A Flat Space Analogue for
the Quantum Origin of Structure

Daniel Green and Yiwen Huang

Department of Physics,University of California at San Diego,
La Jolla, CA 92093, USA

Abstract

The analytic structure of non-Gaussian correlators in inflationary cosmologies
has recently been proposed as a test of the quantum origin of structure in the
universe. To further understand this proposal, we explore the analogous equal-
time in-in correlators in flat space and show they exhibit the same features
as their cosmological counterparts. The quantum vacuum is uniquely identified
by in-in correlators with a total energy pole and no additional poles at
physical momenta. We tie this behavior directly to the S-matrix and show that
poles at physical momenta always arise from scattering of particles present in
the initial state. We relate these flat-space in-in correlators to the
probability amplitude for exciting multiple Unruh-de Witt detectors.
Localizing the detectors in spacetime, through the uncertainty principle,
provides the energy and momentum needed to excite the vacuum and explains the
connection to cosmological particle production. In addition, the entanglement
of these detectors provides a probe of the entangled state of the underlying
field and connects the properties of the correlators to the range of
entanglement of the detectors.

###### Contents

  1. 1 Introduction
  2. 2 From In-In Correlators and the S-matrix
    1. 2.1 The In-In Formalism
    2. 2.2 Relation to the S-Matrix
    3. 2.3 Resolving and Interpreting Poles at Physical Momenta
  3. 3 Particle Detectors and the In-In Formalism
    1. 3.1 Unruh – de Witt Detectors
    2. 3.2 Particle Detection and Cosmological Correlators
    3. 3.3 Classical Interpretation
  4. 4 Detector Entanglement
    1. 4.1 Position Space Entanglement
    2. 4.2 Momentum Space Detectors and Entanglement
  5. 5 Conclusions
  6. A Classical Hidden Variables and Anti-particles

##  1 Introduction

Quantum mechanics is responsible for a number of physical phenomena that are
impossible in classical physics. Characterizing these unique properties of
quantum systems is a problem of wide scientific interest. Bell famously
provided one such example [1] where the measurements of a small number of
entangled spins (qubits) cleanly distinguishes quantum from local classical
physics [2, 3]. Physical properties of real world systems are far more complex
and therefore isolating their uniquely quantum behavior can be more
challenging [4, 5, 6]. One particularly interesting example is the origin of
the initial density fluctuations in the universe. They are believed to have
arisen from quantum fluctuations during an inflationary epoch [7, 8, 9, 10,
11], however, this hypothesis has been difficult to test observationally. The
universe is classical on cosmological scales and one cannot easily apply
Bell’s inequality to the density fluctuations directly [12, 13] (but see e.g.
[14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] for ongoing
work). Instead, one is lead to ask if quantum mechanics was important in
establishing the (statistical) initial conditions for our classical
cosmological observations.

Quantum effects play a crucial role in the dynamics of the early universe in
many inflationary models. While the resulting observational signals can often
be traced to quantum mechanics, the challenge is showing no classical
mechanism could produce the same signal [29, 30, 31, 32, 33, 34]. One proposal
of this kind was made in [35], where it was shown that the analytic structure
of (non-Gaussian) correlation functions is different for classical and quantum
theories when the correlations are produced by local evolution. The origin of
this difference arises from the non-zero number of particles needed to produce
classical density fluctuations, as illustrated in Figure 1. Creation of
particles from the quantum vacuum violates energy conservation but is allowed
because of the uncertainty principle. In contrast, physical particles will
scatter and decay in an interacting theory while conserving energy, giving
rise to poles at physical momentum for classical fluctuations. These results
are consistent with a number of results relating scattering to the analytic
structure of cosmological correlators [36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49]. These features of the correlators are also directly tied to
the prospects of observing the signal [50, 51, 52].

The relationship between correlators and scattering is, of course, best
understood in flat space. The LSZ reduction formula [53] gives a rigorous map
between in-out correlators and S-matrix elements. While measurements of flat
space correlators are not subject to the same limitations as cosmology, we can
still ask if the analytic structure of flat space in-in correlators encodes
the quantum vacuum in the same way. Furthermore, one could hope to use LSZ to
connect the difference between the analytic structure of classical and quantum
correlators directly to scattering of particles in the initial state.

On a purely theoretical level, flat space provides a testing ground for our
understanding of cosmological correlators. Yet, if in-in correlators are
encoding the physics of the quantum vacuum, one would naturally like to
understand how they are related to measurable quantities. Cosmological
expansion is essential in producing fluctuations from the vacuum during
inflation and does not occur in flat space. To make sense of their flat space
analogues, we must introduce particle detectors localized in the spacetime
(Unruh-de Witt detectors [54, 55]). The very act of measuring the state of the
quantum field at a localized point in spacetime introduces the energy and
momentum needed to excite the vacuum (breaking the time and space
translations). This observation was essential for making sense of Unruh
radiation [54] (i.e. the Rindler temperature), another example of particle
production in flat space. In that case, a thermal distribution of particles is
seen by a constantly accelerating (Rindler) observer because of the energy and
momentum needed to accelerate the detector in the first place [56, 57].
Similarly, measuring correlators at a fixed time also requires the injection
of energy (by the uncertainty principle) and naturally explains the apparent
particle production from nothing which is encoded in the in-in correlators.

The non-local correlations of a field in the quantum vacuum also generates
entanglement between the various particle detectors used to detect them [58,
59, 60, 61, 62, 63, 64, 65, 66, 67, 68]. As such, the detector entanglement
represents a probe of the entanglement of the interacting vacuum of the fields
themselves. Famously, the entanglement entropy of the fields on a finite
region is expected to be proportional to the area for the quantum vacuum [69,
70] and the volume for a generic excited state (see e.g. [71] for review).
While the entanglement entropy is not a quantity we can easily measure (or
calculate), naturally one would like to understand if the non-Gaussian
signature of the quantum vacuum state is related.

In this paper, we will expose the connection between flat space scattering,
entanglement, and cosmological observables through the properties of in-in
correlators. While they are a less natural observables in flat space than the
S-matrix, we show that there is a precise link between the structure of poles
in the in-in correlators and the associated scattering processes, as
illustrated in Figure 1. This provides a robust demonstration that the poles
appearing at physical momenta in classical states are directly tied to the
decay or scattering of particles in the initial state. The poles are absent in
the quantum vacuum because it contains no particles, connecting the analysis
of [35] to flat space amplitudes.

Figure 1: Illustration of the difference between non-Gaussian in-in
correlations of $\phi$ for quantum vacuum fluctuations (left) and classical
fluctuations (right). Measuring quantum fluctuations of $\phi$ at three
spacelike separated points corresponds to the creation of three particles from
the vacuum, producing a total energy pole in the correlator. In contrast,
classical fluctuations only occur in a state containing particles. Any local
classical process that produces a total energy pole will also cause particles
in the initial state to decay, producing additional three point correlations
with poles at physical momenta.

To make physical sense of these results, we show that the in-in correlators
can be interpreted as an the amplitudes to excite multiple UdW detectors
localized at space-like separated points. These detectors then provide a
natural connection between cosmological Bell-type tests and more typical
characterizations in terms of entanglement. The entanglement of these
detectors shares many similarities with the entanglement of the underlying
field; yet, we can directly connect these properties to the underlying in-in
correlators. We will see that the analytic structure of the correlators is
directly related to short or long ranged entanglement of the detectors.

This paper is organized as follows: in Section 2, we will discuss the
relationship between the S-matrix and the in-in correlators, demonstrating our
main results about the analytic structure of these correlators. In Section 3,
we show how to interpret our results in terms of UdW detectors. We then show
how these detectors are entangled in Section 4, and conclude in Section 5.
Appendix A contains additional details about the relationship between
enforcing causality of our classical theory and the existence of anti-
particles.

##  2 From In-In Correlators and the S-matrix

The process of creating particles from the vacuum is an inherently quantum
mechanical phenomenon. It gives rise to structure in inflationary models [7,
8, 9, 10, 11] and Hawking radiation from black holes [72]. In flat space, such
a process is forbidden by energy conservation, but the amplitude is still
formally well-defined as it is related to physical scattering processes by
crossing symmetry. This statement can be made rigorously through the LSZ
reduction formula [53].

We would like to understand how quantum fluctuation can be distinguished from
classical (e.g. thermal) fluctuations. Classical fluctuations may occur in any
spacetime and thus we can ask this question in flat space as well. We will
show in this section that an isolated total energy pole in an equal-time
correlator (in-in or in-out) is precisely a reflection of the amplitude for
production of particles from the vacuum. We will then show that for classical
fluctuations, additional poles arise from the on-shell scattering processes of
particles in the initial state. This difference between quantum vacuum
fluctuations and classical fluctuations is illustrated in Figure 1.

###  2.1 The In-In Formalism

Cosmological correlators are described by (equal time) in-in correlations
functions. In perturbation theory, these are defined as [73, 74]

$\langle{\rm in}|Q(t)|{\rm
in}\rangle=\left\langle\bar{T}\exp\left[i\int_{-\infty(1+i\epsilon)}^{t}H_{\mathrm{int}}(t^{\prime})dt^{\prime}\right]\,Q_{\mathrm{int}}(t)\,T\exp\left[-i\int_{-\infty(1-i\epsilon)}^{t}H_{\mathrm{int}}(t^{\prime})dt^{\prime}\right]\right\rangle\
,$ (2.1)

where $H_{\mathrm{int}}(t)=\int d^{3}x\sqrt{-g}{\cal H}_{\rm
int}({\vec{x}},t)$, $g$ is the determinant of the metric, ${\cal H}_{\rm
int}({\vec{x}},t)$ is the Hamiltonian density of the interaction Hamiltonian,
and $Q_{\rm int}(t)$ is the operator $Q(t)$ in terms of the interaction
picture fields. Computed in a quasi-de Sitter background for super-horizon
modes (i.e. points separated by super-horizon distances or fourier modes with
super-horizon wavelengths) the in-in correlators give the classical
statistical correlations of the initial density fluctuations.

It is occasionally useful to express the in-in correlators in terms of
commutators (although, technically, it only applies when $\epsilon=0$).
Expanding the time-ordered exponentials, one finds

$\displaystyle\langle{\rm in}|Q(t)|{\rm in}\rangle$
$\displaystyle=\sum_{N=0}^{\infty}i^{N}\int_{-\infty}^{t}dt_{N}\int_{-\infty}^{t_{N}}dt_{N-1}\cdots\int_{-\infty}^{t_{2}}dt_{1}$
$\displaystyle\hskip 60.0pt\times\Big{\langle}\big{[}H_{{\rm
int}}\big{(}t_{1}\big{)}\big{[}H_{{\rm
int}}\big{(}t_{2}\big{)}\cdots\big{[}H_{{\rm
int}}\big{(}t_{N}\big{)},Q_{\mathrm{int}}(t)\big{]}\cdots\big{]}\big{]}\Big{\rangle}\
.$ (2.2a)

This representation is useful for two reasons. First, this expression makes
causality manifest as the commutators must vanish outside the lightcone.
Second, the non-zero commutator is the defining characteristic of a quantum
theory and thus this representation is useful in isolating the quantum nature
of the correlators.

Our goal in this paper is to understand what aspects of the in-in correlator
reflect truly quantum fluctuations and what other aspects could arise purely
classically. With this in mind, we will focus on the fluctuations of a single
scalar field $\phi$, which may be represented as a quantum mechanical operator
or a classical stochastic variable. Following [35], we can describe the free
classical or quantum theories using the mode expansion

$\phi({\vec{x}},t)=\int\frac{d^{3}k}{(2\pi)^{3}}e^{i{\vec{k}}\cdot{\vec{x}}}\frac{1}{\sqrt{2k}}[a^{\dagger}_{-{\vec{k}}}\,e^{ikt}+a_{{\vec{k}}}e^{-ikt}]\
,$ (2.3)

where $k\equiv|\vec{k}|$. The distinction between quantum and classical is how
we interpret $a_{\vec{k}}$ and $a^{\dagger}_{\vec{k}}$.

In the quantum theory, $a_{\vec{k}}$ and $a_{\vec{k}}^{\dagger}$ are operators
satisfying

$[a_{\vec{k}},a^{\dagger}_{{\vec{k}}^{\prime}}]=(2\pi)^{3}\delta({\vec{k}}-{\vec{k}}^{\prime})\qquad
a_{\vec{k}}|0\rangle=0\,,$ (2.4)

where $|0\rangle$ is the vacuum of the free theory. In contrast, in the
classical theory they are only random variables obeying the statistics

$\langle
a^{\dagger}_{\vec{k}}a_{{\vec{k}}^{\prime}}\rangle_{c}=\frac{1}{2}(2\pi)^{3}\delta({\vec{k}}-{\vec{k}}^{\prime})=\langle
a_{{\vec{k}}^{\prime}}a^{\dagger}_{\vec{k}}\rangle_{c}\,\ .$ (2.5)

In the free theory, these choices give the same equal-time correlators which
are (essentially) the only cosmological observable. Of course, in flat space,
we would be free to directly measure the commutators of the operators or non-
equal time correlators to expose the difference between classical and quantum
mechanics, but we will restrict ourselves to equal-time to parallel the
cosmological correlators.

