# Quantum Field Theory based Quantum Information: Measurements and
Correlations

Charis Anastopoulos (a) , Bei-Lok Hu (b) , and Konstantina Savvidou (a)

(a) Department of Physics, University of Patras, 26500 Greece
(b) Maryland Center for Fundamental Physics and Joint Quantum Institute,
University of Maryland, College Park, Maryland 20742-4111 U.S.A.
anastop@upatras.grblhu<EMAIL_ADDRESS>
###### Abstract

This is the first in a series of papers aiming to develop a relativistic
quantum information theory in terms of unequal-time correlation functions in
quantum field theory. In this work, we highlight two formalisms which together
can provide a useful theoretical platform suitable for further developments:
1) Quantum field measurements using the Quantum Temporal Probabilities (QTP)
method; 2) Closed-Time-Path (CTP) formalism for causal time evolutions. QTP
incorporates the detector into the quantum description, while emphasising that
the records of measurement are macroscopic and can be expressed in terms of
classical spacetime coordinates. We first present a new, elementary derivation
of the QTP formulas for the probabilities of n measurement events. We then
demonstrate the relation of QTP with the Closed-Time-Path formalism, by
writing an explicit formula that relates the associated generating
functionals. We exploit the path integral representation of the CTP formalism,
in order to express the measured probabilities in terms of path integrals.
After this, we provide some simple applications of the QTP formalism. In
particular, we show how Unruh-DeWitt detector models and Glauber’s
photodetection theory appear as limiting cases. Finally, with quantum
correlation being the pivotal notion in relativistic quantum information and
measurements, we highlight the role played by the CTP two-particle irreducible
effective action which enables one to tap into the resources of non-
equilibrium quantum field theory for our stated purpose.

##  1 Introduction

Quantum information theory (QIT) is a quantum extension of classical
information theory. The primary force behind the explosive developments of QIT
during the last decades is entanglement, the uniquely quantum feature which
provides the new and powerful resource for quantum computing, quantum
communication, quantum metrology and more.

While potential applications of quantum information technologies have drawn
close attention of government and private research and development, the
theoretical foundation of QIT is surprisingly lacking in a major portion. Our
understanding of QIT is far lagging behind the fully developed quantum theory
of nature, namely, quantum field theory (QFT), which has proven its validity
and worth in the full range of physical sciences from particle-nuclear physics
to atomic, optical and condensed matter physics, from quarks to black holes
and the early universe. So far, quantum information theory has been largely
developed in the context of non-relativistic quantum mechanics, which is a
small corner of full QFT. It is ostensibly inadequate when basic relativistic
effects like locality, causality and spacetime covariance, need be accounted
for.

###  1.1 QITs not based on QFT are incomplete

QFT is a quantum theory that also includes axioms about the effect of
spacetime structure on the properties of quantum systems, especially regarding
the causal propagation of signals. In contrast, current quantum information
theories do not incorporate the latter axioms. Their notion of causality,
based on the sequence of successive operations on a quantum system, lacks a
direct spacetime representation. As a result, current QITs cannot make crucial
relativistic distinctions, for example, between timelike and spacelike
correlations, it does not describe real-time signal propagation, and it
ignores relativistic constraints on permissible measurements. A genuinely
relativistic QIT must overcome such limitations [2].

Furthermore, experiments that study causal information transfer or
gravitational interaction in multi-partite quantum systems require a QFT
treatment of interactions for consistency. A non-QFT description may severely
misrepresent the theoretical modeling of the system or the physical
interpretation of the results. This point is crucial for tests of foundational
issues of quantum theory invoking quantum information concepts such as
entanglement and decoherence or methods such as signaling and communication.
This is especially so for quantum information experiments in space [3, 4] and
for experiments designed to explore quantum effects from gravity [5, 6, 7, 8].

The introduction of the key concepts of spacetime covariance and causality in
QIT forces us to address problems that originate from the foundations of QFT.
We draw an incomplete list of noteworthy issues below.

Quantum States. In set-ups that involve more than two quantum measurements,
the standard state update rule implies that the quantum state is genuinely
different when recorded from different Lorentz frames [9, 10, 11]. There is no
problem with the theory’s physical predictions that are expressed in terms of
(multi-time) probabilities[12]. However, the usual notions of quantum
information (entropy, entanglement) are defined through the quantum state, and
as such, they are ambiguous in relativistic measurement set-ups.

Local Operations. It is a challenging problem to formalize the notion of a
localized quantum system in QFT. There are powerful theorems demonstrating
that even unsharp localization in a spatial region leads to faster than light
signals [13, 14]. Hence, expressing the crucial quantum informational notion
of a local operation in terms of spatial localization can lead to conflicts
with relativistic causality.

Projective Measurements. There are strong arguments that ideal (i.e.,
projective) measurements in QFT are incompatible with causality [15, 16],
essentially because they change the quantum state over a full Cauchy surface.
However, existing quantum information notions (even the very notion of a
qubit) presuppose maximal extraction of information through ideal
measurements.

We contend that the development of consistent relativistic QIT requires a
measurement theory that (i) respects causality and locality, and (ii) it is
expressed in terms of quantum fields111The explicit formulation in terms of
QFT is the point of divergence of our approach from past works on formulating
a relativistic QIT [17, 18, 19, 20].. Furthermore, this measurement theory
ought to be practical, i.e., it must provide non-trivial predictions for
experiments that are accessible now or in the near future. For other reasons
why a QFT measurement theory is needed, see Ref. [21].

###  1.2 Past work on QFT measurements

To the best of our knowledge, the earliest discussion of measurements on
quantum fields was by Landau and Peierls [22]. They derived an inequality for
the localization of particles. Bohr and Rosenfeld criticized some of their
assumptions [23], and proved that the measurement of field properties requires
a test particle of macroscopic scale: the particle’s charge $Q$ must be much
larger than the electron charge e.

Arguably the first explicit model for QFT measurements was Glauber’s
photodetection theory [24, 25], which was developed as a quantum
generalization of the classical theory of electromagnetic coherence. Glauber’s
theory defines unnormalized probabilities for photon detection in terms of the
electric field operators ${\bf E}(x)$ and the field’s quantum state
$|\psi\rangle$. The joint probability density $P(x_{1},x_{2},\ldots,x_{n})$
for the detection of a photon at each of the spacetime points
$x_{1},x_{2},\ldots,x_{n}$ is given by

$\displaystyle
P(x_{1},x_{2},\ldots,x_{n})=\langle\psi|E^{(-)}(x_{1})E^{(-)}(x_{2})\ldots
E^{(-)}(x_{n})E^{(+)}(x_{n})\ldots E^{(+)}(x_{2})E^{(+)}(x_{1})|\psi\rangle,$
(1)

where $E^{(+)}$ is the positive-frequency component and $E^{(-)}$ the
negative-frequency component of the projected field vector field ${\bf
n}\cdot{\bf E}(x)$. The probability density (1) is essential for the
definition of higher-order coherences of the electromagnetic field, and
consequently, for the description of phenomena like the Hanbury-Brown-Twiss
effect, photon bunching and anti-bunching [26].

The success of Glauber’s theory in quantum optics has been immense. However,
its applicability is limited to measurements of photons. Another problem is
that the field splitting into positive and negative frequency components is
non-local. The probability density (1) assumes the Rotating Wave Approximation
for the interaction of the field to the detector [27, 28]. This approximation
misrepresents the retarded propagation of the electromagnetic field. This
means that the model may face problems with causality in set-ups that involve
propagation along long distances [29, 30, 31].

A very common class of models for QFT measurements are based on the notion of
Unruh-DeWitt detectors [32, 33]. These models first appeared in the study of
the Unruh effect, in order to demonstrate the effects of acceleration on the
quantum field vacuum. In an Unruh-DeWitt detector, the quantum field is
coupled to a point-like system that moves along a pre-determined spacetime
trajectory $x(\tau)$, where $\tau$ is the trajectory proper-time. An Unruh-
DeWitt detector model essentially describes a 0+1 field theory interacting
with a 3+1 field theory through an interaction that is defined by and
embedding of the 0+1 spacetime to the 3+1 spacetime (i.e., the detector’s
trajectory). Hence, the detector is kinematically pointlike: physical
observables by necessity are defined along the trajectory. However, the
dynamics may well involve field couplings that correspond to an extended
system [34, 35].

While a first-principles derivation of the detector’s response for non-
inertial motion is still missing, Unruh-DeWitt detector models are the
simplest models for describing QFT measurements [36], and they have found
several applications—for a sampling of the latter, see Ref. [37]. Unruh-DeWitt
detectors are limited in that detector degrees of freedom are not described by
a QFT. Non-pointlike Unruh-DeWitt detectors may lead to non-causal signals in
set-ups that involve multiple detectors. For different perspectives about
causality in Unruh-DeWitt detectors, see Refs. [38, 39, 40].

Measurement models have also been constructed in the context of algebraic QFT
[41, 42, 43, 44, 45, 46], primarily in order to address issues of causality
and locality. For example, Ref. [44] considers a system and a probe /
apparatus, both described by a QFT. The two field systems start separated and
interact within a bounded spacetime region. In Minkowski spacetime, this
interaction is described by an S matrix, and it leads to correlations between
observables on the system and records on the apparatus. Then, one defines
probabilities for the latter in terms of operators that are well defined on
the probe’s Hilbert space. The overall analysis is fully consistent with QFT.
Furthermore, it also works for curved spacetimes and it is not tied to the
existence of an S-matrix, i.e., the existence of asymptotic in-out regions. It
is straightforwardly generalized to sequences of measurements [47]. However,
this approach has not yet been developed into a practical tool capable of
concrete physical predictions.

###  1.3 Our approach

In this work, we describe QFT measurements using the Quantum Temporal
Probabilities method [48, 49, 50]. The original motivation of this method was
to provide a general framework for temporally extended quantum observables
[51, 52, 53], hence, the name. The key idea in QTP is to distinguish between
the time parameter of Schrödinger’s equation from the time variable associated
to particle detection [54, 55]. The latter is then treated as a macroscopic
quasi-classical variable associated to the detector degrees of freedom. Here,
we use the word ‘quasi-classical’ as in the decoherent histories approach to
quantum theory [56, 57, 58, 59, 60, 62]. A quasi-classical variable is a
coarse-grained quantum variable that satisfies appropriate decoherence
conditions, and as a consequence its dynamics can be approximated by classical
evolution equations [60, 62]. Hence, the detector admits a dual description:
in microscopic scales it is described by quantum theory, but its macroscopic
records are expressed in terms of classical spacetime coordinates.
Furthermore, in QTP the detector is also described in terms of quantum fields.
Its interactions with the detector respect causality, because the interaction
Hamiltonian is a local functional of quantum fields.

QTP provides room for sufficient freedom when dealing with the model for the
apparatus, when compared with other approaches. However, at the current
practical applications of the framework, it converges with the results of
other approaches. The current emphasis is on the derivation of probabilities
for measurements, while incorporating the effects of the apparatus in a small
number of phenomenological quantities that can be related to experiments.

Glauber’s detection theory and Unruh-DeWitt detector models emerge from QTP as
limiting cases that characterize specific regimes: Glauber’s theory appears at
the limit of very small characteristic timescales for the detector, while
Unruh-DeWitt models appear at the limit of very short length-scales. In
comparison to the algebraic QFT approaches to measurements, QTP provides the
same results to leading order in perturbation theory, but allows for the
definition of observables for the spacetime observables, and it is embedded
within a nuanced analysis of the quantum-classical transition in the detector
through the decoherent histories approach.

A key insight from QTP is that probabilities for measurements are defined in
terms on specific unequal-time correlation functions of the quantum field.
This is particularly important, as such correlation functions are a staple of
QFT. Powerful methods have been developed for their calculation and the
analysis of their properties. The specific correlation functions relevant to
QTP are not the usual ones of S-matrix theory (in-out formalism), but they
appear in the Closed-Time-Path (CTP) (Schwinger-Keldysh or ‘in-in’) formalism
[63, 64, 65, 66, 67]. The CTP formalism improves over the in-out formalism, in
that it allows for causal equations of motion, and it has found many
applications in early universe cosmology [68, 69], nuclear-particle process
[67, 70, 71, 72] and condensed matter physics [73, 74].

The connection that we establish between the two formalisms allows us to
translate between the concepts of quantum measurement theory and of quantum
field theory. The former is based on operational notions like Positive-
Operator-Valued measures (POVMs) and effects, while the latter employs
functional methods like path integrals in order to express its predictions in
a manifestly covariant language. In this work, we show, for example, how to
express POVMs for particle detection in terms of path-integrals.

A key point of our analysis is the central role played by the unequal-time
correlation functions. In the QTP description, they contain all information
about measured probabilities. The detection probability for $N$ events depends
on an $2N$-unequal time correlation function. This has the following
implication. Two-event measurements typically characterize bipartite systems,
and as such the corresponding probabilities are related to entanglement.
Hence, field four-point functions contain all information that pertains to
bipartite entanglement. A relation at the level of non-relativistic fields was
shown in [50].

An analysis at the level of the correlation functions brings us closer to the
main ideas of non-equilibrium QFT, where irreversibility is implemented by the
truncation of the hierarchy of field correlation functions, and the slaving of
higher correlation functions to the two-point function. Again, there is a
natural relation between QTP and non-equilibrium formalisms that are based on
CTP. We argue that the QTP probabilities function as a registrar of
information for the quantum field, as it keeps track of how much information
resides in which level of correlation functions, and how it flows from one
level to the other during dynamical evolution.

We believe that the scheme presented here has good potential to systemize
quantum information in QFT, and to identify the parts of this information that
is relevant to the field’s statistical, stochastic and thermodynamic behavior.
Hence, this formalism could provide a concrete method for defining quantum
information in QFT via the correlation hierarchy, as has been proposed in
[75]. Such a definition would be very different from definitions of
information in standard QIT that is based on the properties of the single-time
quantum state.

