# General quantum measurements in relativistic quantum field theory

Adam Bednorz  Faculty of Physics, University of Warsaw, ul. Pasteura 5,
PL02-093 Warsaw, Poland<EMAIL_ADDRESS>
###### Abstract

Single particle detection is described in a limited way by simple models of
measurements in quantum field theory. We show that a general approach, using
Kraus operators in spacetime constructed from natural combinations of fields,
leads to an efficient model of a single particle detector. The model is free
from any auxiliary objects as it is defined solely within the existing quantum
field framework. It can be applied to a large family of setup where the time
resolution of the measurement is relevant, such as Bell correlations or
sequential measurement. We also discuss limitations and working regimes of the
model.

##  I Introduction

Quantum field theory makes predictions about scattering and decays of
particles that can be measured. In contrast to quantum optics and condensed
matter, where low energies allow very efficient detectors, high energy
experiments especially in the ultrarelativistic limit, cope with practical
limitations, e.g. not all particles are detected pdg .

Textbook approach is based on the relation between the transition probability
between incoming and outgoing particles and scattering matrix peskin . This is
completely nonlocal as the states are considered in momentum representation.
Exact modeling of real detectors is impractical because of enormous technical
complexity. Instead, simplified models like Unruh-deWitt have beed proposed,
involving an auxilary particle travelling on its worldline, imitating a
detector unruh ; witt ; rsg ; rove ; lin ; costa ; martinez ; anas ; brody ,
reducing the notion of particle to what a particle detector detects scul .
Such models were fine in the early days of high energy physics when the actual
low efficiency was not a particular problem. It was sufficient to map the
collection statistics to the theoretical predictions about scattering and
decays. However, the Unruh-deWitt and all existing constructions are unable to
model a $\sim 100\%$ efficient detection of a single particle as a click click
. The real measurements in modern experiments have also to be localized in
time and space. Firstly, Bell-type tests of nonlocality epr ; bell ; chsh ;
chi ; eber  require time resolution and high efficiency, achieved in low
energy optics hensen ; nist ; vien ; munch . Higher energy attempts to make
similar tests have failed so far belh . Secondly, sequential measurements
cannot destroy the particle after detection, allowing to measure it once
again. This kind of measurement is useful to reveal incompatibility between
the measured quantities incom . It is possible for immobile solid state
objects solid  and only recently for photons phot . The high energy analogues
are still awaited. In some cases one can use the space-like momentum to
suppress the vacuum noise bb22  but it does not help to increase the
efficiency.

In this work, we propose a different approach to measurement models in quantum
field theory. Instead of auxiliary particles, we shall directly define
measurement Kraus operators kraus , an element of positive operator-valued
measure (POVM) peres ; breuer , within the existing quantum field Hilbert
space, replacing the old fashioned projection, either direct or in the
auxiliary detector’s space. The Kraus operators are functionals of already
existing fields. We show that a properly defined functional can correctly
represent the measurement. However, not all classes of such operators can
serve as single particle detection models. To detect single particles (in a
clickwise fashion), a nonlinear functional is necessary and it has a limited
efficiency outside the safe energy regime. The best option is a universal
measurement of energy-momentum density, which models an almost perfect
measurement in a wide range of parameters. The model is able to map the
outcomes onto almost dichotomic events, with well separated absence and
presence of the particle, when a continuous flux of incoming particles arrives
at the detector. Although the model is in principle perturbative in the
detection strength, we are able to control potential higher order deviation
and identify the working regime.

The paper is organized as follows. We start from standard definition of
generalized measurement operators, mapping them onto the quantum field theory
framework. Next, we present several natural classes of such measurement,
pointing out their weaknesses and advantages. Finally, we discuss the
applicability of the models to Bell test and sequential measurements, and
confront them with other options like Unruh-deWitt model. Lengthy calculations
are left in Appendices.

##  II General quantum measurements

From the obsolete projections, through auxiliary detectors, modern quantum
measurements evolved to a description that requires no extra objects. They are
defined within the system’s space by a set of Kraus operators $\hat{K}$ such
that $\sum\hat{K}^{\dagger}\hat{K}=\hat{1}$ and the state $\hat{\rho}$ after
the measurement is transformed into
$\hat{\rho}^{\prime}=\hat{K}\hat{\rho}\hat{K}^{\dagger}$ (no longer
normalized) with the probability $\mathrm{Tr}\,\hat{\rho}^{\prime}$ kraus ;
peres ; breuer . It is straightforward to generalize it to a time sequence of
operators $\hat{K}_{j}$. Then $\sum\hat{K}^{\dagger}_{j}\hat{K}_{j}=\hat{1}$
while the probability reads

$\mathrm{Tr}\,\hat{K}_{n}\cdots\hat{K}_{1}\hat{\rho}\hat{K}_{1}^{\dagger}\cdots\hat{K}_{n}^{\dagger}$
(1)

The choice of the sets $\hat{K}_{j}$ is quite arbitrary, but a continuous time
limit leads to the most natural choice, a Gaussian form
$\hat{K}(a)\propto\exp(-\lambda(a-\hat{A})^{2}$ for some operator $\hat{A}$
and the outcome $a$ bfb . In quantum field theory this definition can be
adapted by expressing $\hat{A}$ in terms of local field operators (see
Appendix A for the standard quantum field notation conventions). What is even
more helpful, Kraus operators can be also incorporated into the path integral
framework and closed time path (CTP) formalism schwinger ; kaba ; matsubara ,
see Appendix A. The CTP consists of there parts: thermal Matsubara part
(imaginary) and two flat parts (real), forward and backward. We shall
distinguish them denoting $x_{\pm}=x\pm i\epsilon$ for an infinitesimally
small real positive epsilon. Then the Kraus operators need the field with the
part specified, i.e. $\hat{K}(x)\to K(x_{+})$, $\hat{K}^{\dagger}(x)\to
K(x_{-})$, see details in Appendix A. In other words, $\hat{K}$, expressed by
some field $\phi(x)$ must be placed on the proper part of CTP. For a moment,
the relation between $K$ and $\phi$ is a completely general functional, can be
nonlinear and nonlocal, but in short we will heavily reduce this freedom.

The simplest example is the Gaussian form

$\hat{K}(a)\propto\exp\left[-\lambda\left(\int
f(x)\hat{\phi}(x)dx-a\right)^{2}\right],$ (2)

ignoring the $\lambda-$dependent global normalization. We can adopt it to the
path integral form, combining both $\hat{K}$ and $\hat{K}^{\dagger}$ into a
single form, distinguishing the CTP parts

$K(a)\propto\exp\left[-\lambda\sum_{\pm}\left(\int
f(x)\phi(x_{\pm})dx-a\right)^{2}\right]$ (3)

where $f(x)$ is a real-valued function localized in spacetime (nonzero only
inside a finite region of the spacetime) such that $f(x_{+})=f(x_{-})=f(x)$ on
the flat part of the CTP. The apparent nonlocality of the above construction
can be removed by the Fourier transform

$\displaystyle K(a)\propto\int
d\xi_{+}d\xi_{-}\;e^{ia(\xi_{-}-\xi_{+})-(\xi_{+}^{2}+\xi_{-}^{2})/4\lambda}\times$
$\displaystyle\exp\int if(x)(\xi_{+}\phi(x_{+})-\xi_{-}\phi(x_{-}))dx$ (4)

which has an interpretation of two independent random external fields
$\xi_{\pm}$ for upper and lower part of the contour.

We shall apply the measurement to the simple bosonic field with the Lagrangian
density

$2\mathcal{L}(x)=\partial\phi(x)\cdot\partial\phi(x)-m^{2}\phi^{2}(x)$ (5)

To measure the field in vacuum we can define the probability in terms of path
integrals

$p(a)=Z\int\mathcal{D}\phi K(a)\exp\int i\mathcal{L}(z)dz,$ (6)

where $Z$ is the path integral normalization, see Appendix B. The
normalization of the probability can be checked by the identity

$\int daK(a)=\exp\left[-\frac{\lambda}{2}\left(\int
f(x)\phi_{q}(x)dx\right)^{2}\right]$ (7)

with $\phi_{q}(y)=\phi(y_{+})-\phi(y_{-})$ and
$2\phi_{c}(x)=\phi(x_{+})+\phi(x_{-})$. By Wick theorem wick , all
correlations can by expressed in terms of products of two-point propagators,
$\langle\phi(x)\phi(y)\rangle$. Simple, translation-invariant correlations
read

$\langle\phi(x)\phi(y)\rangle=Z\int\mathcal{D}\phi\phi(x)\phi(y)\exp\int
i\mathcal{L}(z)dz$ (8)

with special cases defined on the flat part (Im $x\to 0$)

$\displaystyle
S(x,y)=\langle\phi(x_{+})\phi(y_{+})\rangle=\langle\phi(x_{-})\phi(y_{-})\rangle^{\ast},$
$\displaystyle B(x,y)=\langle\phi(x_{+})\phi(y_{-})\rangle,$ $\displaystyle
C(x,y)=\langle\phi_{c}(x)\phi_{c}(y)\rangle,$ $\displaystyle
G(x,y)=\langle\phi(x_{\pm})\phi_{q}(y)\rangle,$ (9)

for $2\phi_{c}(x)=\phi(x_{+})+\phi(x_{-})$ (symmetrized field, classical
counterpart) and $\phi_{q}(y)=\phi(y_{+})-\phi(y_{-})$ (antisymmetric, quantum
susepctibility to external influence). Here $S$ is known as Feynman
propagator, $C$ is symmetric correlation (real) while $G$, causal Green
function, is imaginary and satisfies $G=0$ for $y^{0}>x^{0}$, and
$\langle\phi_{q}(x)\phi_{q}(y)\rangle=0$. To shorten notation, using the
translation invariance (also for complex times), we shall identify
$X(x,y)\equiv X(x-y)$ for $X=S,B,C,G$. Note that using the identity
$F(x_{\pm})=F_{c}(x)\pm F_{q}(x)/2$ we get $B(x)=C(x)+(G(-x)-G(x))/2$ and
$S(x)=C(x)+(G(x)+G(-x))/2$. The function $G$ is responsible for causality,
i.e. $G(x)\neq 0$ only if $x\cdot x\geq 0$ and $x^{0}\geq 0$. Now the right
hand side of (7) is $1$ because $\phi(x_{+})\equiv\phi(x_{-})$ at the _latest_
time $x^{0}$ (they meet at the returning tip of the time path).

