{\rm int}\,{\rangle\rangle}}$ is zero. The simplest case
is when ${{\langle\langle}\,H_{\rm int}\,{\rangle\rangle}}\neq 0$, because
then perturbation theory alone can justify the mean-field approximation. This
can be seen because the leading (linear) contribution in powers of $H_{\rm
int}$ is necessarily a mean-field result (because ${\cal V}$ first enters
expressions like (4.8) at second order). Neutrino interactions with matter
inside the Sun or Earth provide practical examples of this type, for which the
mean-field description follows as a consequence of the extreme feebleness of
the weak interactions Burgess:1996mz ; Bamert:1997jj .

Things are more subtle when ${{\langle\langle}\,H_{\rm
int}\,{\rangle\rangle}}=0$, however, because then the leading contribution of
$\overline{V}(t)$ to eq. (4.8) arises at the same order in $\delta H_{\rm
int}$ as does the diffuse contribution ${\cal V}(t)$. This is typically what
happens for the interactions of photons with transparent dielectric materials,
for example, and in this case perturbation theory in $\delta H_{\rm int}$
itself is insufficient for mean-field methods to dominate.777For photons it is
instead a large-$N$ argument based on coherence that justifies mean-field
methods (see e.g. EFTBook ). This is also the regime appropriate to the
hotspot, and in what follows we identify possible control parameters for mean-
field methods by comparing for the hotspot the relative size of the real and
imaginary parts of $\overline{H}_{\rm int}$.

###  4.2 Application to the hotspot

The above is made concrete by specializing to the specific hotspot
interactions of the previous sections. Consider first the case of non-
negligible hotspot size, $\xi$, where the underlying coupling has the form
given in (2.7):

$H_{\rm int}(t)=\int_{S_{\xi}}{\hbox{d}}^{2}x\;G_{a}\,\chi^{a}(t,{{\bf
x}})\,\phi(t,{{\bf x}})\,,$ (4.14)

with $S_{\xi}$ the 2-sphere of radius $\xi$ centred on the origin. For the
purposes of the present discusion the coupling $G_{\phi}\,\phi^{2}$ can be
regarded as part of the unperturbed Hamiltonian because it does not couple the
$\phi$ and $\chi^{a}$ sectors to one another.

####  4.2.1 Nonlocal mean-field Hamiltonian

Since $\langle\chi^{a}\rangle=0$ in the thermal bath the mean of the
interaction Hamiltonian vanishes, ${{\langle\langle}\,H_{\rm
int}(t)\,{\rangle\rangle}}=0$, and so the leading term in (4.10) is the
second-order contribution, giving

$\overline{H}_{\rm
int}(t)\simeq-i\widetilde{G}^{2}\int_{0}^{t}{\hbox{d}}\tau\int_{S_{\xi}}{\hbox{d}}^{2}x\int_{S_{\xi}}{\hbox{d}}^{2}x^{\prime}\;\mathscr{W}_{\beta}(t,{{\bf
x}};\tau,{{\bf x}}^{\prime})\,\phi(t,{{\bf x}})\,\phi(\tau,{{\bf
x}}^{\prime})\,,$ (4.15)

where (as before) ${{\langle\langle}\,\chi^{a}(t,{{\bf
x}})\,\chi^{b}(\tau,{{\bf
x}}^{\prime})\,{\rangle\rangle}}=\delta^{ab}\,\mathscr{W}_{\beta}(t,{{\bf
x}};\tau,{{\bf x}}^{\prime})$ and
$\widetilde{G}^{2}:=G_{a}G_{b}\,\delta^{ab}=NG^{2}$ is the coupling after
summing over environmental fields. Evaluating the correlator using (2.2.2)
gives the explicit form

$\displaystyle\mathscr{W}_{\beta}(t,{{\bf x}};\tau,{{\bf x}}^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{8\pi\beta|\mathbf{x}-\mathbf{x}^{\prime}|}\left\\{\coth\left[\dfrac{\pi}{\beta}\left(t-\tau+|\mathbf{x}-\mathbf{x}^{\prime}|-i\delta\right)\right]\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.-\coth\left[\dfrac{\pi}{\beta}\left(t-\tau-|\mathbf{x}-\mathbf{x}^{\prime}|-i\delta\right)\right]\right\\}\,,$

Expression (4.15) for the mean-field interaction Hamiltonian is explicitly
nonlocal, in two different ways. First, it is nonlocal in space because the
correlator $\langle\chi^{a}(t,{{\bf x}})\,\chi^{b}(t^{\prime},{{\bf
x}}^{\prime})\rangle$ has support for arbitrary pairs of points ${{\bf x}}$
and ${{\bf x}}^{\prime}$ on the localized interaction region ${\cal S}_{\xi}$.
No interactions at all (local or nonlocal) arise in $\overline{H}_{\rm
int}(t)$ away from ${\cal S}_{\xi}$ as a consequence of the absence of
$\chi^{a}$ fields anywhere in ${\cal R}_{+}$ away from the interaction
surface. This does not preclude the field $\phi$ from acquiring nontrivial
autocorrelations away from ${\cal S}_{\xi}$ in response to these interactions,
however, such as those seen in (2.2.1). The spatial nonlocality has a
relatively simple form in the limit where the times of interest are more
widely separated than the light-crossing time for the hotspot itself:
$|t-\tau|\geq|{{\bf x}}-{{\bf x}}^{\prime}|\gg 2\xi$. In this case eq. (4.2.1)
shows that the effective interaction (4.15) becomes

$\overline{H}_{\rm
int}(t)\simeq\frac{i\tilde{g}^{2}}{4\beta^{2}}\int_{0}^{t}{\hbox{d}}\tau\;\hat{\phi}_{\ell=0}(t,\xi)\,\hat{\phi}_{\ell=0}(\tau,\xi)\;{\rm
csch}^{2}\left[\dfrac{\pi}{\beta}\left(t-\tau-i\delta\right)\right]\,,$ (4.17)

which uses the relation (2.4) (i.e. $g_{a}=4\pi\xi^{2}G_{a}$) and defines the
projector onto the $\ell=0$ spherical harmonic of the field $\phi({{\bf
x}},t)$

$\hat{\phi}_{\ell=0}(t,\xi):=\frac{1}{4\pi}\int_{0}^{4\pi}{\hbox{d}}^{2}\Omega\;\phi(t,r=\xi,\theta,\varphi)\,,$
(4.18)

with the integration ${\hbox{d}}^{2}\Omega$ being over $4\pi$ solid angle.
This form, local in angular-momentum space (and so nonlocal in position
space), is also seen in other applications PPEFTDis .

The second source of nonlocality is in time, although this dies off
exponentially quickly once $\pi|t-\tau|\gg\beta$. This nonlocality is only
consistent with the approximation that led to (4.17) if $\beta\gg\xi$ since
otherwise it is impossible to satisfy both $|t-\tau|\mathrel{\raise
1.29167pt\hbox{$<$\kern-8.50006pt\lower 4.30554pt\hbox{$\sim$}}}\beta/\pi$ and
$|t-\tau|\gg 2\xi$. It should be noticed in this context that in the black
hole analogy $\beta=4\pi\xi$ and so choosing $|t-\tau|\gg 2\xi$ would also
imply $|t-\tau|\gg\beta/\pi$.