###  1.4 Our results

Our results are the following.

First, we present a new derivation of the QTP formulas for the probabilities
of $n$ measurement events. The derivation is intuitive and pedagogical, as it
only requires elementary perturbation theory. It reproduces the results of the
more rigorous QTP analysis to leading order with respect to the system-
apparatus coupling. This suffices for most applications. Then we analyze the
mathematical structure of the probability formulas. In particular, we show how
they provide an explicit connection between the concepts of quantum
measurement theory and QFT concepts, namely, unequal-time field correlation
functions.

Second, we demonstrate the relation of QTP with the Closed-Time-Path
formalism, by deriving an explicit formula that relates the associated
generating functionals. The Closed-Time-Path formalism has a natural path
integral formulation, which allows us to express the QTP probabilities in
terms of path integrals. Conversely, QTP provides a theory of observables
appropriate to the in-in formalism.

Third, we provide some demonstrative applications of the formalism. We briefly
review the construction of time-of-arrival probabilities for relativistic
particles. Then we show how the commonly employed models of Glauber’s
photodetection and Unruh-DeWitt detectors appear as limiting cases of the QTP
formalism. This implies, in particular, that there exists regimes in which QTP
provides different predictions from Glauber’s theory, and these differences
may be experimentally distinguishable.

Fourth, we explore the links between QTP and non-equilibrium QFT. For field
states with a large number of particles, QTP probabilities provide natural
definitions for Boltzmann-type observables. They also allow us to explore how
information flows between the different levels of the Schwinger-Dyson
hierarchy of correlation functions.

This paper is structured as follows. In Sec. 2, we present the main ideas of
the QTP method, and we derive the probability formula for $n$-detection
events. In Sec. 3, we derive the connection with the CTP formalism, and also
derive a path integral expression for QTP probabilities. In Sec. 4, we present
some applications of the formalism, and in Sec. 5, we explore the links to
non-equilibrium QFT. In Sec. 6, we summarize and discuss our results. The
Appendix A contains a conceptually more rigorous derivation of the QTP
probability formula.

##  2 Probabilities for QFT measurements

In this section, we present the QTP approach to quantum field measurements.
First, we explain in more detail the need for a QFT measurement theory. Then,
we derive the QTP probability formulas and we analyze their structure. The key
feature of the probability formula is that probability densities pertaining to
$n$ measurement events are linear functionals of specific $2n$ unequal-time
correlation functions.

###  2.1 The need for a QFT measurement theory

Most QFT predictions involve set-ups with a single state preparation and a
single-detection event, which can be described in terms of the S-matrix. For
example, S-matrix amplitudes determine scattering cross-sections; S-matrix
poles determine the spectrum of composite particles and decay rates. This
gives the general impression that there is no further need for an elaborate
measurement theory.

However, this over-simplified view is deceptive. There are at least two cases
where the S-matrix analysis is insufficient. First, in quantum optics, we need
joint detection probabilities in order to describe phenomena like photon
bunching and anti-bunching [26]. A first-principles calculation of joint
probabilities requires a non-trivial description of quantum measurements. In
non-relativistic physics, such a calculation involves the use of the state-
update rule, i.e., quantum state reduction. In QFT, a universal rule for
reduction is missing, it is absent even from axiomatizations222Rules for
reduction can be obtained a posteriori if a QFT measurement scheme provides
probability distributions for multiple measurements: see [49] for such
definitions in QTP, [36] for the Unruh-DeWitt detectors and [44, 76] in the
algebraic QFT approach.. In practice, joint detection probabilities relevant
to experiments are constructed through photodetection models, like Glauber’s,
whose derivation is rather heuristic. However, planned experiments in deep
space [3, 4] that involve measurement of electromagnetic field correlations at
large separations will arguably require a first-principles analysis of joint
probabilities in order to take into account the relative motion of detectors
and delayed propagation.

The second case where S matrix theory does not suffice is when the expectation
values of physical quantities at finite moments of time are called for, rather
than just the asymptotic amplitudes offered in the ‘in-out’ S-matrix theory. A
notable example is the construction of dynamical equations for the evolution
of spacetime metric in the early universe and in black holes from the
solutions to the semiclassical Einstein equation [77, 78]; these equations
require the evaluation of the expectation values of the stress-energy and its
fluctuations. Wherever causal time evolution is needed, rather than
transitions between asymptotic states, this ‘in-in’ formulation of QFT, as in
the CTP formalism, is indispensable. This is certainly true for the
description of non-equilibrium processes such as quantum transport in quantum
many-body systems. Powerful functional techniques have been developed to deal
with such problems. Indeed, in this paper we demonstrate a connection between
those techniques and our account of QFT measurements.

A crucial challenge in incorporating quantum informational concepts in QFT is
the lack of a common mathematical language. QIT focuses on the Hilbert space
aspects of quantum theory, for example, the geometry of the space of quantum
states, the characterization of entanglement and the identification of
physically permissible state transformations. It relies crucially on the way
quantum theory describes the extraction of information through measurements.
In contrast, QFT is commonly described through functional methods that lead to
the evaluation of unequal-time field correlation functions. The latter contain
all information about the system, and they can be easily manipulated to derive
quantities like perturbative scattering amplitudes, expectations of stress-
energy tensors and single-particle Wigner functions.

The mathematical difference is not only technical, as it pertains to
fundamental notions like locality and causality. In QIT, these issues are
usually expressed through the Local Operations and Classical Communication
(LOCC) paradigm [79]: The Hilbert space of a quantum informational system is
split up as a tensor product $\otimes_{i}{\cal H}_{i}$, where ${\cal H}_{i}$
is the Hilbert space of the $i$-th subsystem. A local operation on the $i$-th
subsystem is a set of completely positive maps ${\cal C}^{(i)}(a)$ on states
of ${\cal H}_{i}$, such that $\sum_{a}{\cal C}^{(i)}(a)=\hat{I}$, where $a$
denotes measurement outcomes. Then, the causal structure of the system
involves the notion of classical communication. An operation ${\cal
C}^{(i)}(a)$ on a subsystem $i$ may depend on the outcome $b$ of an operation
${\cal D}^{(j)}(b)$ on a subsystem $j$, if the outcome can be communicated to
$i$ through a classical channel prior to the operation ${\cal C}^{(i)}(a)$.
QIT usually does not deal with real-time quantum signal propagation between
disconnected subsystems.

In contrast, QFTs focus on spacetime propagation of information, as expressed
in the theory’s dynamics and spacetime symmetries. These properties are nicely
captured by the unequal-time correlation functions, which can be chosen to be
manifestly spacetime covariant. Causality is implemented by the requirement
that specific correlation functions vanish if their arguments are spacelike
separated333To be precise, causality is incorporated in the cluster
decomposition property [80]: there is specific hierarchy of correlation
functions $G_{n}(x_{1},x_{2},\ldots,x_{n})$, where $n=0,1,2,\ldots$ such that
$G_{n+m}(x_{1},\ldots,x_{n},x^{\prime}_{1},\ldots,x^{\prime}_{m})=G_{n}(x_{1},x_{2},\ldots,x_{n})G_{n}(x^{\prime}_{1},x^{\prime}_{2},\ldots,x^{\prime}_{m})$,
if the cluster of events $x_{1},\ldots,x_{n}$ is spacelike separated from the
cluster $x^{\prime}_{1},\ldots,x^{\prime}_{m}$. .

###  2.2 The Quantum Temporal Probabilities Approach: key points

The key features of the QTP approach to measurements on quantum fields are the
following. First, the apparatus is fully incorporated into the quantum
description and it is also treated via QFT. The interaction between the
measured system and the apparatus local and causal, in the sense that it is
governed by an interaction Hamiltonian that is a local functional of quantum
fields.

The measurement apparatus is a macroscopic system that exhibits classical
behavior. We describe this behavior through the decoherent histories approach
to emergent classicality [57, 60]. We assume that the pointer variable is a
highly coarse-grained observable, so that histories for measurement outcomes
satisfy appropriate decoherence conditions.

QTP treats measurements events as localized in space and in time, as is the
case for all physical measurement outcomes. For example, consider a solid-
state detector that is elementary in the sense that can record only a single
event. The detector has a fixed location in a lab, and it records an event at
a moment of time that is determined with finite accuracy. In principle, both
position and time can be random variables. When directing a single particle
towards an array of elementary detectors, both the specific detector that
records the particle (i.e., the locus of the record) and the time of recording
vary from one run of the experiments to the other. Hence, physical predictions
are expressed in terms of probability densities

$\displaystyle P(x_{1},q_{1};x_{2},q_{2},\ldots,x_{n},q_{n}),$ (2)

for multiple detection events. In Eq. (2), $x_{i}$ stand for spacetime points,
$q_{i}$ stand for any other recorded observable and $P$ is a probability
density with respect to both $x_{i}$ and $q_{i}$.

Since there is no self-adjoint operator for time, there are no ideal (i.e.,
projective) measurements for time. It follows that none of the probabilities
densities (2) corresponds to an ideal measurement, not even ones that involve
single-measurement event. The probabilities (2) are defined through Positive
Operator Valued Measures (POVMs).

###  2.3 Detection probability for a single detector

Consider a QFT on Minkowski spacetime $M$, with Heisenberg-picture fields
$\hat{\phi}_{r}(x)$ defined on a Hilbert space ${\cal F}$ that carries a
unitary representation of the Poincaré group. The index $r$ runs over
spacetime and internal indices.

We denote the Hilbert space associated to an apparatus by ${\cal K}$. We
assume that the apparatus follows a world tube ${\cal W}=\mathbb{R}\times U$
in Minkowski spacetime, for some $U\subset\mathbb{R}^{3}$. We assume that the
size of the apparatus is much larger than the scale of microscopic dynamics.
We will elaborate on the properties of the apparatus later.

We assume a field-apparatus coupling with support in a small spacetime region
around a point $x$. The finite spacetime extent of the interaction mimics the
effect of a detection record localized at $x$. Working in the interaction
picture, we express the coupling term as

$\displaystyle\hat{V}_{x}=\int F_{x}(y)\hat{C}_{a}(y)\otimes\hat{J}^{a}(y),$
(3)

where $\hat{C}_{a}(x)$ is a composite operator on ${\cal F}$ that is local
with respect to the field $\hat{\phi}_{r}(x)$ and $a$ runs over spacetime and
internal indices. The current operators $\hat{J}^{a}(x)$ are defined on ${\cal
K}$. The switching functions $F_{x}(y)$ are dimensionless. They vanish outside
the interaction region and they depend on the motion of the apparatus. The
spacetime volume associated to a switching function is $\upsilon=\int
dYF^{2}_{x}(y)$.

The problem with the interaction term (3) is that it is not Poincaré
covariant, because of the presence of the switching function. Indeed, physical
interactions are not switched on and off in measurements. The switching
function originates from von Neumann’s modeling of measurements [61], and its
role is to localize the system-apparatus interaction in spacetime. Since the
switching function is a priori fixed, time cannot be an random variable, in
apparent contradiction to statements in Sec. 2.2. Indeed, the QTP method, does
not require a switching function, and it treats time explicitly as a random
variable. However, the resulting expressions for the probabilities are the
same to leading order in perturbation theory, modulo some assumptions that are
explicitly stated in Sec. 2.4. Since the treatment with the switching
functions employs more familiar techniques and it is computationally easier,
we chose to employ this for the construction of probabilities in the main
text. For completeness, we present the QTP derivation of the same probability
formula in the Appendix. The difference between the two derivations is not be
important in the context of the QIT applications considered here, but
certainly it is important for any discussion of the relativistic quantum
measurement problem, and related foundational issues.

The S-matrix associated to Eq. (3) is $\hat{S}_{x}={\cal T}\exp[-i\int
d^{4}yF_{x}(y)\hat{C}_{a}(y)\otimes\hat{J}^{a}(y)]$, where ${\cal T}$ stands
for time ordering. To leading order in the interaction,

$\displaystyle\hat{S}_{x}=\hat{I}-i\hat{V}_{x}.$ (4)

Let the initial state of the system be $|\psi\rangle\in{\cal F}$ and the
initial state of the apparatus be $|\Omega\rangle$. A particle record appears
if the detector transitions from $|\Omega\rangle$ to its complementary
subspace ${\cal K}^{\prime}$. Furthermore, on ${\cal K}^{\prime}$, we measure
a property of the particle through a pointer observable $q$. The latter is
described by a family of positive operators $\hat{\Pi}(q)$, such that
$\sum_{q}\hat{\Pi}(q)=\hat{I}-|\Omega\rangle\langle\Omega|$. The pointer
observable is typically very coarse, and we assume that it is stationary under
spacetime translations by the self-dynamics of the detector, so that the
record is preserved after the end of the measurement.

Then, we evaluate the probability

$\displaystyle\mbox{Prob}(x,q)=\langle\psi,\Omega|\hat{S}^{\dagger}_{x}[\hat{I}\otimes\hat{\Pi}(q)]\hat{S}_{x}|\psi,\Omega\rangle$
(5)

that the detector is excited and records a value $q$ to leading order in
perturbation theory

$\displaystyle\mbox{Prob}(x,q)=\int
d^{4}y_{1}d^{4}y_{2}F_{x}(y_{1})F_{x}(y_{2})G_{ab}(y_{1},y_{2})\langle\Omega|\hat{J}^{a}(y_{1})\hat{\Pi}(q)\hat{J}^{b}(y_{2})|\Omega\rangle,$
(6)

where

$\displaystyle
G_{ab}(x,x^{\prime})=\langle\psi|\hat{C}_{a}(x)\hat{C}_{b}(x^{\prime})|\psi\rangle,$
(7)

is a correlation function for the composite operator.