It is convenient to introduce the concept of generating function

$S(\chi)=\int dap(a)e^{i\chi a}.$ (10)

which allows express moments and cumulants

$\langle
a^{n}\rangle=\left.\frac{d^{n}S}{d(i\chi)^{n}}\right|_{\chi=0},\>\langle\langle
a^{n}\rangle\rangle=\left.\frac{d^{n}\ln
S(\chi)}{d(i\chi)^{n}}\right|_{\chi=0}.$ (11)

They are related, in particular,

$\displaystyle\langle\langle a\rangle\rangle=\langle a\rangle,\;\langle\langle
a^{2}\rangle\rangle=\langle\delta a^{2}\rangle,$ $\displaystyle\langle\langle
a^{3}\rangle\rangle=\langle\delta a^{3}\rangle,\;\langle\langle
a^{4}\rangle\rangle=\langle\delta a^{4}\rangle-3\langle\langle
a^{2}\rangle\rangle^{2}$ (12)

for $\delta a=a-\langle a\rangle$. Gaussian distribution have only nonzero
first two cumulants. It is helpful to introduce convolution

$p(a)=(2\lambda/\pi)^{1/2}\int
d\bar{a}\,e^{-2\lambda(a-\bar{a})^{2}}p_{\cdot}(\bar{a})$ (13)

which leads to $\ln S(\chi)=\ln S_{\cdot}(\chi)-\chi^{2}/8\lambda$ so only
$\langle\langle a^{n}\rangle\rangle=\langle\langle
a^{n}\rangle\rangle_{\cdot}$ for $n\neq 2$ while $\langle\langle
a^{2}\rangle\rangle=\langle\langle a^{2}\rangle\rangle_{\cdot}+1/4\lambda$. In
other words we can separate the measurement statistics into the Gaussian
detection noise of the variance $1/4\lambda$, divergent in the limit
$\lambda\to 0$ and the bare function which turns out to be a quasiprobability.
It is normalized and has well defined moments but lacks general positivity.

In the case of our measurement we have a Gaussian function

$\displaystyle S_{\cdot}(\chi)=Z\int\mathcal{D}\phi\exp\int
i\mathcal{L}(z)dz\times$ $\displaystyle\exp\left[-\frac{\lambda}{2}\left(\int
f(x)\phi_{q}(x)\right)^{2}\right]\exp\int i\chi f(y)\phi_{c}(y)dy.$ (14)

Therefore $\langle a\rangle=0$ in the vacuum and

$\displaystyle\langle a^{2}\rangle_{\cdot}=Z\int\mathcal{D}\phi\exp\int
i\mathcal{L}(x)dx\times$ $\displaystyle\exp\left[-\frac{\lambda}{2}\left(\int
f(x)\phi_{q}(x)dx\right)^{2}\right]\left(\int f(y)\phi_{c}(y)dy\right)^{2}$
(15)

Now, the term $\phi(x_{+})-\phi(x_{-})$ must be contracted with some $\phi(y)$
with $y^{0}>x^{0}$, otherwise it vanishes. It leaves essentially only a few
terms

$\displaystyle\langle a^{2}\rangle_{\cdot}=\int dxdyf(x)f(y)C(x-y)$
$\displaystyle-\lambda\left(\int dxdyf(x)f(y)G(x-y)\right)^{2}.$ (16)

Suppose now that we perturbed the vacuum. This is the natural physical
situation when a beam of particles is sent to the detector. Let

$\exp\int idz\mathcal{L}(z)\to\mathcal{P}[\phi]\exp\int idz\mathcal{L}(z)$
(17)

where $\mathcal{P}$ denotes perturbation

$\mathcal{P}[\phi]=\exp\int ig(w)\phi_{q}(w)dw$ (18)

by the shift of the field induced by $g(w)$ (localized in spacetime). In
principle we should discuss not only the particle detector but also generator.
Since we prefer to focus on the detection part we stay at the minimal
description with the single perturbation function $g$. Almost all the above
formulas remain valid, i.e. probability has the same $g$-independent
normalization and is Gaussian. It only gets the nonzero average. We calculate

$\displaystyle S_{\cdot}(\chi)=Z\int\mathcal{D}\phi\exp\int
i\mathcal{L}(z)dz\times$ $\displaystyle\exp\int ig(w)\phi_{q}(w)dw\times$
$\displaystyle\exp\left[-\frac{\lambda}{2}\left(\int
f(x)\phi_{q}(x)dx\right)^{2}\right]\exp\int i\chi f(y)\phi_{c}(y)dy.$ (19)

The average is $\lambda$-independent while the higher cumulants remain
unaffected,

$\langle a\rangle_{\cdot}=\int dwdyf(y)g(w)iG(y-w).$ (20)

As we see, the linear measurement gives simply a Gausssian statistics and
cannot be used to model single particle detection with the Poissonian clicks.
Nevertheless, already the selfconsistency of the above construction is a
promising signature that the approach by Kraus operators is correct.

##  III Nonlinear measurement

We shall define a quadratic Kraus operator and apply it to the vacuum and
continuous plane wave, showing that it exhibits features of Poisson statistics
in contrast to the Gaussian linear case.

###  III.1 Quadratic measurement

Let us define Kraus operators in terms of path integrals

$K(a)\propto\exp\left[-\lambda\sum_{\pm}\left(\int
f(x)\phi^{2}(x_{\pm})dx-a\right)^{2}\right],$ (21)

which can be made local as previously by a Fourier transform. Analogously to
the field measurement, we can write down the formal expression for the
generating function, namely

$\displaystyle S_{\cdot}(\chi)=Z\int\mathcal{D}\phi\exp\int
i\mathcal{L}(z)dz\times$ $\displaystyle\exp\left[-\frac{\lambda}{2}\left(\int
f(x)\phi^{2}_{q}(x)dx\right)^{2}\right]\exp\int i\chi f(y)\phi^{2}_{c}(y)dy$
(22)

where $\phi^{2}_{c}=(\phi_{c})^{2}+(\phi_{q})^{2}/4$ and
$\phi^{2}_{q}=2\phi_{c}\phi_{q}$.

Our aim is to calculate $\langle a^{n}\rangle$ and show that for some linear
perturbation $g$, there exists a regime (some $f$) where the distribution is
Poissonian,

$p(a=n\eta)=e^{-\alpha}\alpha^{n}/n!,$ (23)

proving approximate dichotomy $a=0,\eta$ from $\langle
a^{2}(a-\eta)^{2}\rangle\simeq 0$. Poisson distribution has simple cumulants
(11) $\langle\langle a^{n}\rangle\rangle=\alpha\eta^{n}$ but they coincide
with the moments for $\alpha\ll 1$. We will attempt to expand $\langle
a^{n}\rangle$ in powers $\lambda$ and estimate upper bound for the higher
terms of the expansion using Wick theorem.

###  III.2 The vacuum

We shall apply the quadratic measurement to the vacuum. It is then useful to
calculate moments,

$\displaystyle\langle a^{n}\rangle_{\cdot}=Z\int\mathcal{D}\phi\exp\int
i\mathcal{L}(z)dz\times$ $\displaystyle\exp\left[-\frac{\lambda}{2}\left(\int
f(x)\phi^{2}_{q}(x)dx\right)^{2}\right]\left(\int
f(y)\phi^{2}_{c}(y)dy\right)^{n}$ (24)

expanding in powers of $\lambda$. Problems arise when $\phi^{2}$ contains two
fields at the same time so e.g.
$\langle\phi^{2}(x)\rangle=\langle\phi^{2}_{c}(x)\rangle\to\infty$ It must be
_renormalized_ e.g. by subtracting counteraverage for a fictitious mass
$M\to\infty$. We can subtract large masses i.e.

$\langle\phi^{2}_{c}(x)\rangle\to\langle\phi^{2}_{c}(x)\rangle+\sum_{j}\epsilon_{j}\langle\phi^{2}_{c}(x)\rangle_{m\to
M_{j}}=\Lambda$ (25)

where $M_{j}\gg m$ are large renormalization masses (Pauli-Villars) pavi ;
peskin  while $\epsilon_{j}$ are some numbers (e.g. $\pm 1$) not too large.
The constant $\Lambda$ is an unobservable callibration shift. From now on we
also make this shift in $a$, i.e. $a\to a-\Lambda\int f(x)dx$.

In the lowest order of $\lambda$

$\langle a^{2}\rangle_{\cdot}=\int
f(x)f(y)\langle\phi^{2}_{c}(x)\phi^{2}_{c}(y)\rangle dxdy$ (26)

Denoting $\tilde{f}(k)=\int dxe^{ik\cdot x}f(x)/(2\pi)^{D+1}$ with help of

$\displaystyle\langle\phi^{2}_{c}(x)\phi^{2}_{c}(y)\rangle=\langle\phi^{2}_{c}(x)\phi^{2}_{c}(y)\rangle-\langle\phi^{2}_{q}(x)\phi^{2}_{q}(y)\rangle/4$
$\displaystyle=\langle\phi^{2}(x_{+})\phi^{2}(y_{-})+\phi^{2}(x_{-})\phi^{2}(y_{+})\rangle/2$
(27)

we get (see Appendix C),

$\langle a^{2}\rangle_{\cdot}=\int W(q)|\tilde{f}(q)|^{2}dq$ (28)

with

$W(q)=\frac{\pi^{2}(q\cdot q/4-m^{2})^{(D-2)/2}S_{D}}{\sqrt{q\cdot q}}$ (29)

for $q\cdot q>4m^{2}$ and $0$ otherwise. Here $S_{D}=2\pi^{D/2}/\Gamma(D/2)$
is the sufrace of a unit ball in $D$ dimensions. It shows that nonlinear
fluctuations of the field need pair creation. For $f$ varying over time/length
much larger than $1/m$, they are negligible, allowing low-noise measurement in
the vacuum.