Both of these sources of nonlocality have their roots in the fluctutation of
$\chi^{a}$ and so exist only in regions where the $\chi^{a}$’s have support.
This is why all of the nonlocality mentioned above is restricted to the world-
tube swept out by the interaction surface ${\cal S}_{\xi}$, from the point of
view of an external observer in ${\cal R}_{+}$.

The leading contribution to the diffuse evolution describing deviations from
the mean-field limit is given by (4.13), which (because
${{\langle\langle}\,H_{\rm int}\,{\rangle\rangle}}=0$ in the present
instance), becomes

${\cal V}(t)\simeq-i\int_{0}^{t}{\hbox{d}}\tau\;H_{\rm
int}(\tau)=-i\int_{0}^{t}{\hbox{d}}\tau\;\int_{S_{\xi}}{\hbox{d}}^{2}x\;G_{a}\,\chi^{a}(\tau,{{\bf
x}})\,\phi(\tau,{{\bf x}})\,.$ (4.19)

As mentioned above, second-order contributions from this generically compete
with first-order contributions from (4.15) since both arise at second order in
$H_{\rm int}$.

####  4.2.2 The local limit

The above mean-field interaction simplifies and becomes approximately local in
the special case that $\xi$ and $\beta$ are both microscopic scales, showing
how locality ultimately re-emerges in the long-wavelength limit. To examine
this limit explicitly we use small $\xi$ to integrate out the hotspot’s
spatial size, leading to the effective point-like coupling given in (2.8).
Using this to compute the mean-field Hamiltonian then gives

${\overline{H}}_{\mathrm{int}}(t)=\frac{\lambda}{2}\,{\phi}^{2}(t,\mathbf{0})-i\tilde{g}^{2}{\phi}(t,\mathbf{0})\int_{0}^{t}\;{\hbox{d}}s\;\mathscr{W}_{\beta}(s)\,{\phi}(t-s,\mathbf{0})\,,$
(4.20)

where (as before)
$\langle\chi^{a}(t,\mathbf{0})\,\chi^{b}(t-s,\mathbf{0})\rangle=\delta^{ab}\,\mathscr{W}_{\beta}(s)$,
with

$\mathscr{W}_{\beta}(s):=-\frac{1}{4\beta^{2}}\;{\rm
csch}^{2}\left[\frac{\pi}{\beta}(s-i\delta)\right]\,.$ (4.21)

Two simplifications follow from the observation that $\mathscr{W}_{\beta}(s)$
is peaked exponentially sharply around $s=0$, with width of order $\beta/\pi$.
First, the upper integration limit can be taken to infinity at the expense of
errors that are $\sim e^{-2\pi t/\beta}$ and so are exponentially small in the
regime $t\gg\beta/\pi$. Second, for fields varying on scales long compared
with $\beta$ we can expand
$\phi(t-s,\mathbf{0})\simeq\phi(t,\mathbf{0})-s\,\partial_{t}\phi(t,\mathbf{0})+\cdots$
inside the integrand to get

${\overline{H}}_{\mathrm{int}}(t)=\frac{\lambda}{2}\,{\phi}^{2}(t,\mathbf{0})+{\cal
A}\,{\phi}^{2}(t,\mathbf{0})+{\cal
B}\,{\phi}\,\partial_{t}{\phi}(t,\mathbf{0})+\ldots\,,$ (4.22)

where

${\cal
A}=-i\tilde{g}^{2}\int_{0}^{\infty}\;{\hbox{d}}s\;\mathscr{W}_{\beta}(s)\quad\hbox{and}\quad{\cal
B}=i\tilde{g}^{2}\int_{0}^{\infty}\;{\hbox{d}}s\;s\,\mathscr{W}_{\beta}(s)\,,$
(4.23)

and so on.

These reveal the coefficient ${\cal A}$ to be a renormalization of the
effective coupling $\lambda$, while the coefficient ${\cal B}$ multiplies a
new effective interaction proportional to $\phi\,\partial_{t}\phi$. This last
interaction plays no role in the physics to follow because (depending on the
operator ordering) it either involves the commutator of $\phi$ with its
canonical momentum (and so is a divergent contribution to an irrelevant field-
independent piece in $\overline{H}_{\rm int}$) or it involves a total time
derivative, $\partial_{t}\phi^{2}$ (and so can be eliminated using an
appropriate canonical transformation).

The integrals giving the coefficients ${\cal A}$ and ${\cal B}$ can be
evaluated explicitly to give

${\cal
A}=-\tilde{g}^{2}\left[\frac{1}{4\pi^{2}\delta}+\frac{i}{4\pi\beta}\right]+{\cal
O}(\delta)\quad\hbox{and}\quad{\cal
B}=i\tilde{g}^{2}\left[\frac{1}{4\pi^{2}}\log\left(\frac{2\pi\delta}{\beta}\right)-\frac{i}{8\pi}\right]+{\cal
O}(\delta)\,,$ (4.24)

where the infinitesimal $\delta$ is meant to be taken to zero. The divergences
arise because $\mathscr{W}_{\beta}\sim s^{-2}$ as $s\to 0$, and so can be
regarded as being ultraviolet in origin. They arise here as divergences when
$\delta\to 0$, which just means that this infinitesimal – which was introduced
for other reasons – is playing double duty; providing here also a near-hotpsot
regularization for the singular integration (which could indeed have been
regulated in other ways). To the extent that they contribute to non-redundant
interactions these divergences can be renormalized into effective couplings,
such as $\lambda$, thereby underlining that such self-couplings are
generically always present in the effective theory.

What is not simply a small change to the effective description, even in the
local limit, is the generation of an imaginary part of ${\cal A}$ seen in
(4.24). This cannot be absorbed into $\lambda$ without changing the reality
properties of $\lambda$, and its appearance is a manifestation of the general
probability loss away from the mean-field sector into its ‘diffuse’
complement. Since the relative size of $\tilde{g}^{2}/\beta$ to $\lambda$
provides a measure of the relative importance of the real and imaginary parts
of this coupling we should also only expect the mean-field description to be a
good approximation when $\lambda\gg\tilde{g}^{2}/\beta$. The above discussion
also suggests that if the $\phi$ self-coupling $\lambda$ at the hotspot is
ultimately induced by the microscopic $\phi$–$\chi$ coupling, then its natural
size is $\lambda\sim\tilde{g}^{2}/\xi$. If true, this would suggest the mean-
field limit should appear to work best in the regime $\xi\ll\beta$.