The probability $\mbox{Prob}(x,q)$ of Eq. (6) is not a density with respect to
$x$, because $x$ appears as a parameter of the switching function. We define
an unnormalized probability density $W(x,q)$ with respect to $x$ by dividing
$\mbox{Prob}(x,q)$ with the effective spacetime volume $\upsilon$,

$\displaystyle W(x,q)=\upsilon^{-1}\mbox{Prob}(x,q).$ (8)

To further proceed in our analysis, we have to make specific assumptions about
the detector. First, we assume that the detector carries a representation of
the spacetime translation group with generators $\hat{p}^{\mu}$. At a
fundamental level, the detector Hilbert space also carries a representation of
the Poincaré group. However, the state $|\Omega\rangle$ is not the Poincaré
invariant vacuum444 An actual detector involves a macroscopically large number
of fermions—to be precise, it is described by the subspace of the fermionic
fields for leptons and baryons that corresponds to macroscopically large,
constant, values of the leptonic and baryonic quantum numbers.; it defines a
preferred reference system at which its center of momentum has zero three-
momentum. The fact that the apparatus cannot be described by the vacuum breaks
the covariance of the measurement model.

We choose a reference point $x_{0}$ in the detector’s world-tube, and we write

$\displaystyle\hat{J}^{a}(y)=e^{-i\hat{p}\cdot(y-x_{0})}\hat{J}^{a}(x_{0})e^{i\hat{p}\cdot(y-x_{0})}.$
(9)

It is convenient to take $|\Omega\rangle$ to be approximately translation
invariant. Intuitively, this corresponds to the idea that the apparatus is
prepared in an initial state that is homogeneous at the length scales that
correspond to position sampling and approximately static at the time scales
that correspond to time sampling555In non-relativistic von Neumann
measurements, one often takes the initial state of the detector to be the
ground state of the detector Hamiltonian, so it is exactly static. If the
initial state is only an approximate eigenstate, its time evolution is usually
viewed as part of the measurement’s noise.. In the present context,
approximate translation invariance is the requirement that

$\displaystyle\int
d^{4}xF_{x}(x^{\prime})\hat{J}^{a}(x^{\prime})|\Omega\rangle\simeq\int
d^{4}xF_{x}(x^{\prime})e^{-i\hat{p}\cdot(x^{\prime}-x_{0})}\hat{J}^{a}(x_{0})|\Omega\rangle.$
(10)

With this assumption, we can write
$\langle\Omega|\hat{J}^{a}(y_{1})\hat{\Pi}(q)\hat{J}^{b}(y_{2})|\Omega\rangle=R^{ab}(y_{2}-y_{1},q)$,
where

$\displaystyle
R^{ab}(x,q):=\langle\Omega|\hat{J}^{a}(x_{0})\sqrt{\hat{\Pi}}(q)e^{-i\hat{p}\cdot
x}\sqrt{\hat{\Pi}}(q)\hat{J}^{b}(x_{0})|\Omega\rangle$ (11)

We will refer to $R^{ab}(y,q)$ as the detector kernel.

The simplest choice for the switching functions $F_{x}$ are Gaussians,

$\displaystyle F_{x}(y)=\exp[-\frac{1}{2}D(x,y)],$ (12)

where $D$ a Euclidean distance function on Minkowski spacetime. Such functions
are defined in terms of a dimensionless Euclidean metric $g^{(E)}_{\mu\nu}$ on
Minkowski spacetime. A simple and physically relevant metric is determined by
the timelike vector field $u_{\mu}$ normal along the world-tube, by

$\displaystyle
g_{\mu\nu}=\frac{1}{\delta_{t}^{2}}u_{\mu}u_{\nu}+\frac{1}{\delta_{x}^{2}}(u_{\mu}u_{\nu}+\eta_{\mu\nu}).$
(13)

where $\delta_{t}$ is the temporal accuracy and $\delta_{x}$ is the special
accuracy of the detector. As these quantities correspond to the sampling of
the detection event, they are both macroscopic scales.

Gaussian switching functions satisfy the identity

$\displaystyle
f(x)f(x^{\prime})=f^{2}\left(\frac{x+x^{\prime}}{2}\right)\sqrt{f}(x-x^{\prime}).$
(14)

The spacetime volume $\upsilon$ of the interaction region is
$\upsilon=\pi^{2}\delta_{t}\delta_{x}^{3}$. We note that the function
$\sigma(x):=\frac{1}{\upsilon}f^{2}(x)$ is a normalized probability density on
$M$. Then, we write

$\displaystyle W(x,q)=\int
d^{4}x^{\prime}\sigma(x-x^{\prime})P(x^{\prime},q),$ (15)

where

$\displaystyle P(x,q)=\int
d^{4}y\sqrt{f}(y)R^{ab}(y,q)G_{ab}(x-\frac{1}{2}y,x+\frac{1}{2}y),$ (16)

The probability distribution $W(x,q)$ is the convolution of $P(x,q)$ with the
probability density $\sigma(x)$ that incorporates the accuracy of our
measurements. If $P(x,q)$ is non-negative and the scale of variation in $x$ is
much larger than both $\delta_{t}$ and $\delta_{x}$, we can treat $P(x,q)$ as
a finer-grained version of $W(x,q)$ and employ this as our probability density
for detection.

The kernel $R^{ab}(x,q)$ is typically characterized by some correlation
length-scale $\ell$ and some correlation time-scale $\tau$, such that
$R^{ab}(x,q)\simeq 0$ if $|t(\xi)|\gg\tau$ or $|{\bf x}(\xi)|\gg\ell$. Both
scales $\ell$ and $\tau$ are microscopic and characterize the constituents of
the apparatus and their dynamics. If $\ell\ll\delta_{x}$ and
$\tau\ll\delta_{t}$, then $R^{ab}(x,q)\sqrt{f}(x)\simeq R^{ab}(x,q)$ and we
obtain an expression for the probability density $P(x,q)$ that is sampling-
independent

$\displaystyle P(x,q)=\int
d^{4}yR^{ab}(y,q)G_{ab}(x-\frac{1}{2}y,x+\frac{1}{2}y).$ (17)

The probability densities (17) are not normalized to unity. In general, the
total probability of detection $P_{det}=\sum_{q}\int_{\cal W}d^{4}xP(q,x)$
must be a small number, for perturbation theory to be applicable. There is
always a probability $P(\emptyset)=1-P_{det}$ of no detection. We normalize
probabilities by dividing $P(x,q)/P_{det}$, i.e., by conditioning the
probability densities $P(q,x)$ with respect to the existence of a detection
record.

###  2.4 Remarks

1\. The definition (8) of a spacetime density with respect to time is fully
justified in classical probability theory, but it is not rigorous for quantum
probabilities. There reason is that it involves a combination of probabilities
defined with respect to different experimental set-ups, i.e., different
switching functions for the Hamiltonians. Nonetheless, Eq. (8) can be derived
as a genuine probability density in the context of the QTP method [50, 81], to
leading order in the field-apparatus coupling. This derivation is reproduced
in the Appendix.

2\. Probabilities are defined here using the Born rule for the pointer
variable. Note that, strictly speaking, the probabilities in von Neumann
measurements are defined at a time after the function has been switched off,
and not at the time when the switching is on. The derivation in the appendix
employs a probability assignment for histories which incorporates both the
Born rule and the state reduction rule.

3\. In the proper QTP derivation of detection probabilities, the interaction
is present at all times, as the total description must be time-translation
invariant. The smearing functions $F_{x}(y)$ are not interpreted in terms of a
switching-on of the interaction, but they describe the sampling of the
spacetime point. Hence, the spacetime volume $\hat{\upsilon}$ is a measure of
coarse-graining, i.e., of the inaccuracy in the determination of the spacetime
point. This point is important for a proper derivation, because probabilities
can only be defined for histories that satisfy a decoherence condition, for
which coarse-graining is a prerequisite. In principle, a preferred value of
$\upsilon$ that corresponds to the coarse-graining scale at which
probabilities are well-defined is determined from first-principles—see [48]
for explicit calculations in simple models. This means that not all sampling
functions $F_{x}(y)$ are acceptable: their support cannot be made arbitrarily
small. Such constraints cannot be seen in the derivation that we presented
here.

4\. QTP leads to different predictions beyond the lowest order in the system-
apparatus interaction. However, in many set-ups only the lowest order terms
can be said to provide a meaningful signal, i.e., a correlation between
observables in the measured field and pointer variables in the apparatus.
Higher order interaction terms often degrade such correlations, and for this
reason they can be conceptualized as noise. This is the case, for example, in
Glauber’s photodetection theory.

5\. In general, the field-apparatus couplings are fixed by the standard model
of particle physics. Even in idealized models for detectors, we have to
specify the physical process through which detection takes place. This
information is encoded in the composite operators $\hat{C}(x)$. Consider, for
example, the case of a QFT with a single scalar field $\hat{\phi}(x)$. The
choice $\hat{C}(x)=\hat{\phi}(x)$ corresponds to detection of particles
through absorption. The choice $\hat{C}(x)=:\hat{\phi}^{2}(x):$ corresponds to
particle detection through scattering, through terms that involve one creation
and one annihilation operator. This observation generalizes to fields of
arbitrary spin.

###  2.5 Detection probability for multiple detectors

We proceed to the derivation of a probability formula for the case that the
field interacts with $n$ detectors. Each detector corresponds to a Hilbert
space ${\cal K}_{i}$, $i=1,2,\ldots,n$. The total Hilbert space of the system
is ${\cal F}\otimes{\cal K}_{1}\otimes{\cal K}_{2}\otimes\ldots\otimes{\cal
K}_{n}$. The coupling operator of the $i$-th detector is

$\displaystyle\hat{V}^{(i)}_{x}=\int
d^{4}yF_{x}(y)\hat{C}^{(i)}_{a}(y)\otimes\hat{I}\otimes\ldots\hat{J}_{(i)}^{a}(y)\otimes\ldots\otimes\hat{I},$
(18)

where $\hat{J}_{(i)}^{a}(y)$ is a current operator for the $i$-th detector and
$\hat{C}_{a}^{(i)}$ is the associated composite operator.

We denote the S-matrix for the $i$-th detector by $\hat{S}^{(i)}_{x}$. Again,
to leading order in the interaction
$\hat{S}^{(i)}_{x}=\hat{I}-i\hat{V}^{(i)}_{x}$. We also denote the initial
state of the $i$-th detector by $|\Omega_{i}\rangle$, and the measurement
operators as $\hat{\Pi}^{(i)}(q_{i})$.

The key point is that the S matrix for the total interaction of the field is
defined by time ordering with respect to the spacetime points
$x_{1},x_{2},\ldots,x_{n}$ associated to the detectors,

$\displaystyle\hat{S}_{x_{1},x_{2},\ldots,x_{n}}={\cal
T}\left[\hat{S}^{(1)}_{x_{1}}\hat{S}^{(2)}_{x_{2}}\ldots\hat{S}^{(n)}_{x_{n}}\right];$
(19)

here, ${\cal T}$ stands for time ordering.

To leading order in perturbation theory, the probability density for $n$
measurement events is

$\displaystyle W_{n}(x_{1},q_{1};x_{2},q_{2};\ldots;x_{n},q_{n})=\int
d^{4}x^{\prime}_{1}\ldots
d^{4}x^{\prime}_{n}\sigma^{(1)}(x_{1}-x^{\prime}_{1})\ldots\sigma^{(n)}(x_{n}-x^{\prime}_{n})$
$\displaystyle
P(x^{\prime}_{1},q_{1};x^{\prime}_{2},q_{2};\ldots;x^{\prime}_{n},q_{n})$ (20)

where

$\displaystyle P_{n}(x_{1},q_{1};x_{2},q_{2};\ldots;x_{n},q_{n})=\int
d^{4}y_{1}\ldots
d^{4}y_{n}\sqrt{f^{(1)}}(y_{1})\ldots\sqrt{f^{(n)}}(y_{n})R_{(1)}^{a_{1}b_{1}}(y_{1},q_{1})\ldots$
$\displaystyle\times\ldots R_{(n)}^{a_{n}b_{n}}(y_{n},q_{n})G_{a_{1}\ldots
a_{n},b_{1}\ldots
b_{n}}(x_{1}-\frac{1}{2}y_{1},\ldots,x_{n}-\frac{1}{2}y_{n};x_{1}+\frac{1}{2}y_{1},\ldots,x_{n}-\frac{1}{2}y_{n}).\hskip
5.69046pt$ (21)

Here $R^{(i)}(x,q)$ is the measurement kernel for the $i$-th detector. The
field correlation function $G_{a_{1}\ldots a_{n},b_{1}\ldots
b_{n}}(x_{1},\ldots,x_{n};x_{1}^{\prime},\ldots,x_{n}^{\prime})$ is given by

$\displaystyle G_{a_{1}\ldots a_{n},b_{1}\ldots
b_{n}}(x_{1},\ldots,x_{n};x_{1}^{\prime},\ldots,x_{n}^{\prime})=\langle\psi|{\cal
T}^{*}[\hat{C}_{b_{1}}^{(1)}(x^{\prime}_{1})\ldots\hat{C}_{b_{n}}^{(n)}(x^{\prime}_{n})]$
$\displaystyle\times{\cal
T}[\hat{C}_{a_{n}}^{(n)}(x_{n})\ldots\hat{C}_{a_{1}}^{(1)}(x_{1})]|\psi\rangle$
(22)

where ${\cal T}^{*}$ stands for reverse time ordering.