We can start with a simple example for illustration: given a massless scalar
$\phi$ with a cubic self interaction, ${\cal H}_{\rm
int}=\frac{1}{3!}\mu\phi^{3}$, the in-in three-point function in terms of
fourier modes (the bispectrum) is given by

$\displaystyle\langle\phi(t,{\vec{k}}_{1})\phi(t,{\vec{k}}_{2})\phi(t,{\vec{k}}_{3})\rangle$
$\displaystyle=$ $\displaystyle 2{\rm
Im}\int^{t}_{-\infty}dt^{\prime}\frac{\mu}{8k_{1}k_{2}k_{3}}e^{-i(k_{1}+k_{2}+k_{3})(t-t^{\prime})}$
(2.6) $\displaystyle=$
$\displaystyle-\frac{\mu}{4k_{1}k_{2}k_{3}(k_{1}+k_{2}+k_{3})},$ (2.7)

where $\langle Q(t)\rangle\equiv\langle\Omega|Q(t)|\Omega\rangle$ is the
correlation in the interacting vacuum, $|\Omega\rangle$. We have only fourier
transformed the spatial coordinates by analogy with a typical cosmological
correlator. Just like a cosmological correlator, we see that this in-in
correlation functions exhibits a pole only in the total energy
$k_{t}=(k_{1}+k_{2}+k_{3})$. The presence of such a pole is a unique signature
of the quantum vacuum and therefore we would like to better understand the
physical significance of this correlation.

For classical fluctuations, the appearance of additional poles can be seen by
perturbatively solving the equations of motion with the same cubic
interaction, ${\cal H}_{\rm int}=\frac{1}{3!}\mu\phi^{3}$, such that

$\phi^{(2)}({\vec{k}},t)=\frac{\mu}{2}\int^{t}dt^{\prime}G(k;t-t^{\prime})\int\frac{d^{3}p}{(2\pi)^{3}}\phi({\vec{p}},t^{\prime})\phi({\vec{k}}-{\vec{p}},t^{\prime})$
(2.8)

where $G(k,t)=\sin kt/k$ is the causal Green’s function. Using this to
calculate the bispectrum we find

$\langle\phi(t,{\vec{k}}_{1})\phi(t,{\vec{k}}_{2})\phi(t,{\vec{k}}_{3})\rangle_{c}=\frac{\mu}{16k_{1}k_{2}k_{3}}\left(\frac{3}{k_{t}}+\sum_{i=1}^{3}\frac{1}{k_{t}-2k_{i}}\right)\
,$ (2.9)

where we assumed the contribution from $t\to-\infty$ vanishes111For vacuum
correlators, this is equivalent to the $i\epsilon$ prescription. For excited
states, this will ultimately be tied to how the poles at physical momentum are
resolved.. We see that the classical example has poles both in the total
energy and in folded configurations where $k_{1}=k_{2}+k_{3}$ and permutations
therefore. In this respect, we see that flat space in-in correlators exhibit
the same structure as the cosmological counter-parts 222In [35], it was shown
that this conclusion is an inevitable consequence of causality and Lorentz
invariance. In Appendix A, we extend this argument to show that it is
equivalent to the need for anti-particles in a relativistic quantum theory..

Naturally, we would also like to understand how one interpolates between the
quantum mechanical and classical result. After all, we certainly live in a
quantum universe and would like to understand how classical fluctuations would
arise. We can make this connection by taking the quantum theory in the limit
where every momentum state is highly occupied,

$|n_{{\vec{k}}}\rangle=\frac{1}{\sqrt{n!}}\left(a^{\dagger}_{{\vec{k}}}\right)^{n}|0\rangle\quad\to\quad|n\rangle\equiv\bigotimes_{{\vec{k}}_{i}}|n_{{\vec{k}}}\rangle.$
(2.10)

Suppose we have a real field $\phi$ is in an $n$-particle state, $|n\rangle$,
we see that the operator $\hat{\phi}$ acts schematically

$\displaystyle\phi({\vec{k}},t)|n\rangle$ $\displaystyle\to$
$\displaystyle\sqrt{n_{-{\vec{k}}}+1}f(k,t)|n_{-{\vec{k}}}+1\rangle|\hat{n};-{\vec{k}}\rangle+\sqrt{n_{{\vec{k}}}}f^{*}(k,t)|n_{{\vec{k}}}-1\rangle|\hat{n};{\vec{k}}\rangle\,$
(2.11) $\displaystyle\langle n|\phi({\vec{k}},t)$ $\displaystyle\to$
$\displaystyle\sqrt{n_{{\vec{k}}}+1}f^{*}(k,t)\langle
n_{{\vec{k}}}+1|\langle\hat{n};{\vec{k}}|+\sqrt{n_{-{\vec{k}}}}f(k,t)\langle
n_{-{\vec{k}}}-1|\langle\hat{n};-{\vec{k}}|,$ (2.12)

where $f(k,t)=e^{ikt}/\sqrt{2k}$ is the positive frequency classical solution,
and we defined

$|\hat{n};{\vec{q}}\rangle\equiv|n\rangle\equiv\bigotimes_{{\vec{k}}_{i}\neq{\vec{q}}}|n_{{\vec{k}}_{i}}\rangle\
.$ (2.13)

The two point function in this state is therefore

$\langle\phi({\vec{k}},t)\phi({\vec{k}}^{\prime},t^{\prime})\rangle=\left((n+1)f^{*}(k,t)f(k^{\prime},t^{\prime})+nf(k,t)f^{*}(k^{\prime},t^{\prime})\right)(2\pi)^{3}\delta({\vec{k}}+{\vec{k}}^{\prime})\
,$ (2.14)

which reproduces our classical statistical when taking $n\to\infty$ with
$f({\vec{k}},t)\sqrt{n}$ fixed. In this limit, the two point function becomes

$\langle\phi({\vec{k}},t)\phi({\vec{k}}^{\prime},t^{\prime})\rangle\to
n\left(f(k,t)f^{*}(k^{\prime},t^{\prime})+f^{*}(k,t)f(k^{\prime},t^{\prime})\right)(2\pi)^{3}\delta({\vec{k}}+{\vec{k}}^{\prime})\
.$ (2.15)

This expression is symmetric in $t\leftrightarrow t^{\prime}$ and thus shows
that $\phi({\vec{k}},t)$ and $\phi({\vec{k}}^{\prime},t^{\prime})$ commute, as
we would expect for a classical variable and not for a quantum mechanical
operator.

###  2.2 Relation to the S-Matrix

Now we want to understand how the poles in our in-in correlators are related
to physical scattering processes.

#### Quantum Vacuum

The most direct relationship between scattering and correlation functions is
the LSZ reduction formula [53]. Given a time-ordered (quantum) vacuum
correlation function in flat space, we can extract the associated S-matrix
elements via

$\displaystyle\left\langle\\{p_{j}\\}_{\text{out}}\right|\\{q_{i}\\}_{\text{in}}\rangle=$
$\displaystyle\int\prod_{i=1}^{m}\left\\{\mathrm{d}^{4}x_{i}\frac{ie^{iq_{i}\cdot
x_{i}}\left(-\square_{x_{i}}+m^{2}\right)}{(2\pi)^{\frac{3}{2}}Z^{\frac{1}{2}}}\right\\}\prod_{j=1}^{n}\left\\{\mathrm{d}^{4}y_{j}\frac{ie^{-ip_{j}\cdot
y_{j}}\left(-\square_{y_{j}}+m^{2}\right)}{(2\pi)^{\frac{3}{2}}Z^{\frac{1}{2}}}\right\\}$
$\displaystyle\times\left\langle\Omega\left|\mathrm{T}\phi\left(x_{1}\right)\ldots\phi\left(x_{m}\right)\phi\left(y_{1}\right)\ldots\phi\left(y_{n}\right)\right|\Omega\right\rangle\
,$ (2.16)

where we are using the metric signature $(-+++)$. The scattering states that
appear on the left of this expression are defined by

$|\\{q_{i}\\}_{\text{in}}\rangle=\lim_{t\to-\infty}\prod_{i}\sqrt{2\omega_{{\vec{q}}_{i}}}a^{\dagger}_{{\vec{q}}_{i}}|\Omega\rangle\qquad\left\langle\\{p_{j}\\}_{\text{out}}\right|=\lim_{t\to+\infty}\langle\Omega|\prod_{j}\sqrt{2\omega_{{\vec{p}}_{j}}}a^{\dagger}_{{\vec{p}}_{j}}\
.$ (2.17)

Because the vacuum, $|\Omega\rangle$, is annihilate by $a_{\vec{k}}$,
isolating the positive frequency via the Fourier transform (i.e. integrating
the correlation function with $\int dte^{-i\omega t}$ for $\omega>0$) also
isolates a particle in the initial state.

In perturbation theory, the time-ordered and in-in correlators are closely
related, allowing us to directly related the poles in each to the associated
S-matrix elements. To calculate the in-out correlators, we need the time
ordered Green’s function

$\langle
0|T(\phi(t_{1},{\vec{k}})\phi(t_{2},{\vec{k}}^{\prime})|0\rangle=\frac{1}{2k}e^{-ik|t_{1}-t_{2}|}\,(2\pi)^{3}\delta({\vec{k}}+{\vec{k}}^{\prime})\
.$ (2.18)

We can calculate an equal time in-out correlator for our example with a cubic
interaction to find the same result as the in-in correlator

$\displaystyle\ \langle
T\phi(0,{\vec{k}}_{1})\phi(0,{\vec{k}}_{2})\phi(0,{\vec{k}}_{3})\rangle$
$\displaystyle=$
$\displaystyle-i\int^{\infty}_{-\infty}dt\frac{\mu}{8k_{1}k_{2}k_{3}}e^{-i(k_{1}+k_{2}+k_{3})|t|}$
(2.19) $\displaystyle=$
$\displaystyle-\frac{\mu}{4k_{1}k_{2}k_{3}(k_{1}+k_{2}+k_{3})}\ .$ (2.20)

To see the connection to the S-matrix elements, we need to consider unequal
times such that

$\displaystyle\langle
T\phi(t_{1},{\vec{k}}_{1})\phi(t_{2},{\vec{k}}_{2})\phi(t_{3},{\vec{k}}_{3})\rangle=$
$\displaystyle-i\int^{\infty}_{-\infty}dt\frac{\mu}{8k_{1}k_{2}k_{3}}e^{-ik_{1}|t_{1}-t|}e^{-ik_{2}|t_{2}-t|}e^{-ik_{3}|t_{3}-t|}$
(2.21) $\displaystyle=$
$\displaystyle-\frac{\mu}{8k_{1}k_{2}k_{3}}\bigg{(}\frac{e^{-ik_{2}(t_{2}-t_{1})-ik_{3}(t_{3}-t_{1})}}{k_{1}+k_{2}+k_{3}}$
(2.22)
$\displaystyle+\frac{e^{ik_{1}(t_{1}-t_{2})-ik_{3}(t_{3}-t_{2})}-e^{-ik_{2}(t_{2}-t_{1})-ik_{3}(t_{3}-t_{1})}}{-k_{1}+k_{2}+k_{3}}$
(2.23)
$\displaystyle+\frac{e^{ik_{1}(t_{1}-t_{3})+ik_{2}(t_{2}-t_{3})}-e^{ik_{1}(t_{1}-t_{2})-ik_{3}(t_{3}-t_{2})}}{-k_{1}-k_{2}+k_{3}}$
(2.24)
$\displaystyle+\frac{e^{ik_{1}(t_{1}-t_{3})+ik_{2}(t_{2}-t_{3})}}{k_{1}+k_{2}+k_{3}}\bigg{)}\
,$ (2.25)

where we have assume $t_{1}<t_{2}<t_{3}$ without loss of generality.