A more explicit expression for the real part of $\overline{H}_{\rm int}$ is
given by

$\displaystyle{\mathfrak{H}}(t)$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\Bigl{[}\overline{H}_{\rm int}(t)+\overline{H}_{\rm
int}^{\star}(t)\Bigr{]}$ $\displaystyle\simeq$
$\displaystyle\frac{\lambda}{2}\,{\phi}^{2}(t,\mathbf{0})-\frac{i\tilde{g}^{2}}{2}\int_{0}^{t}{\hbox{d}}s\;\Bigl{(}\mathscr{W}_{\beta}(s)\,\phi(t,\mathbf{0})\,\phi(t-s,\mathbf{0})-\mathscr{W}^{\ast}_{\beta}(s)\,\phi(t-s,\mathbf{0})\,\phi(t,\mathbf{0})\Bigr{)}$
$\displaystyle=$
$\displaystyle\frac{\lambda}{2}\,{\phi}^{2}(t,\mathbf{0})-\frac{i\tilde{g}^{2}}{2}\int_{0}^{t}{\hbox{d}}s\;\mathrm{Re}[\mathscr{W}_{\beta}(s)]\;\Bigl{[}\phi(t,\mathbf{0})\,,\phi(t-s,\mathbf{0})\Bigr{]}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{\tilde{g}^{2}}{2}\int_{0}^{t}{\hbox{d}}s\;\mathrm{Im}[\mathscr{W}_{\beta}(s)]\;\Bigl{\\{}\phi(t,\mathbf{0})\,,\phi(t-s,\mathbf{0})\Bigr{\\}}$
$\displaystyle=$
$\displaystyle\frac{\lambda}{2}\,{\phi}^{2}(t,\mathbf{0})+\frac{\tilde{g}^{2}}{4\pi}{\cal
I}_{+}\int_{0}^{t}{\hbox{d}}s\;\mathrm{Re}[\mathscr{W}_{\beta}(s)]\;\delta^{\prime}(s)+\frac{\tilde{g}^{2}}{8\pi}\int_{0}^{t}{\hbox{d}}s\;\delta^{\prime}(s)\;\Bigl{\\{}\phi(t,\mathbf{0})\,,\phi(t-s,\mathbf{0})\Bigr{\\}}$

where the last equality uses the free-field commutator, $[\phi(t,{{\bf
x}})\,,\phi(t^{\prime},{{\bf x}})]=i\delta^{\prime}(t-t^{\prime}){\cal
I}_{+}/(2\pi)$, computed in (B.12), and that the imaginary part of the free
thermal Wightman function is
$\mathrm{Im}[\mathscr{W}_{\beta}(t)]=\delta^{\prime}(t)/(4\pi)$, as computed
in (B.27).

Integrating by parts, the above formula becomes (noting that the $\delta(t)$
factors vanish for $t>0$)

$\displaystyle{\mathfrak{H}}(t)$ $\displaystyle=$
$\displaystyle\frac{\lambda}{2}\,{\phi}^{2}(t,\mathbf{0})+\frac{\tilde{g}^{2}}{4\pi}\,{\cal
I}_{+}\bigg{(}-\mathrm{Re}[\mathscr{W}_{\beta}(0)]\;\delta^{\prime}(0)-\mathrm{Re}[\mathscr{W}^{\prime}_{\beta}(0)]\bigg{)}$
$\displaystyle\qquad\qquad\qquad\qquad-\frac{\tilde{g}^{2}}{8\pi}\bigg{(}\delta(0)\cdot
2\phi^{2}(t,\mathbf{0})+\phi(t,\mathbf{0})\partial_{t}\phi(t,\mathbf{0})+\partial_{t}\phi(t,\mathbf{0})\phi(t,\mathbf{0})\bigg{)}\,.$

For the present purposes we may drop any terms that are proportional to ${\cal
I}_{+}$, since these do not contribute to the dynamics because they drop out
of commutators with fields. We can also (as always) omit redundant operators
like $\phi\,\partial_{t}\phi$ — see the logic given below equation (4.24) —
and after doing so we have

${\mathfrak{H}}(t)\simeq\frac{1}{2}\left[\lambda-\frac{\tilde{g}^{2}}{2\pi}\,\delta(0)\right]\,{\phi}^{2}(t,\mathbf{0})\,,$
(4.27)

showing once more how the real part of the mean-field Hamiltonian serves to
renormalize888Note that the shift shown in (4.27) matches the shift in (4.24),
when one interprets $\delta(0)=1/(\pi\delta)$ (which follows from writing
$\frac{1}{x-i\delta}=1/x+i\pi\delta(x)$. the self-interaction parameter
$\lambda$, with

$\lambda\;\to\;\lambda_{{\scriptscriptstyle
R}}:=\lambda-\frac{\tilde{g}^{2}}{2\pi}\,\delta(0)\,.$ (4.28)

As usual we henceforth suppress the subscript ‘$R$’.

The imaginary part of $H_{\rm int}(t)$ is similarly given by

$\displaystyle{\mathfrak{I}}(t)$ $\displaystyle:=$
$\displaystyle\frac{1}{2i}\Bigl{[}\overline{H}_{\rm int}(t)-\overline{H}_{\rm
int}^{\star}(t)\Bigr{]}$ (4.29) $\displaystyle=$
$\displaystyle-\frac{\tilde{g}^{2}}{2}\int_{0}^{t}{\hbox{d}}s\;\Bigl{(}\mathscr{W}_{\beta}(t-s)\,\phi(t,\mathbf{0})\,\phi(s,\mathbf{0})+\mathscr{W}^{\ast}_{\beta}(t-s)\,\phi(s,\mathbf{0})\,\phi(t,\mathbf{0})\Bigr{)}\,.$

Although this can also be written in terms of commutators and anti-commutators
of $\phi$, it turns out to be less useful to do so. After renormalizing the
self-interaction parameter using (4.28) and combining terms, the complete
mean-field Hamiltonian can therefore be written as

$\displaystyle\overline{H}_{\rm int}(t)$ $\displaystyle=$
$\displaystyle{\mathfrak{H}}(t)+i\,{\mathfrak{I}}(t)$ $\displaystyle=$
$\displaystyle\frac{\lambda}{2}\phi^{2}(t,\mathbf{0})-\frac{i\tilde{g}^{2}}{2}\int_{0}^{t}{\hbox{d}}s\;\Bigl{(}\mathscr{W}_{\beta}(t-s)\,\phi(t,\mathbf{0})\,\phi(s,\mathbf{0})+\mathscr{W}^{\ast}_{\beta}(t-s)\,\phi(s,\mathbf{0})\,\phi(t,\mathbf{0})\Bigr{)}\
.$

###  4.3 Mean-field $\phi$ correlation function

The virtue of the hotspot model is that it can be solved exactly, making a
comparison with mean-field predictions instructive about the latter’s domain
of validity. This comparison is most easily made using the
$\langle\phi\,\phi\rangle$ correlation function, since this has a known form
Hotspot  — given explicitly at late times by (2.2.1) in the perturbative
limit. To make this comparison we now evaluate the $\langle\phi\,\phi\rangle$
two-point function within the mean-field limit.

####  4.3.1 Mean-field contribution

Keeping in mind that the similarity transformation relating the Heisenberg and
interaction pictures is $\phi_{\scriptscriptstyle H}(t,{{\bf
x}})=V^{\star}(t)\,\phi(t,{{\bf x}})\,V(t)$ where $V(t)=U_{0}^{-1}(t)\,U(t)$
(as before), the correlation function can be written

$\mathrm{Tr}\Bigl{[}{\phi}_{\scriptscriptstyle H}(t,{{\bf
x}})\,{\phi}_{\scriptscriptstyle H}(t^{\prime},{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle
H}\Bigr{]}=\mathrm{Tr}\Bigl{[}{V}^{\star}(t)\,{\phi}(t,{{\bf
x}})\,{V}(t)\,{V}^{\star}(t^{\prime})\,{\phi}(t^{\prime},{{\bf
x}}^{\prime})\,{V}(t^{\prime})\rho(0)\Bigr{]}\,,$ (4.31)

in which we also use that the two pictures agree at the initial time, so
$\rho_{\scriptscriptstyle H}=\rho(0)$ given by (2.9). The mean-field result is
obtained by using in this expression the approximate form

${V}(t)\simeq\overline{V}(t)\otimes{\cal I}_{-}$ (4.32)

with $\overline{V}(t)$ given by (4.1.1).