Again we note that in the appropriate regime, the probability becomes
independent of the switching functions, and equal to

$\displaystyle P_{n}(x_{1},q_{1};x_{2},q_{2};\ldots;x_{n},q_{n})=\int
d^{4}y_{1}\ldots d^{4}y_{n}R_{(1)}^{a_{1}b_{1}}(y_{1},q_{1})\ldots
R_{(n)}^{a_{n}1b_{n}}(y_{n},q_{n})$ $\displaystyle\times G_{a_{1}\ldots
a_{n},b_{1}\ldots
b_{n}}(x_{1}-\frac{1}{2}y_{1},\ldots,x_{n}-\frac{1}{2}y_{n};x_{1}+\frac{1}{2}y_{1},\ldots,x_{n}+\frac{1}{2}y_{n})$
(23)

###  2.6 The detector kernel

The contribution of each detector to the probability is determined by the
detector kernel $R^{ab}(x,q)$, defined by Eq. (11). The detector kernel
coincides with the matrix elements of a unitary operator,

$\displaystyle R^{ab}(x,q)=\langle a,q|e^{i\hat{p}\cdot(x-x_{0})}|b,q\rangle,$
(24)

where $|a,q\rangle=\sqrt{\hat{\Pi}}(q)\hat{J}^{a}(x_{0})|\Omega\rangle$. If
$\hat{\Pi}$ is a projector, then vectors $|a,q\rangle$ with different values
of $q$ are orthogonal: $\langle a,q|b,q^{\prime}\rangle=0$ for $q\neq
q^{\prime}$.

The Fourier transform of the detector kernel

$\displaystyle\tilde{R}^{ab}(\xi,q)=\int d^{4}xe^{-i\xi\cdot x}R^{ab}(x,q),$
(25)

is given by

$\displaystyle\tilde{R}^{ab}(\xi,q)=(2\pi)^{4}e^{i\xi\cdot x_{0}}\langle
a,q|\hat{E}_{\xi}|b,q\rangle,$ (26)

where $\hat{E}_{\xi}=\delta^{4}(\hat{p}-\xi)$ is the projector onto the
subspace with four-momentum $\xi^{\mu}$, $\langle
a,q|\hat{E}_{\xi}|a,q\rangle\geq 0$. The momentum four vector associated to
the detector is timelike and the associated energy $p^{0}$ is positive. This
means that $\tilde{R}^{ab}(p,q)=0$ for spacelike $p$, or if $p^{0}<0$.

For a scalar composite operator $\hat{C}(x)$, we can drop the indices
$a,b,\ldots$ from the detector kernel and write simply $R(x,q)$. If we record
no other observable, we simply write
$R(x):=\langle\omega|e^{i\hat{p}\cdot(x-x_{0})}|\omega\rangle$, where
$|\omega\rangle=\hat{J}^{a}(x_{0})|\Omega\rangle$. Hence, the detector kernel
is determined by the energy-momentum distribution of the ‘threshold’ state
$\omega$.

###  2.7 Index notation for probabilities

It is convenient to express the probability densities (23) using an abstract
notation. We use small Greek indices $\alpha,\beta,\gamma\ldots$ for the pairs
$(x,a)$ where $x$ is a spacetime point and $a$ the internal index for the
composite operators $\hat{C}_{a}$. All indices in a time-ordered product are
upper, and all indices in an anti-time-ordered product are lower. Hence, we
write the correlation functions (22) as

$\displaystyle
G^{\alpha_{1}\alpha_{2}\ldots\alpha_{n}}_{\beta_{1}\beta_{2}\ldots\beta_{n}}$

We denote by $z$ the pairs $(x,q)$, where $x$ is a spacetime point and $q$ any
other recorded observable or the event $\emptyset$ of no detection. If we
denote by $\Gamma$ the set of possible values for $q$, then $z$ takes values
on the set $Z:=M\times\Gamma\cup\\{\emptyset\\}$. We will write the kernel

$\displaystyle\sigma[x-\frac{1}{2}(y+y^{\prime})]\sqrt{f}(y-y^{\prime})R^{ab}(y-y^{\prime},q)$

as $R_{\alpha}^{\beta}(z)$ where $\alpha$ stands for $(y,a)$, $\beta$ for
$(y^{\prime},b)$ and $z$ for $(x,q)$. We will use the same symbol for the
approximate expression
$\delta[x-\frac{1}{2}(y+y^{\prime})]R^{ab}(y-y^{\prime},q)$. We use the
Einstein summation convention over Greek indices, to denote sum over the
discrete index $a$ and spacetime integral.

Using the index notation, we write the probability formula (23) as

$\displaystyle
P_{n}(z_{1},z_{2},\ldots,z_{n})=G^{\alpha_{1}\alpha_{2}\ldots\alpha_{n}}_{\beta_{1}\beta_{2}\ldots\beta_{n}}\;\;{}^{(1)}R_{\alpha_{1}}^{\beta_{1}}(z_{1})\;{}^{(2)}R_{\alpha_{2}}^{\beta_{2}}(z_{2})\ldots\;{}^{(n)}R_{\alpha_{n}}^{\beta_{n}}(z_{n}).$
(27)

There are two ways of viewing Eq. (27). On one hand, it corresponds to a POVM
defined on the Hilbert space ${\cal F}$ of the quantum field, with
alternatives in the set $Z^{n}$. On the other hand, we can consider a vector
space ${\cal S}$ of smearing functions $f_{a}$ to the composite operators
$\hat{C}^{a}(x)$, so that $\hat{C}(f)=\int d^{4}x\hat{C}^{a}(x)f_{a}(x)$. The
detector kernel is an element of ${\cal S}\otimes{\cal S}^{*}$, where ${\cal
S}^{*}$ stands for the dual vector space of ${\cal S}$. Then, Eq. (27) defines
a positive linear functional on $({\cal S}\otimes{\cal S}^{*})^{n}$ that
corresponds to the correlation functions
$G^{\alpha_{1}\alpha_{2}\ldots\alpha_{n}}_{\beta_{1}\beta_{2}\ldots\beta_{n}}$

##  3 Relation of the QTP approach to the Closed-Time-Path formalism

The probability density (21) for $n$ measurement events is a linear functional
of the $2n$-point unequal-time correlation function (22). This correlation
function has $n$ time-ordered arguments and $n$ anti-time-ordered arguments.
It does not appear in the usual S-matrix description of QFT; the correlation
functions in the S-matrix description involve only time-ordered arguments.
Rather, the correlation function (22) appears in the Schwinger-Keldysh or
Closed-Time-Path (CTP) formalism of QFT.

In what follows, we will review the motivation and function of the CTP
formalism, and then we will show how it relates to the QTP probability
assignment.

###  3.1 S-matrix vs CTP correlation functions

By the Dyson formula, the S-matrix for any QFT involves Feynman (i.e., time-
ordered) correlation functions of local composite operators $\hat{C}(x)$ of
the fields. We can generate the time-ordered correlation functions of a
composite operator from the variational derivatives of a generating functional
$Z[J]=\langle 0|\hat{U}[J]|0\rangle$, where

$\displaystyle\hat{U}[J]={\cal T}\exp\left[i\int
d^{4}XJ^{a}(X)\hat{C}_{a}(X)\right].$ (28)

The generating functional admits the path integral representation

$Z[J]=\int~{}D\phi~{}e^{i\\{S[\Phi]+\int d^{4}xJ(x)C(x)\\}},$ (29)

where $S[\phi]$ is the classical action functional and $C(x)$ is the classical
counterpart of the composite operator $\hat{C}(x)$.

It is convenient to work with the generating functional $W[J]=-i\ln Z[J]$. In
perturbation theory, the coefficients of the series expansion of $W$ with
respect to $J$ are the connected Feynman graphs. In particular, we may
introduce the ‘background’ or ‘mean’ field

$C_{a}(x)={\frac{\delta W[J]}{\delta J^{a}(x)}}$ (30)

The Legendre transformation of $W$ provides a more efficient representation of
the correlation functions

$\Gamma[C]=W[J]-\int d^{4}x~{}J^{a}(x)C_{a}(x)$ (31)

since the series expansion of $\Gamma$ involves only one-particle irreducible
Feynman graphs. Despite the formal similarity, the inverse of (30),

${\frac{\delta\Gamma}{\delta C_{a}(x)}}=-J^{a}(x).$ (32)

is not an equation of motion for the mean field, because it is complex-valued
even for hermitian composite operators. This is due to the fact that the
‘mean’ field is not a true expectation value, but rather a matrix element of
the composite operator between in and out states [82].

To define a proper mean-field equation of motion we adopt the Closed-Time-Path
formalism of Schwinger [63] and Keldysh[64]. To this end, we couple the field
to two different external sources $J^{a}(x)$ and $\bar{J}^{a}(x)$, and we
define the CTP generating functional

$\displaystyle
Z_{CTP}[J,\bar{J}]=\langle\psi_{0}|\hat{U}^{\dagger}[\bar{J}]\hat{U}[J]|\psi_{0}\rangle,$
(33)

By definition, $Z[J,J]=1$ and $Z^{*}[J,\bar{J}]=Z[\bar{J},J]$. The state
$|\psi_{0}\rangle$ is defined in the distant past, i.e., prior to any time at
which $J(x)$ has support—it is an in state. Note that a large class of initial
states, relevant to particle experiments, can be reconstructed by the action
of source terms of the form (28). Hence, for a large class of problems, the
generating function $Z_{CTP}$ for the QFT vacuum $|0\rangle$ contains all
physically relevant information.

The CTP generating functional describes correlation functions with $n$ time-
ordered and $m$ anti-time-ordered entries,

$\displaystyle G^{n,m}_{a_{1}\ldots a_{m},b_{1}\ldots
b_{n}}(x_{1},\ldots,x_{n};x_{1}^{\prime},\ldots,x_{n}^{\prime})=\langle\psi_{0}|{\cal
T}^{*}[\hat{C}_{b_{1}}^{(1)}(x^{\prime}_{1})\ldots\hat{C}_{b_{m}}^{(n)}(x^{\prime}_{m})]$
$\displaystyle\times{\cal
T}[\hat{C}_{a_{n}}^{(i)}(x_{n})\ldots\hat{C}_{a_{1}}^{(1)}(x_{1})]|\psi_{0}\rangle$
(34)

as functional derivatives of $Z_{CTP}[J,\bar{J}]$,

$\displaystyle G^{n,m}_{a_{1}\ldots a_{m},b_{1}\ldots
b_{n}}(x_{1},\ldots,x_{n};x_{1}^{\prime},\ldots,x_{n}^{\prime})=i^{n-m}\left(\frac{\partial^{n+m}Z_{CTP}}{\delta
J^{a}_{1}(x_{1})\ldots\delta
J^{a_{n}}(x_{n})\delta\bar{J}^{b}_{1}(x^{\prime}_{1})\ldots\delta\bar{J}^{b_{m}}(x^{\prime}_{n})}\right)_{J=\bar{J}=0}.$

For a vacuum initial state, the generating functional has a path integral
expression

$Z_{CTP}[J,\bar{J}]=\int~{}D\phi~{}D\bar{\phi}\;e^{i\\{S[\phi]-S[\bar{\phi}]+\int
d^{4}x~{}[J^{a}(x)C_{a}(x)-\bar{J}(x)\bar{C}_{a}(x)]\\}},$ (35)

where $\bar{C}$ is defined a functional of $\bar{\phi}$. Thus we formally
double the degrees of freedom and we integrate over pairs of histories
$(\phi,\bar{\phi})$ that vanish in the asymptotic past and future.

We can actually define two background fields $C_{a}(x)$ and $\bar{C}_{a}(x)$
by taking variations with respect to both sources,

$\displaystyle C_{a}(x)=\frac{\delta W}{\delta J^{a}(x)}\hskip
28.45274pt\bar{C}_{a}(x)=-\frac{\delta W}{\delta\bar{J}^{a}(x)}$ (36)

where $W[J,\bar{J}]=-i\ln Z_{CTP}[J,\bar{J}]$. The closed time-path effective
action is defined as the double Legendre transform of $W$

$\Gamma[C,\bar{C}]=W[J,\bar{J}]-\int d^{4}xJ^{a}(x)C_{a}(x)+\int
d^{4}x\bar{J}^{a}(x)\bar{C}_{a}(x).$ (37)

The equations of motion now form a coupled system

$\frac{\delta\Gamma}{\delta C_{a}(x)}=J_{a}(x)\hskip
28.45274pt\frac{\delta\Gamma}{\delta\bar{C}_{a}(x)}=\bar{J}_{a}(x)$ (38)

These equations coincide when $J=J^{\prime}$ is the physical external source,
and their solution $C=\bar{C}$ is the physical mean field. Of course, both
background fields are identified only after the computation of the variational
derivative [68]. Unlike the in-out formulation, the equations of motion (38)
are always real and causal. The cases where the in-out effective action
becomes complex correspond to dissipative evolution for the mean field. The
doubling of degrees of freedom is essential for the description of
dissipation, because no action functional of a single mean field alone could
possibly lead to the right equations of motion.

The CTP effective action formalism can be used to treat fully non-equilibrium
dynamics of open quantum systems, going far beyond the traditional linear
response theory which are confined to near-equilibrium conditions. In this
context, we are often interested in calculating mean-field expectation values
$\langle\hat{C}_{a}(x)\rangle$ that may describe the evolution of
classicalized macroscopic field observables.

