University of Amsterdam

MSc Physics
Theoretical Physics Track

Master Thesis

Violations of the Null Energy Condition in QFT and their Implications

by
Dimitrios Krommydas
10867473

January 2017
60 ECs
September 2015 - January 2017

_
Supervisor:_

Ben Freivogel

_
Examiner:_
J.P. van der Schaar

String Theory Group
Cosmology Group
Institute of Physics

###### Abstract

We study violations of the Null Energy Condition (NEC) in Quantum Field Theory
(QFT) and their implications. For the first part of the project, we examine
these violations for classes of already known and novel (first discussed here)
QFT states. Next, we discuss the implications of these violations focusing on
the example of Wormhole Traversability. After reviewing the current literature
on the existing restrictions on these violations, we conjecture that NEC
violating states are incompatible with the Semi-Classical Gravity
approximation. We argue that this conjecture provides the only way out of the
problems introduced by the violations of NEC in this regime. Building on this,
we propose a bound that should hold for all QFT states. Finally, we show that
both our conjecture and bound hold for some relevant classes of QFT states.

###### Contents

  1. 1 Introduction
  2. 2 Null Energy Condition
    1. 2.1 Energy Conditions
    2. 2.2 Raychaudhuri Equations
      1. 2.2.1 Raychaudhuri Equation for Null Geodesics
      2. 2.2.2 The focusing Theorem
    3. 2.3 Penrose Singularity Theorem
    4. 2.4 Other GR results
  3. 3 Wormholes
    1. 3.1 Wormhole Traversability
      1. 3.1.1 Schwarzschild Non-Traversable Wormhole
      2. 3.1.2 Ellis Traversable Wormhole
    2. 3.2 The Null Flaring-out Condition
  4. 4 Violations of NEC in QFT
    1. 4.1 The form of NEC in QFT
    2. 4.2 1+1 CFT
    3. 4.3 Vacuum plus 2 particles
      1. 4.3.1 Framework
      2. 4.3.2 Calculation
    4. 4.4 Casimir
    5. 4.5 Perturbed vacuum states
  5. 5 Local and Non-Local Constraints
    1. 5.1 QNEC
    2. 5.2 ANEC
    3. 5.3 Quantum Inequalities
      1. 5.3.1 ER=EPR and the Lost Wormhole
    4. 5.4 QFC
  6. 6 Semi-Classical Gravity
    1. 6.1 Regime of validity of semi-classical gravity
    2. 6.2 Conjecture
      1. 6.2.1 Fluctuations of the Stress Tensor
      2. 6.2.2 Very Quantum States
      3. 6.2.3 The Probability distribution
    3. 6.3 Smearing
      1. 6.3.1 The Smeared Stress Energy Tensor
      2. 6.3.2 Fluctuations of Smeared Stress Tensor
      3. 6.3.3 Testing the Conjecture
      4. 6.3.4 Proposed Bound
  7. 7 Conclusions

##  1 Introduction

The purpose of this thesis is to explore the violations of the Null Energy
Condition (NEC) in Quantum Field Theory (QFT). First, we introduce the Energy
Conditions in the framework of General Relativity (GR), their implications and
a number of theorems they are used to prove (sec. 2.1, 2.2.2, 2.3, 2.4). But
before even defining the Energy Conditions, it is imperative to understand why
there was a need for imposing them. The beauty and the revolutionary nature of
GR can be summarized in one fundamental equation, the Einstein equation111In
truth these are many equations, one for each component of $G_{\mu\nu}$.
However, they can be written in this nice compact form.:

$G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G_{N}T_{\mu\nu}\;.$
(1.1)

Here $G_{\mu\nu}$ is the Einstein tensor, a combination of the Ricci tensor
$R_{\mu\nu}$ and Ricci scalar $R$ and describes the (intrinsic) curvature of
spacetime (pseudo-Riemannian manifold). $G_{N}$ is the Newton’s constant and
$T_{\mu\nu}$ is the stress energy tensor, the conserved current of spacetime
translations, which describes the matter content of spacetime. A famous
sentence which aspires to explain relation (1.1) is: Spacetime tells matter
how to move, and matter tells spacetime how to curve. But for all its beauty
and intuitive simplicity, the Einstein equation leaves an important question
unanswered. What properties should matter have, to be considered physical? Of
course, Einstein already knew this when he created his theory. His concerns
can be summarized in the following quote [1]:

It [general relativity] is similar to a building, one wing of which is made of
fine marble (left part of the equation), but the other wing of which is built
of low-grade wood (right side of the equation). The phenomenological
representation of matter is, in fact, only a crude substitute for a
representation which would do justice to all known properties of matter

An easy way to understand the gravity of the issue is the following: without
any restrictions on $T_{\mu\nu}$, any metric is a valid solution to the
Einstein equations. Just pick your favorite metric, compute $G_{\mu\nu}$, plug
it in the Einstein equations and derive the $T_{\mu\nu}$ for this spacetime.
The stress energy tensor you calculated will be automatically conserved by the
Bianchi identity. Thus, to limit the arbitrariness of $T_{\mu\nu}$, the Energy
Conditions are imposed, and it is believed that at least at the classical
level, these are conditions that the physical matter satisfies.

And in fact, these restrictions seem to work pretty well. They save us from a
lot of awkward situations, a short list of which is the following: violations
of the second law of thermodynamics, violation of “cosmic censorship”,
production of naked singularities, singularity avoidance inside Black Holes,
the breakdown of “positive mass” theorems in GR, the breakdown of Penrose
Singularity theorem, breakdown of Hawking area theorem, the existence of warp
drives & traversable wormholes, which in turn imply faster than light travel
and closed timelike curves, etc. (sec 2.4 ). The traversability of Wormholes
(ch. 3) is the main implication of the violations of NEC that we will study
throughout this thesis.

Since NEC plays such a central role in classical relativistic physics, one
might suspect that it is more than a mere assumption. Jan Pieter van der
Schaar and Maulik Parikh showed that the NEC can arise in the target space
(spacetime) of a bosonic string, from the Virasoro constraint on the world-
sheet [2]. In later work, M.Parikh et al. showed that NEC can be also derived
by assuming the second law of thermodynamics and applying it to the Bekenstein
entropy [3]. The above findings, suggest that NEC is a fundamental restriction
imposed by physical theories and not an ad hoc postulate.