###  III.3 Measurement of the plane wave

We want the perturbation $g$ to generate enveloped wave of a particular
frequency, i.e. of the form

$g(x)=e^{iE_{p}x^{0}}h(x^{1}+L)+e^{-iE_{p}x^{0}}h^{\ast}(x^{1}+L)$ (30)

with a function $h(y)$ localized at $|y|\ll L$ and $E_{p}>m$, which should
generate a plane wave in $x^{1}$ direction. It corresponds to a constant
coherent flux of free particles in $x^{1}$ direction. Our measurement model is
able to capture single particles in the flux in contrast to vacuum and its
fluctuations. The effect of the perturbation can be described by

$G_{g}(x)=\int dyG(x-y)g(y)$ (31)

in the limit $L\to\infty$, where it reads (see Appendix D)

$G_{g}(x)=2i\mathrm{Im}Ae^{i|p|x^{1}-iE_{p}x^{0}}$ (32)

or equivalently

$\displaystyle G_{g}(x)=G_{g+}(x)+G_{g-}(x),$ $\displaystyle
G_{g\mp}=(iA_{i}\pm A_{r})e^{\mp ip\cdot x},$ (33)

for $p=(E_{p},|p|,0,0)$ and $2A=\tilde{h}(-|p|)/|p|$.

For a linear perturbation $g$ we can now determine the measurement statistics,
inserting (18) into (24).

The measurement function $f(x)$ will vary at the scale much longer than
$1/E_{p}$. Defining

$F(x)=\sum_{\mp}e^{\mp ip\cdot x}F_{\mp}(x),$ (34)

for $F=G,C,S,B$, and expanding $k=k^{\prime}\mp p$ in (73) for small
$k^{\prime}$

$\displaystyle
C_{\mp}(x)=\int\frac{dk^{\prime}}{2(2\pi)^{D}}\delta(2k^{\prime}\cdot
p)e^{ik^{\prime}\cdot x},$ $\displaystyle
G_{\mp}(x)=\int\frac{idk^{\prime}}{(2\pi)^{D+1}}\frac{e^{ik^{\prime}\cdot
x}}{\mp 2p\cdot k^{\prime}_{+}},$ (35)

explicit calculation gives

$\displaystyle B_{+}(x)=\delta(x^{\perp})\delta(x^{0}v-x^{1})/2E_{p},$
$\displaystyle C_{\mp}(x)=B_{+}(x)/2,$ $\displaystyle
G_{\mp}(x)=\pm\theta(x^{0})B_{+}(x),$ $\displaystyle S_{\mp}(x)=\theta(\pm
x^{0})B_{+}(x),$ (36)

and $B_{-}=0$ with $x^{\perp}=(x^{2},x^{3},\dots)$ and the speed of the field
$v=|p|/E_{p}$ (in the units of speed of light). This is a very intuitive
physical picture since the dynamics is concentrated on the lines of the
propagation at constant speed $v$.

In the lowest order of $g$, the average
$\langle(g\phi)(g\phi)(f\phi^{2})\cdots(f\phi^{2})\rangle$ turns out to be a
sum of Feynman graphs with part of vertices on $+$ side and part on $-$ side
of CTP, see Fig. 1.

Figure 1: A chain of propagators $G$, $S$ and $B$. The arrow points into the
time direction positive and negtive imaginary sides denoter by $+$ and $-$,
respectively.

We shall now express the first $\langle a^{n}\rangle_{0}$, $n=1,2,3,4$ in
terms of the just derived functions in the lowest order of $\lambda$, we get

$\displaystyle\langle a^{n}\rangle_{\cdot}\simeq 2E_{p}|A|^{2}\times$
$\displaystyle\int d\boldsymbol{x}\left(\int
dx^{0}f(x^{0},x^{1}+vx^{0},x^{\perp})/E_{p}\right)^{n},$ (37)

where we used the shift $x^{1}\to x^{1}-vx^{0}$ in the integrals. The factor
$2E_{p}$ appears beacuse each $\phi^{2}$ contains $2$ fields $\phi$ giving $2$
per vertex and $2^{n}$ in total. On the other hand
$2\phi^{2}=\phi^{2}(x_{+})+\phi^{2}(x_{-})$ so we get a factor $2^{-n}$ to
cancel out with the above one. Each line ($S_{\pm}$ and $B_{+}$) has the
factor $(2E_{p})^{-1}$ giving $(2E_{p})^{1-n}$ in total. For $n$ vertices we
have all possible decompositions into $n_{+}$ and $n_{-}=n-n_{+}$ vertices
with $n_{+}=0\dots n$. All combinations give the total factor
$(1+1)^{n}=2^{n}$ (by binomial formula). There are no additional factors
$n_{\pm}!$ because these factors cancel out ($n_{\pm}$ permutations cancel out
with $\theta(\pm x^{0})$ ordering).

To get the Poisson statistics, it is sufficient that

$\int
dx^{0}f(x^{0},x^{1}+vx^{0},x^{\perp})/E_{p}=\left\\{\begin{array}[]{ll}\eta&\mbox{
if }\boldsymbol{x}\in V\\\ 0&\mbox{ if }\boldsymbol{x}\notin
V\end{array}\right.$ (38)

where $V$ is a certain volume in $D$ dimensional space. In other words, we
need constant integral of the measuring function $f$ along line of speed $v$,
i.e. $x^{1}=vx^{0}$ see Figure 2. We have also $\alpha=2E_{p}|A|^{2}V$. The
approximation is valid as long as variation lengthscale of $f$, say $\ell$ is
much larger that the wavelength $1/p$, i.e. $p\ell\gg 1$, see figure 3.

Figure 2: Projecting of the measurement lines onto the $D$-dimensional spatial
base Figure 3: Lengthscales in measurement. The wavelength $1/p$ is shorter
than the variation length $\ell$ of the envelope function $f$.

Finally, higher order terms in $\lambda$ expansion will contain
$\phi^{2}_{q}=\phi^{2}(x_{+})-\phi^{2}(x_{-})$. Fortunately, in our
approximation (36) these terms cancel. This is because inserting such points
in the existing graph gives always two opposite expressions, see Figure 4.

Figure 4: Approximate cancellation of disturbances caused by the measurement.
Insertion of $\phi^{2}_{q}$ at $x_{+}$ and $x_{-}$ in the chains of
propagators (36) on the line of propagation gives two exactly opposite terms.

###  III.4  Perturbative estimates

We shall calculate $\langle a\rangle$ including order $\lambda$. In the
leading order

$\langle a\rangle=2|A|^{2}\int dxf(x^{0},x^{1}+vx^{0},x^{\perp})=2|A|^{2}\int
dxf(x)$

because the shift is irrelevant. This result is exact in the leading order,
$\lambda^{0}$. Now, the next order term, $\lambda$ would be $0$ if we make our
approximations on $G$, $S$, $B$. In order to find a nonzero contribution, we
need to estimate the lowest deviations. The Feynman-Schwinger graph is
depicted in Figure 5.

Figure 5: The Feynman-Schwinger graph contributing to the $\lambda$ correction
to $\langle a\rangle$

The deviation of $G$ reads

$\Delta G_{\mp}(x)=\int\frac{dk^{\prime}}{(2\pi)^{D+1}}\frac{k^{\prime}\cdot
k^{\prime}e^{ik^{\prime}\cdot x}}{i(2p\cdot k^{\prime}_{+})^{2}}$ (39)

for $k^{\prime 0}=k^{\prime 1}|p|/E_{p}$ and $\partial=\partial_{x}$. Hence,

$\Delta G_{\mp}(x)=\partial\cdot\partial[\theta(x^{0})x^{0}B_{+}(x)]/2iE_{p}$
(40)

The derivatives can be moved to $f$ so the graphs will contain products
$(\partial f)(\partial ff$ or $(\partial\partial f)ff$.

We shall estimate the corrections for

$f(x)=FI_{0}(x^{0}-x^{1}/v-w(\boldsymbol{x}))I(\boldsymbol{x})]$ (41)

where $I_{0}(t)=\theta(t)\theta(L_{0}-t)$, and

$I(\boldsymbol{x})=\left\\{\begin{array}[]{ll}1&\mbox{ if }\boldsymbol{x}\in
V\\\ 0&\mbox{ if }\boldsymbol{x}\notin V\end{array}\right.$ (42)

of linear size $L$ for some function (measurement start time) $w$ defined
inside $V$. In this case the derivatives are nonzero only on the boundary. In
the case of two first derivatives, it gives essentially a product of two
$\delta$ functions, pinning $f$ to the boundary. It turns out that the case of
second derivative is actually of the same order. Formally we would get
$\delta^{\prime}$ which is infinite, so in both cases we need to regularize
$\theta$ functions. Let us assume that $\theta$ and $I$ change from $0$ to $1$
smoothly over the distance $\ell\ll L,L_{0}$. The total correction to $\langle
a\rangle$ is $\sim\lambda\langle a^{3}\rangle_{0}L_{0}/L\ell E_{p}$ since the
numerator contains also $x^{0}\sim L_{0}$. Therefore, the Poisson statistics
will be a good approximation in the limit $\lambda\eta^{2}\ll E_{p}\ell
L/L_{0}$ with $\eta=FL_{0}/E_{p}$. On the other hand the detection noise
cannot blur the distribution giving another constraint
$1\ll\lambda(\alpha\eta)^{2}$. If we want maximally a single click, then
additionally $\alpha\ll 1$. Summarizing, the single click statistics occurs if

$1\ll\alpha^{-2}\ll\lambda\eta^{2}\ll\ell LE_{p}/L_{0}$ (43)

Finally, the previously calculated vacuum fluctuation should also be small.
They are almost completely negligible (exponentially) if $m\ell\gg 1$, i.e.
the measurement shape function varies over scales impossible to generate pairs
of particles. However, in the massless (or low mass) case, the fluctuation are
estimated by $W\sim(q\cdot q)^{(D-3)/2}$ giving $|q|\sim 1/\ell$, i.e.
$\langle a^{2}\rangle_{0}\sim\ell^{2-2D}\tilde{f}^{2}(0)$ and
$\tilde{f}(0)\sim VFL_{0}$. At $D=1$ they are actually divergent
logarithmically with the shrinking mass $m$. For $D>1$ we need also
$VE_{p}/\ell^{D-1}\ll\alpha$. Together with the previous condition it implies
$\ell LE_{p}/L_{0}\gg 1\gg VE_{p}/\ell^{D-1}$ which gives the impossible
requirement $\ell^{D}\gg VL_{0}/L$. The reason of this failure is that
$\phi^{2}$ is not a conserved quantity and the measurement can easily change
it locally (above the mass threshold). We shall resolve this problem replacing
$\phi^{2}$ by the conserved energy-momentum density $T_{\mu\nu}$.