The CTP effective action satisfies $\Gamma[C,C]=0$ and
$\Gamma[C,\bar{C}]=-\Gamma[\bar{C},C]^{*}$. Therefore, if we separate the CTP
effective action in its real and imaginary parts, the former is an odd
function of $C_{a}$ and $\bar{C}_{a}$ and the latter is even. Let
$C_{a}^{(0)}=\bar{C}_{a}^{(0)}$ be an extremum of the CTP effective action,
which corresponds to a solution of the mean field equations of motion with
zero sources. We take linearized perturbations $\Delta
C_{a}=C_{a}-C_{a}^{(0)}$ and $\Delta\bar{C}_{a}=\bar{C}_{a}-C_{a}^{(0)}$.
Then, the CTP effective action must take the form

$\displaystyle\Gamma[\Delta
C,\Delta\bar{C}]={\frac{1}{2}}\int~{}d^{d}xd^{d}x^{\prime}\\{-[\Delta
C_{a}(x)](x){\cal D}^{ab}(x,x^{\prime})\\{\Delta C_{b}\\}(x^{\prime})$
$\displaystyle+i[\Delta C_{a}](x)N^{ab}(x,x^{\prime})[\Delta
C_{b}](x^{\prime})\\}$

where $[\Delta C_{a}]=(\Delta C_{a}-\Delta\bar{C}_{a})$, $\\{\Delta
C_{a}\\}=(\Delta C_{a}+\Delta\bar{C}_{a})$, and ${\cal D}^{ab}$ and $N^{ab}$
are two non-local kernels. Superficially, $N^{ab}$ appears redundant, because
it does appear in the mean-field equations of motion. The latter take the
simple form

$\int~{}d^{d}x^{\prime}{\cal D}^{ab}(x,x^{\prime})\Delta
C_{b}(x^{\prime})=J^{a}(x),$ (40)

where $J^{a}(x)$ is an external source. The kernel $N^{ab}$ contains
information about the departure of the actual evolution from the mean-field
equation (40). This deviation reflects the fact that the mean field is coupled
to a stochastic external source [83, 84, 85, 86], rather than a deterministic
one like $J^{a}(x)$ in Eq. (40). This means that the effective classical
equations of motion are of the form

$\int~{}d^{d}x^{\prime}{\cal D}^{ab}(x,x^{\prime})\Delta
C_{b}(x^{\prime})=J^{a}(x)+\xi^{a}(x),$ (41)

where $\xi^{a}(x)$ has vanishing expectation—thus justifying Eq. (40)—but non-
zero correlator
$\langle\xi(x)^{a}\xi^{b}(x^{\prime})\rangle=N^{ab}(x,x^{\prime})$. This is
why $N$ is referred to as a the noise kernel. Hence, CTP allows us to explore
how higher-order correlation functions are manifested as noise in the mean-
field (classical) equations of motion.

###  3.2 Measurements and the CTP generating functional

The relation between the QTP description of measurements and the CTP formalism
is more transparent, if we use the index notation of Sec. 2.7. For
consistency, the sources $J$ have a lower Greek index, and the sources
$\bar{J}$ have an upper Greek index. Then, we write the CTP generating
functional as

$\displaystyle
Z_{CTP}[J,\bar{J}]=\sum_{n,m=0}^{\infty}\frac{i^{m-n}}{n!m!}G^{\alpha_{1}\ldots\alpha_{n}}_{\beta_{1}\ldots\beta_{m}}J_{\alpha_{1}}\ldots
J_{\alpha_{n}}\bar{J}^{\beta_{1}}\ldots\bar{J}^{\beta_{m}},$ (42)

or, conversely,

$\displaystyle
G^{\alpha_{1}\ldots\alpha_{n}}_{\beta_{1}\ldots\beta_{m}}=i^{n-m}\left(\frac{\partial^{n+m}Z_{CTP}[J,\bar{J}]}{\partial
J_{\alpha_{1}}\ldots\partial
J_{\alpha_{n}}\partial\bar{J}^{\beta_{1}}\ldots\partial\bar{J}^{\beta_{n}}}\right)_{J=\bar{J}+0}.$
(43)

The probability densities (27) involve balanced correlation functions, i.e.,
correlation functions with an equal number of upper and lower indices. We can
construct a generating functional that contains only such functions. The key
observation is that such correlations contribute to the sum only through
products of the form $J_{\alpha}\bar{J}^{\beta}$. Hence, the natural source
for a diagonal generating functional $Z^{d}_{CTP}$ that only involves balanced
correlation functions is a ‘tensor’ $L_{\alpha}^{\beta}$. We define

$\displaystyle
Z^{d}_{CTP}[L]=\sum_{n=0}^{\infty}\frac{1}{n!}G^{\alpha_{1}\ldots\alpha_{n}}_{\beta_{1}\ldots\beta_{m}}L_{\alpha_{1}}^{\beta_{1}}\ldots
L_{\alpha_{N}}^{\beta_{N}}.$ (44)

Suppose now that we consider only measurements of a single type, i.e., all
detector kernels $R_{\alpha}^{\beta}(z)$ are identical. Then, we can define a
moment-generating functional for all probability densities (27), in terms of
sources $j(z)$,

$\displaystyle
Z_{QTP}[j]=\sum_{n=0}^{\infty}\sum_{z_{1},z_{2},\ldots,z_{n}}\frac{1}{n!}P_{n}(z_{1},z_{2},\ldots,z_{n})j(z_{1})\ldots
j(z_{n}).$ (45)

It is straightforward to show that

$\displaystyle Z_{QTP}[j]=Z^{d}_{CTP}[R\cdot j],$ (46)

where $(R\cdot j)_{A}^{B}=\sum_{z}R_{A}^{B}(z)J(z)$.

Eq. (46) is a fundamental relation for quantum measurements in QFT, as it
relates the moment generating functional for a hierarchy of measured
probability densities to the generating functional of unequal-time correlation
functions

It is straightforward to write a path integral expression for $Z^{d}_{CTP}[L]$
for a vacuum initial state

$\displaystyle
Z^{d}_{CTP}[L]=\int~{}D\phi~{}D\bar{\phi}\;e^{iS[\phi]-iS[\bar{\phi}]+\int
d^{4}xd^{4}x^{\prime}C_{a}(x)\bar{C}_{b}(x^{\prime})L^{ab}(x,x^{\prime})}.$
(47)

To obtain a simple path integral expression for a broader class of states, we
recall that many field initial states can be obtained from the action of an
external source $\zeta(x)$ on the vacuum, i.e., they are of the form
$|\psi_{0}\rangle=\hat{U}[\zeta]|0\rangle$, where now we write
$\hat{U}[\zeta]={\cal T}\exp\left[i\int
d^{4}X\zeta^{k}(X)\hat{A}_{k}(X)\right]$ in terms of composite operators
$\hat{A}_{k}(x)$ that differ, in general from $\hat{C}_{a}(x)$.

To see this, consider the case of a free scalar field $\hat{\phi}(x)$.
Choosing $\hat{A}(x)=\hat{\phi}(x)$, $\hat{U}[\zeta]$ is a Weyl operator, and
$|\psi_{0}\rangle$ is a field coherent state. For single-event measurements,
and for any state with a fixed number $N$ of particles, the probability
density (17) depends only on the one-particle reduced density matrix
$\rho_{1}({\bf k},{\bf
k^{\prime}})=N^{-1}\langle\psi_{0}|\hat{a}^{\dagger}_{\bf k}\hat{a}_{{\bf
k}^{\prime}}|\psi_{0}\rangle$, where $\hat{a}_{\bf k}$ and
$\hat{a}^{\dagger}_{\bf k}$ are the creation and annihilation operators of the
field. It is straightforward to show that one can reproduce any pure
$\rho_{1}$ with an appropriate choice of $\zeta(x)$. For
$\hat{A}(x)=:\hat{\phi}(x)^{2}:$, $\hat{U}[\zeta]$ is a product of a
Bogoliubov transformation with a Weyl operator, and, hence, $|\psi_{0}\rangle$
is a squeezed state. We can reproduce a large class of two-particle reduced
density matrices $\rho_{2}({\bf k}_{1},{\bf k}_{2};{\bf k^{\prime}}_{1},{\bf
k}^{\prime}_{2})=N^{-2}\langle\psi_{0}|\hat{a}^{\dagger}_{{\bf
k}_{1}}\hat{a}^{\dagger}_{{\bf k}_{2}}\hat{a}_{{\bf
k}^{\prime}_{2}}\hat{a}_{{\bf k}^{\prime}_{1}}|\psi_{0}\rangle$, including
entangled ones, with appropriate choices of $\zeta$.

Hence, for a quantum state that is obtained from an external source
$\zeta_{k}$, we write the path integral expression

$\displaystyle
Z^{d}_{CTP}[f,L]=\int~{}D\phi~{}D\bar{\phi}\;e^{iS[\phi]-iS[\bar{\phi}]+i\int
d^{4}x~{}\zeta^{k}(x)[A_{k}(x)-\bar{A}_{k}(x)]+\int
d^{4}xd^{4}x^{\prime}C_{a}(x)\bar{C}_{b}(x^{\prime})L^{ab}(x,x^{\prime})},$
(48)

where we must assume that the spacetime support of the kernel $L^{ab}$ is
later than the support of $\zeta$ (state preparation is prior to measurement).
By Eq. (46)

$\displaystyle
Z_{QTP}[f,j]=\int~{}D\phi~{}D\bar{\phi}\;e^{iS[\phi]-iS[\bar{\phi}]+i\int
d^{4}x~{}\zeta^{k}(x)[A_{k}(x)-\bar{A}_{k}(x)]}$ $\displaystyle\times
e^{\sum_{z}\int
d^{4}xd^{4}x^{\prime}C_{a}(x)\bar{C}_{b}(x^{\prime})R^{ab}(x,x^{\prime};z)j(z)}$
(49)

The probability densities for $n$ measurement events are obtained from
functional variation of $Z_{QTP}[\zeta,j]$ with respect to $j$ at $j=0$ For
example, the single-event probability density $\hat{P}_{1}(x,z)$ of Eq. (17)
is given by the path integral

$\displaystyle P_{1}(x,q)=\int~{}D\phi~{}D\bar{\phi}\left(\int
d^{4}yC_{a}(x+\frac{1}{2}y)\bar{C}_{b}(x-\frac{1}{2}y)K^{ab}(y,q)\right)$
$\displaystyle\times e^{iS[\phi]-iS[\bar{\phi}]+i\int
d^{4}x~{}\zeta^{k}(x)[A_{k}(x)-\bar{A}_{k}(x)]}.$

Expressions such as the above provide an explicit link between concepts of
quantum measurement theory like POVMs and the practical and highly successful
functional language of QFT. We believe that this link is essential for a local
and covariant definition of quantum informational notions in QFT.

##  4 Applications

Next, we present three immediate applications of the QTP formalism. First, we
review how the QTP formalism leads to the construction of time of arrival
probabilities for relativistic particles. Second, we derive Unruh-DeWitt
detectors from the QTP probability assignment at a particular limit. Third, we
identify the regime in which Glauber’s photodetection theory applies.

###  4.1 Time-of-arrival probabilities

The first application of the QTP formalism is the construction of
probabilities for the time-of-arrival. The simplest time-of-arrival
measurement involves a particle prepared on an initial state
$|\psi_{0}\rangle$ that is localized around $x=0$ and has positive mean
momentum. The issue is to construct the probability $P(L,t)dt$ that the
particle is detected at $x=L$ at some moment between $t$ and $t+\delta t$. The
problem is that no unique time-of-arrival probability distribution exists [87,
88]. The absence of a self-adjoint time operator [89] means that we cannot use
Born’s rule for this task.

A time-of-arrival probability distribution can be constructed from Eq. (17),
in absence of any additional observable $q$. The simplest case, corresponding
to a free scalar field $\hat{\phi}(x)$ of mass $m$, was analysed in Ref. [50].
For a scalar composite operator $\hat{C}(X)$, we obtain a probability density
for detection

$\displaystyle P(x)=\int
d^{4}\xi\;R(\xi)G(x-\frac{1}{2}\xi,x+\frac{1}{2}\xi).$ (50)

In a time-of-arrival measurement, a detector is placed at a macroscopic
distance $L$ from the particle source. If $L$ is much larger than the size of
the detector, only particles with momenta along the source-detector axis are
recorded. Hence, the problem is reduced to two spacetime dimensions. Hence, we
express the spacetime coordinate $x=(t,L)$. Then, Eq. (50) becomes

$\displaystyle
P(t,L)=\int\frac{dpdp^{\prime}}{2\pi}\frac{\rho(p,p^{\prime})}{2\sqrt{\epsilon_{p}\epsilon_{p^{\prime}}}}\;\tilde{R}\left(\frac{p+p^{\prime}}{2},\frac{\epsilon_{p}+\epsilon_{p^{\prime}}}{2}\right)e^{i(p-p^{\prime})L-i(\epsilon_{p}-\epsilon_{p^{\prime}})t},$
(51)

where $\epsilon_{p}=\sqrt{p^{2}+m^{2}}$, and $\tilde{R}(E,p)$ is the Fourier
transform of $R(x)=R(t,L)$.

We normalize Eq. (51) by treating $L$ as a parameter and $t$ as a random
variable. The procedure is described in Ref. [50]. The end result is a
probability distribution

$\displaystyle
P(t,L)=\int\frac{dpdp^{\prime}}{2\pi}\tilde{\rho}(p,p^{\prime})\sqrt{v_{p}v_{p^{\prime}}}S(p,p^{\prime})e^{i(p-p^{\prime})L-i(\epsilon_{p}-\epsilon_{p^{\prime}})t},$
(52)

where $v_{p}=p/\epsilon_{p}$ is the particle velocity. $S(p,p^{\prime})$ are
the matrix elements $\langle p|\hat{S}|p^{\prime}\rangle$ of the localization
operator $\hat{S}$, defined by

$\displaystyle\langle
p|\hat{S}|p^{\prime}\rangle:=\frac{\tilde{R}\left(\frac{p+p^{\prime}}{2},\frac{\epsilon_{p}+\epsilon_{p^{\prime}}}{2}\right)}{\sqrt{\tilde{R}(p,\epsilon_{p})\tilde{R}(p^{\prime},\epsilon_{p^{\prime}})}}.$
(53)

By definition, $\langle p|\hat{S}|\hat{p^{\prime}}\rangle\geq 0$ and
$S(p,p)=1$. Its name originates from the fact that $\hat{S}$ describes the
localization of an elementary measurement event.