Although all observed forms of classical matter obey at least NEC, this is not
the case for all quantum matter. There are simple examples of quantum field
theory states that violate NEC, even for free field theories in flat
spacetime. Some examples are: squeezed vacuum states [5], superpositions of
the vacuum and an excited mode (sec. 4.3), Casimir effect (sec. 4.4), moving
mirrors [6] and even superpositions of 1+1 CFT states (sec. 4.2). Finally, a
notorious example in which NEC violations occur is the Hawking evaporation
process [67].

In the first part of this project, we define the energy conditions (sec. 2.1)
and we present their form for the popular perfect fluid example. We then
derive the Raychaudhuri equations (sec. 2.2), which combined with the Einstein
equations and NEC, lead to a formulation of the Null Focusing Theorem (sec.
2.2.2). Next, we use the null focusing theorem to interpret the Penrose
Singularity Theorem (sec. 2.3), which is presented along with a long list of
GR results (sec. 2.4) in order to highlight the importance of NEC.

Another important aspect of NEC in GR is that it protects us from exotic
spacetimes which allow for causality violations, time-travel, and other
troublesome effects. We dedicate ch. 3 to the detailed analysis of one of
these spacetimes, the Traversable Wormhole. We first present the well-studied
case of the non-traversable wormhole which resides in the maximally extended
Schwarzschild spacetime (sec. 3.1.1). Then, we analyze the features of the
famous, traversable Ellis Wormhole (sec.3.1.2) and finally use NEC to show
that this solution is not allowed in GR.

As we mentioned, QFT does not always respect NEC. In ch. 4, we provide
evidence for this lack of respect. We first present one of our results; a
general class of 1+1 CFT states which, as we explicitly show, leads to
violations of NEC (sec. 4.2). We then reproduce a computation of [5]; namely,
we calculate the expectation value of the relevant components of
$\hat{T}_{\mu\nu}$ in the vacuum plus two modes state, and finally show that
it violates a quantum version of NEC. After citing the results for the
expectation value of $\hat{T}_{\mu\nu}$ for a free scalar in the Casimir
state, we show how it violates NEC and explain the significance of the
directionality of the Casimir plates. In the end of ch. 4, we display our
attempt to violate NEC for a general class of states which are close to the
vacuum (sec. 4.5); i.e. $e^{-i\int J(x)d^{d}x\Phi(x)}\lvert 0\rangle$ where
$J$ is an arbitrarily small, real source and $\Phi$ a massless free scalar
field. Surprisingly, we see that no such violation occurs.

In ch. 5, we discuss a collection of the most relevant local and non-local
constraints on the violations of NEC. First, we review an averaged version of
a quantum null energy condition, called the Averaged Null Energy Condition
(ANEC) and mention which GR results can be salvage by imposing it in the
quantum regime (sec. 5.2). Next, we discuss the Quantum Inequalities (QI’s), a
set of bounds on the magnitude of the NEC violating flux which are known to
hold for arbitrary quantum states and all inertial observers in Minkowski
spacetime. More interestingly, QI’s have been shown to hold also for a number
of curved spacetimes. We close ch.5, with a review of some of the most recent
developments in the area of quantum energy conditions, namely the Quantum Null
Energy Condition (sec. 5.1) and the Quantum Focusing Conjecture (sec. 5.4),
introduced by R. Bousso, A. Wall et al. The Quantum Focusing Conjecture is a
proposal for a quantum corrected version of the Classical Focusing Theorem
(sec. 2.2.2). The quantum null energy condition, which is a pure QFT
statement, arises naturally by imposing this conjecture on a specific setup;
moreover, it was proven in separate work [69], the same year when the quantum
focusing conjecture paper [68] appeared. In a nutshell, the quantum null
energy condition says that the quantum stress energy tensor is bounded from
below by the second derivative of some entanglement entropy.

In ch. 6, we study the violations of NEC in semi-classical gravity. Since we
do not have a complete theory of Quantum Gravity yet, it is of utmost
importance to understand these violations in a regime of physics over which we
have more control. After a brief introduction to semi-classical gravity
approximation (ch. 6), we discuss its regime of validity (sec. 6.1). We
explain that there are cases, even in small energy scales compared to the
Planck scale, where the semi-classical approximation cannot be trusted. For
instance, large fluctuations of the metric, induced by large
$\hat{T}_{\mu\nu}$ fluctuations, break the fixed background assumption. In the
next section (sec. 6.2), we conjecture that states for which a quantum version
of NEC is violated, lead to such a breakdown. Hence, at the semi-classical
level NEC still holds, allowing us to generalize our GR theorems and exclude
“exotic” solutions. In sec. 6.2.2, we give a simplified version of these very
quantum states and present some intuition as to why these states are
incompatible with the semi-classical regime. We then suggest, that if one
wishes to make precise statements about the fluctuations of
$\hat{T}_{\mu\nu}$, he/she should study the probability distribution function
of the operator in the state of interest. Since finding all eigenvalues and
their weights is usually an impossible task, one should find and study the
relevant features of the distribution (sec. 6.2.3). As we find out, a
remarkably suitable quantity for this assignment is the variance of the
distribution. Hence, we introduce a dimensionless quantity which contains the
variance and is suitable for quantifying the “quantumness” of a state. We then
use this quantity to test the validity of our conjecture, for some relevant
classes of states (sec. 6.3.3). In the end of this chapter (ch. 6), inspired
by our previous analysis, we propose a QFT bound (sec. 6.3.4) which we believe
should hold for all QFT states. Our bound, namely:

$\langle T^{s}_{\mu\nu}K^{\mu}K^{\nu}\rangle+\sigma\geq 0,$

involves the sum of the expectation of the null projected component of a
smeared stress energy tensor and the standard deviation. The smearing is
important, since it introduces a natural cut-off to the UV (short distance)
divergences present in
$\langle\hat{T}_{\mu\nu}(x)\hat{T}_{\rho\sigma}(x)\rangle$ and transforms the
point-wise defined stress tensor into a quantity more appropriate for
describing physical measurements (sec. 6.3). In the end of sec. 6.3.4, we
explicitly show that our bound holds for some interesting classes of states.