##  IV  Energy-momentum measurement

We shall modify the previous simple quadratic measurement replacing $\phi^{2}$
by energy-momentum density, which is still quadratic in $\phi$ but the
addtional derivatives turn out to suppress unwanted noise in the high energy
limit.

###  IV.1 Energy-momentum tensor

Energy-momentum stress tensor by Noether theorem reads

$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\partial^{\nu}\phi-g^{\mu\nu}\mathcal{L}$
(44)

with $\partial^{\nu}=g^{\nu\tau}\partial_{\tau}$ is equal in our case

$T^{\mu\nu}=\partial^{\mu}\phi\partial^{\nu}\phi-g^{\mu\nu}(g^{\sigma\tau}\partial_{\sigma}\phi\partial_{\tau}\phi-m^{2}\phi^{2})/2$
(45)

We define the energy-momentum measurement

$K(a)\propto\exp\left[-\lambda\sum_{\pm}\left(\int
f_{\mu\nu}(x)T^{\mu\nu}(x_{\pm})dx-a\right)^{2}\right],$ (46)

which is normalized in the same way as the field, with symmetric
$f_{\mu\nu}=f_{\nu\mu}$. The generating function reads

$\displaystyle S_{\cdot}(\chi)=Z\int\mathcal{D}\phi\exp\int
i\mathcal{L}(z)dz\times$ $\displaystyle\exp\left[-\frac{\lambda}{2}\left(\int
f_{\mu\nu}(x)T^{\mu\nu}_{q}(x)dx\right)^{2}\right]\times$
$\displaystyle\exp\int i\chi f_{\mu\nu}(y)T^{\mu\nu}_{c}(y)dy,$ (47)

where we denoted $T^{\mu\nu}_{q}(x)=T^{\mu\nu}(x_{+})-T^{\mu\nu}(x_{-})$ and
$2T^{\mu\nu}_{c}(x)=T^{\mu\nu}(x_{+})+T^{\mu\nu}(x_{-})$. Note also that

$\displaystyle
2T^{\mu\nu}_{q}(x)=\partial^{\mu}\phi_{c}\partial^{\nu}\phi_{q}+\partial^{\mu}\phi_{q}\partial^{\nu}\phi_{c}$
$\displaystyle-g^{\mu\nu}(\partial\phi_{c}\cdot\partial\phi_{q}-m^{2}\phi_{c}\phi_{q}),$
$\displaystyle
T^{\mu\nu}_{c}(x)=\partial^{\mu}\phi_{c}\partial^{\nu}\phi_{c}-g^{\mu\nu}(\partial\phi_{c}\cdot\partial\phi_{c}-m^{2}\phi_{c}^{2})/2$
$\displaystyle+\partial^{\mu}\phi_{q}\partial^{\nu}\phi_{q}/4-g^{\mu\nu}(\partial\phi_{q}\cdot\partial\phi_{q}-m^{2}\phi_{q}^{2})/8.$
(48)

As in the case of $\phi^{2}$, the calculations involve correlations of the
type $\langle(g\phi)\cdots(g\phi)(fT)\cdots(fT)\rangle$. However, there are
dangerous contact terms to be regularized by fermionic ghosts peskin ; fadeev
, see Appendix E. Fortuntately, once identified, we can basically forget about
ghosts, just keep the unitarity constraints when calculating loops, $\langle
T_{q}(x)T_{q}(y)\cdots T_{q}(w)\rangle=0$ as a calculation rule if only
$T_{q}$ are involved. Basic examples of graphs involved in our calculations
are depicted in Fig. 6. From now on we shall subtract the zero-temperature
average from $T$, i.e. $T\to T-\langle T\rangle_{0}$, as it is unobservable,
contains the renormalization parameters, and we are interested only in the
noise and sensitivity of the detector to the incoming particles. By this shift
$\langle T\rangle=0$.

Figure 6: Loops and chains in the graphs involving energy-momentum tensor.
They vanish if solely $T_{q}$ (not $T_{c}$) appears in a loop/chain

###  IV.2 Measurement of the vacuum

In the lowest order of $\lambda$

$\langle a^{2}\rangle_{\cdot}=\int f_{\mu\nu}(x)f_{\xi\eta}(y)\langle
T^{\mu\nu}_{c}(x)T^{\xi\eta}_{c}(y)\rangle dxdy$ (49)

analogously to the $\phi^{2}$. We get (see Appendix F),

$\langle a^{2}\rangle_{\cdot}=\int
dq\,\tilde{f}_{\mu\nu}(q)\tilde{f}_{\xi\eta}(-q)\frac{X^{\mu\nu\xi\eta}(q)W(q)}{(q\cdot
q)^{2}(D+2)D}$ (50)

with vac

$\displaystyle X^{\mu\nu\xi\eta}=P(q^{\mu}q^{\nu}-(q\cdot
q)g^{\mu\nu})(q^{\xi}q^{\eta}-(q\cdot q)g^{\xi\eta})$
$\displaystyle+R[(q^{\mu}q^{\eta}-(q\cdot
q)g^{\mu\eta})(q^{\xi}q^{\nu}-(q\cdot q)g^{\xi\nu})$
$\displaystyle+(q^{\mu}q^{\xi}-(q\cdot q)g^{\mu\xi})(q^{\nu}q^{\eta}-(q\cdot
q)g^{\nu\eta})],$ $\displaystyle P=m^{4}+(D+1)m^{2}q\cdot q/2+(q\cdot
q)^{2}(D^{2}-3)/16,$ $\displaystyle R=m^{4}-m^{2}q\cdot q/2+(q\cdot
q)^{2}/16,$ (51)

and $W$ defined by (29) The case $D=1$ is degenerate, see Appendix F.

###  IV.3 Poisson statistics

We can adapt most of results from the $\phi^{2}$ replacing $f$ with
$f_{\mu\nu}$ and adding factor $p^{\mu}p^{\nu}$ from $T$ in (37). Then e.g.
$\eta=\int
dx^{0}f_{\mu\nu}(x^{0},x^{1}+vx^{0},x^{\perp})p^{\mu}p^{\nu}/E_{p}$. Replacing
further $F$ with $F_{\mu\nu}$ in (41) we have
$\eta=F_{\mu\nu}p^{\mu}p^{\nu}L_{0}/E_{p}$. We only need to include deviation
from derivatives in $T$ that can act on variables in $f$ considering potential
disturbance (the $\sim\lambda$ term). Fortunately, if say $F_{00}=F$ and $0$
otherwise (assuming we measure energy density, not momentum), their
contribution to the disturbance is negligible, and we stay with (43).

What is qualitatively different, vacuum fluctuations remain small in the
massless limit. From our above discussion we have then $\langle
a^{2}\rangle\sim\ell^{-2(D+1)}\tilde{F}^{2}(0)$. If we want
$\ell^{-D-1}\tilde{F}\ll\alpha\eta$ then $V/\ell^{D+1}E_{p}\ll\alpha$. In
contrast to the $\phi^{2}$ case increasing $E_{p}$ helps to satisfy this and
the other requirements, due to the fact that we work with the conserved
quantity.

##  V Conclusion

We have presented a self-consistent measurement model suitable for high energy
particle detectors. Defined completely withing existing framework of quantum
field theory combine with POVM and Kraus functionals, it allows to identify a
particle as a click with almost perfect efficiency, in constrast to Unruh-
deWitt model. The presented examples stress the importance of nonlinearity and
connection with conservation principles to choose a proper Kraus functional.
The strengh of the measurement, parameter $\lambda$, must be neither too low,
to keep the detection noise smalle, nor too high, to keep the backaction
small. Only an intermediate regime, depending on the energy, time and length
scales, allows to $\sim 100\%$ efficiency of the detetion. Although we
analyzed a simple bosonic field, the ideas are quite general an can be easily
extended to fermions and compound particles. We believe that further work on
such models will help to establish a Bell-type family of experiments in high
energy regime, identifying the main technical challenges. One can also explore
completely different detector’s functions $f$, e.g. an analogue of an
accelerating observer like in the original Unruh model or a sequence of
measurements. Note also that our model is a theoretical idealization and it
may need practical adjustments taking into account specific experiments.

## Acknowledgements

I thank W. Belzig and P. Chankowski for many fruitful discussions on the
subject and pointing out important issues.

##  Appendix A Closed time path formalism

We shall summarize the notation in quantum field theory of a scalar field, in
$D+1$ dimensions ( $D$ spatial dimensions and $1$ time, with $D=3$ in full
space but also $D=1$ for simple illustrative cases)
$x=(x^{0}=ct,\boldsymbol{x})$, with time $t$, speed of light $c$ and spatial
position $\boldsymbol{x}=(x^{1},\dots,x^{D})$. For simplicity $c=\hbar=1$. We
denote partial derivatives $\partial_{\mu}=\partial/\partial x^{\mu}$ and
Minkowski scalar product $A\cdot B=A^{\mu}B_{\mu}=A^{\mu}g_{\mu\nu}B^{\nu}$
with flat metric $g^{\mu\nu}=g_{\mu\nu}=1$ for $\mu=\nu=0$,
$g^{\mu\nu}=g_{\mu\nu}=-1$ for $\mu=\nu=1\dots D$ and
$g^{\mu\nu}=g_{\mu\nu}=0$ for $\mu\neq\nu$. Real scalar field
$\hat{\phi}(\boldsymbol{x})$ with conjugate field $\hat{\pi}(\boldsymbol{x})$
obeys commutation relation

$[\hat{\phi}(\boldsymbol{x}),\hat{\pi}(\boldsymbol{y})]=i\delta(\boldsymbol{x}-\boldsymbol{y})$
(52)

for $[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}$. Relativistic field
Hamiltonian reads

$\hat{H}=\int
d\boldsymbol{x}(\hat{\pi}^{2}(\boldsymbol{x})+|\nabla\hat{\phi}(x)|^{2}+m^{2}\hat{\phi}^{2}(\boldsymbol{x}))/2$
(53)

Here the $\nabla$ term is in fact a sum of partial derivatives

$|\nabla\hat{\phi(x)}|^{2}=\sum_{j=1}^{D}(\partial_{j}\hat{\phi}(\boldsymbol{x}))^{2}.$
(54)

Heisenberg picture transforms the field with time

$\hat{\phi}(x)=e^{i\hat{H}t}\hat{\phi}(\boldsymbol{x})e^{-i\hat{H}t}$ (55)

Translation into path integrals gives

$\langle\Phi^{\prime}|\exp(-i\hat{H}t)|\Phi\rangle=\int\mathcal{D}\phi\exp\int
i\mathcal{L}(x)dx$ (56)

with $\phi(x^{0}=0,\dots)=\Phi$ and $\phi(x^{0}=t,\dots)=\Phi^{\prime}$, where
the Lagrangian density $\mathcal{L}$ is given by (5).