Since our above expression assumes $t_{1}<t_{2}<t_{3}$, applying LSZ while
maintaining this order is consistent if $t_{1}$ is associated with the initial
state and $t_{2},t_{3}$ are the final states. We can calculate the S-matrix
elements by first taking the Fourier transform,

$\langle
T\phi(\omega_{3},\vec{k}_{3})\phi(\omega_{2},\vec{k}_{2})\phi(\omega_{1},\vec{k}_{1})\rangle^{\prime}=\int
dt_{1}dt_{2}dt_{3}e^{i(-\omega_{1}t_{1}+\omega_{2}t_{2}+\omega_{3}t_{3})}\langle
T\phi(t_{3},\vec{k}_{3})\phi(t_{2},\vec{k}_{2})\phi(t_{1},\vec{k}_{1})\rangle\
.$ (2.26)

Performing the integral imposing $t_{1}<t_{2}<t_{3}$ gives the term of
interest

$\displaystyle\langle
T\phi(\omega_{3},\vec{k}_{3})\phi(\omega_{2},\vec{k}_{2})\phi(\omega_{1},\vec{k}_{1})\rangle^{\prime}$
(2.27)
$\displaystyle\supset\frac{i\mu}{8k_{2}k_{2}k_{3}}2\pi\delta(\omega_{2}+\omega_{3}-\omega_{1})\Big{(}-\frac{1}{\omega_{3}-k_{3}}\frac{1}{\omega_{3}-k_{3}+\omega_{2}-k_{2}}\frac{1}{k_{1}+k_{2}+k_{3}}-$
$\displaystyle\frac{1}{\omega_{3}-k_{3}}\frac{1}{\omega_{3}+\omega_{2}-k_{1}}\frac{1}{-k_{1}+k_{2}+k_{3}}+\frac{1}{\omega_{3}-k_{3}}\frac{1}{\omega_{3}-k_{3}+\omega_{2}-k_{2}}\frac{1}{-k_{1}+k_{2}+k_{3}}-$
$\displaystyle\frac{1}{\omega_{3}-k_{1}-k_{2}}\frac{1}{\omega_{3}-\omega_{2}-k_{1}}\frac{1}{-k_{1}-k_{2}+k_{3}}+\frac{1}{\omega_{3}-k_{3}}\frac{1}{\omega_{2}+\omega_{3}-k_{1}}\frac{1}{-k_{1}-k_{2}+k_{3}}$
$\displaystyle-\frac{1}{\omega_{3}-k_{1}-k_{2}}\frac{1}{\omega_{3}+\omega_{2}-k_{1}}\frac{1}{k_{1}+k_{2}+k_{3}}\Big{)}\
.$

Now we want to isolate the part of the correlator that encodes the
${\vec{k}}_{1}\to{\vec{k}}_{2}+{\vec{k}}_{3}$ scattering amplitude. Using LSZ,
we see

$\displaystyle\langle k_{2},k_{3}|k_{1}\rangle$
$\displaystyle=\lim_{\omega_{i}\to
k_{i}}(\omega_{1}-k_{1})(\omega_{2}-k_{2})(\omega_{3}-k_{3})(8k_{1}k_{2}k_{3})\langle
T\phi(\omega_{3},\vec{k}_{3})\phi(\omega_{2},\vec{k}_{2})\phi(\omega_{1},\vec{k}_{1})\rangle$
(2.28)

Since each factor of $(\omega_{i}-k_{i})$ will vanish in the limit
$\omega_{i}\to k_{i}$, it is easy to see that only terms in the in-out
correlator with three poles in on-shell limit will contribute to amplitude. As
result, only two terms from Equation (2.27) contribute to the amplitude

$\displaystyle\langle k_{2},k_{3}|k_{1}\rangle$
$\displaystyle=i\mu(2\pi)\lim_{\omega_{i}\to
k_{i}}\delta(\omega_{2}+\omega_{3}-\omega_{1})\Big{(}-\frac{\omega_{3}-k_{3}}{\omega_{3}-k_{3}}\frac{\omega_{1}-k_{1}}{\omega_{3}+\omega_{2}-k_{1}}\frac{\omega_{2}-k_{2}}{-k_{1}+k_{2}+k_{3}}$
(2.29) $\displaystyle\hskip
142.26378pt+\frac{\omega_{3}-k_{3}}{\omega_{3}-k_{3}}\frac{\omega_{2}-k_{2}}{\omega_{3}-k_{3}+\omega_{2}-k_{2}}\frac{\omega_{1}-k_{1}}{-k_{1}+k_{2}+k_{3}}\Big{)}$
$\displaystyle=-i\mu(2\pi)\delta(k_{2}+k_{3}-k_{1})$

We see that the poles in $k_{1}-k_{2}-k_{3}$ are precisely those that give the
$k_{1}\to k_{2}+k_{3}$ scattering amplitude.

The equal-time in-in and in-out correlation functions are the same, yet in
both cases the poles responsible for a non-zero scattering amplitude vanish at
equal time. This is a reflection of the fact that no scattering process takes
place in the vacuum: the only way to produce the scattering process requires
one of the operators to be placed at $t\to-\infty$ and the other two taken at
$t\to+\infty$. The total energy pole that survives at equal time reflects only
the $0\to 3$ or $3\to 0$ processes that are forbidden by energy conservation.

#### Classical Fluctuations

Time-ordered (in-out) correlation functions are also calculable for classical
statistics. Using the time-ordered Green’s function, we can calculate the
leading correction to $\varphi$ as

$\phi^{(2)}({\vec{k}},t)=\frac{\mu}{2}\int dt^{\prime}G_{\rm
F}(k,t-t^{\prime})\int\frac{d^{3}p}{(2\pi)^{3}}\phi({\vec{p}},t^{\prime})\phi({\vec{k}}-{\vec{p}},t^{\prime})\
,$ (2.30)

where we are using the Feynman propagator

$G_{\rm F}(k,t)=\frac{1}{2k}e^{-ik|t|}\ .$ (2.31)

We can then calculate the time-ordered correlator by substituting this
expression and applying the classical (Gaussian) statistics,

$\displaystyle\langle
T\phi(t,{\vec{k}}_{1})\phi(t,{\vec{k}}_{2})\phi(t,{\vec{k}}_{3})\rangle_{c}$
$\displaystyle=-i\int^{\infty}_{-\infty}dt^{\prime}\frac{\mu}{16k_{1}k_{2}k_{3}}e^{-ik_{1}|t-t^{\prime}|-i(k_{2}+k_{3})(t-t^{\prime})}+{\rm
permutations}$ (2.32)
$\displaystyle=-\frac{\mu}{16k_{1}k_{2}k_{3}}\left(\frac{3}{k_{t}}-\frac{1}{k_{1}-k_{2}-k_{3}}-\frac{1}{k_{2}-k_{3}-k_{1}}-\frac{1}{k_{3}-k_{1}-k_{2}}\right)\
.$

Like the quantum case, this is precisely the same as the in-in correlator and
we see the appearance of poles at physical momenta.

In order to understand these new poles, we first notice that the LSZ formula
does not apply straightforwardly to our classical correlator. Concretely, we
recall that the relation between $\omega>0$ ($\omega<0$) and particles in the
in-state (out-state) relied on the fact that the quantum vacuum is annihilated
by the negative frequency mode, $a_{\vec{k}}|\Omega\rangle=0$. To make sense
of what LSZ would imply for our classical correlations, let us interpret the
classical correlators as arising from highly occupied state, $|n\rangle$. If
we apply LSZ in this state, we get

$\displaystyle\int\mathrm{d}^{4}z\frac{ie^{iq\cdot
z}\left(-\square_{z}+m^{2}\right)}{(2\pi)^{\frac{3}{2}}Z^{\frac{1}{2}}}$
$\displaystyle\left\langle
n_{\text{out}}\left|\mathrm{T}\varphi\left(x_{1}\right)\ldots\varphi\left(x_{m}\right)\varphi\left(z\right)\varphi\left(y_{1}\right)\ldots\varphi\left(y_{n}\right)\right|n_{\text{in}}\right\rangle$
(2.33) $\displaystyle=$ $\displaystyle\sqrt{2\omega_{p}}\left\langle
n_{\text{out}}\left|\mathrm{T}\varphi\left(x_{1}\right)\ldots\varphi\left(x_{m}\right)\varphi\left(y_{1}\right)\ldots\varphi\left(y_{n}\right)a_{p}^{\dagger}\right|n_{\text{in}}\right\rangle$
$\displaystyle-\sqrt{2\omega_{p}}\left\langle
n_{\text{out}}\left|a_{p}^{\dagger}\mathrm{T}\varphi\left(x_{1}\right)\ldots\varphi\left(x_{m}\right)\varphi\left(y_{1}\right)\ldots\varphi\left(y_{n}\right)\right|n_{\text{in}}\right\rangle\
,$

where we have labeled $|n_{\text{out}}\rangle$ and $|n_{\text{in}}\rangle$ to
indicate that they are defined to $t=+\infty$ and $-\infty$ respectively. We
see that the LSZ formula does not isolate a particle in an in- or out-state,
but instead isolated a particle in the in state minus a hole in the out state
(or vice versa) for every field.

A related consequence of this analogue of Equation (2.33) is that equal time
correlators can now exhibit physical poles. Specifically, we can use LSZ to
calculate the S-matrix element,

$\lim_{t\to+\infty}\left(\langle
n_{\text{out}}|a_{k_{2}}a_{k_{3}}a^{\dagger}_{-k_{1}}\right)\,|n_{\text{in}}\rangle=\langle
n_{k_{2}}+1,n_{k_{3}}+1,n_{k_{1}}-1|n_{\text{in}}\rangle\propto{\cal A}_{1\to
2}\delta({\vec{k}}_{1}-{\vec{k}}_{2}-{\vec{k}}_{3})\ .$ (2.34)

This is just the amplitude for the decay of a particle with momentum
${\vec{k}}_{1}$ in the initial state to particles with momentum
${\vec{k}}_{2}$ and ${\vec{k}}_{3}$ in the final state. Since all the
operators acting on $\langle n|$ as $t\to+\infty$ do not vanish, there is no
reason to expect the equal-time correlator to vanish either.

The most straightforward consequence is that the poles at physical momenta
seen in the equal-time in-out (and consequently in-in) correlators, Equation
(2.32), are the same poles responsible for the $1\to 2$ and $2\to 1$ S-matrix
elements determined by LSZ. This can be seen by directly applying the (naive)
LSZ formula to the unequal-time classical in-out correlator. The result
mirrors our quantum calculation. The full expression for the non-equal time
correlator is quite long, but the term responsible for the “S-matrix” element
is now,

$\displaystyle\langle\varphi_{k_{1}}(t_{1})\varphi_{k_{2}}(t_{2})\varphi_{k_{3}}(t_{3})\rangle^{\prime}_{c}\supset$
$\displaystyle\frac{\mu}{16k_{1}k_{2}k_{3}}\Bigg{(}\frac{\cos[k_{2}\left(t_{1}-t_{2}\right)+k_{3}\left(t_{1}-t_{3}\right)]}{k_{1}-k_{2}-k_{3}}$
(2.35)
$\displaystyle+\frac{\cos[k_{1}\left(t_{1}-t_{2}\right)+k_{3}\left(t_{2}-t_{3}\right)]}{-k_{1}+k_{2}+k_{3}}+\frac{\cos\left[k_{1}\left(t_{1}-t_{3}\right)+k_{2}\left(t_{3}-t_{2}\right)\right]}{-k_{1}+k_{2}+k_{3}}\Bigg{)}$
$\displaystyle\xrightarrow[]{t_{1}=t_{2}=t_{3}}\frac{\mu}{16k_{1}k_{2}k_{3}}\frac{1}{-k_{1}+k_{2}+k_{3}}\
.$ (2.36)

In other words, if we were to repeat the LSZ procedure, as in Equation (2.29),
to the first two lines of (2.35), we would recover a non-zero result. The
final line shows that this term survives the equal limit limit, in contrast to
quantum case. This provides a concrete demonstration that the physical poles
in the in-in correlators can be interpreted as the decay of particles in the
initial state, as was argued in [35] for inflationary correlators.

###  2.3 Resolving and Interpreting Poles at Physical Momenta

The connection between in-in correlators and S-matrix elements provides some
useful intuition. However, if the in-in correlators represent a physical
measurements, we would not expect them to have true poles at physical momenta,
as the answers to physical questions are rarely infinite333It is also known
that these divergences are not regulated by standard renormalization
techniques, as was seen from studying the divergences of perturbation theory
in non-Bunch Davies dS vacua [75]. . In the case of inflationary correlators,
it was argued in [35] that the S-matrix elements that are responsible for the
poles also cause the particles to decay, therefore introducing an effective
width. As a result, one might expect the physical poles to be replaced by a
resonance, both in the inflationary context and in flat space. While this
resolution was seen in explicit examples [32, 34], it is less clear that this
the only resolution, particularly in cosmology. In an expanding universe, the
energy density blue shifts in the past and diverges as $t\to-\infty$. As a
result, cosmological correlators are also regulated by the finite duration of
inflation [76, 77, 78, 79, 80, 81].

Flat space provides a very useful testing ground for the regulation of these
physical poles. First, there is no analogue of the blueshift and the energy
does not diverge at early times. As a result, the $t\to-\infty$ limit is not
necessarily unphysical. In addition, in flat space, decays can be forbidden by
energy conservation444As energy is not conserved in cosmology, decays can
always occur. and thus can eliminate the role of a finite width. This is
easily achieved by considering massive scalars in place of our massless
correlators. Using this approach, we will see that it is not strictly
cosmological expansion that is responsible for physical divergences as
$t\to-\infty$.

#### Massive Particles

In flat space, the decays of particles are controlled by energy conservation.
The simplest way to test the connection between the physical poles and decay
is to consider massive particles such that $E_{k}=\sqrt{k^{2}+m^{2}}$ and the
positive frequency mode functions become

$\phi({\vec{k}},t)=\frac{1}{\sqrt{2E_{k}}}e^{iE_{k}t}\ .$ (2.37)

With the modified mode function, we can calculate the equal time in-in
bispectrum as before. For quantum and classical statistics, one finds

$\displaystyle\langle
T\phi(0,{\vec{k}}_{1})\phi(0,{\vec{k}}_{2})\phi(0,{\vec{k}}_{3})\rangle$
$\displaystyle=$
$\displaystyle-\frac{\mu}{4E_{1}E_{2}E_{3}(E_{1}+E_{2}+E_{3})},$ (2.38)

and

$\displaystyle\langle\phi(t,{\vec{k}}_{1})\phi(t,{\vec{k}}_{2})\phi(t,{\vec{k}}_{3})\rangle_{c}=\frac{\mu}{16E_{1}E_{2}E_{3}}$
$\displaystyle\Bigg{(}\frac{3}{E_{t}}-\frac{1}{E_{1}-E_{2}-E_{3}}$ (2.39)
$\displaystyle-\frac{1}{E_{2}-E_{1}-E_{3}}-\frac{1}{E_{3}-E_{1}-E_{2}}\Bigg{)}\
,$

respectively. While we see the poles when $E_{1}+E_{2}=E_{3}$, and
permutations thereof, these poles cannot be reached at physical energies for
the same kinematic reason that the lightest massive particle is stable. As a
result, we see the connection between the stability of these particles and the
absence of poles at physical momentum.