With this replacement – and using the initial conditions (2.9) and (2.10) –
the mean-field correlation function reduces to an in-in expectation in the
observed $\phi$ sector, of the form

$\displaystyle{\mathrm{Tr}}\Bigl{[}{\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}){\phi}_{{\scriptscriptstyle H}}(t^{\prime},{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle H}\Bigr{]}_{\scriptscriptstyle MF}$
$\displaystyle:=$
$\displaystyle{\mathrm{Tr}}_{+}\bigg{[}{\overline{V}}^{\star}(t){\phi}(t,{{\bf
x}}){\overline{V}}(t){\overline{V}}^{\star}(t^{\prime}){\phi}(t^{\prime},{{\bf
x}}^{\prime}){\overline{V}}(t^{\prime})\,\rho_{+}\bigg{]}$ (4.33)
$\displaystyle=$
$\displaystyle\bra{\mathrm{vac}}{\overline{V}}^{\star}(t){\phi}(t,{{\bf
x}}){\overline{V}}(t){\overline{V}}^{\star}(t^{\prime}){\phi}(t^{\prime},{{\bf
x}}^{\prime}){\overline{V}}(t^{\prime})\ket{\mathrm{vac}}$

where the interaction-picture state is evaluated at $t=0$. Evaluating
$\overline{H}_{\rm int}$ using (4.20), we have

${\overline{V}}(t)\simeq{\cal
I}_{+}-i\int_{0}^{t}{\hbox{d}}\tau\;{\overline{H}}_{\mathrm{int}}(\tau)\simeq{\cal
I}_{+}-i\int_{0}^{t}{\hbox{d}}\tau\Bigl{[}{\mathfrak{H}}(\tau)+i\,{\mathfrak{I}}(\tau)\Bigr{]}\,,$
(4.34)

where real and imaginary parts of $\overline{H}_{\rm int}(t)$ are given by
(4.2.2). Inserting (4.34) into (4.33) and working to leading nontrivial order
in $\overline{H}_{\rm int}$ then yields the quantity to be evaluated:

$\displaystyle{\mathrm{Tr}}\Bigl{[}{\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}){\phi}_{{\scriptscriptstyle H}}(t^{\prime},{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle H}\Bigr{]}_{\scriptscriptstyle
MF}\simeq\bra{\mathrm{vac}}{\phi}(t,{{\bf x}}){\phi}(t^{\prime},{{\bf
x}}^{\prime})\ket{\mathrm{vac}}$ (4.35)
$\displaystyle\qquad\qquad+i\int_{0}^{t}{\hbox{d}}\tau\;\bra{\mathrm{vac}}\Bigl{[}{\mathfrak{H}}(\tau),\phi(t,{{\bf
x}})\Bigr{]}{\phi}(t^{\prime},{{\bf
x}}^{\prime})\ket{\mathrm{vac}}+i\int_{0}^{t^{\prime}}{\hbox{d}}\tau\;\bra{\mathrm{vac}}{\phi}(t,{{\bf
x}})\Bigl{[}{\mathfrak{H}}(\tau),{\phi}(t^{\prime},{{\bf
x}}^{\prime})\Bigr{]}\ket{\mathrm{vac}}$
$\displaystyle\qquad\qquad\qquad+\int_{0}^{t}{\hbox{d}}\tau\;\bra{\mathrm{vac}}\Bigl{\\{}{\mathfrak{I}}(\tau),\phi(t,{{\bf
x}})\Bigr{\\}}{\phi}(t^{\prime},{{\bf
x}}^{\prime})\ket{\mathrm{vac}}+\int_{0}^{t^{\prime}}{\hbox{d}}\tau\;\bra{\mathrm{vac}}{\phi}(t,{{\bf
x}})\Bigl{\\{}{\mathfrak{I}}(\tau),{\phi}(t^{\prime},{{\bf
x}}^{\prime})\Bigr{\\}}\ket{\mathrm{vac}}\,.$

####  4.3.2 Aside: $\overline{V}^{\star}$ vs $\overline{V}^{-1}$

We pause the main line of development here to settle a side issue that might
bother the reader at this point. The issue is this: equation (4.35) is derived
by substituting the mean-field expression (4.32) into the general correlator
definition (4.31). For the full theory unitarity ensures $V^{\star}=V^{-1}$
but the same is not true for the mean-field limit, since we have seen
$\overline{V}^{\star}\neq\overline{V}^{-1}$. So although nothing changes if we
replace $V^{\star}\to V^{-1}$ in (4.31), making the replacement (4.32) in the
result instead leads to

${\cal U}(t,{{\bf x}};t^{\prime},{{\bf
x}}^{\prime}):=\bra{\mathrm{vac}}{\overline{V}}^{-1}(t)\,{\phi}(t,{{\bf
x}})\,{\overline{V}}(t){\overline{V}}^{-1}(t^{\prime})\,{\phi}(t^{\prime},{{\bf
x}}^{\prime})\,{\overline{V}}(t^{\prime})\ket{\mathrm{vac}}$ (4.36)

which differs from the right-hand side of (4.33). Using (4.36) would change
(4.35) by replacing the anticommutators $\\{{\mathfrak{I}},(\cdot)\\}$ with
commutators $[{\mathfrak{I}},(\cdot)]$ – an important difference in practice
because the commutator is much easier to evaluate (as we do for completeness
in Appendix B.1).

Which is right? This is partially a matter of definition, since it hinges on
how the full result gets spit into mean-field and diffuse parts. The guiding
principle in §4.1 is to make this split so that observables like (4.8) break
into a sum of mean-field and diffuse pieces, with no interference terms. The
same principle tells us to define the mean-field correlator using (4.33)
rather than (4.36). Specialized to $t=t^{\prime}$ both equations have the same
form as (4.8), and it is only for (4.33) that mean-field and diffuse parts
cleanly split, because $V^{\star}=\overline{V}^{\star}+{\cal V}^{\star}$
divides linearly while $V^{-1}$ does not.

####  4.3.3 Equal-time Limits

With the mean-field/diffuse split in mind, we next specialize the correlation
function to equal times ($t=t^{\prime}$), so that (4.31) agrees with (4.8)
with the choice ${\cal O}(t)=\phi(t,{{\bf x}})\phi(t,{{\bf x}}^{\prime})$. As
described above, this ensures the equal-time correlation function nicely
splits into the sum of mean-field and diffuse parts

${\mathrm{Tr}}\Bigl{[}{\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}){\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle
H}\Bigr{]}={\mathrm{Tr}}\Bigl{[}{\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}){\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle H}\Bigr{]}_{{\scriptscriptstyle
MF}}+{\mathrm{Tr}}\Bigl{[}{\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}){\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle H}\Bigr{]}_{\mathrm{diff}}$ (4.37)

where we define the (equal-time) mean-field correlations as999Note that this
is not the same as taking $t=t^{\prime}$ in the formula (4.34), since
$\overline{V}(t)\overline{V}^{\star}(t)\neq{\cal I}_{+}$.