For the fermionic fields, here use only to generate renormalization
counterterms, one has to replace commutator $[\hat{\phi},\hat{\pi}]$ in (52)
by anticommutator
$\\{\hat{\phi},\hat{\pi}\\}=\hat{\phi}\hat{\pi}+\hat{\pi}\hat{\phi}$ and
introduce Grassmann anticommuting fields $\phi$ in path integrals, i.e.
$\phi(x)\phi(y)=-\phi(y)\phi(y)$, $\int d\phi=0$, $\int\phi d\phi=1$. The
complications, including the signs and order conventions, are thoroughly
described in literature peskin .

Time flow over Schwinger-Kadanoff-Baym-Matsubara closed time path (CTP)
schwinger ; kaba ; matsubara ; weldon ; chou ; landsman ; kapusta  is
parameterized $t(s)$ (optionally with a subscript indicating the specific
point in spacetime, $x^{0}(s_{x})$). The real parameter $s\in[s_{i},s_{f}]$
must satisfy $dt/ds\neq 0$ and $\mathrm{Im}\;dt/ds\leq 0$, the jump
$t(s_{i})-t(s_{f})=i\beta$ for $\beta=1/k_{B}T>0$ (inverse temperature), see
Fig. 7. For fermionic fields, the jump is accompanied by the sign reversal for
each field. In the case $k_{B}T\to 0$ we have $t(s_{\mp})\to\pm i\infty$. For
the convenience the flat part splits into $t\to t_{\pm}=t(s_{\pm})=t\pm
i\epsilon$ ($\epsilon\to 0_{+}$, a small positive number going to $0$ in the
limit) and $x_{\pm}=(t_{\pm},\boldsymbol{x})$ with $s_{+}<s_{-}$.

Figure 7: The time path in the CTP approach in the case of finite temperature
$\beta=1/T$. At zero temperature, the shift $\beta$ stretches to infinity with
$t_{i}\to+i\infty$, $t_{f}\to-i\infty$

We use derivative rule: $\partial_{0}=(dt/ds)^{-1}\partial/\partial s$ and
differential $dx=dx^{0}dx^{1}\cdots dx^{D}$ with $dx^{0}=(dt/ds)ds$ and

$\delta(x-y)=\delta(x^{0}-y^{0})\delta(x^{1}-y^{1})\cdots\delta(x^{D}-y^{D})$
(57)

with $\delta(x^{0}-y^{0})=\delta(s_{x}-s_{y})/(dt/ds)|_{s=s_{x}=s_{y}}$. These
rules allow to establish full compliance of CTP with perturbative relativistic
quantum field theory ab13 ; ab16 .

##  Appendix B Two point correlations

With the definitions in Appendix A and (5) one can calculate all relevant
quantum field theory functions, i.e.

$\displaystyle\langle\phi(x)\phi(y)\cdots\rangle=\langle\mathcal{T}\hat{\phi}(x)\hat{\phi}(y)\cdots\rangle$
$\displaystyle=Z\int\mathcal{D}\phi\phi(x)\phi(y)\exp\int i\mathcal{L}(z)dz$
(58)

with

$Z^{-1}=\int\mathcal{D}\phi\exp\int i\mathcal{L}(z)dz,$ (59)

where $\mathcal{T}$ denotes ordering by $s$, i.e.

$\mathcal{T}\hat{\phi}(x)\hat{\phi}(y)=\left\\{\begin{array}[]{ll}\hat{\phi}(x)\hat{\phi}(y)&\mbox{
if }s_{x}>s_{y},\\\ \hat{\phi}(y)\hat{\phi}(x)&\mbox{ if
}s_{y}>s_{z}.\end{array}\right.$ (60)

Since the formal path functional is Gaussian, all correlations split into
these simple second-order correlations (Wick theorem wick )

$\displaystyle\langle\phi(x_{1})\cdots\phi(x_{n})\rangle=$ $\displaystyle
2^{-n/2}\sum_{\sigma}\prod_{j=1}^{n/2}\langle\phi(x_{\sigma(j)})\phi(x_{\sigma(n/2+j)})\rangle$
(61)

for even $n$ while $0$ for odd $n$, summing over all permutations. For the
fermionic field one has to include also the permutation sign
$\mathrm{sgn}\,\sigma$. In the case of linear perturbation we often use the
identity $\int dae^{-2\lambda
a^{2}+ba}=(\pi/2\lambda)^{1/2}e^{b^{2}/8\lambda}$. Applying functional
derivative

$\displaystyle\delta(x-y)=\frac{\delta\phi(y)}{\delta\phi(x)}=\left\langle\frac{\delta\phi(y)}{\delta\phi(x)}\right\rangle$
$\displaystyle=Z\int\mathcal{D}\phi\frac{\delta\phi(y)}{\delta\phi(x)}\exp\int
i\mathcal{L}(z)dz$ (62)

and integrating by parts

$\displaystyle\delta(x-y)Z^{-1}=\int\mathcal{D}\phi\phi(y)\int\frac{\delta\mathcal{L}(w)}{i\delta\phi(x)}dw\exp\int
i\mathcal{L}(z)dz$ $\displaystyle=\int\mathcal{D}\phi
i\phi(y)(\partial\cdot\partial\phi(x)+m^{2}\phi(x))\exp\int i\mathcal{L}(z)dz$
(63)

we get

$(g^{\mu\nu}\partial_{\mu}\partial_{\nu}+m^{2})\langle\phi(x)\phi(y)\rangle=\delta(x-y)/i$
(64)

with the derivative over $x$. The equation can be solved by Fourier transform
in $\boldsymbol{x}$ and $\boldsymbol{y}$, i.e.

$\phi(x)=\int\frac{d\boldsymbol{k}}{(2\pi)^{D/2}}e^{i\boldsymbol{k}\cdot\boldsymbol{x}}\phi(x^{0},\boldsymbol{k})$
(65)

with the standard scalar product
$\boldsymbol{k}\cdot\boldsymbol{x}=\sum_{j=1}^{D}k^{j}x^{j}$. We obtain

$\displaystyle(\partial_{0}\partial_{0}+\boldsymbol{k}\cdot\boldsymbol{k}+m^{2})\langle\phi(x^{0},\boldsymbol{k})\phi(y^{0},\boldsymbol{q})\rangle$
$\displaystyle=\delta(x^{0}-y^{0})\delta(\boldsymbol{k}-\boldsymbol{q})/i$
(66)

with the standard
$\delta(\boldsymbol{k}-\boldsymbol{q})=\delta(k^{1}-q^{1})\cdots\delta(k^{D}-q^{D})$.
This is a simple linear equation with solution

$\langle\phi(x^{0},\boldsymbol{k})\phi(y^{0},\boldsymbol{q})\rangle=\sum_{\pm}\pm\delta(\boldsymbol{k}-\boldsymbol{q})\frac{e^{\mp
i|x^{0}-y^{0}|E_{k}}}{1-e^{\mp\beta E_{k}}}$ (67)

with $E_{k}=\sqrt{\boldsymbol{k}\cdot\boldsymbol{k}+m^{2}}$ and

$|x^{0}-y^{0}|=\left\\{\begin{array}[]{ll}x^{0}-y^{0}&\mbox{ if
}s_{x}>s_{y}\\\ y^{0}-x^{0}&\mbox{ otherwise }\end{array}\right.$ (68)

In the zero-temperature limit $\beta\to 0$ we get

$\langle\phi(x^{0},\boldsymbol{k})\phi(y^{0},\boldsymbol{q})\rangle=\delta(\boldsymbol{k}-\boldsymbol{q})e^{-i|x^{0}-y^{0}|E_{k}}.$
(69)

Equivalently

$\langle\phi(x)\phi(y)\rangle=\sum_{\pm}\int\frac{\pm
d\boldsymbol{k}}{(2\pi)^{D}2E_{k}}e^{i\boldsymbol{k}\cdot(\boldsymbol{x}-\boldsymbol{y})}\frac{e^{\mp
i|x^{0}-y^{0}|E_{k}}}{1-e^{\mp\beta E_{k}}}$ (70)

or in the zero temperature limit

$\langle\phi(x)\phi(y)\rangle=\int\frac{d\boldsymbol{k}}{2E_{k}(2\pi)^{D}}e^{i\boldsymbol{k}\cdot(\boldsymbol{x}-\boldsymbol{y})}e^{-i|x^{0}-y^{0}|E_{k}}.$
(71)