##  2 Null Energy Condition

In this chapter, we study the Null Energy Condition (NEC) in GR. We start by
defining it, along with other relevant Energy Conditions in sec. 2.1, and then
discuss its role in the formulation and proof of many important GR theorems
(sec. 2.2.2, 2.3, 2.4). Since the first appearance of the energy conditions,
the set of matter distributions/spacetimes that are considered physical has
been greatly enlarged. As a result, the only remaining local condition that is
believed to hold today is NEC. Fortunately, this last stronghold is known to
hold for all observed classical matter.

###  2.1 Energy Conditions

The energy conditions are coordinate-invariant restrictions on the Stress-
Energy Tensor. This invariance is enforced by requiring that the quantities
involved are scalar. The scalar quantities which are usually considered, are
contractions of $T_{\mu\nu}$ with timelike $t^{\mu}$ or null vectors
$k^{\mu}$. Typically, these conditions are discussed for the case of the
perfect fluid. For the purposes of this introductory chapter, we continue this
tradition. For a perfect fluid source, $T_{\mu\nu}$ is:

$T_{\mu\nu}=(\rho+p)U_{\nu}U_{\mu}+pg_{\mu\nu}\;.$ (2.1)

Where $U_{\mu}$ is the fluid four-velocity, $g_{\mu\nu}$ is the spacetime
metric, $\rho$ is the energy density and $p$ is the pressure in the $x^{i}$
direction ($i=1,3,4$). Without further ado, we present a list of classical,
local energy condition following Carroll [7]

######  Definition 2.1.

Null Energy Condition: $T_{\mu\nu}k^{\mu}k^{\nu}\geq 0$ for all null vectors
$k^{\mu}$.

As we shall see shortly, we started with the weakest (in the sense that it is
implied by all the others) among these local energy conditions. However, our
intention is to discuss violations of these conditions. So the less
restrictive the condition is, the harder it is to violate. Hence, NEC is the
hardest to violate, which justifies this seemingly backward order.

For a perfect fluid, NEC translates to $\rho+p\geq 0$, and is implied by the
stronger Weak Energy Condition.

######  Definition 2.2.

Weak Energy Condition (WEC): $T_{\mu\nu}t^{\mu}t^{\nu}\geq 0$ for all timelike
vectors $t^{\mu}$.

When $p$ is isotropic, WEC holds for all timelike vectors if
$T_{\mu\nu}U^{\mu}U^{\nu}\geq 0$ and $T_{\mu\nu}k^{\mu}k^{\nu}\geq 0$ for some
null vector. For a perfect fluid:

$T_{\mu\nu}U^{\mu}U^{\nu}=\rho,$ (2.2)

$T_{\mu\nu}U^{\mu}U^{\nu}=(\rho+p)(U_{\mu}k^{\mu})^{2}.$ (2.3)

WEC implies $\rho\geq 0$ and $\rho+p\geq 0$. When is worried about non-
physical matter, WEC looks like the first thing he/she should consider since
it straightforwardly excludes negative energy from all points in spacetime.
However, as we mentioned, there are matter distributions of spacetimes we
consider physical that do violate it. A renown example is AdS spacetime, which
is filled with negative energy222One need not worry about AdS violating NEC
since the negative energy density is compensated by positive pressure..

######  Definition 2.3.

Dominant Energy Condition (DEC) : $T_{\mu\nu}t^{\mu}t^{\nu}\geq 0$ (WEC) and
$T_{\mu\nu}T^{\nu}_{\phantom{\lambda}\lambda}t^{\mu}t^{\lambda}\leq 0$, for
all timelike vectors $t^{\mu}$.

The second part of the definition is the requirement that $T_{\mu\nu}t^{\mu}$
is a non-spacelike vector. For a prefect fluid, this translates to
$\rho\geq|p\rvert$, i.e. the energy density must be non-negative and greater
or equal than the magnitude of the pressure.

######  Definition 2.4.

Null Dominant Energy Condition (NDEC) : $T_{\mu\nu}k^{\mu}k^{\nu}\geq 0$ (NEC)
and $T_{\mu\nu}T^{\nu}_{\phantom{\lambda}\lambda}k^{\mu}k^{\lambda}\leq 0$,
for all null vectors $k^{\mu}$.

The second part of the definition is the requirement that $T_{\mu\nu}k^{\mu}$
is a non-spacelike vector. NDEC excludes all sources excluded by DEC, apart
from those with negative vacuum energy, i.e. $\rho=-p$.

######  Definition 2.5.

Strong Dominant Energy Condition (SEC) :
$T_{\mu\nu}t^{\mu}t^{\nu}\geq\frac{1}{2}T^{\lambda}_{\phantom{\lambda}\lambda}t^{\sigma}t_{\sigma}$,
for all timelike vectors $t^{\mu}$.

This implies that $\rho+p\geq 0$ and $\rho+3p\geq 0$. In this case SEC implies
NEC and restricts negative pressure from being too large. We should point out
that these conditions are very strong, in the sense that they are local. This
means that they should hold for every point in spacetime and are, thus, far
more restricting than their global counterparts, the Averaged energy
conditions, which we shall discuss in a later chapter (ch. 5.2).

###  2.2 Raychaudhuri Equations

Suppose that one wants to follow the evolution of a collection of bodies
(particles) falling under their own gravity. If no interaction is assumed,
he/she would expect them to attract and tend to converge. A way to formulate
this evolution in the context of general relativity is by the use of the
Raychaudhuri Equation [4]. For the following discussion one needs to define
the notion of a congruence:

######  Definition 2.6.

Congruence: a set of curves in an open region of spacetime, such that every
point in the region lies on precisely one curve.