At first sight, this indeed suggests that the finite width of the particle is
sufficient to avoid poles at physical momenta. At least for the three point
function, eliminating the width also eliminated the pole. However, if we
continue to higher point correlators, even stable particles can lead to poles
as physical momenta. We do not find such poles at four-points with a
$\lambda\phi^{4}$ interaction. However, the classical five point correlator
due to a contact interaction ${\cal H}_{\rm
int}=\frac{1}{5!}\frac{\phi^{5}}{\Lambda}$ takes the form

$\langle\phi(t,{\vec{k}}_{1})..\phi(t,{\vec{k}}_{5})\rangle_{c}=\frac{1}{256\Lambda
E_{1}E_{2}E_{3}E_{4}E_{5}}\bigg{(}\frac{5}{E_{\rm
tot}}+\sum_{i}\frac{3}{(E_{\rm tot}-2E_{i})}+\sum_{i\neq j}\frac{1}{E_{\rm
tot}-2E_{i}-2E_{j}}\bigg{)}$ (2.40)

where $E_{\rm tot}=\sum_{i}E_{i}$. The final term contains a pole that is
consistent with the allowed kinematic region of $2\to 3$ scattering. In this
sense, we can see that nothing prevents us from reaching this pole for
physical momentum. In addition, since the particles don’t decay, there is no
finite width that needs to be included in this calculation. Finally, there is
no analogue of the blue-shifting of energies at early times that demands that
we regulate the early time limit of this calculation. Clearly we need another
physical interpretation for how this pole arises.

#### Finite Time of Interactions

The origin of the physical poles in the classical case can be understood from
the integral expression for the correlator,

$\langle\phi(t,{\vec{k}}_{1})..\phi(t,{\vec{k}}_{5})\rangle_{c}=\frac{1}{16\Lambda
E_{1}..E_{5}}\sum_{i}\int_{-\infty}^{t}dt^{\prime}\sin(E_{i}(t-t^{\prime}))\prod_{j\neq
i}\cos(E_{j}(t-t^{\prime}))$ (2.41)

When we sit on a pole where $E_{\rm tot}-2E_{i}-2E_{j}$, there is a non-
oscillatory contribution to the integrand such that the integral diverges at
$t\to-\infty$. This is a reflection of the fact that there is now an on-shell
process that changes the classical distribution. Specifically, our Gaussian
state $|n\rangle$ is not a stationary configuration in the presence of these
interactions. Instead, the particles can now scatter, exchanging energy,
momentum, and even particle number. Given an infinite amount of time to
interact, we should expect the final distribution of particles to be a
stationary configuration, e.g. a thermal distribution. Stationary states do
not normally exhibit the long range correlations of the quantum vacuum
fluctuations555We are not claiming long range correlations are impossible in
general, but the consistent appearance of these poles in perturbation theory
suggests that it does not arise from a generic local Hamiltonian near a
Gaussian fixed point..

We should therefore create the initial Gaussian state at a finite time in the
past, $t_{i}<t$, such that the distribution at time $t$ is only weakly non-
Gaussian. For simplicity, we return to the bispectrum

$\displaystyle\langle\phi(t,{\vec{k}}_{1})\phi(t,{\vec{k}}_{2})\phi(t,{\vec{k}}_{3})\rangle_{c}=$
$\displaystyle\frac{\mu}{16k_{1}k_{2}k_{3}}\Bigg{(}\frac{3\left(1-\cos(k_{t}\Delta
t)\right)}{k_{t}}$ (2.42)
$\displaystyle+\bigg{(}\frac{1-\cos\left((k_{1}-k_{2}-k_{3})\Delta
t\right)}{k_{1}-k_{2}-k_{3}}+{\rm permutations}\bigg{)}\Bigg{)}$ (2.43)

where $\Delta t=t-t_{i}$. Unlike a mass or width which moves the poles to
complex momenta, now we see that there are no poles at all. Instead, when we
take $k_{1}\to k_{2}+k_{3}$

$\frac{\mu}{16k_{1}k_{2}k_{3}}\frac{1-\cos\left((k_{1}-k_{2}-k_{3})\Delta
t\right)}{k_{1}-k_{2}-k_{3}}\to\frac{\mu}{32(k_{1}+k_{2})k_{2}k_{3}}(\Delta
t)^{2}(k_{1}-k_{2}-k_{3})\ ,$ (2.44)

which vanishes as $k_{1}-k_{2}-k_{3}\to 0$. This correlator gets its largest
contribution when $(k_{1}-k_{2}-k_{3})\approx\Delta t^{-1}$ and is enhanced
relative to the total energy pole by a factor $\Delta tk_{t}$. As a result,
the signal in the folded configurations will still dominate over the
equilateral [82], much like the cosmological setting [35], . This is
consistent with more general expectation about the perturbative structure of
cosmological correlators. On general grounds, even the total energy pole is
expected to vanish for cosmological correlators in a UV complete theory [83,
38]. Yet, in perturbation theory the poles accurately capture the observable
signals [84].

##  3 Particle Detectors and the In-In Formalism

The structure of in-in correlation in flat space is largely the same as in
(quasi-) de Sitter space. However, without cosmological particle production,
we don’t have an obvious interpretation of the correlator in terms of some
physical process. One might worry that this is some mathematical devise that
lacks a physics reality outside of cosmology. In this section, we will show
how the in-in correlator arises in physical models of particle detection. This
will allow us to give a clear physical meaning to the flat space correlators
and their poles.

###  3.1 Unruh – de Witt Detectors

We will first review the Unruh-de Witt (UdW) model for particle detection. The
central idea is that we have a single qubit, that registers whether or not
there was a particle in some localized region of space-time. To do so, we
place it in the zero-state initially ($|0\rangle_{\bm{s}}$). We then couple
this qubit to our field, $\phi({\vec{x}},t)$, for some finite amount of time
and inside some localized region of space. After we turn off the coupling to
$\phi$, we should have a non-zero probability of finding our qubit in the one-
state, $|1\rangle_{\bm{s}}$, if there was a particle (or were particles) in
the detector while it was on.

We implement this model by introducing an interaction Hamiltonian that couples
our qubit to $\phi$. Following [56], the detector is described by a
Hamiltonian

$H_{\rm D}=\lambda\,\epsilon(t)\int
d^{3}x\phi({\vec{x}},t)\left[\psi({\vec{x}}){\hat{\bm{s}}}+\psi^{*}({\vec{x}}){\hat{\bm{s}}}^{\dagger}\right]\
,$ (3.1)

where $\lambda$ is the coupling constant, $\epsilon(t)$ is a function that
defines how we turn on/off the detector, and $\psi({\vec{x}})$ is a function
that defines the spatial resolution of our detector such that $\psi(x)$
vanishes outside the detector. The detector state is defined by

${\hat{\bm{s}}}|0\rangle_{\bm{s}}={\hat{\bm{s}}}^{\dagger}|1\rangle_{\bm{s}}=0\qquad{\hat{\bm{s}}}^{\dagger}|0\rangle_{\bm{s}}=|1\rangle_{\bm{s}}\qquad{\hat{\bm{s}}}|1\rangle_{\bm{s}}=|0\rangle_{\bm{s}}\
,$ (3.2)

and the free field is again given by

$\phi({\vec{x}},t)=\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{\sqrt{2k}}e^{i{\vec{k}}\cdot{\vec{x}}}[a^{\dagger}_{-{\vec{k}}}e^{ikt}+a_{{\vec{k}}}e^{-ikt}]\
.$ (3.3)

Even though we are giving a quantum description of the detector, the
fluctuations of field $\phi$ may be quantum or classical.

Let us check that, at leading order in $\lambda$, this detector works as
promised. We will put the scalar field into a single particle state

$|\phi_{1}({\vec{y}},t_{0})\rangle=\phi({\vec{y}})|0\rangle=\int\frac{d^{3}p}{(2\pi)^{3}}e^{-i{\vec{p}}\cdot{\vec{y}}}f(p,t_{0})a^{\dagger}_{\vec{p}}|0\rangle\
,$ (3.4)

where $f(p,t)=e^{ipt}/\sqrt{2p}$ as before. If we now turn on the detector
interaction, the probability for finding an excited detector at leading order,
${\cal O}(\lambda^{2})$, is

$P_{1}=|{\cal A}_{1;1\to 0}|^{2}+|{\cal A}_{1;1\to 2}|^{2}$ (3.5)

where

$\displaystyle{\cal A}_{1;1\to 0}$ $\displaystyle=$ $\displaystyle\langle
0|\lambda\int dt\epsilon(t)\int
d^{3}x\psi^{*}({\vec{x}})\int\frac{d^{3}p}{(2\pi)^{3}}e^{i{\vec{p}}\cdot({\vec{x}}-{\vec{y}})}f^{*}(p,t)f(p,t_{0})|0\rangle$
(3.6) $\displaystyle=$ $\displaystyle\lambda\int dt\epsilon(t)\int
d^{3}x\psi^{*}({\vec{x}})G_{\rm F}({\vec{x}},t;{\vec{y}},t_{0})$ (3.7)

and

$\displaystyle{\cal A}_{1;1\to 2}$ $\displaystyle=$ $\displaystyle\langle
2|\lambda\int dt\epsilon(t)\int
d^{3}x\psi^{*}({\vec{x}})\int\frac{d^{3}p}{(2\pi)^{3}}e^{i{\vec{p}}\cdot({\vec{x}}-{\vec{y}})}\phi(-{\vec{p}},t)\phi_{0}({\vec{p}})a^{\dagger}_{p}|0\rangle$
(3.8) $\displaystyle=$ $\displaystyle\lambda\int dt\epsilon(t)\int
d^{3}x\psi^{*}({\vec{x}})\langle 2|\phi(x,t)\phi(y,t_{0})|0\rangle\ .$ (3.9)

The first term, ${\cal A}_{1;1\to 0}$, is the amplitude that the particle at
${\vec{y}}$ and time $t_{0}$ is absorbed at time $t$ in the detector located
at ${\vec{x}}$. This term captures the physics of interest, namely the
detection of the particle that we put in the initial state. The second term,
${\cal A}_{1;1\to 2}$, the probability that the detector creates an anti-
particle666The real scalar $\phi$ is its own anti-particle, but this
distinction is helpful for againing intuition. See Appendix A for more
details.  from the vacuum while registering a particle in the detector. This
contribution is not the detection of our initial particle, but is the
detection of a particle created from the vacuum by the detector.

One important aspect of the UdW detector is that it shows concretely that the
act of measuring the field $\phi$ changes the state of the system. In
particular, if we try to measure a particle in the vacuum (of the free
theory), a non-zero amplitude for exciting the detector (at order $\lambda$)
requires the creation of an anti-particle. We can see this, in analogy with
Equation (3.4), by projecting onto a 1-particle final state,

${\cal A}_{1;0\to 1}=\lambda\int dt\epsilon(t)\int
d^{3}x\psi({\vec{x}})\langle 1|\phi(x,t)|0\rangle\ .$ (3.10)

We can interpret this as follows: the act of performing the measurement
absorbs a particle from a particle-anti-particle pair, creating an outgoing
anti-particle of equal and opposite momentum in the process. Importantly, it
is the act of localizing the measurement in time and space that provides the
energy and momentum needed to excite the vacuum. This explanation coincides
with our intuition from the uncertainty principle.

To confirm the interpretation, let us consider what happens as we change the
function $\epsilon(t)$ to be less localized in time, thus corresponding to
smaller energies by the uncertainty principle. If we create the one-particle
state at a time $t_{0}<0$ and measure at $t\approx 0$ with

$\epsilon(t)=\frac{1}{\sqrt{2\pi}\sigma_{t}}e^{-t^{2}/(2\sigma_{t}^{2})}\qquad\psi({\vec{x}})=\delta({\vec{x}})\
,$ (3.11)

then the amplitude becomes

${\cal A}_{1;0\to 1}=\lambda\int
dt\epsilon(t)\frac{1}{2E_{\vec{k}}}e^{iE_{\vec{k}}t}=\frac{\lambda}{2E_{k}}\,e^{-E_{\vec{k}}^{2}\sigma_{t}^{2}/2}\
.$ (3.12)

This is, again, nothing more than the uncertainty principle, as our resolution
in energy is inversely proportional to our resolution in time,
$\sigma_{E}\propto\sigma^{-1}_{t}$. Given that we start in the vacuum (zero
energy), we must have a large uncertainty in energy to create particles. For
example, if we were to work with massive particles such that $E_{\vec{k}}\geq
m$ for all ${\vec{k}}$, the probability of finding a particle in the vacuum is
exponentially suppressed unless $\sigma_{E}\gg m$ or, equivalently,
$\sigma_{t}\ll 1/m$.