The special correlations read

$\displaystyle S(x)=\sum_{\pm}\int\frac{\pm
d\boldsymbol{k}}{(2\pi)^{D}2E_{k}}e^{i\boldsymbol{k}\cdot\boldsymbol{x}}\frac{e^{\mp
i|x^{0}|E_{k}}}{1-e^{\mp\beta E_{k}}},$ $\displaystyle
B(x)=\sum_{\pm}\int\frac{\pm
d\boldsymbol{k}}{(2\pi)^{D}2E_{k}}e^{i\boldsymbol{k}\cdot\boldsymbol{x}}\frac{e^{\pm
ix^{0}E_{k}}}{1-e^{\mp\beta E_{k}}},$ $\displaystyle
C(x)=\int\frac{d\boldsymbol{k}}{(2\pi)^{D}2E_{k}}e^{i\boldsymbol{k}\cdot\boldsymbol{x}}\frac{\cos(x^{0}E_{k})}{\tanh(\beta
E_{k}/2)},$ (72)

with $|x^{0}|$ reduced to usual absolute value and zero-temperature limits

$\displaystyle
S(x)=\int\frac{d\boldsymbol{k}}{2E_{k}(2\pi)^{D}}e^{i\boldsymbol{k}\cdot\boldsymbol{x}}e^{-i|x^{0}|E_{k}}$
$\displaystyle=\int\frac{idk}{(2\pi)^{D+1}}\frac{e^{ik\cdot(x-y)}}{k\cdot
k-m^{2}+i\epsilon},$ $\displaystyle
B(x)=\int\frac{d\boldsymbol{k}}{2E_{k}(2\pi)^{D}}e^{i\boldsymbol{k}\cdot\boldsymbol{x}}e^{ix^{0}E_{k}}$
$\displaystyle=\int\frac{dk}{(2\pi)^{D}}\delta(k\cdot k-m^{2})e^{ik\cdot
x}\theta(k^{0}),$ $\displaystyle
C(x)=\int\frac{d\boldsymbol{k}}{(2\pi)^{D}2E_{k}}e^{i\boldsymbol{k}\cdot\boldsymbol{x}}\cos(x^{0}E_{k})$
$\displaystyle=\int\frac{dk}{2(2\pi)^{D}}\delta(k\cdot k-m^{2})e^{ik\cdot x},$
$\displaystyle
G(x)=\int\frac{d\boldsymbol{k}}{iE_{k}(2\pi)^{D}}e^{i\boldsymbol{k}\cdot\boldsymbol{x}}\sin(x^{0}E_{k})$
$\displaystyle=\int\frac{idk}{(2\pi)^{D+1}}\frac{e^{ik\cdot x}}{k_{+}\cdot
k_{+}-m^{2}}.$ (73)

The causal Green function is independent of temperature and defined only for
$x^{0}\geq 0$ while $G=0$ for $x\cdot x<0$, where $k^{0}_{+}=k^{0}-i\epsilon$
($\epsilon\to 0_{+}$ as previously is necessary to make the integrals well
defined).

Convenient substitution $\boldsymbol{k}=k\boldsymbol{n}$ where
$\boldsymbol{n}$ is a unit vector and $k=m\sinh\eta>0$. Then
$E_{k}=m\cosh\eta$ and
$d\boldsymbol{k}/E_{k}=d\boldsymbol{n}(m\sinh\eta)^{D-1}d\eta$ (in $D=1$ we
have $\sum_{\boldsymbol{n}=\pm 1}$ instead of $\int d\boldsymbol{n}$). Then

$\displaystyle\langle\phi(x)\phi(0)\rangle=\int\frac{(m\sinh\eta)^{D-1}d\eta
d\boldsymbol{u}}{2(2\pi)^{D}}\times$ $\displaystyle
e^{im\sinh\eta\boldsymbol{u}\cdot\boldsymbol{x}}e^{-i|x^{0}|m\cosh\eta}$ (74)

For $x^{1\dots D}=0$ we have

$\langle\phi(x)\phi(0)\rangle=\int\frac{(m\sinh\eta)^{D-1}d\eta
d\boldsymbol{u}}{2(2\pi)^{D}}e^{-i|x^{0}|m\cosh\eta}.$ (75)

The integral $\int d\boldsymbol{u}=S_{D}$ is the surface of $D-1$ unit sphere
(embedded in $D$ dimensions). In particular $S_{1}=2$, $S_{2}=2\pi$,
$S_{3}=4\pi$, or in general $S_{D}=2\pi^{D/2}/\Gamma(D/2)$ ($\Gamma$ \- Euler
Gamma function, here $\Gamma(1/2)=\pi^{1/2}$, $\Gamma(1)=1$ and
$\Gamma(z+1)=z\Gamma(z)$). Substituting $w=\cosh\eta$ we get

$\displaystyle\int\frac{m^{D-1}(w^{2}-1)^{D/2-1}dwd\boldsymbol{u}}{2^{D}\pi^{D/2}\Gamma(D/2)}e^{-i|x^{0}|mw}=$
$\displaystyle\frac{m^{(D-1)/2}K_{(D-1)/2}(i|x^{0}|m)}{(2\pi)^{(D+1)/2}(i|x^{0}|)^{(D-1)/2}}.$
(76)

By Lorentz invariance and analyticity we can write in general

$\displaystyle\langle\phi(x)\phi(0)\rangle=$
$\displaystyle\frac{m^{(D-1)/2}K_{(D-1)/2}(m\sqrt{-x\cdot
x})}{(2\pi)^{(D+1)/2}(-x\cdot x)^{(D-1)/4}}$ (77)

with the complex square root defined so that the real part is positive. The
divergence at $x=0$ is removable because we can make an infinitesimal shift in
imaginary direction.

We shall list the special cases in zero temperature limit prop . Case $D=1$:

$\displaystyle C(x)=\left\\{\begin{array}[]{ll}K_{0}(m\sqrt{-x\cdot
x})/2\pi&\mbox{ for }x\cdot x<0,\\\ -Y_{0}(m\sqrt{x\cdot x})/4&\mbox{ for
}x\cdot x>0,\end{array}\right.$ (80) $\displaystyle
G(x,y)=-iJ_{0}(m\sqrt{x\cdot x})/2\mbox{ for }x\cdot x>0\mbox{ and }x^{0}>0.$
(81)

Case $D=2$. Taking into account that $K_{1/2}(z)=e^{-z}\sqrt{\pi/2z}$ we can
write

$\langle\phi(x)\phi(0)\rangle=\frac{\exp(-\sqrt{-x\cdot
x}m)}{4\pi\sqrt{-x\cdot x}}$ (82)

or in particular

$\displaystyle C(x)=\frac{C^{\prime}(x)}{4\pi\sqrt{|x\cdot x|}},$
$\displaystyle C^{\prime}(x)=\left\\{\begin{array}[]{ll}\exp(-m\sqrt{-x\cdot
x})&\mbox{ for }x\cdot x<0\\\ -\sin(m\sqrt{x\cdot x})&\mbox{ for }x\cdot
x>0\end{array}\right.,$ (85) $\displaystyle G(x)=\frac{\cos(\sqrt{x\cdot
x}m)}{2\pi i\sqrt{x\cdot x}},$ (86)

for $x^{0}>0$ and $x\cdot x>0$. Case $D=3$:

$\displaystyle\langle\phi(x)\phi(0)\rangle=\frac{m}{4\pi^{2}\sqrt{-x\cdot
x}}K_{1}(m\sqrt{-x\cdot x}),$ $\displaystyle
C(x)=\frac{mC^{\prime}(x)}{8\pi^{2}\sqrt{|x\cdot x|}},$ $\displaystyle
C^{\prime}(x)=\left\\{\begin{array}[]{ll}2K_{1}(m\sqrt{-x\cdot x})&\mbox{ for
}x\cdot x<0,\\\ \pi Y_{1}(m\sqrt{x\cdot x})&\mbox{ for }x\cdot
x>0,\end{array}\right.$ (89) $\displaystyle G(x)=\frac{imJ_{1}(\sqrt{x\cdot
x}m)}{4\pi\sqrt{x\cdot x}}+\frac{\delta(x\cdot x)}{2\pi i},$ (90)

with only the last term in the $m\to 0$ case.

The $m\to 0$ case. By expansion of Bessel functions, we have for $D=1$

$-2\pi\langle\phi(x)\phi(0)\rangle\to\ln(-x\cdot x)/2+\ln(m/2)+\gamma$ (91)

with Euler-Mascheroni constant $\gamma$ and $G(x)\to-i/2$ (for $x^{0}>0$ and
$x\cdot x>0$). One encounters infrared divergence of correlation at small
$\boldsymbol{k}$ and $m\to 0$. For $D>1$ we have

$\langle\phi(x)\phi(0)\rangle\to\frac{\Gamma((D-1)/2)}{4\pi^{(D+1)/2}(-x\cdot
x)^{(D-1)/2}}.$ (92)

For $D=2$:

$\displaystyle G(x)\to 1/2\pi i\sqrt{x\cdot x},$ $\displaystyle
C(x)\to\theta(-x\cdot x)/4\pi\sqrt{-x\cdot x}.$ (93)

and $D=3$:

$\displaystyle G(x)\to\delta(x\cdot x)/2\pi i,$ $\displaystyle
C(x)\to-1/4\pi^{2}x\cdot x.$ (94)

Since $C$ diverges at $x\cdot x\to 0$ (also for $m>0$) we have calculate it as
Cauchy principal value.

##  Appendix C Vacuum fluctuations

Here we present the details of the calculation of (28). We begin with

$\displaystyle\langle a^{2}\rangle_{\cdot}=\int
dkdp\tilde{f}(k+p)\tilde{f}(-k-p)\theta(k^{0})\theta(p^{0})\times$
$\displaystyle 8\pi^{2}\delta(k\cdot k-m^{2})\delta(p\cdot p-m^{2}).$ (95)

Note that $k$ and $p$ are forward, i.e. $k\cdot k,p\cdot p>0$ and
$k^{0},p^{0}>0$ so $q=k+p$ is also forward. On the other hand, forward $q$ and
timelike $k,p$ are not sufficient to keep both $k$ and $p$ forward, see Fig.
8.

Figure 8: Relation between $k$, $p$ and $q$ showing that timelike $p=q-k$ is
not necessarily forward, even if $k$ and $q$ are.

Replacing $k+p=q$ and shifting $p$, we get

$\displaystyle\int
dqdp\tilde{f}(q)\tilde{f}(-q)\theta(q^{0}/2+p^{0})\theta(q^{0}/2-p^{0})8\pi^{2}\times$
$\displaystyle\delta((q/2+p)\cdot(q/2+p)-m^{2})\times$
$\displaystyle\delta((q/2-p)\cdot(q/2-p)-m^{2}).$ (96)

Note that

$\displaystyle\delta((q/2+p)\cdot(q/2+p)-m^{2})\times$
$\displaystyle\delta((q/2+p)\cdot(q/2+p)-m^{2})$ $\displaystyle=\delta(2q\cdot
p)\delta(q\cdot q/4+p\cdot p-m^{2}),$ (97)

which gives fixings $q\cdot p=0$ and $q\cdot q/4+p\cdot p=m^{2}$.