If the set of curves is restricted to that of geodesics, then one obtains a
geodesic congruence. It is convenient to think of a geodesic congruence as a
set of paths followed by non-interacting particles moving through spacetime.
If their geodesics cross, a caustic is formed, i.e. the congruence comes to an
end.

The starting point of the derivation is defining a vector field
$U^{\mu}=dx^{\mu}/d\tau$, tangent to a timelike geodesic congruence. Then, one
has to define a separation vector $V^{\mu}$, which points from one geodesic of
the congruence to another in its vicinity. For a general congruence this
vector obeys:

$\frac{DV^{\mu}}{d\tau}\equiv
U^{\nu}\nabla_{\nu}V^{\mu}=B^{\mu}_{\phantom{\nu}\nu}V^{\mu},\quad
B^{\mu}_{\phantom{\nu}\nu}=\nabla_{\nu}U^{\mu}.$

Hence, $B^{\mu}_{\phantom{\nu}\nu}$ tells as how much $V^{\mu}$ fails to be
parallelly transported along the congruence. In other words, it describes how
far the geodesics are from being parallel. The picture (fig. 1) one should
have in mind, is a sphere of test particles centered at a point. At the end of
the day, what one wants to describe is the evolution of the particles with
respect to their central geodesic. A quantitative description of this
evolution can be obtained by following solely the behavior of
$B^{\mu}_{\phantom{\nu}\nu}$.

Figure 1:  (Taken from [76]) A 2-D picture of the geodesic sphere. k is a null
vector along a null geodesic, and the green curves portray both the focusing
and shearing of the null congruence.

At each point $p$ a subspace of the tangent space $T_{p}M$ is considered,
which corresponds to vectors normal to the timelike vector field $U^{\mu}$.
Any vector can be projected to this subspace with the help of a projection
tensor
$P^{\mu}_{\phantom{\nu}\nu}=\delta^{\mu}_{\phantom{\nu}\nu}+U^{\mu}U_{\nu}$.
It is easy to see, using $U^{\mu}U_{\mu}=-1$ (timelike norm) and
$U^{\mu}\nabla_{\mu}U^{\nu}=0$ (geodesic), that $B^{\mu}_{\phantom{\nu}\nu}$
belongs to $T_{p}M$, i.e. $U^{\mu}B_{\mu\nu}=U^{\mu}\nabla_{\nu}U^{\nu}=0$ and
$U^{\nu}B_{\mu\nu}=U^{\nu}\nabla_{\nu}U^{\nu}=0$. At this point, it is useful
to decompose $B_{\mu\nu}$ into a symmetric and an anti-symmetric part
$B_{\mu\nu}=B_{(\mu\nu)}+B_{[\mu\nu]}$. $B_{(\mu\nu)}$ can be further
decomposed into a trace and a trace-free part. Finally one should notice that
since $B_{\mu\nu}$ belongs to the subspace of vectors normal to $U^{\mu}$, the
projection tensor operator can be used to take traces. When the dust settles,
one has:

$B_{\mu\nu}=\frac{1}{3}\theta P_{\mu\nu}+\sigma_{\mu\nu}+\omega_{\mu\nu}\;.$

The quantity of interest here is the scalar $\theta$ (trace of $B_{\mu\nu}$)
which is the expansion of the congruence, or in our sphere picture, the
expansion of the sphere with center the geodesic we considered the deviation
from. $\sigma_{\mu\nu}$ is the shear tensor. It describes the rate by which
the geodesic ball shears into an ellipsoid without any change in volume, and
it is a symmetric (0,2) tensor. This symmetry represents the fact that our
sphere gets distorted in a symmetric way. The last constituent of the equation
is the rotation/vorticity, again without any change in the volume of the
sphere, and it is the anti-symmetric part of $B$. The anti-symmetry accounts
for the usual clockwise and anti-clockwise options.

Finally, we can write down an equation which describes the evolution of our
congruence. To obtain it, we take the covariant derivative of $B_{\mu\nu}$
along $U^{\mu}$:

$\frac{DB_{\mu\nu}}{d\tau}\equiv
U^{\lambda}\nabla_{\lambda}=-B^{\lambda}_{\phantom{\nu}\nu}B_{\mu\lambda}-R_{\sigma\mu\nu\lambda}U^{\sigma}U^{\lambda}.$
(2.4)

Taking the trace of (2.4) we obtain the Raychaudhuri equation for the
expansion:

$\frac{d\theta}{d\tau}=-\frac{1}{3}\theta^{2}-\sigma_{\mu\nu}\sigma^{\mu\nu}+\omega_{\mu\nu}\omega^{\mu\nu}-R_{\mu\nu}U^{\mu}U^{\nu}.$
(2.5)

####  2.2.1 Raychaudhuri Equation for Null Geodesics

Since we are interested in studying NEC, the relevant process for us is the
evolution of null congruences. Hence here, we derive the Raychaudhuri equation
for such congruences, which at first sight seems incompatible with the above
derivation. The problem one faces is that since now the tangent vector
$K^{\mu}=\frac{dx^{\mu}}{d\lambda}$ is a null vector, the subspace of vectors
normal to it will contain itself. What we should study in this case is the
evolution of spatial vectors in a two-dimensional space normal to $K^{\mu}$;
since there is no unique way to define this subspace, we will need to define
an auxiliary null vector which points in the opposite direction and hope that
it will not show up in our final equation. The vector is chosen so that it has
the following normalization:

$l^{\mu}l_{\mu}=0,\leavevmode\nobreak\ l^{\mu}K_{\mu}=-1.$ (2.6)

Next, we write the projection operator333 metric on the tangent space to our
vectors such that it kills $K^{\mu}$ and $l^{\mu}$, and projects everything
else onto the tangent subspace to these vectors:

$h_{\mu\nu}=g_{\mu\nu}+K_{\mu}l_{\nu}+K_{\nu}l_{\mu}\;.$ (2.7)

Again the failure of the deviation vector $V_{\mu}$ to be parallelly
transported, is described by a tensor
$B^{\mu}_{\phantom{\nu}\nu}=\nabla_{\nu}K^{\mu}$. However, the said tensor
contains too much information for our purposes, so the quantity we consider
is:

$B^{\prime\mu}_{\phantom{\nu}\nu}=h^{\mu}_{\phantom{\alpha}\alpha}h^{\beta}_{\phantom{\nu}\nu}B^{\alpha}_{\phantom{\beta}\beta}\;.$
(2.8)

Finally, to obtain the Raychaudhuri equation, we proceed as in the timelike
case:

$\frac{DB^{\prime}_{\mu\nu}}{d\tau}\equiv
K^{\lambda}\nabla_{\lambda}=-B^{\lambda}_{\phantom{\nu}\nu}B_{\mu\lambda}-h^{\alpha}_{\phantom{\mu}\mu}h^{\beta}_{\phantom{\nu}\nu}R_{\sigma\alpha\beta\lambda}U^{\sigma}U^{\lambda}.$
(2.9)

After taking the trace of (2.9), we obtains:

$\frac{d\theta^{\prime}}{d\lambda}=-\frac{1}{2}\theta^{\prime
2}-\sigma^{\prime}_{\mu\nu}\sigma^{\prime\mu\nu}+\omega^{\prime}_{\mu\nu}\omega^{\prime\mu\nu}-R_{\mu\nu}K^{\mu}K^{\nu},$
(2.10)

which is the desired equation that describes the evolution of the null
expansion. Finally we note that our expression is indeed independent of
$l_{\mu}$, as required.

Figure 2: (Taken from [25])Causal diamond on a 4-D Minkowski spacetime, where
one spatial dimension has been omitted: starting with a null congruence B on a
fixed time slice, we draw the light-cones at each point. Four null
hypersurfaces arise, from which $F_{1}$ and $F_{3}$ form the causal diamond.
The expansion of B along hypersurface $F_{1}$ is negative (ingoing rays) and
along $F_{2}$ positive (outgoing rays).

####  2.2.2 The focusing Theorem

We are free to choose the vector $U^{\mu}$ to be hypersurface orthogonal, such
that $\omega_{\mu\nu}=0$. $\sigma^{\prime}_{\mu\nu}\sigma^{\prime\mu\nu}\geq
0$ since it is a spatial tensor. Notice that we have gotten this far without
the using the Einstein equation, so up to this point, the derivation was very
general. To extract results on how such a congruence should behave in the
presence of gravity, we now make use of the Einstein equations:

$G_{\mu\nu}=8\pi G_{N}T_{\mu\nu}\;.$ (2.11)

And after rewriting them in a form that suits our purposes, we contract with
the null vector:

$\begin{split}R_{\mu\nu}&=8\pi
G(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu})K^{\mu}K^{\nu}\\\ &=8\pi
G_{N}T_{\mu\nu}\;.\end{split}$ (2.12)

By imposing NEC $T_{\mu\nu}k^{\mu}k^{\nu}\geq 0$, and plugging everything back
to (2.11), we obtain:

$\frac{d\theta}{d\lambda}=-\frac{1}{2}\theta^{2}-\sigma_{\mu\nu}\sigma^{\mu\nu}-T_{\mu\nu}K^{\mu}K^{\nu}\leq
0.$ (2.13)

What above result tells us, is that the rate by which the expansion of our
congruence changes is always negative. If we go back to our sphere picture and
think about what happens to the null rays emanating from it, we will see that
they always try to focus. If we start with $\theta\leq 0$, then the light rays
will keep focusing. If we start with $\theta>0$, then the congruence will
either become stationary ($\theta=0$) and then start focusing or will reach
infinity before that happens (depending on our spacetime). This is what we
would normally expect from gravity to do. Thus here, the NEC translates into
the statement that gravity is attractive. It is useful to formulate the above
observations into a theorem:

######  Theorem 2.1.

The classical (Null) focusing theorem: combining the Raychaudhuri equation,
the Einstein equation and NEC we obtain that the evolution of the expansion
for a null congruence is always
$\frac{d\theta}{d\lambda}\equiv\theta^{\prime}\leq 0$, thus null rays always
tend to focus.

###  2.3 Penrose Singularity Theorem

The Penrose Singularity Theorem is regarded by many as one of the first
genuine post-Einsteinian results of GR. It was derived by R. Penrose in 1965
[18], the same year the CMB was measured, miraculously 50 years after the
derivation of the Einstein equations. The original purpose of Penrose was to
prove that deviations from spherical symmetry are not able to prevent the
formation of singularities, as previously suggested by Oppenheimer [16].

To achieve this goal, Penrose introduced the concepts of incompleteness and
closed trapped surface, which now play a central role in the relativistic
world. Without further ado, let us define these quantities, and introduce the
theorem. In the end of the section, we provide some physical intuition as to
what it means.

The first thing needed for the proof is a sharp definition of a singularity.
There are well-known examples in which the most obvious definition as “the
point where some curvature scalar diverges”, fails444Examples are the Taub-NUT
singularities, which are formed by incomplete geodesics spiraling infinitely
around a topologically closed spatial dimension, while all curvature scalars
are finite [26].. A universal definition is the following:

######  Definition 2.7.

A singularity in a Lorentzian manifold is an incomplete, endless curve [24].

Nearly all singularity theorems, after the Penrose one, prove the existence of
“geodesic incompleteness”. Another crucial notion is that of a closed trapped
surface. Imagine that you have a 2+1 dimensional spacetime. Pick a spacelike
surface (Cauchy slice), draw a circle (congruence) and the light-cones at each
point of the circle. In your drawing, there are four null hyper-surfaces
emanating from the circle, two towards the future and two towards the past.
Drawing another spacelike surface, you can follow the evolution of the circle
in time. On one of each of these future/past null surfaces the circle gets
smaller and on the other is getting bigger (assuming that spacetime is
stationary). A null future/past-trapped surface is a circle, which contracts
on both future/past hyper-surfaces. This is exactly what happens on every null
congruence inside a Schwarzschild horizon. An intuitive example of this
behavior is described in the following fig. 3 .