For a pure state initial state $|\psi\rangle$, Eq. (51) becomes

$\displaystyle
P(t,L)=\langle\psi|\hat{U}^{\dagger}(t,L)\sqrt{|\hat{v}|}\hat{S}\sqrt{|\hat{v}|}\hat{U}(t,L)|\psi\rangle,$
(54)

where $\hat{U}(t,L)$ is the spacetime-translation operator
$\hat{U}(t,L)=e^{i\hat{p}L-i\hat{H}t}$ and $\hat{v}=\hat{p}\hat{H}^{-1}$ is
the velocity operator. Eq. (54) defines a positive probability distribution if
$\hat{S}$ is a positive operator, which is always the case if
$\ln\tilde{R}(p,\epsilon_{p})$ is a convex function of $p$. Then, the Cauchy-
Schwarz inequality applies,

$\displaystyle\langle p|\hat{S}|p^{\prime}\rangle\leq\sqrt{\langle
p|\hat{S}|p\rangle\langle p^{\prime}|\hat{S}|p^{\prime}\rangle}=1.$ (55)

When Eq. (55) is saturated, i.e., $\langle p|\hat{S}|p^{\prime}\rangle=1$, we
have maximum localization. This means that $\hat{S}=\delta(\hat{x})$, where
$\hat{x}=i\frac{\partial}{\partial p}$. The corresponding probability density
(54) was first identified by León [90], in terms of the eigenvectors of a non-
self-adjoint time-of-arrival operator. It generalizes the non-relativistic
time-of-arrival probability distribution of Kijowski [91].

###  4.2 The limit of a pointlike detector

In a pointlike detector, the detector’s world-tube shrinks to a single
timelike curve $x_{0}^{\mu}(\tau)$, parameterized by its proper time $\tau$.
We implement this limit, by expressing the smearing function as

$\displaystyle F_{x}(y)=\int d\tau f(\tau-s)\delta^{4}[y-x_{0}(\tau)],$ (56)

where $f(\tau)$ is a smearing function peaked around $0$, and $s$ is defined
by the condition $x_{0}(s)=x$.

We choose a Gaussian switching function
$f(s)=\exp\left[-\frac{s^{2}}{2\delta_{t}^{2}}\right]$. Then, we obtain a
probability density for detection $W(q,\tau)=\int ds\sigma(\tau-s)P(q,s)$,
where $\sigma(s)=(\sqrt{\pi}\delta_{t})^{-1}f^{2}(s)$, and

$\displaystyle P(q,\tau)=\int
ds\sqrt{f(s)}G_{ab}[x_{0}(\tau-\frac{1}{2}s),x_{0}(\tau+\frac{1}{2}s)]$
$\displaystyle\langle\Omega|\hat{J}^{a}[x_{0}(\tau-\frac{1}{2}s)]\Pi(q)\hat{J}^{b}[x_{0}(\tau+\frac{1}{2}s)]|\Omega\rangle.$
(57)

The Unruh-DeWitt detection models [32, 33] are a special case of pointlike
detectors, in which

$\displaystyle\hat{J}^{a}[x_{0}(\tau)]=e^{i\hat{h}\tau}\hat{J}^{a}[x_{0}(0)]e^{-i\hat{h}\tau},$
(58)

in terms of a Hamiltonian $\hat{h}$ that is defined with respect to the rest
frame of the detector. Such Hamiltonians are well defined if the four-velocity
$\dot{x}^{\mu}_{0}(\tau)$ is a Killing vector of Minkowski spacetime, because
then time evolution with respect to $\tau$ corresponds to the time translation
of a Lorentzian time coordinate. It is not obvious that such a Hamiltonian
exists for non-inertial trajectories. In a general QFT, Hamiltonians that
generate time translations along non inertial time coordinates do not exist
[93, 92]. However, it may be the case that they can be defined as appropriate
limits when the spatial size of the system goes to zero, the vanishing of the
system’s size canceling the ultraviolet divergences that characterize such
Hamiltonians. To the best of our knowledge such an analysis has not been
carried out. In this sense, the defining assumption for Unruh-DeWitt
detectors—for non-inertial trajectories—remains unjustified from first
principles.

Assuming the Unruh-DeWitt condition (58), and taking $|\Omega\rangle$ as the
lowest eigenstate of $\hat{h}$, we find

$\displaystyle P(q,\tau)=\int
ds\sqrt{f(s)}G_{ab}[x_{0}(\tau-\frac{1}{2}s),x_{0}(\tau+\frac{1}{2}s)]\langle\Omega|\hat{\mu}^{a}\Pi(q)e^{i\hat{h}s}\hat{\mu}^{b}|\Omega\rangle,$
(59)

where we wrote $\hat{\mu}^{a}=\hat{J}^{a}[x_{0}(0)]$. If we identify $q$ with
the energy $\epsilon$ in the rest frame of the detector, we find that

$\displaystyle P(\epsilon,\tau)=\int ds\sqrt{f(s)}e^{i\epsilon
s}G_{ab}[x_{0}(\tau-\frac{1}{2}s),x_{0}(\tau+\frac{1}{2}s)]\langle\Omega|\hat{\mu}^{a}|\epsilon\rangle\langle\epsilon|\hat{\mu}^{b}|\Omega\rangle.$
(60)

For a scalar coupling operator, where we drop the indices $a,b$ and so on, and
for energies such that $\epsilon\delta_{t}>>1$, we write

$\displaystyle
P(\epsilon,\tau)=|\langle\Omega|\hat{\mu}|\epsilon\rangle|^{2}\int
dse^{i\epsilon s}G_{ab}[x_{0}(\tau-\frac{1}{2}s),x_{0}(\tau+\frac{1}{2}s)],$
(61)

i.e., we obtain the standard formula for the response of the Unruh-DeWitt
detector.

Hence, the probabilities of the Unruh-DeWitt detectors for inertial path can
be obtained from the QTP probability formula—that is derived from full QFT—at
the limit where the world-tube of the detector shrinks to a single timelike
curve.

###  4.3  Derivation of Glauber’s theory as a limit

Consider a detector in inertial motion, i.e., the timelike normal of its
world-tube is a time-translation vector field in Minkowski spacetime. For
simplicity, we consider a quantum scalar field $\hat{\phi}(x)$, with a
composite operators $\hat{C}(x)=\hat{\phi}(x)$. We consider particles with
$m=0$, i.e., scalar photons.

A coherent state $|z\rangle$ of the scalar field is defined as an eigenvector
of the annihilation operator

$\displaystyle\hat{a}_{k}|z\rangle=z(k)|z\rangle,$ (62)

where $z(k)$ is a square-integrable complex-valued function; here $k=({\bf
k},\omega_{\bf k})$ is an on-shell four-vector.

It is convenient to express the initial state in the $P$ representation

$\displaystyle\hat{\rho}=\int Dzf_{P}(z)|z\rangle\langle z|,$ (63)

where the functional $f_{P}(z)$ is the $P$-symbol of quantum state
$\hat{\rho}$ [94]. The functional integration in Eq. (63) is defined in terms
of the Gaussian measure associated to the Bargmann representation of the
quantum field [95].

For simplicity, we assume that no magnitude $q$ is recorded. The probability
density Eq. (17) is expressed as

$\displaystyle P(x)=\int Dzf_{P}(z)P_{z}(x),$ (64)

where $P_{z}(x)$ is the probability density Eq. (17) evaluated for an initial
coherent state. It is straightforward to show that $P_{z}$ is a sum of three
contributions

$\displaystyle P_{z}(x)=P_{z}^{(0)}(x)+P_{z}^{(1)}(X)+P_{z}^{(2)}(x),$ (65)

where

$\displaystyle P_{z}^{(0)}(x)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\tilde{R}(k)}{2\omega_{\bf
k}},$ $\displaystyle P_{z}^{(1)}(x)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}\sqrt{2\omega_{\bf
k}}}\frac{d^{3}k^{\prime}}{(2\pi)^{3}\sqrt{2\omega_{\bf
k}^{\prime}}}z(k)z^{*}(k^{\prime})e^{i(k-k^{\prime})\cdot
x}\tilde{R}\left(\frac{k+k^{\prime}}{2}\right),$ $\displaystyle
P_{z}^{(2)}(x)$ $\displaystyle=$ $\displaystyle
2\mbox{Re}\;\int\frac{d^{3}k}{(2\pi)^{3}\sqrt{2\omega_{\bf
k}}}\frac{d^{3}k^{\prime}}{(2\pi)^{3}\sqrt{2\omega_{\bf
k}^{\prime}}}z(k)z(k^{\prime})e^{i(k+k^{\prime})\cdot
x}\tilde{R}\left(\frac{k^{\prime}-k}{2}\right),$ (66)

The three terms in Eq. (65) have the following interpretation.

  * •

The term $P_{z}^{(0)}$ is essentially vacuum noise. It is constant and state
independent, and it contributes negligibly to the total detection probability.

  * •

The term $P_{z}^{(1)}$ is generated by the components of the coherent states
that is diagonal with respect to the total particle number.

  * •

The term $P_{z}^{(2)}$ involves contributions from terms with different
numbers of photons.

The term $P_{z}^{(1)}$ in Eq. (65) involves a double momentum integral over
$e^{-i(\omega_{k}-\omega_{k^{\prime}})t}$ (co-rotating terms), while the term
$P_{z}^{(2)}$ involves a double integral over
$e^{-i(\omega_{k}+\omega_{k^{\prime}})t}$ (counter-rotating terms). For
sufficiently large $t$, co-rotating terms typically dominate over counter-
rotating terms, hence, we can invoke the Rotating Wave Approximation (RWA) and
drop the contribution of the latter terms.

The RWA is a defining approximation of Glauber’s photo-detection theory. Its
domain of validity in photodetection has been a matter of some debate.
Photodetection probabilities obtained from the RWA appear to violate causality
at short times [96, 97]. However, it has been suggested that a modified form
of the RWA in photodetection can guarantee causality [98]. Fundamentally, RWA
is unacceptable because the corresponding Hamiltonian is unbounded from below
[99]. For details about the accuracy of the RWA and related approximations,
see, Ref. [100] and references therein.

To examine the validity of the RWA in the QTP probability assignment, we
consider a coherent state that describes a pulse of mean wave-number ${\bf
k}_{0}$ initially ($t=0$) localized around ${\bf x}=0$. We choose a Gaussian

$\displaystyle z(k)=(2\pi)^{3}z_{0}(2\pi\Delta^{2})^{-3/2}e^{-\frac{({\bf
k}-{\bf k}_{0})^{2}}{2\Delta^{2}}},$ (67)

where $z_{0}$ is a constant complex valued vector and $\Delta$ is the spectral
width of the pulse.

First, we evaluate the term $P_{z}^{(1)}(\tau,{\bf Q})$ in Eq. (65) using a
saddle point approximation. We expand $\omega_{k}=\omega_{k_{0}}+{\bf
v}_{k_{0}}\cdot({\bf k}-{\bf k}_{0})$, where ${\bf v}_{k}={\bf k}/|{\bf k}|$
is the three- velocity vector. We also assume that $\tilde{R}$ varies slowly
with $k$, so that the contribution of the detector kernel is a constant
$\tilde{R}(k_{0})$. We obtain

$\displaystyle P_{z}^{(1)}(t,{\bf
x})=\frac{1}{2}|z_{0}|^{2}\tilde{R}(k_{0})e^{-\Delta^{2}({\bf x}-{\bf
v}_{k_{0}}t)^{2}}.$ (68)

We also evaluate the term $P_{z}^{(2)}$ in the saddle-point approximation. We
find

$\displaystyle P_{z}^{(2)}(t,{\bf
x})\sim\mbox{Re}\left[z_{0}^{2}e^{ik_{0}\cdot x}\right].$ (69)

The oscillatory terms in Eq. (69) must be averaged over a spatial region of
size $\delta_{x}^{3}$ and for times of order $\delta_{t}$. For Gaussian
smearing functions, this averaging leads to multiplicative factors of order
$e^{-\delta_{t}^{2}|{\bf k}|_{0}^{2}}e^{-\delta_{x}^{2}\omega_{k_{0}}^{2}}$.
Hence, for sufficiently large ${\bf k}_{0}$, the contribution of the term
$P_{z}^{(2)}$ to the probability density (65) is suppressed, and the RWA is
justified. In general, we expect the RWA fails only for states with
significant contribution from photons in the deep infrared.

Assuming the RWA, the detection probability can be expressed as

$\displaystyle P(x)=\int d^{4}yR(y)G_{RWA}[x-\frac{1}{2}y,x+\frac{1}{2}y],$
(70)

in terms of the correlation function

$\displaystyle
G_{RWA}(x,x^{\prime})=2\langle\Psi|\hat{\phi}^{(-)}(x^{\prime})\hat{\phi}^{(+)}(x)|\Psi\rangle,$
(71)

where $\hat{\phi}^{(+)}$ and $\hat{\phi}^{(-)}$ denote the positive and
negative frequency part of $\hat{\phi}$.

Suppose now that the spacetime spread of $R(x)$ is much smaller than the
spatial and temporal variation of the field. For massless particles, this is a
stronger form of the dipole approximation that is often employed in quantum
optics: we assume that the typical wavelengths and frequencies in the initial
state are much larger than the size of the detector and the associated time
spread of detection. Then, we can approximate $R(x)$ with a delta function, to
obtain

$\displaystyle P(x)\sim G_{RWA}[x,x].$ (72)

This expression coincided with the standard formula of Glauber’s photo-
detection theory.