$\displaystyle{\mathrm{Tr}}\Bigl{[}{\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}){\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle H}\Bigr{]}_{\scriptscriptstyle MF}$
$\displaystyle:=$
$\displaystyle\bra{\mathrm{vac}}\overline{V}^{\star}(t)\phi(t,{{\bf
x}})\phi(t,{{\bf y}})\overline{V}(t)\ket{\mathrm{vac}}$ $\displaystyle=$
$\displaystyle\bra{\mathrm{vac}}{\phi}(t,{{\bf x}})\,{\phi}(t,{{\bf
x}}^{\prime})\ket{\mathrm{vac}}\
+i\int_{0}^{t}{\hbox{d}}\tau\;\bra{\mathrm{vac}}\Bigl{[}{\mathfrak{H}}(\tau),{\phi}(t,{{\bf
x}}){\phi}(t,{{\bf x}}^{\prime})\Bigr{]}\ket{\mathrm{vac}}$
$\displaystyle\qquad+\int_{0}^{t}{\hbox{d}}\tau\;\bra{\mathrm{vac}}\Bigl{\\{}{\mathfrak{I}}(\tau),{\phi}(t,{{\bf
x}}){\phi}(t,{{\bf x}}^{\prime})\Bigr{\\}}\ket{\mathrm{vac}}$

c.f. (4.34), while the diffuse part of the correlations is

${\mathrm{Tr}}\Bigl{[}{\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}){\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle
H}\Bigr{]}_{\mathrm{diff}}:={\mathrm{Tr}}\Bigl{[}{\cal
V}^{\star}(t)\phi(t,{{\bf x}})\phi(t,{{\bf y}}){\cal
V}(t)\rho_{\scriptscriptstyle H}\Bigr{]}\ .$ (4.39)

####  4.3.4 Equal-time mean-field correlation

This section explicitly evaluates the equal-time mean-field correlation
(4.3.3), ending with a final form that reduces the mode sums to a single
integration over explicit elementary functions.

The terms involving the real part ${\mathfrak{H}}$ of the mean-field
Hamiltonian involve commuators are simplest and so all integrals can be
evaluated explicitly. The quantity to be evaluated is

$i\int_{0}^{t}{\hbox{d}}\tau\;\bra{\mathrm{vac}}\Bigl{[}{\mathfrak{H}}(\tau),{\phi}(t,{{\bf
x}}){\phi}(t,{{\bf
x}}^{\prime})\Bigr{]}\ket{\mathrm{vac}}=\frac{i\lambda}{2}\int_{0}^{t}{\hbox{d}}\tau\;\bra{\mathrm{vac}}\Bigl{[}\phi^{2}(\tau,\mathbf{0}),{\phi}(t,{{\bf
x}}){\phi}(t,{{\bf x}}^{\prime})\Bigr{]}\ket{\mathrm{vac}}$ (4.40)

(which is why we could ignore terms in ${\mathfrak{H}}(t)$ proportional to the
${\cal I}_{+}$ in (4.27)). Using the single-field commutator

$\Bigl{[}\phi(\tau,\mathbf{0}),\phi(t,{{\bf x}})\Bigr{]}=\frac{i}{4\pi|{{\bf
x}}|}\bigg{(}\delta\big{[}\tau-(t-|{{\bf
x}}|)\big{]}-\delta\big{[}\tau-(t+|{{\bf x}}|)\big{]}\bigg{)}{\cal I}_{+}\ ,$
(4.41)

evaluated in Appendix A.3 allows this term to be written

$\Bigl{[}\phi^{2}(\tau,\mathbf{0}),\phi(t,{{\bf
x}})\Bigr{]}=\frac{i}{2\pi|{{\bf x}}|}\bigg{(}\delta\big{[}\tau-(t-|{{\bf
x}}|)\big{]}-\delta\big{[}\tau-(t+|{{\bf
x}}|)\big{]}\bigg{)}\phi(\tau,\mathbf{0})\ ,$ (4.42)

which in turn implies

$\displaystyle\Bigl{[}\phi^{2}(\tau,\mathbf{0}),\phi(t,{{\bf x}})\phi(t,{{\bf
x}}^{\prime})\Bigr{]}$ $\displaystyle=$
$\displaystyle\Bigl{[}\phi^{2}(\tau,\mathbf{0}),\phi(t,{{\bf
x}})\Bigr{]}\phi(t,{{\bf x}}^{\prime})+\phi(t,{{\bf
x}})\Bigl{[}\phi^{2}(\tau,\mathbf{0}),\phi(t,{{\bf x}}^{\prime})\Bigr{]}$
$\displaystyle=$ $\displaystyle\frac{i}{2\pi|{{\bf
x}}|}\bigg{(}\delta\big{[}\tau-(t-|{{\bf
x}}|)\big{]}-\delta\big{[}\tau-(t+|{{\bf
x}}|)\big{]}\bigg{)}\phi(\tau,\mathbf{0})\phi(t,{{\bf x}}^{\prime})$
$\displaystyle\qquad\qquad+\frac{i}{2\pi|{{\bf
x}}^{\prime}|}\bigg{(}\delta\big{[}\tau-(t-|{{\bf
x}}^{\prime}|)\big{]}-\delta\big{[}\tau-(t+|{{\bf
x}}^{\prime}|)\big{]}\bigg{)}\phi(t,{{\bf x}})\phi(\tau,\mathbf{0})\ .$

Only $\delta$-functions with singularities at the retarded times $t-|{{\bf
x}}|$ and $t-|{{\bf x}}^{\prime}|$ contribute in the regime of interest, so

$\displaystyle
i\int_{0}^{t}{\hbox{d}}\tau\;\bra{\mathrm{vac}}\Bigl{[}{\mathfrak{H}}(\tau),{\phi}(t,{{\bf
x}}){\phi}(t,{{\bf x}}^{\prime})\Bigr{]}\ket{\mathrm{vac}}$
$\displaystyle\quad=-\frac{\lambda}{4\pi}\int_{0}^{t}{\hbox{d}}\tau\;\bigg{(}\frac{\delta\big{[}\tau-(t-|{{\bf
x}}|)\big{]}}{|{{\bf x}}|}\bra{\mathrm{vac}}\phi(\tau,\mathbf{0})\phi(t,{{\bf
x}}^{\prime})\ket{\mathrm{vac}}+\frac{\delta\big{[}\tau-(t-|{{\bf
x}}^{\prime}|)\big{]}}{|{{\bf x}}^{\prime}|}\bra{\mathrm{vac}}\phi(t,{{\bf
x}})\phi(\tau,\mathbf{0})\ket{\mathrm{vac}}\bigg{)}$
$\displaystyle\quad=-\frac{\lambda}{4\pi|{{\bf x}}|}\,\Theta(t-|{{\bf
x}}|)\bra{\mathrm{vac}}\phi(t-|{{\bf x}}|,\mathbf{0})\phi(t,{{\bf
x}}^{\prime})\ket{\mathrm{vac}}-\frac{\lambda}{4\pi|{{\bf
x}}^{\prime}|}\,\Theta(t-|{{\bf x}}^{\prime}|)\bra{\mathrm{vac}}\phi(t,{{\bf
x}})\phi(t-|{{\bf x}}^{\prime}|,\mathbf{0})\ket{\mathrm{vac}}$
$\displaystyle\quad=-\frac{\lambda\Theta(t-|{{\bf x}}|)}{16\pi^{3}|{{\bf
x}}|\big{(}-(-|{{\bf x}}|-i\delta)^{2}+|{{\bf
x}}^{\prime}|^{2}\big{)}}-\frac{\lambda\Theta(t-|{{\bf
x}}^{\prime}|)}{16\pi^{3}|{{\bf x}}^{\prime}|\big{(}-(|{{\bf
x}}^{\prime}|-i\delta)^{2}+|{{\bf x}}|^{2}\big{)}}$ (4.44)