We need to calculate $W(q)$ equal

$\int dp\theta(|q^{0}|/2-|p^{0}|)\delta(2q\cdot p)\delta(q\cdot q/4+p\cdot
p-m^{2})/2$ (98)

but due to Lorentz invariance we need to do it only for $q^{1\dots D}=0$ and
$q^{0}>0$, which this the easiest. Then $p^{0}=0$ and by substitution
$|\boldsymbol{p}|=\tilde{p}$ it reduces to

$\int\tilde{p}^{D-1}S_{D}d\tilde{p}\delta((q^{0})^{2}/4-\tilde{p}^{2}-m^{2})/4q^{0}.$
(99)

Finally putting $p=\sqrt{(q^{0})^{2}/4-m^{2}}$ and replacing
$q^{0}\to\sqrt{q\cdot q}$, we get (29).

##  Appendix D Plane wave generation

We shall derive the causal Green function in the limit of oscillatin
perturbation that generates a flane wave. We define

$\displaystyle G_{g}(x)=\int dyG(x,y)g(y)=$
$\displaystyle\int_{y^{0}<x^{0}}dy\frac{d\boldsymbol{k}}{iE_{k}(2\pi)^{D}}e^{i\boldsymbol{k}\cdot(\boldsymbol{x}-\boldsymbol{y})}\sin((x^{0}-y^{0})E_{k})g(y).$
(100)

We first integrate over $y^{2,3,...}$ while substituting $y^{0}=x^{0}-t$, and
$y^{1}=y-L$ getting

$\displaystyle\int_{t>0}dtdy\frac{dk}{iE_{k}(2\pi)}e^{ik(x^{1}-y)}e^{ikL}\times$
$\displaystyle\sin(tE_{k})(e^{iE_{p}(x^{0}-t)}h(y)+e^{-iE_{p}(x^{0}-t)}h^{\ast}(y))=$
$\displaystyle\int_{t>0}dtdy\frac{dk}{E_{k}(4\pi)}e^{ik(x^{1}-y)}e^{ikL}\times$
$\displaystyle[(e^{itE_{k}+iE_{p}(x^{0}-t)}-e^{-itE_{k}+iE_{p}(x^{0}-t)})h(y)$
$\displaystyle+(e^{itE_{k}-iE_{p}(x^{0}-t)}-e^{-itE_{k}-iE_{q}(x^{0}-t)})h^{\ast}(y)]$
(101)

with $k$ being 1-dimensional variable and $E_{k}=\sqrt{k^{2}+m^{2}}$. We
integrate over $t$ with damping factor $e^{-0_{+}t}$. Then

$\displaystyle G_{g}(x)=\int\frac{dk}{E_{k}(4\pi)}e^{ik(L+x^{1})}\times$
$\displaystyle\left[\left(\frac{e^{iE_{p}x^{0}}}{0_{+}-iE_{k}+iE_{p}}-\frac{e^{iE_{p}x^{0}}}{0_{+}+iE_{k}+iE_{p}}\right)\tilde{h}(k)\right.$
$\displaystyle\left.+\left(\frac{e^{-iE_{p}x^{0}}}{0_{+}-iE_{k}-iE_{p}}-\frac{e^{-iE_{p}x^{0}}}{0_{+}+iE_{k}-iE_{p}}\right)\tilde{h}^{\ast}(-k)\right]$
(102)

with $\tilde{h}(k)=\int dyh(y)e^{-iky}$. For large $L$ the factor $e^{ikL}$ is
quickly oscillating while the other functions vary slowly. Exceptions are the
first and last denominator at $E_{k}\sim E_{p}$ as they diverge. Only in these
cases we make approximations $E_{k}-E_{p}\simeq(|k|-|p|)|p|/E_{p}$ for
$|p|=\sqrt{E_{p}^{2}-m^{2}}$. The integral concentrates only near two peaks at
$k=\pm|q|$ so we can calculate

$\displaystyle G_{g}(x)\to\sum_{\pm}\int\frac{dk}{4\pi
E_{p}}e^{ik(L+x^{1})}\times$
$\displaystyle\left[\frac{e^{iE_{p}x^{0}}\tilde{h}(\pm|p|)}{0_{+}+i|p|(|p|\mp
k)/E_{p}}-\frac{e^{-iE_{p}x^{0}}\tilde{h}^{\ast}(\mp|p|)}{0_{+}-i|p|(|p|\mp
k)/E_{p}}\right]$
$\displaystyle=\sum_{\pm}\int\frac{dk}{4\pi|p|}e^{ik(L+x^{1})}\times$
$\displaystyle\left[\frac{e^{iE_{p}x^{0}}\tilde{h}(\pm|p|)}{0_{+}+i(|p|\mp
k)}-\frac{e^{-iE_{p}x^{0}}\tilde{h}^{\ast}(\mp|p|)}{0_{+}-i(|p|\mp
k)}\right].$ (103)

The integral over $k$ can be now calculated using residues, giving

$\displaystyle 2|p|G_{g}(x)\to$ $\displaystyle
e^{iE_{p}x^{0}-i|p|x^{1}}\tilde{h}(-|p|)-e^{i|p|x^{1}-iE_{p}x^{0}}\tilde{h}^{\ast}(+|p|)$
(104)

equivalent to (33).

##  Appendix E Contact term problem in energy correlations

For the normalization $\langle 1\rangle_{0}=1$ (unitarity) to hold, we expect
$\langle T^{\mu\nu}_{q}(x)T^{\xi\eta}_{q}(y)\rangle$ to vanish because any
quantity $A_{q}(x)=A_{+}(x)-A_{-}(x)$ should cancel the correlation if $x^{0}$
is the _latest_ time. For $\mu=\nu=\xi=\eta=0$ the above expression will
contain the term

$(\partial^{0}_{x}\partial^{0}_{y}G(x,y))(\partial^{0}_{x}\partial^{0}_{y}G(y,x)),$
(105)

which indeed is $0$ for both $x^{0}>y^{0}$ and $y^{0}-x^{0}$. However,
something strange happens at $x^{0}=y^{0}$. Then taking the definitions of $G$
we get

$\partial^{0}_{x}G(x,y)=\int\frac{d\boldsymbol{k}}{i(2\pi)^{D}}e^{i\boldsymbol{k}\cdot(\boldsymbol{x}-\boldsymbol{y})}\cos((x^{0}-y^{0})E_{k})$
(106)

for $x^{0}>y^{0}$. and $0$ for $x^{0}<y^{0}$. Unfortunately, there is a
discontinuity at $x^{0}=y^{0}$ which gives
$\partial^{0}_{x}\partial^{0}_{y}G(x,y)=i\delta(x-y)$ As a result, we get
$\delta^{2}(x-y)=\delta(0)\delta(x-y)$, with $\delta(0)$ undefined, formally
divergent to $+\infty$. The problem persists at higher (arbitrary) order
correlations since we get e.g.

$\displaystyle\delta(x-y)\delta(y-z)\delta(z-w)\delta(w-x)=$
$\displaystyle\delta(x-y)\delta(x-y)\delta(y-z)\delta(z-w)\delta(0)$ (107)

again diverging. To cure the problem, we need an auxiliary independent
renormalizing field $\phi_{M}$, but with a large mass $M\to\infty$, and
construct $T^{\mu\nu}_{M}(x)$ by replacing $m\to M$ and $\phi\to\phi_{M}$.
Unfortunately, $\phi_{M}$ must be anticommuting (Grassmann, fermionic). More
precisely, there are two fields $\phi_{M}(x)$ and $\phi^{\ast}_{M}(y)$ but
they anitcommute, i.e. $AB=-BA$ for $A,B=\phi_{M}(x,y),\phi_{M}^{\ast}(x,y)$.
The field is called a ghost because it is not physical fadeev . The
renormalizing Lagrangian density reads

$\mathcal{L}_{M}(x)=\partial\phi^{\ast}_{M}(x)\cdot\partial\phi_{M}(x)-m^{2}\phi^{\ast}(x)\phi(x)$
(108)

with the energy-momentum tensor

$\displaystyle
T^{\mu\nu}_{M}(x)=\partial^{\mu}\phi^{\ast}_{M}\partial^{\nu}\phi_{M}+\partial^{\nu}\phi^{\ast}_{M}\partial^{\mu}\phi_{M}$
$\displaystyle-g^{\mu\nu}(g^{\sigma\tau}\partial_{\sigma}\phi^{\ast}_{M}\partial_{\tau}\phi_{M}-M^{2}\phi^{\ast}_{M}\phi_{M}).$
(109)

The basic correlations read $\langle\phi_{M}(x)\phi_{M}(y)\rangle=0$ and

$\displaystyle\langle\phi_{M}(x)\phi^{\ast}_{M}(y)\rangle=$
$\displaystyle\int\frac{d\boldsymbol{k}}{(2\pi)^{D}2E_{k}}e^{i\boldsymbol{k}\cdot(\boldsymbol{x}-\boldsymbol{y})}\left(\frac{e^{-i|x^{0}-y^{0}|E_{k}}}{1+e^{-\beta
E_{k}}}-\frac{e^{i|x^{0}-y^{0}|E_{k}}}{1+e^{\beta E_{k}}}\right)$ (110)

with the zero-temperature limit

$\langle\phi_{M}(x)\phi^{\ast}_{M}(y)\rangle=\int\frac{d\boldsymbol{k}}{(2\pi)^{D}2E_{k}}e^{i\boldsymbol{k}\cdot(\boldsymbol{x}-\boldsymbol{y})}e^{-i|x^{0}-y^{0}|E_{k}}.$
(111)

The Wick decomposition includes now the sign of the permutation

$\displaystyle\langle\phi^{\ast}_{M}(x_{1})\phi_{M}(y_{1})\cdots\phi^{\ast}_{M}(x_{n})\phi_{M}(y_{n})\rangle=$
$\displaystyle\sum_{\sigma}\mathrm{sgn}\,\sigma\langle\phi^{\ast}_{M}(x_{\sigma(1)})\phi_{M}(y_{1})\rangle\cdots\langle\phi^{\ast}_{M}(x_{\sigma(n)})\phi_{M}(y_{n})\rangle.$
(112)

Now, because of the opposite sign the permutation we get

$\langle T^{\mu\nu}_{Mq}(x)T^{\xi\eta}_{Mq}(y)\rangle=-\langle
T^{\mu\nu}_{q}(x)T^{\xi\eta}_{q}(y)\rangle$ (113)

so let us modify $T^{\mu\nu}$ by

$T^{\mu\nu}(x)\to T^{\mu\nu}(x)+T^{\mu\nu}_{M}(x)$ (114)

in our Kraus operator. Then what we get is

$\langle T^{\mu\nu}_{q}(x)T^{\xi\eta}_{q}(y)\rangle=0.$ (115)

Note that finite $M$ would also add some correction to terms containing
$T^{\mu\nu}_{c}$ (which do not spoil unitarity, though) so we keep the limit
$M\to\infty$.