Hence, QTP recovers Glauber’s probabilities for a single measurement event by
assuming (i) the RWA and (ii) $R(y)\sim\delta(y)$. As we showed, for a large
class of initial states, the RWA is not an independent assumption, as it
arises due to the spatial and temporal coarse-graining of the detector. Thus,
the crucial assumption of Glauber’s theory is that the characteristic scales
of the detector kernel are much smaller than any time / length parameter that
characterize the initial state of the field. A more detailed analysis, with an
emphasis on the divergence from Glauber’s theory, will be undertaken
elsewhere.

##  5 Links to non-equilibrium QFT

In Sec. 2, we saw that the QTP probabilities are linear functional of balanced
correlation functions. The measurements do not probe unbalanced correlation
functions. Since the latter include $\langle\hat{C}_{a}(x)\rangle$, QTP
probabilities cannot access mean field information. This limitation is not
fundamental. Remember that the operator $\hat{C}(x)$ appears in the
interaction term with the apparatus. This restriction means that we cannot use
couplings of the form $\int d^{4}x\hat{C}_{a}(x)\otimes\hat{J}^{a}(X)$, in
order to directly measure the operator $\hat{C}_{a}(x)$. At least such
measurements are not possible with weak field-apparatus coupling where
perturbation theory is applicable666It is possible to measure $\hat{C}_{a}(x)$
for non-perturbative couplings [104], but the resulting probabilities are very
different from those of QTP, and they are closer to those of von Neumann
measurements..

Suppose, for example that $\hat{C}$ coincides with the field operator
$\hat{\phi}$—we drop the index $a$ for simplicity. Then, single-detector
probabilities record only local information about particles. Let the field be
in a state characterized by a macroscopically large number of particles; then,
it can be viewed as a thermodynamic system. Then the single-detector
probability essentially coincides with a particle-number density function. If
we also measure the recorded particle’s momentum $k$, the QTP probability
density $P(x,k)$ is an operationally defined version of Boltzmann’s
distribution function. By Eq. (17), $P(x,k)$ is a linear functional of the
correlation function
$G(x,x^{\prime})=\langle\hat{\phi}(x)\hat{\phi}(x^{\prime})\rangle$, which is
usually taken to satisfy the Baym-Kadanoff equations.

From the above analysis, it follows that Boltzmann’s thermodynamic entropy,
defined on a Cauchy surface $\Sigma$,

$\displaystyle S_{B}(\Sigma)=-\int_{\Sigma}d^{3}xd^{3}kP(x,k)\ln P(x,k)$ (73)

is a Shannon-type entropy for single-detection measurements. This means that
quantum informational quantities, defined through measurements, have a direct
application to non-equilibrium QFT. Furthermore, $n$-detector QTP
probabilities probe higher-order correlation functions of the quantum field,
thus allowing an analysis that is not accessible by traditional methods.

Such an analysis is beyond the scope of this paper. In this section, we will
only describe some structural similarities and connections between the QTP
analysis and methods of non-equilibrium QFT. A detailed analysis will be taken
up in other publications.

###  5.1 Stochastic correlation dynamics from two-particle irreducible
effective action

As shown in Sec. 3.2, the generating functional of QTP correlation functions
is defined in terms of non-local source terms $L^{ab}(x,x^{\prime})$. It is
structurally similar to the two-particle irreducible effective action (2PIEA)
[67, 107] that has found many applications in non-equilibrium QFT—see, for
example, [70, 71, 72].

We proceed to a brief review of the 2PIEA formalism following [109, 108],
emphasizing how it can be used to define evolution equations with noise from
higher-order correlation functions.

For ease of notation, we use a version of DeWitt’s condensed notation, where
capital indices $A$ correspond to both the spacetime dependence and the branch
of the CTP field (forward or backward in time, $\phi$ or $\bar{\phi}$). Hence,
we will be writing $\phi_{A}$, $C_{A}$, and so on. The action in the CTP
generating functional will be $S[\phi_{A}]=S[\phi_{a}]-S[\bar{\phi}_{a}]$. In
the two-particle irreducible representation, the (two-point) correlation
function stands is an independent variable, not a functional of the mean
field. Thus there is a separate source $K^{AB}$ driving $C_{A}C_{B}$ over the
usual $J^{A}C_{A}$ term in the one-particle irreducible representation—see the
similarity to Eq. (49).

From the generating functional

$Z\left[K^{AB}\right]=e^{iW\left[K^{AB}\right]}=\int
D\phi_{A}\;e^{i\left(S+\frac{1}{2}K^{AB}C_{A}C_{B}\right)}$ (74)

we have

$G^{AB}=\left\langle\hat{C}_{A}\hat{C}_{B}\right\rangle=2\left.\frac{\delta
W}{\delta K^{AB}}\right|_{K=0}$ (75)

and

$\left.\frac{\delta^{2}W}{\delta K^{AB}\delta
K^{CD}}\right|_{K=0}=\frac{i}{4}\left\\{\left\langle\hat{C}_{A}\hat{C}_{B}\hat{C}_{C}\hat{C}_{D}\right\rangle-\left\langle\hat{C}_{A}\hat{C}_{B}\right\rangle\left\langle\hat{C}_{C}\hat{C}_{D}\right\rangle\right\\}.$
(76)

Suppose that we want to express the effective dynamics of $G_{AB}$ in a closed
form, but to go beyond the Baym-Kadanoff equations, by taking into account
noise from higher-order correlations. For an non-equilibrium system, we seek a
formulation in terms of a new object ${\bf G}_{AB}$. This is a stochastic
correlation function whose expectation value over the noise average gives the
usual two point functions. The fluctuations of ${\bf G}_{AB}$ reproduce the
quantum fluctuations in the binary products of field operators. The simplest
assumption is to take ${\bf G}_{AB}$ as a Gaussian process, defined by

$\left\langle{\bf
G}_{AB}\right\rangle=\left\langle\hat{C}_{A}\hat{C}_{B}\right\rangle;\qquad\left\langle{\bf
G}_{AB}{\bf
G}_{CD}\right\rangle=\left\langle\hat{C}_{A}\hat{C}_{B}\hat{C}_{C}\hat{C}_{D}\right\rangle$
(77)

The Legendre transform of $W$ is the two-particle irreducible effective
action,

$\Gamma_{2PI}\left[G_{AB}\right]=W\left[K^{AB}\right]-\frac{1}{2}K^{AB}{G}_{AB};\qquad
K^{AB}=-2\frac{\delta\Gamma}{\delta{G}_{AB}}$ (78)

The Schwinger-Dyson equation for the propagators is simply,
$\frac{\delta\Gamma_{2PI}}{\delta G_{AB}}=0$. When including the stochastic
source ${\bf G}_{AB}$, it becomes

$\frac{\delta\Gamma_{2PI}}{\delta{\bf G}_{AB}}=-\frac{1}{2}\kappa^{AB}$ (79)

where $\kappa_{ab}$ is a stochastic nonlocal Gaussian source defined by

$\left\langle\kappa^{AB}\right\rangle=0;\qquad\left\langle\kappa^{AB}\kappa^{CD}\right\rangle=4i\left[\frac{\delta^{2}\Gamma_{2PI}}{\delta
G_{AB}\delta G_{CD}}\right]^{\dagger}$ (80)

The noiseless Eq. (79) ($\kappa=0$) provides the basis for the derivation of
transport equations in the near equilibrium limit. Indeed, for a
$\lambda\phi^{4}$ theory, we obtain the Boltzmann equation for a distribution
function $f$ defined from the Wigner transform of $G^{ab}$. The full
stochastic equation (79) leads, in the same limit, to a Boltzmann - Langevin
equation [108].

###  5.2 Correlation Histories

The two-particle irreducible formalism can be extended to an n-particle
irreducible formalism, for any $n$. There is an effective action
$\Gamma_{nPI}$ for each $n$, from which all effective actions for
$n^{\prime}<n$ can be derived. Taking $n\rightarrow\infty$, we obtain a master
effective action. The functional variation of the master effective action
yields the hierarchy of Schwinger-Dyson equations [109].

To obtain effective closed dynamics for the correlations at order $n$, we must
truncate the Schwinger-Dyson hierarchy upon this order. Truncation renders the
master effective action complex. Its imaginary part arises from correlation
functions of order higher than $n$, the fluctuations of which Calzetta and Hu
define as correlation noises [108] at order $n$. For example, the noise
$\kappa^{AB}$ in Eq. (79) is the correlation noise of order two.

Calzetta and Hu defined the notion of correlation histories [110], in analogy
to the decoherent histories program. A fine-grained correlation history
corresponds to the full Schwinger-Dyson hierarchy of correlation functions.
When we truncate the hierarchy at finite order $n$, we treat only correlation
functions of order $n$ as independent. Higher -order correlations are ignored
or slaved to the lowest-order ones. A truncated hierarchy defines a coarse-
grained correlation history. For example, mean field theory studies coarse-
grained correlation histories at order $n=1$; the Baym-Kadanoff equation, or
Boltzmann equation and their stochastic generalizations refer to coarse-
grained correlation histories of order $n=2$.

The key point is that the truncation of the master effective action always
leads to dissipation and noise for the coarse-grained histories. Any truncated
theory is an effective field theory in the correlation hierarchy formulation.
This effective field theory does not carry the full information, this loss of
information being expressed as correlation noise. The higher-order
correlations are analogous to an environment in the theory of open quantum
systems [111]. This noise may lead to decoherence of correlation histories
[110], i.e., to the classicalization of the effective description.

The QTP approach demonstrates that the different levels of correlation
histories can be accessed by the measurement of $n$-detector joint
probabilities. Eq. (27) assigns to each initial state $|\psi\rangle$ of the
field a hierarchy of joint probability distributions
$P_{n}(z_{1},z_{2},\ldots,z_{n})$.

In classical probability theory, a hierarchy of correlation functions defines
a classical stochastic process, if it satisfies the Kolmogorov additivity
condition,

$\displaystyle P_{n-1}(z_{1},\ldots,z_{n-1})=\int
dz_{n}P_{n}(z_{1},z_{2},\ldots,z_{n})$ (81)

Quantum probability distributions for sequential measurements do not satisfy
this condition [112]. Hence, the violation of Eq. (81) is a genuine signature
of quantum dynamics; it cannot be reproduced by classical physics, including
classical stochastic processes. It is rather different from the Leggett-Garg
inequalities [113] that also refer to the behavior of quantum multi-time
probabilities. Some authors refer to the violation of Eq. (81) as ”temporal
non-locality” [114]. In contrast, if measurements on a quantum field
approximately satisfy Eq. (81), then the measurement outcomes can be simulated
by a stochastic process with $n$-time probabilities given by the probability
distributions (81). Then the generating functional $Z_{QTP}$ corresponds to a
stochastic process, i.e., it is obtained as the functional Laplace transform
of a classical stochastic probability measure.

Hence, the hierarchy $P_{n}(z_{1},z_{2},\ldots,z_{n})$ provides a natural and
unambiguous classicality criterion777For a different classicality criterion,
based on the decoherent histories approach, that applies to the special case
of measurements of the field operator, see [115].. Given the relation between
QTP probabilities and QFT correlation functions, this criterion can be used to
probe the information content of different levels for correlation histories.
For example, the validity of Eq. (81) is necessary for deriving deterministic
or classical stochastic dynamics for $P_{1}(z)$, i.e., for deriving the
Boltzmann or the Boltzmann-Langevin equation. The failure of (81) means that
the four-point correlation functions are too ‘quantum’ to allow effective
classical stochastic dynamics for the two-point correlation function.

Conversely, the failure of Eq. (81) can be used to provide a measure of
irreducibly quantum information at the level $n=2$. An example of such a
measure is

$\displaystyle S_{Q}=\int dz_{2}\left|\int
dz_{1}P_{2}(z_{1},z_{2})-P_{1}(z_{2})\right|.$ (82)

A second informational quantity is the correlation of the probability
distribution, i.e., a measure of the deviation of $P_{2}(z_{1},z_{2})$ from
$P_{1}(z_{1})P_{2}(z_{2})$888The factorization assumption of the joint
probability for two events is equivalent to Boltzmann’s Stosszahlansatz in the
derivation of Boltzmann’s equation through the
Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy [116]. The joint probability
density $P_{2}$ is then slaved to $P_{1}$. This information is typically
quantified by the correlation entropy

$\displaystyle S_{C}=\int
dz_{1}dz_{2}P_{2}(z_{1},z_{2})\ln\frac{P_{2}(z_{1},z_{2})}{P_{1}(z_{1})P_{1}(z_{2})}.$
(83)

In general, the correlation entropy will contain information for both quantum
correlations (if $S_{Q}\neq 0$) and classical stochastic ones. Indeed, $S_{C}$
may not have the usual properties of correlation entropy if $S_{Q}\neq 0$, and
other measures that will distinguish will be more convenient. The third
relevant informational quantity in Boltzmann’s entropy (73), defined in terms
of $P_{1}(z_{1})$. These three quantities are the most important for
describing the flow of information at the level of the 2PIEA.

Hence, the QTP hierarchy functions as a registrar of information of the
quantum system, keeping track of how much information resides in what order,
and how it flows from one order to another through the dynamics. There is a
good potential for this scheme to systemize quantum information in QFT999Of
course, several issues remain to be addressed. The probability distributions
$P_{n}$ depend on the measurement apparatus through the detector kernels. We
would like to avoid strong apparatus dependence when deriving informational
quantities. We may have to consider ideal, maximal information, apparatuses
(like the ones appearing in the study of the time-of-arrival), or look for
quantities that can be minimized with respect to all physical detector
kernels. : keeping track of the contents and the flow of information and
measuring the degree of coherence in a quantum system.