In the limit that the transients have passed – i.e. once $t-|{{\bf x}}|>0$ and
$t-|{{\bf x}}^{\prime}|>0$ – this simplifies to

$\displaystyle
i\int_{0}^{t}{\hbox{d}}\tau\;\bra{\mathrm{vac}}\Bigl{[}{\mathfrak{H}}(\tau),{\phi}(t,{{\bf
x}}){\phi}(t,{{\bf
x}}^{\prime})\Bigr{]}\ket{\mathrm{vac}}=-\frac{\lambda}{16\pi^{3}|{{\bf
x}}||{{\bf x}}^{\prime}|\big{(}|{{\bf x}}|+|{{\bf x}}^{\prime}|)}$ (4.45)

which agrees with the $\lambda$-dependent part of equal-time correlation
functions computed in (2.2.1) and (3.74), (evaluated using the renormalized
coupling $\lambda=\lambda_{{\scriptscriptstyle R}}$).

The more complicated contribution involves the imaginary part of
$\overline{H}_{\rm int}$, in which we evalaute ${\mathfrak{I}}$ using (4.29)
to get

$\displaystyle\int_{0}^{t}{\hbox{d}}\tau\;\bra{\mathrm{vac}}\Bigl{\\{}{\mathfrak{I}}(\tau),{\phi}(t,{{\bf
x}}){\phi}(t,{{\bf x}}^{\prime})\Bigr{\\}}\ket{\mathrm{vac}}$ (4.46)
$\displaystyle\quad=-\int_{0}^{t}{\hbox{d}}\tau\int_{0}^{\tau}{\hbox{d}}\tau^{\prime}\;\bra{\mathrm{vac}}\bigg{\\{}\text{\scalebox{0.85}{$\frac{\tilde{g}^{2}\Bigl{(}\mathscr{W}_{\beta}(\tau-\tau^{\prime})\phi(\tau,\mathbf{0})\phi(\tau^{\prime},\mathbf{0})+\mathscr{W}^{\ast}_{\beta}(\tau-\tau^{\prime})\phi(\tau^{\prime},\mathbf{0})\phi(\tau,\mathbf{0})\Bigr{)}}{2}$}},{\phi}(t,{{\bf
x}}){\phi}(t,{{\bf x}}^{\prime})\bigg{\\}}\ket{\mathrm{vac}}$
$\displaystyle\quad=-\frac{\tilde{g}^{2}}{2}\int_{0}^{t}{\hbox{d}}\tau\int_{0}^{t}{\hbox{d}}\tau^{\prime}\;\mathscr{W}_{\beta}(\tau-\tau^{\prime})\bra{\mathrm{vac}}\Bigl{\\{}\phi(\tau,\mathbf{0})\phi(\tau^{\prime},\mathbf{0}),{\phi}(t,{{\bf
x}}){\phi}(t,{{\bf x}}^{\prime})\Bigr{\\}}\ket{\mathrm{vac}}\,,$

where the last equality changes integration variables
$\tau\leftrightarrow\tau^{\prime}$ in one of the two terms and uses the
property $\mathscr{W}_{\beta}^{\ast}(t)=\mathscr{W}_{\beta}(-t)$ of the
Wightman function. The formula (4.3.3) therefore takes the final form

$\displaystyle{\mathrm{Tr}}\Bigl{[}{\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}){\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle H}\Bigr{]}_{\scriptscriptstyle MF}$
$\displaystyle\simeq$ $\displaystyle\frac{1}{4\pi^{2}|{{\bf x}}-{{\bf
x}}^{\prime}|^{2}}-\frac{\lambda}{16\pi^{3}|{{\bf x}}||{{\bf
x}}^{\prime}|\big{(}|{{\bf x}}|+|{{\bf x}}^{\prime}|)}$
$\displaystyle\quad-\frac{\tilde{g}^{2}}{2}\int_{0}^{t}{\hbox{d}}\tau\int_{0}^{t}{\hbox{d}}\tau^{\prime}\;\mathscr{W}_{\beta}(\tau-\tau^{\prime})\bra{\mathrm{vac}}\Bigl{\\{}\phi(\tau,\mathbf{0})\phi(\tau^{\prime},\mathbf{0}),{\phi}(t,{{\bf
x}}){\phi}(t,{{\bf x}}^{\prime})\Bigr{\\}}\ket{\mathrm{vac}}$

in the regime $t-|{{\bf x}}|>0$ and $t-|{{\bf x}}^{\prime}|>0$ (after
transients have passed from the switch-on of couplings at $t=|{{\bf x}}|=0$).

The matrix element in this last expression is evaluated in Appendix B.2 with
the result

$\displaystyle{\mathrm{Tr}}\Bigl{[}{\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}){\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle H}\Bigr{]}_{\scriptscriptstyle
MF}\simeq\frac{1}{4\pi^{2}|{{\bf x}}-{{\bf
x}}^{\prime}|^{2}}\left[1-\frac{\tilde{g}^{2}}{4\pi^{2}}\left(\dfrac{\zeta(3)t}{\pi\beta^{3}}-{\displaystyle\int_{0}^{\infty}}{\hbox{d}}p\;p\;\dfrac{{\hbox{d}}{\cal
D}_{\beta}(p,\delta)}{{\hbox{d}}p}\right)\right]$
$\displaystyle\qquad\qquad-\frac{\lambda}{16\pi^{3}|{{\bf x}}||{{\bf
x}}^{\prime}|\big{(}|{{\bf x}}|+|{{\bf
x}}^{\prime}|)}-\frac{\tilde{g}^{2}}{16\pi^{4}|{{\bf x}}||{{\bf
x}}^{\prime}|}\int_{0}^{\infty}{\hbox{d}}p\;\sin(k|{{\bf
x}}|)\int_{0}^{\infty}{\hbox{d}}k\;\sin(k|{{\bf x}}^{\prime}|)$ (4.48)
$\displaystyle\qquad\qquad\qquad\times\left\\{\frac{{\cal C}_{\beta}(p)+{\cal
C}_{\beta}(k)}{p+k}\sin\big{[}(p+k)t\big{]}-\frac{{\cal
D}_{\beta}(p,\delta)+{\cal
D}_{\beta}(k,\delta)}{p+k}\Bigl{[}1+\cos\big{[}(p+k)t\big{]}\Bigr{]}\right\\}$

where the Riemann-Zeta function evaluates to $\zeta(3)\simeq 1.202$ and the
functions ${\cal C}_{\beta}$ and ${\cal D}_{\beta}$ are given by

${\cal C}_{\beta}=\frac{p}{4\pi}\coth\left(\frac{\beta
p}{2}\right)\quad\hbox{and}\quad{\cal
D}_{\beta}=\frac{p}{2\pi^{2}}\log\left(\frac{2\pi
e^{\gamma}\delta}{\beta}\right)+\frac{p}{2\pi^{2}}\mathrm{Re}\left[\psi^{(0)}\left(-i\;\frac{\beta
p}{2\pi}\right)\right]\ ,$ (4.49)

where $\psi^{(0)}$ is the digamma function defined by
$\psi^{(0)}(z)=\frac{\mathrm{d}}{\mathrm{d}z}\log\Gamma(z)$, and where
$\delta$ is (as usual) to be taken to zero at the end (after renormalization).