###  3.2 Particle Detection and Cosmological Correlators

Now suppose we want to measure correlations of these vacuum fluctuations using
our particle detector777Here we are using particle detectors in flat space,
but it is interesting to also consider these detectors as a probe of inflation
directly [85].. We can imagine placing $N$ UdW detectors at distinct points in
space ${\vec{x}}_{i}$ such that the detectors do not overlap,
$\psi({\vec{x}}-{\vec{x}}_{i})\psi({\vec{x}}-{\vec{x}}_{j})=0$ for $i\neq j$.
Furthermore, we will assume that $\epsilon(t)$ is sufficiently localized in
time such that the detectors are all space-like separated when they are on. We
denote $|\Omega\rangle$ as the interacting vacuum of $\phi$ when $\lambda=0$,
so that the amplitude for all $N$ detectors to be in the $|1\rangle_{\bm{s}}$
state and for $\phi$ to remain in the vacuum is

${\cal A}_{N;\Omega}=\left(\prod_{i}\int
dt^{\prime}_{i}d^{3}x^{\prime}_{i}\lambda\epsilon(t-t^{\prime}_{i})\psi({\vec{x}}_{i}-{\vec{x}}_{i}^{\prime})\right)\langle\Omega|\phi(t^{\prime}_{1},{\vec{x}}^{\prime}_{1})..\phi(t_{N}^{\prime},{\vec{x}}^{\prime}_{N})|\Omega\rangle+{\rm
local}\ .$ (3.13)

We see the this amplitude is proportional to the in-in correlator, convolved
with the detector. Because the detectors are spacelike separated, from
Equation (2.1) we see that any additional terms associated with the commutator
of the interaction Hamiltonian, $H_{\rm int}$, with the detector Hamiltonian,
$H_{\rm D}$, will give purely local terms (i.e. these contributions are
equivalent to field redefinitions of $\phi$ and do not produce a total energy
pole). We can write this in terms of the in-in correlation function in fourier
space as

${\cal A}_{N;\Omega}=\left(\lambda^{N}\prod_{i}\int
dt^{\prime}_{i}d^{3}k_{i}e^{-i{\vec{k}}_{i}\cdot({\vec{x}}_{i}-{\vec{x}}_{i}^{\prime})}\tilde{\epsilon}(t-t^{\prime}_{i})\tilde{\psi}({\vec{k}}_{i})\right)\langle\Omega|\phi(t^{\prime}_{1},{\vec{k}}_{1})..\phi(t_{N}^{\prime},{\vec{k}}_{N})|\Omega\rangle$
(3.14)

where we used the notation where $\tilde{\psi}({\vec{k}})$ is the fourier
transforms of $\psi({\vec{x}})$. We see that the fourier modes that contribute
to this amplitude are only those that appear in the detector itself via
$\tilde{\psi}(k)$.

In the case where $\epsilon(t)\approx\delta(t)$ and
$\psi({\vec{x}})\approx\delta({\vec{x}})$ with $t^{\prime}_{i}\to t$, the
amplitude becomes proportional to the equal time in-in correlator in position
space:

${\cal
A}_{N;\Omega}\to\lambda^{N}\langle\Omega|\phi(t,{\vec{x}}_{1})..\phi(t,{\vec{x}}_{N})|\Omega\rangle\
.$ (3.15)

For quantum vacuum fluctuations, the amplitude for three coincident particles
becomes

$\displaystyle{\cal A}_{3;\Omega}$ $\displaystyle=$
$\displaystyle\lambda^{3}\langle\phi({\vec{x}}_{1},t)\phi({\vec{x}}_{2},t)\phi({\vec{x}}_{3},t)\rangle$
(3.16) $\displaystyle=$
$\displaystyle-\lambda^{3}\frac{\mu}{4}\left[\prod_{i}\int\frac{d^{3}k_{i}}{(2\pi)^{3}}e^{i{\vec{k}}_{i}\cdot{\vec{x}}_{i}}\right]\frac{1}{(k_{1}+k_{2}+k_{3})}(2\pi)^{3}\delta({\vec{k}}_{1}+{\vec{k}}_{2}+{\vec{k}}_{3})$
(3.17) $\displaystyle=$
$\displaystyle-\lambda^{3}\frac{\mu}{4}\int\frac{d^{3}k_{1}}{(2\pi)^{3}}e^{i{\vec{k}}_{1}\cdot{\vec{x}}_{13}}\int\frac{d^{3}k_{2}}{(2\pi)^{3}}e^{i{\vec{k}}_{2}\cdot{\vec{x}}_{23}}\frac{1}{(k_{1}+k_{2}+k_{3})k_{1}k_{2}k_{3}}\
,$ (3.18)

where we defined ${\vec{x}}_{ij}\equiv\vec{x}_{i}-\vec{x}_{j}$. In order to
make the ${\vec{k}}_{i}$-integrals manageable, let us assume $x_{2}$ is far
from $x_{1}$ and $x_{3}$ so that $x_{13}\ll x_{23}$. We can then expand in
$k_{1}\approx k_{3}\gg k_{2}$ to find

$\displaystyle\langle\phi({\vec{x}}_{1},t)\phi({\vec{x}}_{2},t)\phi({\vec{x}}_{3},t)\rangle$
$\displaystyle\approx-\frac{\mu}{4}\int\frac{d^{3}k_{1}}{(2\pi)^{3}}e^{i{\vec{k}}_{1}\cdot{\vec{x}}_{13}}\int\frac{d^{3}k_{2}}{(2\pi)^{3}}e^{i{\vec{k}}_{2}\cdot{\vec{x}}_{23}}\frac{1}{2k_{1}^{3}k_{2}}$
(3.19) $\displaystyle\approx\frac{\mu}{32\pi^{4}}\frac{\log
x_{13}}{x_{23}^{2}}\ .$ (3.20)

For comparison, the two point function of the massless field $\phi$ will fall
off like $1/x^{2}$ ($\phi$ is dimension one in 3+1 dimensions). As we separate
the detectors, it is important that the contribution to the amplitude will
decay with the same power of the distance as the it would in the free theory.

In contrast, let us see what happens to the detector in the presence of
classical fluctuations. We again take the in-in three point function and we
will focus on the contribution from a physical pole (the total energy pole we
give the same result as the quantum theory),

$\displaystyle\langle\phi({\vec{x}}_{1},t)\phi({\vec{x}}_{2},t)\phi({\vec{x}}_{3},t)\rangle_{c}\supset$
$\displaystyle\frac{\mu}{16}\left[\prod_{i}\int\frac{d^{3}k_{i}}{(2\pi)^{3}}e^{i{\vec{k}}_{i}\cdot{\vec{x}}_{i}}\right]\frac{1}{k_{1}k_{2}k_{3}(k_{1}+k_{2}-k_{3})}(2\pi)^{3}\delta({\vec{k}}_{1}+{\vec{k}}_{2}+{\vec{k}}_{3})$
$\displaystyle\supset$
$\displaystyle\frac{\mu}{16}\int\frac{d^{3}k_{1}}{(2\pi)^{3}}e^{i{\vec{k}}_{1}\cdot{\vec{x}}_{13}}\int\frac{d^{3}k_{2}}{(2\pi)^{3}}e^{i{\vec{k}}_{2}\cdot{\vec{x}}_{23}}\frac{1}{k_{1}k_{2}k_{3}}\frac{1}{k_{1}+k_{2}-k_{3}}\
,$ (3.21)

where in the second line $k_{3}=|{\vec{k}}_{1}+{\vec{k}}_{2}|$. Again we will
assume the point $x_{2}$ is far from $x_{1}$ and $x_{3}$. As such we consider
the limit $x_{13}\ll x_{23}$ by expanding in $k_{1}\approx k_{3}\gg k_{2}$.

$\displaystyle\langle\phi({\vec{x}}_{1},t)\phi({\vec{x}}_{2},t)\phi({\vec{x}}_{3},t)\rangle_{c}$
$\displaystyle\approx$
$\displaystyle\frac{\mu}{16}\int\frac{d^{3}k_{1}}{(2\pi)^{3}}e^{i{\vec{k}}_{1}\cdot{\vec{x}}_{13}}\int\frac{d^{3}k_{2}}{(2\pi)^{3}}e^{i{\vec{k}}_{2}\cdot{\vec{x}}_{23}}\frac{1}{k_{1}^{2}k_{2}^{2}(1-\cos\theta)}$
(3.22) $\displaystyle\approx$
$\displaystyle\frac{\mu}{64\pi^{4}}\frac{1}{x_{13}x_{23}}\left(-\log\left(\frac{\theta_{\rm
min}^{2}}{2}\right)\,f(\hat{x}_{23}\cdot\hat{x}_{13})\right)\ ,$ (3.23)

where $\theta_{\rm min}$ is the minimum angle between ${\vec{k}}_{1}$ and
${\vec{k}}_{2}$ and $f(\hat{x}_{23}\cdot\hat{x}_{13})$ is a function of the
angle between $\vec{x}_{13}$ and $\vec{x}_{23}$ with $f(1)=1$. This formula is
noteworthy for two reasons: first, it has a pole as $x_{13}\to 0$ which
enhances the size of the non-Gaussian signal. Second, it falls off more slowly
than the Gaussian two-point function and will therefore give the dominant
contribution to any long distance correlations.

###  3.3 Classical Interpretation

We have seen how the detector model provides a physical interpretation of the
quantum in-in correlators. For quantum fluctuations, we can naturally
understand the role of the detector in exciting the vacuum and giving rise to
particles. The uncertainty principle tells us that localizing a measurement in
time will mean that we are no longer in an energy eigenstate (in this case,
the vacuum). This is an inevitable feature of a quantum-measurement involving
an operator that doesn’t commute with the Hamiltonian.

Stated this way, it is the interpretation of the classical measurement that
requires explanation. Classical measurements do not have to disturb the state
and therefore the response of our detector should be a fundamental property of
the classical system. On the other hand, our derivation of the amplitude for
exciting the detector did not assume the in-in correlator was calculated in
the vacuum and would be equally applicable in the classical limit using

$\langle\phi({\vec{k}}_{1})..\phi({\vec{k}}_{N})\rangle_{c}=\lim_{n\to\infty}2{\rm
Im}\langle n|\phi({\vec{k}}_{1})..\phi({\vec{k}}_{N})\int^{t}dt^{\prime}H_{\rm
int}(t^{\prime})|n\rangle\ .$ (3.24)

For this to be consistent with something classical, it must be how we
interpret the state of the detector that has to change. For states close to
the quantum vacuum, we demonstrate that the UdW detector is designed to be
excited when a particle is present in the initial state. In the classical
case, we have lots of particles in the initial state, even in the absence of
fluctuations, and therefore the response of the detector to this state
requires more care.

Consider what happens as we evolve the state $|n\rangle$ in the presence of
the detector Hamiltonian, $H_{D}$. Working to linear order in $\lambda$, we
get at time $t$,

$\displaystyle|0\rangle_{\bm{s}}\,|n\rangle\to$
$\displaystyle|0\rangle_{\bm{s}}\,|n\rangle+i\lambda|1\rangle_{\bm{s}}\int
d^{3}x\psi(x)\hat{\phi}({\vec{x}},t)|n\rangle$ (3.25) $\displaystyle=$
$\displaystyle|0\rangle_{\bm{s}}\,|n\rangle$ (3.26)
$\displaystyle+i\lambda|1\rangle_{\bm{s}}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\psi(-{\vec{k}})}{\sqrt{2E_{k}}}\left(e^{ikt}\sqrt{n_{-{\vec{k}}}+1}|n_{-{\vec{k}}}+1\rangle|\hat{n};-{\vec{k}}\rangle+e^{-ikt}\sqrt{n_{{\vec{k}}}}|n_{{\vec{k}}}-1\rangle|\hat{n};{\vec{k}}\rangle\right)\
.$

The second term shows that the detector will register a “particle” for both an
increase and decrease in the total number of particles. In addition, we notice
that in the limit $n_{k}\to\infty$, this is approximately

$\displaystyle\lim_{n\to\infty}|0\rangle_{\bm{s}}\,|n\rangle\to$
$\displaystyle|0\rangle_{\bm{s}}\,|n\rangle+i\lambda|1\rangle_{\bm{s}}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\psi(-{\vec{k}})}{\sqrt{2E_{k}}}\sqrt{n_{k}}\cos(kt)|n\rangle\
.$ (3.27)

Concretely, our state is responding to the classical (real) oscillations of
$\phi$ around the mean density.