Figure 3:  (Taken from [24]) The evolution of a Null congruence in 4-D
spacetimes, where one spatial dimension is omitted: the first diagram depicts
this evolution for a stationary spacetime. An initial null congruence (yellow
circle) will either expand (blue circle) or contract (orange circle),
depending on whether light is ingoing or outgoing (fig. 2). However, if
spacetime is contracting fast enough, the congruence on a later slice will
contract in both the ingoing and outgoing case.

A closed trapped surface is in addition compact, without boundary. Having
acquired the required knowledge, we present the Penrose Singularity Theorem:

######  Theorem 2.2.

If the space-time contains a non-compact Cauchy hyper-surface $\Sigma$ and a
closed future-trapped surface, and if the Null Energy Condition holds for null
$K^{\mu}$, then there are future incomplete null geodesics.

The assumption of having a well-defined Cauchy evolution problem, goes also by
the name global hyperbolicity. It is the requirement that there are no
timelike nor null closed curves and that spacetime does not have any holes
through which information can go in and out off. Furthermore, one can
interpret the non-compactness as the requirement that space is infinite.

Using this new intuition and the focusing theorem (th. 2.1) implied by NEC, it
is almost already clear that a singularity, in the Penrose sense, will form.
We know that a closed future-trapped surface will form eventually, and the
only problem we need to solve is that our geodesics may cross (form a caustic)
instead of terminating on a singularity. This problem is solved by assuming
global hyperbolicity, which implies that the topology of spacetime cannot
change. Thus, at least one geodesic terminates and cannot be extended any
further.

###  2.4 Other GR results

In previous section (sec. 2.3), we focused on explaining the Penrose
Singularity Theorem and described the role of NEC in its formulation. To make
the importance of NEC even more pronounced, we cite a detailed but still not
exhaustive list of results in GR, for which NEC is a central assumption [28].

  1. 1.

Singularity formation, after a gravitational collapse, in spatially open
spacetimes [18].

  2. 2.

Singularity formation, in asymptotically flat spacetimes with a non-simply
connected Cauchy surface [29] [30].

  3. 3.

Event horizon formation, after gravitational collapse [18] [19].

  4. 4.

Apparent horizons, trapped and marginally trapped surfaces must reside in
asymptotically flat black holes [20].

  5. 5.

Second Law of black-hole mechanics [31].

  6. 6.

Second Law of generalized black-hole mechanics

  7. 7.

Inability of asymptotically predictable black holes to bifurcate [20].

  8. 8.

The domain of outer communication of a stationary, asymptotically flat
spacetime is simply connected, if the domain is globally hyperbolic [33] [35].

  9. 9.

A stationary, asymptotically flat black hole has topology $\mathcal{S}^{2}$,
if the domain of outer communication is globally hyperbolic and the closure of
the black hole is compact [34] [33].

  10. 10.

Most formulations of the black hole “no Hair” theorem for asymptotically flat
black holes [36], [37], [21], [22], [38], [39], [40], [41], [42], [43], [44],
[45].

  11. 11.

Generalized black holes are regions of “no escape” [46].

  12. 12.

Limits on energy extraction by gravitational radiation from colliding
asymptotically flat black holes [31].

  13. 13.

The Generalized Second Law of Thermodynamics [47].

  14. 14.

Bousso’s covariant entropy bound [47].

  15. 15.

The Shapiro “time-delay” is always a delay, never an advance [48].

  16. 16.

Classical Chronology Protection Conjecture [49].

##  3 Wormholes

There is a vast amount of literature on wormholes, from science fiction books
and comics to rigorous mathematical papers. In principle, the existence of
these objects does not lead to problems in physics, nor to strange time
paradoxes. That is because in most cases wormholes are not traversable. For
example, a specific class of non-traversable wormholes the are the Einstein
Rosen (ER) bridges, which can be found in the simplest of spacetimes (sec.
3.1). However, when wormholes can be traversed, even by a single light signal,
things start to get out of hand. Causality violations, time travel and all the
paradoxes that come with them, escape the Pandora’s Box that is traversable
wormholes, threatening to overthrow our current physical understanding of
nature.

###  3.1 Wormhole Traversability

The first people to our knowledge to rigorously study traversable wormhole
spacetimes, were Morris and Thorne [11]. In their work, they start by
discussing several already existing solutions to the Einstein field equations
that contained wormholes and argue against their use for interstellar travel.
Furthermore, they discuss the type of properties the solution must have so
that a “traveler” can have a pleasant trip and introduced their own solutions,
whose only caveat is the “Exotic” (NEC violating) stress energy tensor.
Although it is known that classically, the existence of traversable wormholes
is prevented by the NEC, the non-compliance of QFT to NEC allows for their
solution to be realized in the semi-classical gravity regime. In this project,
we are less ambitious than our predecessors, in the sense that we are not
studying wormholes for the purposes of human interstellar travel. We settle
with studying the simple case of light. In other words, we investigate the
conditions for super-luminal messaging through the wormhole. Hereon, we follow
the structure of their discussion, reformulated in a way that fits the needs
of our project. The first step, is to define what we mean by a traversable
wormhole:

######  Definition 3.1.

Traversable Wormhole: a Wormhole spacetime through which light signal can be
sent back and forth.

####  3.1.1 Schwarzschild Non-Traversable Wormhole

First, we present a surprising example of a non-traversable wormhole. The
wormhole which exists in the maximally extended Schwarzschild spacetime. Here,
we briefly explain a few relevant to the discussion features of the said
spacetime:

$ds^{2}=-(1-\frac{2G_{N}M}{r})dt^{2}+(1-\frac{2G_{N}M}{r})^{-1}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}),$ (3.1)

where $M$ is the mass of the Black Hole and $r$ is the radial distance from
its center. This is the unique, spherically symmetric, static, vacuum solution
of the Einstein equations, which has a Horizon (outer-trapped surface) at
$r=2G_{N}M$ and a singularity
($R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}=\infty$) at $r=0$. We use the
Kruskal–Szekeres coordinates, which have the advantage of covering the entire
spacetime manifold of the maximally extended Schwarzschild solution, and are
well-behaved everywhere outside the physical singularity $r=0$. In these
coordinates, the metric takes the form:

$ds^{2}=\frac{32G_{N}^{3}M^{3}}{r}(-dT^{2}+dR^{2})+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}),$ (3.2)

where the new coordinates are:

$T=\pm(\frac{r}{2G_{N}M}-1)e^{\frac{r}{4G_{N}M}}\mbox{sinh}(\frac{t}{4G_{N}M}),$
(3.3)

$R=\pm(\frac{r}{2G_{N}M}-1)e^{\frac{r}{4G_{N}M}}\mbox{cosh}(\frac{t}{4G_{N}M}).$
(3.4)

The minus sign describes the interior $0<r<2G_{N}M$, the plus the outside
$2G_{N}M<r$ and the event horizon sits at $T=\pm R$.