Although the $\tilde{g}^{2}$-independent terms in (4.3.4) agree with the
perturbative and Markovian results (2.2.1) and (3.74), those that include
$\tilde{g}^{2}$ do not. This difference is due to the contributions of the
diffuse evolution first entering at this order, as we now show.

####  4.3.5 Including diffuse correlations

The above calculation omits the diffuse correlations (4.39), and to the order
we work it suffices to use the lowest-order expression (4.13) for ${\cal V}$:

${\cal V}(t)\simeq-i\int_{0}^{t}{\hbox{d}}\tau\;\delta H_{\rm
int}(\tau)=-i\int_{0}^{t}{\hbox{d}}\tau\;g_{a}\,\phi(\tau,\mathbf{0})\otimes\chi^{a}(\tau,\mathbf{0})\,.$
(4.50)

Using this in (4.39) then gives

$\mathrm{Tr}\Bigl{[}{\phi}_{\scriptscriptstyle H}(t,{{\bf
x}}){\phi}_{\scriptscriptstyle H}(t,{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle H}\Bigr{]}_{\rm
diff}\simeq\tilde{g}^{2}\int_{0}^{t}{\hbox{d}}\tau\int_{0}^{t}{\hbox{d}}\tau^{\prime}\;\mathscr{W}_{\beta}(\tau-\tau^{\prime})\;\bra{\mathrm{vac}}{\phi}(\tau,\mathbf{0}){\phi}(t,{{\bf
x}}){\phi}(t,{{\bf
x}}^{\prime}){\phi}(\tau^{\prime},\mathbf{0})\ket{\mathrm{vac}}\,.$ (4.51)

We now show that adding (4.51) to the mean-field result (4.3.4) reproduces the
perturbative expression for the full correlator given in (2.2.1). Summing
these mean-field and diffuse contributions gives the result

${\mathrm{Tr}}\Bigl{[}{\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}){\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle H}\Bigr{]}\simeq\frac{1}{4\pi^{2}|{{\bf
x}}-{{\bf x}}^{\prime}|^{2}}-\frac{\lambda}{16\pi^{3}|{{\bf x}}||{{\bf
x}}^{\prime}|\big{(}|{{\bf x}}|+|{{\bf x}}^{\prime}|)}+{\cal
I}_{\beta}(t,{{\bf x}},{{\bf x}}^{\prime})$ (4.52)

where the last term is given by the following combination of matrix elements

$\displaystyle{\cal I}_{\beta}(t,{{\bf x}},{{\bf x}}^{\prime})$
$\displaystyle:=$
$\displaystyle-\frac{\tilde{g}^{2}}{2}\int_{0}^{t}{\hbox{d}}\tau\int_{0}^{t}{\hbox{d}}\tau^{\prime}\;\mathscr{W}_{\beta}(\tau-\tau^{\prime})\bra{\mathrm{vac}}\Bigl{\\{}\phi(\tau,\mathbf{0})\phi(\tau^{\prime},\mathbf{0}),{\phi}(t,{{\bf
x}}){\phi}(t,{{\bf x}}^{\prime})\Bigr{\\}}\ket{\mathrm{vac}}$
$\displaystyle\qquad\qquad+\tilde{g}^{2}\int_{0}^{t}{\hbox{d}}\tau\int_{0}^{t}{\hbox{d}}\tau^{\prime}\;\mathscr{W}_{\beta}(\tau-\tau^{\prime})\;\bra{\mathrm{vac}}{\phi}(\tau,\mathbf{0}){\phi}(t,{{\bf
x}}){\phi}(t,{{\bf
x}}^{\prime}){\phi}(\tau^{\prime},\mathbf{0})\ket{\mathrm{vac}}\ .$

These can be usefully (but tediously) re-written as a double commutator plus a
remainder,

$\displaystyle{\cal I}_{\beta}(t,{{\bf x}},{{\bf x}}^{\prime})$
$\displaystyle=$ $\displaystyle{\cal P}_{\beta}(t,{{\bf x}},{{\bf
x}}^{\prime})+{\cal Q}_{\beta}(t,{{\bf x}},{{\bf x}}^{\prime})$ (4.54)

where the double commutator is

${\cal P}_{\beta}(t,{{\bf x}},{{\bf
x}}^{\prime}):=-\tilde{g}^{2}\int_{0}^{t}{\hbox{d}}\tau\int_{0}^{t}{\hbox{d}}\tau^{\prime}\;\mathscr{W}_{\beta}(\tau-\tau^{\prime})\bra{\mathrm{vac}}\Bigl{[}\phi(\tau,\mathbf{0}),{\phi}(t,{{\bf
x}})\Bigr{]}\Bigl{[}\phi(\tau^{\prime},\mathbf{0}),{\phi}(t,{{\bf
x}}^{\prime})\Bigr{]}\ket{\mathrm{vac}}$ (4.55)

while the remainder becomes

$\displaystyle{\cal Q}_{\beta}(t,{{\bf x}},{{\bf x}}^{\prime})$
$\displaystyle:=$
$\displaystyle\frac{\tilde{g}^{2}}{2}\int_{0}^{t}{\hbox{d}}\tau\int_{0}^{t}{\hbox{d}}\tau^{\prime}\;\mathscr{W}_{\beta}(\tau-\tau^{\prime})\bra{\mathrm{vac}}\bigg{(}\Bigl{[}\phi(\tau,\mathbf{0}),{\phi}(t,{{\bf
x}})\Bigr{]}\phi(\tau^{\prime},\mathbf{0}){\phi}(t,{{\bf x}}^{\prime})$
$\displaystyle\qquad-\phi(\tau,\mathbf{0})\Bigl{[}\phi(\tau^{\prime},\mathbf{0}),{\phi}(t,{{\bf
x}})\Bigr{]}{\phi}(t,{{\bf x}}^{\prime})+{\phi}(t,{{\bf
x}})\Bigl{[}\phi(\tau,\mathbf{0}),{\phi}(t,{{\bf
x}}^{\prime})\Bigr{]}\phi(\tau^{\prime},\mathbf{0})$
$\displaystyle\qquad\qquad\qquad-{\phi}(t,{{\bf
x}})\phi(\tau,\mathbf{0})\Bigl{[}\phi(\tau^{\prime},\mathbf{0}),{\phi}(t,{{\bf
x}}^{\prime})\Bigr{]}\bigg{)}\ket{\mathrm{vac}}\ .$