In the free theory, there was no problem working in this very carefully
defined excited state, $|n\rangle$. Including interactions not only produces
higher-point correlations, it also takes the state away from the static
initial state we took in the Gaussian theory. Since the detectors are
essentially registering changes in the state away from the Gaussian initial
state, this classical evolution alone should be sufficient to excite the
detectors. Evolving the state according the interacting Hamiltonian, we find

$\displaystyle|n\rangle\to$ $\displaystyle|n\rangle+\int dt^{\prime}H_{\rm
int}(t^{\prime})|n\rangle$ (3.28) $\displaystyle=$
$\displaystyle|n\rangle+\frac{\mu}{3!}\int dt^{\prime}\int
d^{3}x\phi^{3}({\vec{x}},t^{\prime})|n\rangle$ (3.29) $\displaystyle=$
$\displaystyle|n\rangle+\frac{\mu}{3!}\int
dt^{\prime}\bigg{(}\prod_{i}\int\frac{d^{3}k_{i}}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{k_{i}}}}\Big{(}e^{ik_{i}t^{\prime}}\sqrt{n_{-{\vec{k}}_{i}}+1}|n_{-{\vec{k}}_{i}}+1\rangle|n_{{\vec{k}}_{i}}\rangle$
(3.30)
$\displaystyle\qquad+e^{-ik_{i}t^{\prime}}\sqrt{n_{{\vec{k}}_{i}}}|n_{{\vec{k}}_{i}}-1\rangle|n_{-{\vec{k}}_{i}}\rangle\Big{)}\bigg{)}\delta^{(3)}(\sum_{i}{\vec{k}}_{i})|\hat{n};{\vec{k}}_{1},{\vec{k}}_{2},{\vec{k}}_{3},-{\vec{k}}_{1},-{\vec{k}}_{2},-{\vec{k}}_{3}\rangle\
.$ (3.31)

We see that the time evolution of state changes the number (density) of
particles of different momenta. This change to the density is precisely of the
form that our UdW detector will register as a “particle”. Classically, the
detector is not responsible for exciting the fluctuations, those were already
created by the classical-time evolution of $\phi$.

An additional confusion with this description is the meaning of the total
energy pole. The poles at physical momenta capture the decay of the particles
in the initial state which leads to density fluctuations by changing the
number densities of particles with different momenta. The total energy pole
does not have such a simple description classically. In the quantum theory, we
interpreted this pole as a violation of energy conservation, via the
uncertainty principle, which would be forbidden in a classical theory.
However, the quantum interpretation was needed because the quantum vacuum is
an energy eigenstate of the full interacting Hamiltonian. In contrast, what we
see in Equation (3.31) is that the state $|n\rangle$ is not a stationary state
of the classical Hamiltonian. In the presence of the interaction, the system
wants to evolve towards equilibrium as explained in Section 2.3. The total
energy pole is therefore not a violation of energy conservation but a
reflection that our classical probabilistic system included states of
different energies initially.

##  4 Detector Entanglement

Entanglement has become an increasingly valuable probe of fundamental physics.
It can reveal the structure of quantum field theories and states of matter in
flat space [71]. In curved space times, it is central to our understanding of
black holes [86, 87, 88] and particle production [89]. Entanglement is even
thought to encode the causal structure of spacetime itself [90, 91, 92].

In the quantum field theory, the entanglement entropy in vacuum is expected to
follow an area law [69, 70], while it should scale as volume in a generic
excited state. This offers a different starting point for understanding the
nature of the quantum vacuum than the structure of poles in cosmological
correlators. In this section, we will explore to what degree these properties
are related.

In order to make the comparison, we will consider $N$ UdW-detectors and some
free field $\phi$ such that the state of the detectors is

$\displaystyle\left|\Psi_{\phi,{\rm UdW}}\right\rangle$
$\displaystyle=(\mathbf{1}-C)|0_{\\{i\\}}\rangle-i\sum_{j}\Phi_{j}|\Omega\rangle|1_{j},0_{\\{i,\hat{j}\\}}\rangle-\sum_{j,k}\Phi_{j}\Phi_{k}|\Omega\rangle|1_{j}1_{k},0_{\\{i,\hat{j},\hat{k}\\}}\rangle+O\left(\lambda^{3}\right)\
,$ (4.1)

where $\Phi_{i}=\lambda\int dt\epsilon_{i}(t)\int
d^{3}x\psi_{i}(\vec{x})\phi(\vec{x},t)$, $C$ is a normalization constant, and
the detector states are defined by

$|1_{j},0_{\\{i,\hat{j}\\}}\rangle\equiv{\hat{\bm{s}}}^{\dagger}_{j}|0\rangle_{\bm{s}_{j}}\bigotimes_{i\neq
j}|0\rangle_{\bm{s}_{i}}\qquad|1_{j}1_{k},0_{\\{i,\hat{j},\hat{k}\\}}\rangle\equiv{\hat{\bm{s}}}^{\dagger}_{j}|0\rangle_{\bm{s}_{k}}\otimes{\hat{\bm{s}}}^{\dagger}_{k}|0\rangle_{\bm{s}_{k}}\bigotimes_{i\neq
j,k}|0\rangle_{\bm{s}_{i}}\ .$ (4.2)

This setup is the $N$-particle generalization of the detector configuration
described in [59, 60].

We will be interested in understanding the entanglement of the individual
detectors as a probe of the state of $\phi$. As the detectors already encode
the cosmological correlators, this offers a simple approach to connecting
these correlations with the entanglement. Our approach will be qualitatively
similar entanglement harvesting [64, 65]. However, given our cosmological
motivations, we will not be interested in whether the detectors themselves are
in a uniquely quantum state (cosmological observations are always classical
after all). Instead we want to use it as a probe of the quantum nature of
$\phi$ itself.

We will assume that we know that $\phi$ is in the interacting vacuum. This is
a useful assumption as we can project the state in Equation (4.1) onto the
vacuum, so that the state of the detector becomes

$\displaystyle\left|\Psi_{\rm
UdW}\right\rangle\equiv\langle\Omega\left|\Psi_{\phi,{\rm UdW}}\right\rangle$
$\displaystyle=(\mathbf{1}-C)|0_{\\{i\\}}\rangle-\sum_{j,k}\langle\Omega|\Phi_{i}^{+}\Phi_{j}^{+}|\Omega\rangle|1_{j}1_{k},0_{\\{i,\hat{j},\hat{k}\\}}\rangle+O\left(\lambda^{3}\right)\
.$ (4.3)

As a theoretical tool, this projection has the advantage that it leaves the
detectors in a pure quantum state. In other words, we have projected onto the
product state so that the density matrix takes the form $\rho_{\phi,{\rm
UdW}}=\rho_{{\rm UdW}}\bigotimes|\Omega\rangle\langle\Omega|$. As a result,
the entanglement between the detector and the field degrees of freedom has
been set to zero, $S_{\text{detector}}=S_{\phi}=0$. Note that this procedure
does not represent a true measurement, as the purpose of the detectors is to
measure the state of the field $\phi$, which we presumably to not know
directly.

###  4.1 Position Space Entanglement

First, we will assume that each detector is localized at a point in space-time
so that the detector is described by
$\psi_{i}({\vec{x}})\approx\delta({\vec{x}}-{\vec{z}}_{i})$ for some point
${\vec{z}}_{i}$ and $\epsilon(t)\approx\delta(t)$. Note that with these
choices, $\lambda$ has units of distance. We now want to use the entanglement
between the UdW detectors as a proxy for the entanglement in the underlying
state of $\phi$. Having localized the detectors in space, it is natural to
consider entanglement between the detectors localized in two regions $A$ and
$B$.

For simplicity, let us take the region $A$ as a sphere of radius $R$ and
$B=\overline{A}$ is the region outside the sphere. Now we want to define the
state of the detectors located inside $A$ and $B$, $\vec{x}_{i}\in A$ and
$\vec{y}_{j}\in B$, as follows:

$\displaystyle|0,0\rangle$ $\displaystyle\equiv|0_{\\{i\\}}\rangle$ (4.4)
$\displaystyle|\\{{\vec{x}}_{n}\\},0\rangle$
$\displaystyle\equiv|1_{{\vec{x}}_{1}}..1_{{\vec{x}}_{n}},0_{\\{i,\hat{x}_{1},..,\hat{x}_{n}\\}}\rangle$
(4.5) $\displaystyle|0,\\{{\vec{y}}_{N}\\}\rangle$
$\displaystyle\equiv|1_{{\vec{y}}_{1}}..1_{{\vec{y}}_{N}},0_{\\{i,\hat{y}_{1},..,\hat{y}_{N}\\}}\rangle$
(4.6) $\displaystyle|\\{{\vec{x}}_{n}\\},\\{{\vec{y}}_{N}\\}\rangle$
$\displaystyle\equiv|1_{{\vec{x}}_{1}}..1_{{\vec{x}}_{n}},1_{{\vec{y}}_{1}}..1_{{\vec{y}}_{N}},0_{\\{i,\hat{x}_{1},..,\hat{x}_{n},\hat{y}_{1},..,\hat{y}_{N}\\}}\rangle$
(4.7)

so that

$\displaystyle\left|\Psi_{\rm UdW}\right\rangle=$
$\displaystyle|0,0\rangle+\sum_{\\{x_{n}\\}\neq\varnothing}a_{\\{{\vec{x}}_{n}\\}}|\\{{\vec{x}}_{n}\\},0\rangle+\sum_{\\{{\vec{y}}_{N}\\}\neq\varnothing}b_{\\{{\vec{y}}_{N}\\}}|0,\\{{\vec{y}}_{N}\\}\rangle$
(4.8)
$\displaystyle+\sum_{\\{{\vec{x}}_{n}\\},\\{{\vec{y}}_{N}\\}\neq\varnothing}c_{\\{{\vec{x}}_{n}\\},\\{{\vec{y}}_{N}\\}}|\\{{\vec{x}}_{n}\\},\\{{\vec{y}}_{N}\\}\rangle\
.$

The entanglement between $A$ and $B$ can be determined from reduced density
matrix of the detectors in $A$ (inside the sphere) tracing over the detectors
in $B$ (outside the sphere). The contributions from $a_{\\{{\vec{x}}_{n}\\}}$
and $b_{\\{{\vec{y}}_{N}\\}}$ can be removed by a change of basis (as they do
not entangle detectors in $A$ with detectors in $B$). The contribution to a
non-trivial density matrix comes instead from

$\rho_{\\{{\vec{x}}_{n}\\},\\{{\vec{x}}_{m}\\}}=\sum_{\\{{\vec{y}}_{N}\\}}c_{\\{{\vec{x}}_{n}\\},\\{{\vec{y}}_{N}\\}}c^{\dagger}_{\\{{\vec{x}}_{m}\\},\\{{\vec{y}}_{N}\\}}|\\{{\vec{x}}_{n}\\}\rangle\langle\\{{\vec{x}}_{m}\\}|\
.$ (4.9)

In the free theory, these coefficients are determined by Wick contractions
alone,

$c_{\\{{\vec{x}}_{n}\\},\\{{\vec{y}}_{N}\\}}=\prod_{i,j}\langle\Phi_{{\vec{x}}_{i}}\Phi_{{\vec{y}}_{j}}\rangle+{\rm
permutations}\ .$ (4.10)

For a massive theory, these two point functions are exponentially suppressed
and therefore

$c_{{\vec{x}}_{i},{\vec{y}}_{j}}\propto e^{-m|{\vec{x}}_{i}-{\vec{y}}_{j}|}\
.$ (4.11)

Any non-trivial entanglement between the detectors in $A$ and $B$ will
therefore be exponentially suppressed as well.

For massless fields, correlations scale like powers for the distance and thus
are more entangled than the massive case. Expanding in $\lambda$, the leading
contribution is from

$c_{{\vec{x}}_{i},{\vec{y}}_{j}}=\langle\Phi_{{\vec{x}}_{i}}\Phi_{{\vec{y}}_{j}}\rangle=\frac{\lambda^{2}}{|\vec{x}_{i}-\vec{y}_{j}|^{2}}\
,$ (4.12)

where we assumed we are in $3+1$ dimensions. The key question we want to
understand is how $B$ and $A$ are entangled. We are particularly interested in
how the reduced density matrix of the detectors within $A$ depends on their
proximity to the boundary with $B$. In massive QFT, entanglement is short
ranged and is responsible for the famous area-law in the quantum vacuum. We
now want to understand what happens for the range of entanglement of our
detectors. For massive fields, the exponential decay of the two point
correlators ensures the entanglement is short range.

To understand the range of entanglement of the detectors for massless $\phi$,
we will consider a small number of localized detectors with $A$ entangled with
detectors in $B$. The simplest such possibility is two detectors, one in $A$
and one in $B$. The state $|\Psi_{\rm UdW}\rangle$ of these detectors gives us
a density matrix at ${\cal O}(\lambda^{4})$

$\rho^{(AB)}=\left(\begin{array}[]{cccc}1-c_{{\vec{x}},{\vec{y}}}^{2}&0&0&-c_{{\vec{x}},{\vec{y}}}\\\
0&0&0&0\\\ 0&0&0&0\\\
-c_{{\vec{x}},{\vec{y}}}&0&0&c_{{\vec{x}},{\vec{y}}}^{2}\end{array}\right)\ ,$
(4.13)

where the matrix indices are $|0_{\vec{x}}0_{\vec{y}}\rangle$,
$|1_{\vec{x}}0_{\vec{y}}\rangle$, $|0_{\vec{x}}1_{\vec{y}}\rangle$ and
$|1_{\vec{x}}1_{\vec{y}}\rangle$. This is, of course, the density matrix of a
pure state888Notice this density matrix differs from those in similar two-
detector models as in [59, 60]. The difference arises because we have
projected onto the ground state of $\phi$ to arrive at a pure state, rather
than tracing over $\phi$, which results in a mixed state.. Tracing over $B$
produces the reduced density matrix for the detector in $A$,

$\rho^{(A)}={\rm
Tr}_{B}\rho^{(AB)}=\left(\begin{array}[]{cc}1-c_{{\vec{x}},{\vec{y}}}^{2}&0\\\
0&c_{{\vec{x}},{\vec{y}}}^{2}\end{array}\right)\ ,$ (4.14)

and entanglement entropy

$S^{A}_{\rm ent}=-{\rm
Tr}\rho^{(A)}\log\rho^{(A)}=-c_{{\vec{x}},{\vec{y}}}^{2}(\log
c_{{\vec{x}},{\vec{y}}}^{2}-1)+{\cal O}(\lambda^{6})\ .$ (4.15)

We see that in the limit $y\to\infty$, holding ${\vec{x}}$ fixed, $S^{A}_{\rm
ent}\propto y^{-4}\log y\to 0$.