##  6 Conclusions

In this work, we presented the QTP formalism for measurements in quantum
fields, and we explored its relations to the Closed-Time-Path description of
QFT. These relations provide a direct translation between the operational
language of measurement theory (POVMs, effects, and so on) to the manifestly
covariant description of QFT through functional methods. This is an essential
step towards formulating a general theory of relativistic quantum information.

Our work highlights the central role of the hierarchy of unequal-time
correlation functions. On one hand, POVMs for measurement are linear
functional of such correlation functions, on the other hand, these correlation
functions are crucial for the implementation of perturbation theory, but also
give real causal dynamics in the CTP approach. When working at the level of
non-equilibrium QFT, correlation functions are employed in order to define
both observables and effective irreversible dynamics.

Equally important is the fact that correlation functions are covariant and
causal objects. For this reason, we contend that a relativistic QIT that
respects both causality and spacetime symmetry must define all informational
quantities in terms of such correlation functions. This contrasts the standard
approach of QIT that is based on properties of single-time quantum states,
like von Neumann entropy or entanglement.

Building a sound theoretical foundation for relativistic QIT is not only
important for theoretical completeness. It is also needed for describing
quantum experiments in space [3, 4] that will explore the effects of non-
inertial motion (acceleration, rotation) and gravity on quantum resources like
entanglement. Corrections to standard photodetection models, like the ones
predicted by QTP, may be measurable in experiments that involve long
separations or large relative velocities between detectors.

## Acknowledgements

Research was supported by grant JSF-19-07-0001 from the Julian Schwinger
Foundation.

##  Appendix A Derivation of the QTP probability formula

In this section, we present a brief account of the QTP method, leading to the
derivation of Eq. (15), as a genuine probability density.

###  A.1 Amplitudes of detection

We consider a composite physical system that consists of a microscopic and a
macroscopic component. The microscopic component is the quantum system to be
measured and the macroscopic component is the measuring device.

Let ${\cal H}$ be the Hilbert space of the composite system, and $\hat{H}$ the
associated Hamiltonian operator. In QTP, we model a measurement event as a
transition between two complementary subspaces of ${\cal H}$. To this end, we
split ${\cal H}$ in two subspaces: ${\cal H}={\cal H}_{+}\oplus{\cal H}_{-}$.
The subspace ${\cal H}_{+}$ describes the accessible states of the system
given that the event under consideration is realized. For example, if the
event is a detection of a microscopic particle by an apparatus, ${\cal H}_{+}$
includes all states of the apparatus with a definite macroscopic record of
detection. We denote the projection operator onto ${\cal H}_{+}$ as $\hat{P}$
and the projector onto ${\cal H}_{-}$ as $\hat{Q}:=1-\hat{P}$. We also assume
that a pointer variable $\lambda$ of the measurement apparatus takes a
definite value after the transition has occurred. The corresponding positive
operators $\hat{\Pi}(\lambda)$ satisfy $\hat{\Pi}(\lambda)$ satisfy
$\sum_{\lambda}\hat{\Pi}(\lambda)=\hat{P}$.

First, we construct the probability amplitude
$|\psi;\lambda,[t_{1},t_{2}]\rangle$ that, given a state
$|\psi_{0}\rangle\in{\cal H}_{-}$ at $t=0$, a transition occurs during the
time interval $[t_{1},t_{2}]$ and a value $\lambda$ is obtained for the
pointer variable. For a vanishingly small time interval, we set $t_{1}=t$ and
$t_{2}=t+\delta t$, and we keep only leading-order terms with respect to
$\delta t$. At times prior to $t$, the state lies in ${\cal H}_{-}$. This is
taken into account by evolving the initial state $|\psi_{0}\rangle$ with the
restricted propagator in ${\cal H}_{-}$,

$\displaystyle\hat{S}_{t}=\lim_{N\rightarrow\infty}(\hat{Q}e^{-i\hat{H}t/N}\hat{Q})^{N}.$
(84)

By assumption, the transition occurs at some instant within the time interval
$[t,t+\delta t]$, hence, there is no constraint in the propagation from $t$ to
$t+\delta t$; the propagation is implemented by the unrestricted evolution
operator $e^{-i\hat{H}\delta t}\simeq 1-i\delta t\hat{H}$. At time $t+\delta
t$, the event corresponding to $\hat{\Pi}_{\lambda}$ is recorded, so the
amplitude is transformed by the action of $\hat{\Pi}_{\lambda}$
($\sqrt{\hat{\Pi}_{\lambda}}$, if $\hat{\Pi}_{\lambda}$ is not a projector).
For times greater than $t+\delta t$, there is no constraint, so the amplitude
evolves as $e^{-i\hat{H}(T-t)}$ until some final moment $T$.

At the limit of small $\delta t$, the successive operations above yield

$\displaystyle|\psi_{0};\lambda,[t,t+\delta t]\rangle=-i\,\delta
t\,\,e^{-i\hat{H}T}\hat{C}(\lambda,t)|\psi_{0}\rangle.$ (85)

where

$\displaystyle\hat{C}(\lambda,t):=e^{i\hat{H}t}\sqrt{\hat{\Pi}}(\lambda)\hat{H}\hat{S}_{t}.$
(86)

is a history operator. Since the amplitude (85) is proportional to $\delta t$,
it defines a density with respect to time:
$|\psi_{0};\lambda,t\rangle:=\lim_{\delta t\rightarrow 0}\frac{1}{\delta
t}|\psi_{0};\lambda,[t,t+\delta t]\rangle$. From Eq. (85)

The total amplitude that the transition occurred at some moment within a time
interval $[t_{1},t_{2}]$ is

$\displaystyle|\psi;\lambda,[t_{1},t_{2}]\rangle=-ie^{-i\hat{H}T}\int_{t_{1}}^{t_{2}}dt\hat{C}(\lambda,t)|\psi_{0}\rangle.$
(87)

Eq. (87) involves the restricted propagator $\hat{S}_{t}$. This quantity may
be difficult to evaluate in practice. Furthermore, if $\hat{S}_{t}$ is
literally defined by Eq. (84), it is unitary in ${\cal H}_{-}$, and may lead
to quantum-Zeno-type problems in the definition of probabilities. In Ref.
[118] it was shown that these problems can be avoided by employing alternative
definitions of the restricted propagator, that involve a regularization time
scale. However, this renders $\hat{S}_{t}$ model dependent.

However, for the type of measurements considered here, the problems with the
evaluation of $\hat{S}_{t}$ can be avoided. We consider a Hamiltonian
$\hat{H}=\hat{H_{0}}+\hat{H_{I}}$ where $[\hat{H}_{0},\hat{P}]=0$, and $H_{I}$
is a perturbing interaction. To leading order in the interaction,

$\displaystyle\hat{C}(\lambda,t)=e^{i\hat{H}_{0}t}\sqrt{\hat{\Pi}}(\lambda)\hat{H}_{I}e^{-i\hat{H}_{0}t},$
(88)

with no dependence on $\hat{S}_{t}$.

###  A.2 Probabilities of detection

We define the probability $\mbox{Prob}(\lambda,[t_{1},t_{2}])\/$ that at some
time in the interval $[t_{1},t_{2}]$ a detection with outcome $\lambda$
occurred by taking the squared modulus of the amplitude Eq. (87),

$\displaystyle\mbox{Prob}(\lambda,[t_{1},t_{2}]):=\langle\psi;\lambda,[t_{1},t_{2}]|\psi;\lambda,[t_{1},t_{2}]\rangle=\int_{t_{1}}^{t_{2}}\,dt\,\int_{t_{1}}^{t_{2}}dt^{\prime}Tr\left[\hat{C}(\lambda,t)\hat{\rho_{0}}\hat{C}^{\dagger}(\lambda,t^{\prime})\right],$
(89)

where $\hat{\rho}_{0}=|\psi_{0}\rangle\langle\psi_{0}|$.

The quantities $\mbox{Prob}(\lambda,[t_{1},t_{2}])\/$ do not define a
probability measure with respect to time, because they do not satisfy the
Kolmogorov additivity condition
$\mbox{Prob}(\lambda,[t_{1},t_{3}])=\mbox{Prob}(\lambda,[t_{1},t_{2}])+\mbox{Prob}(\lambda,[t_{2},t_{3}])$
fails, unless the quantity

$\displaystyle{\cal
D}(t_{1},t_{2},t_{3}):=2Re\left[\int_{t_{1}}^{t_{2}}\,dt\,\int_{t_{2}}^{t_{3}}dt^{\prime}Tr\left(\hat{C}(\lambda,t)\hat{\rho_{0}}\hat{C}^{\dagger}(\lambda,t^{\prime})\right)\right]$
(90)

vanishes. Clearly, ${\cal D}(t_{1},t_{2},t_{3})$ does not vanish identically.
However, in a macroscopic system (or in a system with a macroscopic component)
one expects that Eq. (90) holds with a good degree of approximation, given a
sufficient degree of coarse-graining [60, 62]—see also the explicit
measurement models in [48]. Thus, a necessary condition for the time of
transition to be associated to macroscopic records in a measurement apparatus
is the existence of a coarse-graining time-scale $\sigma$, such that $|{\cal
D}(t_{1},t_{2},t_{3})|$ is negligible for $|t_{2}-t_{1}|>>\sigma$ and
$|t_{3}-t_{2}|>>\sigma$. Then, Eq. (89) does define a probability measure when
restricted to intervals of size larger than $\sigma$.

We can express the time-of-transition probabilities in terms of densities with
respect to a continuous time variable by smearing the amplitudes Eq. (87) with
respect to a time-scale $\delta_{t}>>\sigma$. To this end, we introduce a
family of functions $f(s)$ of width $\delta_{t}$, localized around $s=0$, and
normalized so that $\lim_{\delta_{t}\rightarrow 0}f(s)=\delta(s)$. Then, we
define the smeared amplitude

$\displaystyle|\psi_{0};\lambda,t\rangle_{\delta_{t}}:=\int
ds\sqrt{f(s-t)}|\psi_{0};\lambda,s\rangle=\int
ds\sqrt{f(s-t)}\hat{C}(\lambda,s)|\psi_{0}\rangle.$ (91)

The modulus- squared amplitudes

$\displaystyle
W(t,\lambda)={}_{\sigma}\langle\psi_{0};\lambda,t|\psi_{0};\lambda,t\rangle_{\sigma}=\int
dsds^{\prime}\sqrt{f(s-t)f(s^{\prime}-t)}Tr\left[\hat{C}(\lambda,s)\hat{\rho}_{0}\hat{C}^{\dagger}(\lambda,s^{\prime})\right]$
(92)

define a probability measure: they are of the form
$Tr[\hat{\rho}_{0}\hat{\Pi}(\lambda,t)]$, where

$\displaystyle\hat{\Pi}(\lambda,t)=\int
dsds^{\prime}\sqrt{f_{\sigma}(s-t)f_{\sigma}(s^{\prime}-t)}\hat{C}^{\dagger}(\lambda,s^{\prime})\hat{C}(\lambda,s)$
(93)

is a density with respect to both variables $\lambda$ and $t$.

The positive operator

$\displaystyle\hat{\Pi}_{\tau}(N)=1-\int_{0}^{\infty}dt\int
d\lambda\hat{\Pi}_{\tau}(\lambda,t),$ (94)

corresponds to the alternative $N$ that no detection took place in the time
interval $[0,\infty)$. Since the detector is a quantum system, there exists a
non-zero probability that the microscopic particle excites no transition, and
thus, no record is left.

The operator $\hat{\Pi}_{\tau}(N)$ together with the positive operators Eq.
(93) define a Positive-Operator-Valued Measure (POVM). The POVM Eq. (93)
determines the probability density that a transition took place at time $t$,
and that the outcome $\lambda$ for the value of an observable has been
recorded.

Consider a Hilbert space of the form ${\cal F}\otimes{\cal K}$, where ${\cal
F}$ describes a quantum field and ${\cal K}$ describes the apparatus. We
assume the same set-up as in Sec. 2.3, except for the fact that the
interaction Hamiltonian is Poincaré invariant, i.e., it does not involve a
switching function, $\hat{H}_{I}=\int d^{3}x\hat{C}_{a}({\bf
x})\hat{J}^{a}({\bf x})$. Then, the probability density (92) contains a kernel
$\langle\Omega|\hat{J}^{a}({\bf x},t)\hat{\Pi}(\lambda)\hat{J}^{b}({\bf
x}^{\prime},t^{\prime})|\Omega\rangle$. We take the pointer variable $\lambda$
to describe the position ${\bf X}$ of the detection record and another
observable $q$. Since $\hat{\Pi}(\lambda)$ is a highly coarse-grained
observable, we can approximate it as a product $\hat{\Pi}({\bf
X})\hat{\Pi}(\lambda)$, i.e., a product of two commuting POVMs, for position
and for the observable $q$. Then, we can write
$\hat{\Pi}(\lambda)=\sqrt{\hat{\Pi}({\bf
X})}\hat{\Pi}(\lambda)\sqrt{\hat{\Pi}({\bf X})}$.

Suppose we sample position with accuracy $\delta_{x}$. Then we expect that
$\sqrt{\hat{\Pi}({\bf X})}\hat{J}^{a}({\bf x},t)|\Omega\rangle$ vanishes if
$|{\bf X}-{\bf x}|>>\delta_{x}$. Hence, we can write $\sqrt{\hat{\Pi}({\bf
X})}\hat{J}^{a}({\bf x},t)|\Omega\rangle=C\sqrt{g}({\bf X}-{\bf
x})\hat{J}^{a}({\bf x},t)|\Omega\rangle$, where $g$ is a positive sampling
function of width $\delta_{x}$ and $C$ is a multiplicative constant. With this
substitution, we recover Eq. (15), only now rather than the switching function
$F(x-x^{\prime})$, we have the sampling functions $\sqrt{f(t-t^{\prime})g({\bf
x}-{\bf x^{\prime}})}$.

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