These integrals evaluate (see Appendix B.3) in the regime $t-|{{\bf x}}|>0$
and $t-|{{\bf x}}^{\prime}|>0$ to

${\cal P}_{\beta}(t,{{\bf x}},{{\bf
x}}^{\prime})=-\frac{\tilde{g}^{2}}{64\pi^{2}\beta^{2}|{{\bf x}}||{{\bf
x}}^{\prime}|\sinh^{2}\left[\frac{\pi}{\beta}(-|{{\bf x}}|+|{{\bf
x}}^{\prime}|-i\delta)\right]}\ ,$ (4.57)

and

${\cal Q}_{\beta}(t,{{\bf x}},{{\bf
x}}^{\prime})=\frac{\tilde{g}^{2}}{64\pi^{4}|{{\bf x}}||{{\bf
x}}^{\prime}|}\bigg{[}\frac{1}{(|{{\bf x}}|-|{{\bf
x}}^{\prime}|+i\delta)^{2}}-\frac{1}{(|{{\bf x}}|+|{{\bf
x}}^{\prime}|)^{2}}\bigg{]}\ .$ (4.58)

When we use these in (4.52) the overall correlation function is therefore

$\displaystyle{\mathrm{Tr}}\Bigl{[}{\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}){\phi}_{{\scriptscriptstyle H}}(t,{{\bf
x}}^{\prime})\rho_{\scriptscriptstyle H}\Bigr{]}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{4\pi^{2}|{{\bf x}}-{{\bf
x}}^{\prime}|^{2}}-\frac{\lambda}{16\pi^{3}|{{\bf x}}||{{\bf
x}}^{\prime}|\big{(}|{{\bf x}}|+|{{\bf x}}^{\prime}|)}$
$\displaystyle\qquad\qquad-\frac{\tilde{g}^{2}}{64\pi^{2}\beta^{2}|{{\bf
x}}||{{\bf x}}^{\prime}|\sinh^{2}\left[\frac{\pi}{\beta}(-|{{\bf x}}|+|{{\bf
x}}^{\prime}|-i\delta)\right]}+\frac{\tilde{g}^{2}}{16\pi^{4}}\;\frac{1}{(|{{\bf
x}}|^{2}-|{{\bf x}}^{\prime}|^{2})^{2}}\ .$

This last expressions agrees perfectly with the perturbative Heisenberg-
picture result computed in Hotspot , once this is evaluated in the equal-time
limit – see eq. (2.2.1) – inside the future light-cone of the event at
$t=|{{\bf x}}|=0$ where the coupling switches on.

####  4.3.6 Domain of validity of mean-field methods

We see from these calculations that the mean-field correlator does not in
general agree with the Heisenberg-picture result, even if this comparison is
only made at leading order in $\lambda$ and $\tilde{g}^{2}$. The comparison of
the previous section shows that the difference between the mean-field and
Heisenberg-picture answers is precisely given by the diffuse contribution that
must be small for mean-field methods to apply.

The difference between mean-field result (4.3.4) and the corresponding
Heisenberg-picture answer (4.3.5) lies completely in their $\tilde{g}^{2}$
dependence; the term involving the self-coupling $\lambda$ is identical in
both cases. Since $\tilde{g}$ and $\lambda$ both have dimensions of length,
another scale must appearin the comparison of $\lambda$ and $\tilde{g}^{2}$,
and the explicit evaluation – e.g. (4.3.5) – shows this scale to be either
$\beta$ or a combination of $|{{\bf x}}|$ and $|{{\bf x}}^{\prime}|$.

Because our interest is typically where $|{{\bf x}}|$ and $|{{\bf
x}}^{\prime}|$ are much larger than $\beta$ the relative size of the
$\lambda$-dependent term and the largest of $\tilde{g}^{2}$ corrections is set
by the relative size of $\lambda$ and $\tilde{g}^{2}/\beta$, suggesting that
mean-field methods provide a reliable approximation in the regime
$\lambda\gg\tilde{g}^{2}/\beta$.

In summary, we see that mean-field methods can apply to the hotspot problem,
but only in some parts of parameters space such as when
$\lambda\gg\tilde{g}^{2}/\beta$. Where it does apply the resulting effective
Hamiltonian can be nonlocal, both in the angular directions of ${\cal
S}_{\xi}$ and in time, due to the nonlocality of the $\chi$ correlations with
which the external $\phi$ field interacts.

## Acknowledgements

We thank Sarah Shandera for useful conversations, and KITP Santa Barbara for
hosting the workshop (during a pandemic) that spawned this work. (Consequently
this research was supported in part by the National Science Foundation under
Grant No. NSF PHY-1748958.) CB’s research was partially supported by funds
from the Natural Sciences and Engineering Research Council (NSERC) of Canada.
Research at the Perimeter Institute is supported in part by the Government of
Canada through NSERX and by the Province of Ontario through MRI.

##  Appendix A Useful intermediate steps

This appendix gathers together many intermediate steps not given in the main
text, including the evaluation of several of the integrals encountered there.
Our goal is to be as explicit as possible.

###  A.1 Kernel equations for the gaussian ansatz

This section evaluations the implications of the master equation (3.1.3) for
the gaussian kernels in the ansatz (3.39).

First we compute the time-derivative of the above ansatz

$\displaystyle\frac{\partial\sigma_{{\scriptscriptstyle
S}}(t,\varphi_{1},\varphi_{2})}{\partial t}$ $\displaystyle=$
$\displaystyle\bigg{[}\frac{\partial_{t}{\cal N}(t)}{{\cal
N}(t)}-\int{\hbox{d}}^{3}{{\bf x}}\int{\hbox{d}}^{3}{{\bf
y}}\;\bigg{\\{}\frac{1}{2}\partial_{t}\mathcal{A}_{1}({{\bf x}},{{\bf
y}};t)\varphi_{1}({{\bf x}})\varphi_{1}({{\bf y}})$
$\displaystyle\quad\quad\quad\quad+\frac{1}{2}\partial_{t}\mathcal{A}_{2}({{\bf
x}},{{\bf y}};t)\varphi_{2}({{\bf x}})\varphi_{2}({{\bf y}})+\partial_{t}{\cal
B}({{\bf x}},{{\bf y}};t)\varphi_{1}({{\bf x}})\varphi_{2}({{\bf
y}})\bigg{\\}}\bigg{]}\sigma_{{\scriptscriptstyle
S}}(t,\varphi_{1},\varphi_{2})\ .$

and we note the RHS of equation (3.1.3)

RHS of (3.1.3) $\displaystyle=$
$\displaystyle-\frac{i}{2}\int{\hbox{d}}^{3}{{\bf
x}}\bigg{[}-\frac{\delta^{2}}{\delta\varphi_{1}({{\bf
x}})^{2}}+\big{|}\boldsymbol{\nabla}\varphi_{1}({{\bf
x}})\big{|}^{2}+\frac{\delta^{2}}{\delta\varphi_{2}({{\bf
x}})^{2}}-\big{|}\boldsymbol{\nabla}\varphi_{2}({{\bf
x}})\big{|}^{2}\bigg{]}\sigma_{{\scriptscriptstyle