Now let us consider the generalization of this case where we take a single
detector in $A$ at location ${\vec{x}}$ with a large number of detectors in
$B$, $N_{B}$, that are uniformly distributed in space. The entanglement
between any the detector in $A$ and any detector $B$ is the same as our above
example, such that the reduced density matrix is again diagonal with
$\rho_{00}=1-\rho_{11}$ and

$\rho_{11}\approx\lambda^{4}\sum_{{\vec{y}}_{j}}\frac{1}{|{\vec{x}}-{\vec{y}}_{j}|^{4}}\
.$ (4.16)

First let us place the detector in $A$ near then center of the sphere so that
$x\ll R$. Using the large $N_{B}$ limit to replace the sum by an integral, the
leading contribution to the reduced density matrix becomes

$\rho_{11}\approx\frac{4\pi\lambda^{4}N_{B}}{V}\int_{R}^{\infty}r^{2}dr\frac{1}{r^{4}}=\frac{4\pi\lambda^{4}}{R}n_{B}\
,$ (4.17)

where $n_{B}=N_{B}/V$ is the number density of detectors in $B$. In contrast,
a point near the boundary gets a much larger contributions. E.g. if $x\approx
R$ we can write

$\rho_{11}\approx\frac{2\pi\lambda^{4}N_{B}}{V}\int^{\infty}_{1/\Lambda}r^{2}dr\frac{1}{r^{4}}=2\pi\lambda^{4}\Lambda
n_{B}\ ,$ (4.18)

where $\Lambda^{-1}$ is the smallest distance between a detector in $B$ and
${\vec{x}}$. So far, the entanglement of a few detectors inside $A$, for a
free theory of $\phi$, are consistent with the intuition that entanglement
mostly arises from point near the boundary of $A$, or that entanglement in the
vacuum is short ranged999We can also repeat this argument for generalized free
field (e.g. [93]) by substituting
$|{\vec{x}}-{\vec{y}}|^{-2}\to|{\vec{x}}-{\vec{y}}|^{-2\Delta}$ in Equation
(4.12), where $\Delta$ is the scaling dimension of the operator. It is
interesting to note that the unitarity bound $\Delta\geq 1$ ensures that
entanglement of the detectors is short ranged..

Figure 2: Entanglement between detector in region $A$, located at ${\vec{x}}$,
and two detectors in $B$, located at ${\vec{y}}_{1}$ and ${\vec{y}}_{2}$,
produced by the three point interaction ${\cal H}_{\rm int}=\mu\phi^{3}$. Time
runs in the vertical direction such that the correlation between detectors on
the equal time surface is due to interactions in the past.

Now we would like to understand how the range of entanglement changes with
interactions. We will again focus on the cases where the underlying QFT has a
non-trivial three-point function. The contributions of interest will arise
from entanglement induced between a single detector in $\vec{x}\in A$, now
with two detectors in $B$ at positions ${\vec{y}}_{1}$ and ${\vec{y}}_{2}$.
The interaction introduces a non-trivial amplitude for exciting the three
detectors simultaneously,

$c_{{\vec{x}},\\{{\vec{y}}_{1},{\vec{y}}_{2}\\}}=\lambda^{3}\langle\Omega|\Phi({\vec{x}})\Phi({\vec{y}}_{1})\Phi({\vec{y}}_{2})\rangle\Omega\rangle\
,$ (4.19)

where we have used Equation (3.16) to relate the in-in correlator and the
entanglement between detectors. This contribution to the state of the
detectors is illustrated in Figure 2. Tracing over the detectors in $B$ we
arrive again at a diagonal density matrix for the single detector in $A$ with
$\rho^{(A)}_{00}=1-\rho^{(A)}_{11}$, but now with

$\rho^{(A)}_{11}=c_{{\vec{x}},{\vec{y}}_{1}}^{2}+c_{{\vec{x}},{\vec{y}}_{2}}^{2}+c_{{\vec{x}},\\{{\vec{y}}_{1},{\vec{y}}_{2}\\}}^{2}\
.$ (4.20)

Since the three point functions are different for classical and quantum
statistics of $\phi$, we must consider the entanglement separately as well.
First, we consider the case where $\phi$ is in the interacting (quantum)
vacuum with a non-zero three-point function given by Equation (3.20). We are
specifically interested in the case where ${\vec{x}}$ is located near the
center of the sphere and the two detectors of $B$ are close together so that
$|{\vec{y}}_{1}-{\vec{y}}_{2}|\ll|{\vec{x}}-{\vec{y}}_{1}|\approx|{\vec{x}}-{\vec{y}}_{2}|$.
In this limit, we find

$c^{\rm(quantum)}_{{\vec{x}},\\{{\vec{y}}_{1},{\vec{y}}_{2}\\}}\approx\lambda^{3}\frac{\mu}{32\pi^{4}}\frac{\log|{\vec{y}}_{1}-{\vec{y}}_{2}|}{|{\vec{x}}-{\vec{y}}_{1}|^{2}}\
.$ (4.21)

We notice that the contribution from the interaction is suppressed relative to
the contribution from the free theory. Concretely, in the limit
$y_{1},y_{2}\to\infty$, we find the scaling behavior

$c^{\rm(quantum)}_{{\vec{x}},\\{{\vec{y}}_{1},{\vec{y}}_{2}\\}}\propto\lambda^{3}\mu
y_{1}^{-2}\ll c_{{\vec{x}},{\vec{y}}_{1}}\propto\lambda^{2}y_{1}^{-2}\ ,$
(4.22)

where we used $\mu\lambda\ll 1$, which is required for perturbative control.
The entanglement in the region $A$ is therefore dominated by the free theory
and is again short ranged. This is consistent with our expectations from QFT.

Now let use consider classical (excited) state for $\phi$ where the three-
point correlator is given by Equation (2.32). When we compute the reduced
density matrix with the detectors again near the center of the sphere, we can
use Equation (3.23) to find

$c^{\rm(classical)}_{{\vec{x}},\\{{\vec{y}}_{1},{\vec{y}}_{2}\\}}\approx\lambda^{3}\frac{\mu}{64\pi^{4}}\frac{1}{|{\vec{x}}-{\vec{y}}_{1}||{\vec{y}}_{1}-{\vec{y}}_{2}|}\left(-\log\left(\frac{\theta_{\rm
min}^{2}}{2}\right)\,f(\hat{x}\cdot\hat{y}_{12})\right)\ ,$ (4.23)

where $\vec{y}_{12}={\vec{y}}_{1}-{\vec{y}}_{2}$, and $\hat{x}$ and
$\hat{y}_{12}$ are the unit vectors associated with ${\vec{x}}$ and
${\vec{y}}_{12}$. We again take the limit in the limit $y_{1},y_{2}\to\infty$
but instead find

$c^{\rm(classical)}_{{\vec{x}},\\{{\vec{y}}_{1},{\vec{y}}_{2}\\}}\propto\lambda^{3}\mu
y_{1}^{-1}|{\vec{y}}_{1}-{\vec{y}}_{2}|^{-1}\gg
c_{{\vec{x}},{\vec{y}}_{1}}\propto\lambda^{2}y_{1}^{-2}$ (4.24)

The range of entanglement is clearly been increased by the interaction.
Furthermore, if we were to repeat the calculation in Equation (4.17) with a
large number of detectors in $B$, the integrals would no longer converge. This
long range entanglement between detectors at large distances for the classical
case, which is absent in the quantum vacuum, is suggestive of the area versus
volume law behavior know to distinguish the two cases.

This difference between classical and quantum correlators also manifests
itself in the relative entropy [94], a measure of distance between two states.
Since we are interested in distinguishing the quantum vacuum from the
classical (excited) state, consider the relative entropy for detector A,
$S(\rho_{A}|\rho^{\prime}_{A})=\text{tr}\rho_{A}\log\rho_{A}-\text{tr}\rho_{A}\log\rho^{\prime}_{A}$,
where $\rho_{A}$ is the density matrix for detector A in the quantum vacuum
and $\rho^{\prime}_{A}$ is the same density matrix in a classical state. Here
we can write $\rho^{\prime}_{A}=\rho_{A}+\delta\rho$, where

$\delta\rho=\left(\begin{array}[]{cc}(c_{{\vec{x}},\\{{\vec{y}}_{1},{\vec{y}}_{2}\\}}^{\rm(classical)})^{2}-(c_{{\vec{x}},\\{{\vec{y}}_{1},{\vec{y}}_{2}\\}}^{\rm(quantum)})^{2}&0\\\
0&-(c_{{\vec{x}},\\{{\vec{y}}_{1},{\vec{y}}_{2}\\}}^{\rm(classical)})^{2}+(c_{{\vec{x}},\\{{\vec{y}}_{1},{\vec{y}}_{2}\\}}^{\rm(quantum)})^{2}\end{array}\right)\
,$ (4.25)

Since both $\rho_{A}$ and $\delta\rho$ are diagonal,
$[\rho_{A},\delta\rho]=0$, we can expand the relative entropy

$\displaystyle S(\rho_{A}|\rho^{\prime}_{A})$
$\displaystyle=\text{tr}\rho_{A}\log\rho_{A}-\text{tr}\rho_{A}\log(\rho_{A}+\rho_{A}^{-1}\delta\rho-\frac{1}{2}\rho_{A}^{-2}\delta\rho^{2})$
(4.26) $\displaystyle=\frac{1}{2}\text{tr}\rho_{A}^{-1}\delta\rho^{2}\ .$
(4.27)

As a result, the leading order contribution to the relative entropy is

$S(\rho_{A}|\rho^{\prime}_{A})=\frac{c^{-2}_{{\vec{x}},{\vec{y}}}}{4}((c_{{\vec{x}},\\{{\vec{y}}_{1},{\vec{y}}_{2}\\}}^{\rm(classical)})^{2}-(c_{{\vec{x}},\\{{\vec{y}}_{1},{\vec{y}}_{2}\\}}^{\rm(quantum)})^{2})^{2}\
.$ (4.28)

This provides further confirmation that the difference between correlators
manifests itself a robust difference in the states of the detectors used to
observe them.

###  4.2 Momentum Space Detectors and Entanglement

Although entanglement is more often discussed in position space, momentum
space entanglement [95, 96, 97, 98, 99, 100] offers another useful window into
the nature of cosmological correlators. Cosmological correlators are easier to
calculate and represent in momentum space. Likewise, entanglement in momentum
space is easier to calculate because the vacuum of the free theory is a tensor
product in the momentum basis. As a result, entanglement between states of
different momenta is uniquely a property of the interacting theory and thus is
a window into the non-Gaussian nature of the correlations.

A natural approach that combines these benefits is to work with UdW detectors
that register particles in specific momentum eigenstates, rather than a
specific positions. We can achieve this by modifying the detector response so
that

$\psi_{i}({\vec{x}})=e^{-i{\vec{k}}_{i}\cdot{\vec{x}}}\ ,$ (4.29)

and therefore

$\Phi_{i}=\lambda\phi({\vec{k}}_{i},t)\ .$ (4.30)

By measuring momentum eigenstates at a fixed time, our detectors are
responding to $\phi({\vec{k}},t)$ which is precisely what appears in our
cosmological correlators.