##  Appendix F Energy-momentum fluctuations

We define

$\tilde{f}_{\mu\nu}(k)=\int dxe^{ik\cdot x}f_{\mu\nu}(x)/(2\pi)^{D+1}$ (116)

and note that

$\displaystyle\langle T^{\mu\nu}_{c}(x)T^{\xi\eta}_{c}(y)\rangle=$
$\displaystyle\langle T^{\mu\nu}_{c}(x)T^{\xi\eta}_{c}(y)\rangle-\langle
T^{\mu\nu}_{q}(x)T^{\xi\eta}_{q}(y)\rangle/4$
$\displaystyle=\langle(T^{\mu\nu}(x_{+})T^{\xi\eta}_{c}(y_{-})+T^{\mu\nu}(x_{-})T^{\xi\eta}_{c}(y_{+}))\rangle/2$
(117)

Using $B(x,y)$ and $B(y,x)$ we get

$\displaystyle\langle a^{2}\rangle_{\cdot}=\int
dkdp\tilde{f}_{\mu\nu}(k+p)\tilde{f}_{\xi\eta}(-k-p)\times$
$\displaystyle(k^{\mu}p^{\nu}-g^{\mu\nu}(k\cdot p+m^{2})/2)\times$
$\displaystyle(k^{\xi}p^{\eta}-g^{\xi\eta}(k\cdot p+m^{2})/2)\times$
$\displaystyle 8\pi^{2}\theta(k^{0})\theta(p^{0})\delta(k\cdot
k-m^{2})\delta(p\cdot p-m^{2}).$ (118)

Replacing $k+p=q$ and shifting $p$, we get

$\langle a^{2}\rangle_{\cdot}=\int
dq\tilde{f}_{\mu\nu}(q)\tilde{f}_{\xi\eta}(-q)X^{\mu\nu\xi\eta}(q)$ (119)

with

$\displaystyle X^{\mu\nu\xi\eta}(q)=\int
dp\theta(|q^{0}|/2-|p^{0}|)\delta(2q\cdot p)4\pi^{2}\times$
$\displaystyle\delta(q\cdot q/4+p\cdot p-m^{2})\times$
$\displaystyle(q^{\mu}q^{\nu}/4-p^{\mu}p^{\nu}-g^{\mu\nu}q\cdot q/4)\times$
$\displaystyle(q^{\xi}q^{\eta}/4-p^{\xi}p^{\eta}-g^{\xi\eta}q\cdot q/4).$
(120)

We can observe that (a) $X$ is $0$ is $q\cdot q<0$ (b) $X$ is symmetric under
interchanging $\mu\leftrightarrow\nu$ or $\xi\leftrightarrow\eta$ or
$\mu\nu\leftrightarrow\xi\eta$ (c) satisfies $q_{\mu}X^{\mu\nu\xi\eta}(q)=0$
(Ward identity or energy conservation) (d) is Lorentz covariant (there is no
preferred frame). Even more, if we take $q^{1\dots D}=0$ then the constraints
lead to $p^{0}=0$ and $|\boldsymbol{p}|^{2}=(q^{0})^{2}/4=m^{2}$ so we have
bound $q\cdot q>4m^{2}$. Therefore the expected form of $X$ is

$\displaystyle X^{\mu\nu\xi\eta}=\theta(q\cdot q-4m^{2})4\pi^{2}\times$
$\displaystyle[P^{\prime}(q^{\mu}q^{\nu}-(q\cdot
q)g^{\mu\nu})(q^{\xi}q^{\eta}-(q\cdot q)g^{\xi\eta})$
$\displaystyle+R^{\prime}[(q^{\mu}q^{\eta}-(q\cdot
q)g^{\mu\eta})(q^{\xi}q^{\nu}-(q\cdot q)g^{\xi\nu})$
$\displaystyle+(q^{\mu}q^{\xi}-(q\cdot q)g^{\mu\xi})(q^{\nu}q^{\eta}-(q\cdot
q)g^{\nu\eta})]].$ (121)

We only need to find two functions $P^{\prime}(q\cdot q)$ and
$R^{\prime}(q\cdot q)$, which is the simplest by contraction

$g_{\mu\nu}g_{\xi\eta}X^{\mu\nu\xi\eta}=(q\cdot
q)^{2}D(DP^{\prime}+2R^{\prime})$ (122)

and

$g_{\mu\xi}g_{\nu\eta}X^{\mu\nu\xi\eta}=(q\cdot
q)^{2}D[P^{\prime}+R^{\prime}(D+1)]$ (123)

giving

$\displaystyle(D-1)(D+2)D(q\cdot q)^{2}P^{\prime}=$
$\displaystyle(D+1)g_{\mu\nu}g_{\xi\eta}X^{\mu\nu\xi\eta}-2g_{\mu\xi}g_{\nu\eta}X^{\mu\nu\xi\eta},$
$\displaystyle(D-1)(D+2)D(q\cdot q)^{2}R^{\prime}=$ $\displaystyle
Dg_{\mu\xi}g_{\nu\eta}X^{\mu\nu\xi\eta}-g_{\mu\nu}g_{\xi\eta}X^{\mu\nu\xi\eta}.$
(124)

At $D=1$ both equations give $P^{\prime}+2R^{\prime}$ but it is not a problem,
as the $P^{\prime}$ and $R^{\prime}$ terms are actually the same. It is clear
from the general rule $X^{0\nu\xi\eta}=X^{1\nu\xi\eta}q^{1}/q^{0}$ and
similarly for all indices so $X$ is fixed by just one term $X^{1111}$ (not
true at $D>1$).

We can express the contraction by previous quantities

$\displaystyle g_{\mu\nu}g_{\xi\eta}X^{\mu\nu\xi\eta}=$ $\displaystyle\int
dp\theta(|q^{0}|/2-|p^{0}|)\delta(2q\cdot p)\delta(q\cdot q/4+p\cdot
p-m^{2})\times$ $\displaystyle(q\cdot q/4-p\cdot p-(D+1)q\cdot q/4)^{2}$
$\displaystyle=((D-1)q\cdot q/4+m^{2})^{2}W(q)$ (125)

and

$\displaystyle g_{\mu\xi}g_{\nu\eta}X^{\mu\nu\xi\eta}=$ $\displaystyle\int
dp\theta(|q^{0}|/2-|p^{0}|)\delta(2q\cdot p)\delta(q\cdot q/4+p\cdot
p-m^{2})\times$ $\displaystyle((q\cdot q)^{2}/16+(p\cdot p)^{2}+(D+1)(q\cdot
q)^{2}/16$ $\displaystyle-(q\cdot p)^{2}/2-(q\cdot q)^{2}/8+(p\cdot p)(q\cdot
q)/2)$ $\displaystyle=((q\cdot q)^{2}(D-1)/16+m^{4})W(q)$ (126)

for $W$ given by (29) so that

$\displaystyle(q\cdot q)^{2}(D+2)DP^{\prime}=$
$\displaystyle(m^{4}+(D+1)m^{2}q\cdot q/2+(q\cdot q)^{2}(D^{2}-3)/16)W(q),$
$\displaystyle(q\cdot q)^{2}(D+2)DR^{\prime}=$
$\displaystyle(m^{4}-m^{2}q\cdot q/2+(q\cdot q)^{2}/16)W(q)$ (127)

with $P^{\prime}=PW$, $R^{\prime}=RW$ in (51).

In the case $D=1$ we can set $R^{\prime}=0$ and

$P^{\prime}=\pi^{2}m^{4}(q\cdot q)^{-5/2}(q\cdot q/4-m^{2})^{-1/2}$ (128)

In the limit $m\to 0$ also $P^{\prime}\to 0$ but only at $q\cdot q>0$. The
limit $q\cdot q\to 0$ has to be considered separately Note that the final
integral

$\displaystyle\int dq\theta(q\cdot q/4-m^{2})m^{4}(q\cdot q)^{-5/2}\times$
$\displaystyle(q\cdot q/4-m^{2})^{-1/2}|(q^{\mu}q^{\nu}-g^{\mu\nu}(q\cdot
q))\tilde{f}_{\mu\nu}(q)|^{2}\pi^{2}$ (129)

can be transformed using change of variables $q^{0}=m\sqrt{w}\cosh u$,
$q^{1}=m\sqrt{w}\sinh u$ into

$\displaystyle\int_{4}^{\infty}dw\int
duw^{-5/2}(w/4-1)^{-1/2}m^{4}w^{2}\times$
$\displaystyle|(U^{\mu}U^{\nu}-g^{\mu\nu})\tilde{f}_{\mu\nu}(m\sqrt{w}U)|^{2}\pi^{2}$
(130)

with $U^{0}=\cosh u$, $U^{1}=\sinh u$. Suppose $\tilde{f}(q)$ is a regular
function that decays to $0$ if $q\to\infty$. Then the limit $m\to 0$ reduces
the integral essentially to the lines of light, i.e. $q^{0}=\pm q^{1}$, for
$|u|\gg 1$. We make then approximation $U=e^{u}U_{\pm}/2$ with
$\lambda=m\sqrt{w}e^{u}$ and $U_{\pm}=(1,\pm 1)$. Then we get

$\langle
a^{2}\rangle_{0}=\sum_{\pm}\int_{0}^{\infty}d\lambda\lambda^{3}|U_{\pm}^{\mu}U_{\pm}^{\nu}\tilde{f}_{\mu\nu}(\lambda
U_{\pm})|^{2}\pi^{2}/3\cdot 2^{5}$ (131)

since

$\int_{4}^{\infty}dww^{-5/2}(w/4-1)^{-1/2}=1/6.$ (132)

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