Now we can again split our detectors into two groups, $A$ and $B$, where
momenta $\vec{k}\in A$ and $\vec{p}\in B$. We will define group $A$ as the
detectors with momenta below a fixed cutoff, $k_{i}\leq\Lambda$, and the $B$
detectors have momenta above the cutoff, $p_{j}>\Lambda$. The state of the
detectors is again represented in analogy with Equation (4.8), now defining

$\displaystyle|0,0\rangle$ $\displaystyle\equiv|0_{\\{i\\}}\rangle$ (4.31)
$\displaystyle|\\{\vec{k}_{n}\\},0\rangle$
$\displaystyle\equiv|1_{{\vec{k}}_{1}}..1_{{\vec{k}}_{n}},0_{\\{i,\hat{k}_{1},..,\hat{k}_{n}\\}}\rangle$
(4.32) $\displaystyle|0,\\{\vec{p}_{N}\\}\rangle$
$\displaystyle\equiv|1_{{\vec{p}}_{1}}..1_{{\vec{p}}_{N}},0_{\\{i,\hat{p}_{1},..,\hat{p}_{N}\\}}\rangle$
(4.33) $\displaystyle|\\{\vec{k}_{n}\\},\\{\vec{p}_{N}\\}\rangle$
$\displaystyle\equiv|1_{{\vec{k}}_{1}}..1_{{\vec{k}}_{n}},1_{{\vec{p}}_{1}}..1_{{\vec{p}}_{N}},0_{\\{i,\hat{k}_{1},..,\hat{k}_{n},\hat{p}_{1},..,\hat{p}_{N}\\}}\rangle\
.$ (4.34)

As a result, the state of our momentum detectors is given by

$\displaystyle\left|\Psi_{\rm UdW}\right\rangle=$
$\displaystyle|0,0\rangle+\sum_{\\{{\vec{k}}_{n}\\}\neq
0}a_{\\{{\vec{k}}_{n}\\}}|\\{{\vec{k}}_{n}\\},0\rangle+\sum_{\\{{\vec{p}}_{N}\\}\neq
0}b_{\\{{\vec{p}}_{N}\\}}|0,\\{p_{N}\\}\rangle$ (4.35)
$\displaystyle+\sum_{\\{{\vec{k}}_{n}\\},\\{{\vec{p}}_{N}\\}\neq
0}c_{\\{{\vec{k}}_{n}\\},\\{{\vec{p}}_{N}\\}}|\\{{\vec{k}}_{n}\\},\\{{\vec{p}}_{N}\\}\rangle\
.$

In the free theory, the Hilbert space of $\phi$ can be decomposed into a
tensor product over momentum eigenmodes via the Fock space, ${\cal
H}=\otimes_{\vec{p}}{\cal H}_{\vec{p}}$. Since each detector is tied only to a
single momentum scale, the combined state of the detectors in $A$ and $B$ with
the fields are similarly described by a tensor product in momentum space. More
dramatically, when we project onto the vacuum of $\phi$, which is a zero
momentum state, we will find the detector is never excited unless we have two
detectors with equal and opposite momentum (i.e. $k$ and $-{\vec{k}}$ or
${\vec{p}}$ and $-{\vec{p}}$). This reflects the fact that, in the free
theory, measuring a particle at momentum ${\vec{k}}$ or ${\vec{p}}$ requires
the production of an (anti)-particles with momentum $-{\vec{k}}$ or
$-{\vec{p}}$. Without a second detector, this is a real particle and therefore
is not in the vacuum. Furthermore, since $k$ and $-{\vec{k}}$ are in $A$ and
${\vec{p}}$ and $-{\vec{p}}$ are in $B$, we cannot generate entanglement
between $A$ and $B$ in the free theory. Therefore, We need interactions in
order to generate entanglement in momentum space.

We can easily determine the detector state in the interacting theory for a
single detector in $A$ with momentum ${\vec{k}}$ and two detectors in $B$ with
momenta ${\vec{p}}_{1}$ and ${\vec{p}}_{2}$. Like the free theory, the
quadratic terms vanish,
$c_{k,{\vec{p}}_{1}}=c_{k,{\vec{p}}_{2}}=c_{{\vec{p}}_{2},{\vec{p}}_{1}}=0$,
such that the leanding non-trivial contribution to the state is

$c_{{\vec{k}},\\{{\vec{p}}_{1},{\vec{p}}_{2}\\}}=\lambda^{3}\langle\phi({\vec{k}},t)\phi({\vec{p}}_{1},t)\phi({\vec{p}}_{2},t)\rangle\
.$ (4.36)

We can now trace over the detectors in $B$ to arrive at a diagonal reduced
density matrix for $A$ with $\rho^{(A)}_{00}=1-\rho_{11}$ and

$\rho^{(A)}_{11}=|c_{{\vec{k}},\\{{\vec{p}}_{1},{\vec{p}}_{2}\\}}|^{2}\ .$
(4.37)

Again, the entanglement entropy of $A$ is given in terms of this single
coefficient

$S^{A}_{\rm ent}=-{\rm
Tr}\rho^{(A)}\log\rho^{(A)}=-|c_{{\vec{k}},\\{{\vec{p}}_{1},{\vec{p}}_{2}\\}}|^{2}(\log|c_{{\vec{k}},\\{{\vec{p}}_{1},{\vec{p}}_{2}\\}}|^{2}-1)+{\cal
O}(\lambda^{8})\ .$ (4.38)

From Equation (4.36), we see that the entanglement entropy is determine by the
in-in correlators.

Now we want to compare the nature of the momentum space entanglement for our
two types of statistics. These are just our in-in correlatators in moemtnum
space, so in the quantum theory we have

$c^{\rm(quantum)}_{{\vec{k}},\\{{\vec{p}}_{1},{\vec{p}}_{2}\\}}=-\lambda^{3}\frac{\mu}{4kp_{1}p_{2}(k+p_{1}+p_{2})}\
,$ (4.39)

and in the classical theory

$c^{\rm(classical)}_{{\vec{k}},\\{{\vec{p}}_{1},{\vec{p}}_{2}\\}}=-\lambda^{3}\frac{\mu}{16kp_{1}p_{2}}\left(\frac{3}{k+p_{1}+p_{2}}+\frac{1}{k-p_{1}-p_{2}}+\frac{1}{p_{1}-k-p_{2}}+\frac{1}{p_{2}-k-p_{1}}\right)\
.$ (4.40)

The presence of poles at physical momenta means that $S_{\rm ent}^{(A),{\rm
classical}}\gg S_{\rm ent}^{(A),{\rm quantum}}$. This again reflects the
underlying fact that creating a state with classical fluctuations introduces
much stronger correlations between scales than occur in the vacuum.

We also note that, in this case, the structure of entanglement of the UdW
detectors in momentum space is proportional to the entanglement of $\phi$ in
momentum space [96]. The origin of this relationship is that entanglement in
momentum space is trivial in the free theory. Therefore, the perturbative
construction of the detector states is consistent with the perturbative nature
of the entanglement in the vacuum. In position space, the reduced density
matrix is non-trivial in the free theory and is less naturally organized as a
perturbative expansion in correlators (although see e.g. [101, 102, 103] for
developments).

##  5 Conclusions

The nature of the vacuum in quantum field theory is unlike any classical
statistical state. The quantum vacuum is the lowest-energy state and therefore
dictates that fluctuations only have positive energies. This fact, built into
the structure of perturbative QFT, manifests itself in the structure of
correlation functions and gives rise to the LSZ reduction formula relating
correlators and S-matrix elements [53]. Classical (e.g. thermal) fluctuations
are always around a positive energy state and thus can both increase and
decrease the energy.

In cosmology, these kinds of vacuum fluctuation are thought to be responsible
for structure in the universe. Yet, it remains a viable possibility that
structure arose from thermal fluctuation [29, 30, 31, 32, 33, 34]. One cannot
perform experiments on the state of the universe to isolate their quantum
nature and resolve this question [16]. Instead, we must rely on statistical
properties of the initial conditions to infer how structure was created.
Concretely, it was proposed in [35] that the difference in the analytic
structure of correlators for quantum and classical fluctuations is both
completely general and observable.

In this paper, we demonstrated that this proposed cosmological Bell-like test
has a flat space analogue. The same analytic structure seen in cosmological
correlators appears in both in-in and in-out correlators in flat space, and is
responsible for the LSZ reduction for quantum vacuum correlators. For
classical correlators, the additional poles seen in inflationary correlators
is a direct consequence of scattering processes involving particles that are
necessarily present in the initial state.

The meaning of the in-in correlator is less clear in flat space than it is in
cosmology. In cosmology, we interpret these correlators as a signal of
cosmological particle production. In flat space, the interacting (quantum)
vacuum is a well-defined energy eigenstate containing no particles and, yet,
has non-zero in-in correlations. Instead, we show the flat space in-in
correlator contributes to the amplitude for exciting a localized (Unruh - de
Witt) particle detector. The particle production arises in flat space because
of the uncertainty principle: a particle detector that is sufficiently
localized in space and time to make such a measurement breaks translations and
excites particles from the vacuum. We additionally show that one can use the
entanglement of these detectors as probes to the entanglement of the
underlying field.

Much is known about the unique properties of quantum mechanical systems in
flat space. Naturally one would hope these insights could be applied to
cosmology, particularly in light of some analogous structure of the
correlators. The central challenge is that cosmological observables are
classical, for all practical purposes; one cannot simply expose the quantum
nature of cosmology through the direct measurement of non-commuting
observables. Yet, the question of what kinds of initial conditions can only be
prepared in a quantum universe is closely related to the problem of quantum
state preparation on a quantum computer. One might hope that intuition from
quantum computing could shed further light on cosmology, or vice versa [104,
105].

##### Acknowledgements

We are grateful to Daniel Baumann, Jonathan Braden, Dick Bond, Tim Cohen, John
McGreevy, Mehrdad Mirbabayi, Eva Silverstein, and Rafael Porto for helpful
discussions. The authors were supported by the US Department of Energy under
grant no. DE-SC0009919.

##  Appendix A Classical Hidden Variables and Anti-particles

In our classical in-in correlators, the appearance of poles at physical
momenta can be traced to the form of the causal propagator,

$G(k,t)=\frac{\sin(kt)}{k}=\frac{1}{2ik}\left(e^{ikt}-e^{-ikt}\right).$ (A.1)

The appearance of both positive and negative frequency modes in this
expression is responsible for the additional poles in the classical
correlators. If we were to simply remove the negative frequency term in this
expression in the limit $t\to 0$ the propagator is non-analytic in $k$,
$G(k,t=0)\propto 1/k$. This would imply a violation of causality as the
propagator is non-zero at space-like separated points. This is the precise
sense in which replicating the quantum signal requires a non-local theory [1,
2, 3].

The need for a negative frequency mode in the causal propagator is essentially
the same reason that causality demands anti-particles. To see this connection,
we will work with a complex scalar, $\varphi$, such that

$\varphi({\vec{x}},t)=\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{\sqrt{2k}}e^{-i{\vec{k}}\cdot{\vec{x}}}[a_{\vec{k}}e^{-ikt}+b^{\dagger}_{-{\vec{k}}}e^{ikt}]\
.$ (A.2)

In contrast to a real scalar, we have the option of setting $b_{-{\vec{k}}}=0$
so that we propagate only positive frequency modes.

To make sense of the classical case, we will consider the quantum theory at
large occupation number, as we did in Section 2.1. Suppose the field $\phi$ is
in an $n$-particle and $n$-anti-state at each momenta, as defined in Equation
(2.10). Schematically, the field would act on this state as

$\displaystyle\varphi({\vec{k}},t)|n,n\rangle$ $\displaystyle\to
Ae^{-ikt}|n_{\vec{k}}-1,n\rangle+B^{*}e^{ikt}|n,n_{-{\vec{k}}}+1\rangle$ (A.3)
$\displaystyle\bar{\varphi}({\vec{k}},t)|n,n\rangle$ $\displaystyle\to
A^{*}e^{ikt}|n_{-{\vec{k}}}+1,n\rangle+Be^{-ikt}|n,n_{{\vec{k}}}-1\rangle$
(A.4) $\displaystyle\langle n,n|\varphi({\vec{k}},t)$ $\displaystyle\to\langle
n,n_{{\vec{k}}}-1|B^{*}e^{ikt}+\langle n_{-{\vec{k}}}+1,n|Ae^{-ikt}$ (A.5)
$\displaystyle\langle n,n|\bar{\varphi}({\vec{k}},t)$ $\displaystyle\to\langle
n,n_{-{\vec{k}}}+1|Be^{-ikt}+\langle n_{{\vec{k}}}-1,n|A^{*}e^{ikt}$ (A.6)

where $|n,m\rangle$ is an $n$ particle $m$ anti-particle state, and $A$ and
$B$ are time-independent functions of $k$ that are determined by how
$a_{\vec{k}}$ and $b_{\vec{k}}$ act on these states. Concretely, if $B=0$ it
would imply $b$ and $b^{\dagger}$ annihilate the state $|n,n\rangle$.

Using how the operators act on the states, we can write the two point
functions as

$\langle\varphi({\vec{k}},t)\bar{\varphi}({\vec{k}}^{\prime},t^{\prime})\rangle=|B|^{2}e^{ik(t-t^{\prime})}\langle
n,n_{{\vec{k}}}-1|n,n_{-{\vec{k}}^{\prime}}-1\rangle+|A|^{2}e^{-ik(t-t^{\prime})}\langle
n_{{\vec{k}}^{\prime}}+1,n|n_{-{\vec{k}}}+1,n\rangle$ (A.7)
$\langle\bar{\varphi}({\vec{k}},t)\varphi({\vec{k}}^{\prime},t^{\prime})\rangle=|A|^{2}e^{ik(t-t^{\prime})}\langle
n_{{\vec{k}}}-1,n|n_{-{\vec{k}}^{\prime}}-1,n\rangle+|B|^{2}e^{-ik(t-t^{\prime})}\langle
n,n_{-{\vec{k}}}+1|n,n_{{\vec{k}}^{\prime}}+1\rangle.$ (A.8)

In the local model $A=B=1/(\sqrt{2k})$. We argued from the structure of the
causal Green’s function that $B=0$ is nonlocal, which eliminates the
possibility of locally propagating only the positive frequency modes
classically.

We can see the connection between $B=A$, locality, and antiparticles using
time-order (in-out) correlators. We start by noting that $a^{\dagger}$ creates
a positive energy state when acting on the right and negative energy state
when acting on the right. However, since the time ordering can be changed by a
Lorentz boost, $\varphi$ must create a positive energy state when acting on
either the right or the left. As a result, $\varphi$ creates a particle when
acting on the right and an anti-particle when acting on the left. This is only
possible with $|A|=|B|$.

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