Figure 4:  (Taken from [7])In these diagram T is the vertical and R the
horizontal axis. Because the angular directions are suppressed, one should
think of each point on this diagram as a 2-sphere of radius the value of
coordinate $r$ at that point.

To see the wormhole, we slice the Kruskal diagram (fig. 4) in spacelike slices
of constant $T$, and draw a picture of each slice (fig. 5) restoring the
angular coordinates. The pictures demonstrate the two asymptotically flat
regions I and III, which are initially disconnected at the white hole’s anti-
horizon, then they connect through a wormhole of maximal radius $2G_{N}M$, and
finally become completely disconnected.

Figure 5: (Taken from [7]) The spacelike slices of (fig. 4); Snap-shots of
space, which show the formation and vanishing of the Schwarzschild dynamical
wormhole.

The first who noticed this “dynamical wormhole” of Schwarzschild spacetime,
was Ludwig Flamm [17], only a year after Einstein formulated his field
equations. Although it is intriguing that a wormhole shows up already in one
of the plainest and well-studied GR’s solutions, the connection between
spacetimes I and III is spacelike. Thus, there is neither a null nor a
timelike path that connects them, which means that the Schwarzschild wormhole
is not traversable.

####  3.1.2 Ellis Traversable Wormhole

Although we discussed only the Schwarzschild case, there are many more
examples of spacetimes which contain non-traversable wormholes. Thus, it is
more prudent to introduce a list of desired properties the wormhole spacetime
should have, to be considered traversable. Following [11], we add some
auxiliary properties that make the solution sufficiently easy to handle:

1) The spacetime metric should be spherically symmetric and static. This is
purely for practical reason since having this amount of symmetry will greatly
simplify our calculations.

2) The solution must obey everywhere the Einstein equations

3) The solution must have a throat that connects two asymptotically flat
regions of spacetime. Asymptotic flatness can, and has been relaxed, in
several solutions of this form [52].

4) There should be no horizon, since it will prevent a two-way communication.

5) The solution should be perturbatively stable, meaning that it will not
collapse when our photons try to cross its throat. The stability the wormholes
defined here and more, has been studied in [50], [53], [54].

6) The photons must be able to go through the wormhole’s throat and back, in a
reasonable amount of time. More specifically, the proper time it takes for the
signal to reach an observer, measured by that observer, must be small.

7) The curvature should be low, compared to the Planck scale. High curvatures,
as we discuss in sec. 6.1, signal the breakdown of the semiclassical gravity
approximation. If that happens, one needs to consider Quantum Gravity effects
which would push the discussion beyond the scope of this work.

8) There must be no singularity in our solution. Although this might seem a
condition similar to the above, finiteness of the curvature scalars everywhere
is only part of this restriction, as discussed in def. 2.7.

Now that we have our criteria, we are ready to write down a metric that
satisfies them. To our knowledge, the following metric is the simplest
traversable wormhole metric one can consider:

$ds^{2}=-dt^{2}+dl^{2}+(b^{2}_{0}+l^{2})(d\theta^{2}+sin^{2}\theta
d\phi^{2}).$ (3.5)

This solution’s name is Ellis Wormhole, and it is a special solution of the
Ellis drain-hole [15]. The coordinates have the following ranges:
$-\infty<t<+\infty$, $-\infty<l<+\infty$, $0\leq\theta\leq\pi$, $0\leq\phi\leq
2\pi$ and $b_{0}$ is a constant. $t$ is the proper time a static observer
measures, $\theta,\phi$ are the angular spherical coordinates, and $l$ is the
proper radial distance at a fixed $t$. As required the spacetime is static,
spherically symmetric and does not have horizons or singularities. The
asymptotically flat regions lie at $l=\infty$ and $l=-\infty$.

If one wishes to obtain some visual confirmation of the wormhole in this
spacetime, he/she can fix time and the $\theta$ angular coordinate and embed
this 2-D spacelike slice in a 3-D Euclidean space. The resulting diagram is
shown in fig. 6.

Figure 6: (Taken from [11]) A wormhole connecting two asymptotically flat
regions of spacetime. The diagram depicts this geometry for constant time
$t=t_{0}$ and $\theta=\pi/2$

Finally, we study the most relevant quantity to this work; the stress energy
tensor which corresponds to this spacetime. Since we have the metric of the
manifold, its computation is a rather trivial matter. We first compute the
Einstein tensor $G_{\mu\nu}$, and then use the Einstein equations to obtain
$T_{\mu\nu}$. The components of the Einstein tensor after the usual simple but
long calculation, are the following:

$\begin{split}G_{tt}&=-\frac{b^{2}_{0}}{(b^{2}_{0}+l^{2})^{2}}\;,\\\
G_{ll}&=-\frac{b^{2}_{0}}{(b^{2}_{0}+l^{2})^{2}}\;,\\\
G_{\theta\theta}&=\frac{b^{2}_{0}}{(b^{2}_{0}+l^{2})}\;,\\\
G_{\phi\phi}&=\frac{b^{2}_{0}sin^{2}\theta}{(b^{2}_{0}+l^{2})}\;.\end{split}$
(3.6)

Notice, that the corresponding $T_{\mu\nu}$ has some strange properties.
Namely:

$\begin{split}T_{tt}&=-4\pi G_{N}\frac{b^{2}_{0}}{(b^{2}_{0}+l^{2})^{2}}\;,\\\
T_{ll}&=-4\pi G_{N}\frac{b^{2}_{0}}{(b^{2}_{0}+l^{2})^{2}}\;,\end{split}$
(3.7)

have negative signs. In sec. 2.1 we saw that NEC requires
$T_{\mu\nu}K^{\mu}K^{\nu}\geq 0$, but one can see that for a
$K^{\mu}=(1,1,0,0)$ (in the $l$ direction) $T_{\mu\nu}K^{\mu}K^{\nu}\leq 0$.
We also mentioned that reasonable, physical classical matter satisfies this
condition. In light of this unusual nature, this type of matter content was
coined by Morris and Thorne “exotic”. The authors of [11] cite one of their
discussions with Don Page, where it is argued that any wormhole spacetime must
contain such exotic matter in order to be traversable. This co-dependence of
exotic matter and traversability takes the form of a condition, which goes by
the name flaring-out condition. To honor the legacy of [11], we keep this
name, although our definition of the condition is slightly different.

###  3.2 The Null Flaring-out Condition

To see how the traversability of wormhole spacetimes restricts $T_{\mu\nu}$,
we use the Focusing Theorem (th. 2.1). We follow the evolution of a null
congruence through the wormhole geometry depicted in fig. 6. The light rays on
this diagram travel in the $l$ direction of the co-dimension one surface which
includes the throat. Light rays start their journey through the throat as
ingoing, following a hypersurface on which the expansion scalar is negative.
However, the geometry of the surface requires that the expansion at the throat
becomes zero, and then starts to increase; otherwise, the light rays would
never escape. Hence, the light rays need to defocus, i.e. the expansion scalar
needs to be positive555 Bear in mind that no caustics form on the throat, so
the congruence has no reason to end..

######  Definition 3.2.

Flaring-out Condition: Any traversable wormhole spacetime requires an increase
of the expansion scalar: $\frac{d\theta}{d\lambda}>0$.

This behavior clearly violates the focusing theorem, which states that
$\frac{d\theta}{d\lambda}\leq 0$ should always hold. As a consequence, a
violation of NEC is implied. To show this, we use (2.13) :

$\frac{d\theta}{d\lambda}=-\frac{1}{2}\theta^{2}-\sigma_{\mu\nu}\sigma^{\mu\nu}-T_{\mu\nu}K^{\mu}K^{\nu}\leq
0.$ (3.8)

The evolution of the expansion can become positive if one of the above
quantities becomes positive enough. Since the only quantity that is not a
square of a real number is $T_{\mu\nu}K^{\mu}K^{\nu}$, only this term is
capable of changing the overall sign of the r.h.s. of (3.8). But NEC tells us
that $T_{\mu\nu}K^{\mu}K^{\nu}\geq 0$, hence the traversability of wormholes
requires the existence of NEC violating matter. This, as we saw, is true for
our simple example of the Ellis wormhole, and for all traversable classical
and semi-classical wormhole solutions we have seen in literature.

As we already mentioned, this exotic $T_{\mu\nu}$ can be generated by a number
of otherwise well-behaved states in QFT. Only, in this case, the classical
Einstein equations cannot be used anymore, but a semiclassical version of
them, in which the l.h.s. is the usual, fixed spacetime geometry described by
$G_{\mu\nu}$ and the r.h.s is a quantum matter source given by $\langle
T_{\mu\nu}\rangle$.

##  4 Violations of NEC in QFT

In this chapter, we discuss violations of the NEC in QFT. As we have already
mentioned, if one wants to study such violations, the first problem he/she
encounters is to define a version of NEC in the context of QFT (sec. 4.1).
Thereafter, we study some explicit examples in which these violations occur
(sec. 4.2, 4.3, 4.4), and discuss their implications. The acute reader might
have already guessed what fundamental feature of QFT leads to her rebellious
behavior; nonetheless, we leave this discussion for later.

###  4.1 The form of NEC in QFT

First, one should notice that the $T_{\mu\nu}$ that appears in a QFT, is not
the same object that appears in the classical Einstein equations. In quantum
theories, the fundamental objects that carry the information about the state
of our system are wave-functions/states. The way to extract information about
any observable is to act on these states with the corresponding operators.

In QFT, things are a bit more subtle, but the main idea is the same: if we
want $T_{\mu\nu}$ to make sense, we need to promote it to an operator, while
if we want to extract the observable quantities it describes, we need to act
with it on a physical state. More explicitly, when the operator acts on a
state which is its eigenstate, the result is the eigenstate times its
eigenvalue, which we then read off as the value of the observable. So what
happens if the state we consider is not an eigenstate of the operator we are
interested in? Any state can be decomposed in the basis of eigenvectors of our
operator. The action of the operator will give us the sum of these superposed
eigenstates, each multiplied by its energy eigenvalue times a weight
(c-number). This should come as no surprise to us, since it is a direct
manifestation of the quantum nature of the theory.

Now, if one is determined to extract a single value, he can take the
expectation value of the operator in this state. This is essentially the above
sum with the following differences: a) the squared weights appear and they
correspond to the probability with which each eigenvalue occurs, b) since the
basis of eigenvectors is orthonormal, cross-terms vanish and diagonal terms
give the identity, leaving us with a sum of pure numbers.

So every expectation value is an implicit sum of all possible eigenvalues of
the operator, each multiplied by the appropriate probability. Clearly, it is
uncertain if one should trust this “average” as the value of the observable he
wanted to compute. An example in which one could imagine that this quantity
makes sense, is a state for which the probability distribution of the
eigenvalues is sharply peaked around a single value. Then, one would have good
reason to believe that the leading contribution is indeed given by that
eigenvalue of the operator666In later chapter, we will see that not only the
sharpness but the overall form of the probability distribution function
matters. For example for a function with a long tail with no cut-off, a big
contribution to the expectation value may come from some very big (compared to
the mean) eigenvalue at the end of the tail (heavy tail)..

An obvious counter-example is a state in which the probability is evenly
distributed among two eigenvalues, and thus the expectation value is not close
to an actual eigenvalue of the operator, but some intermediate value. We shall
see that the distinction between the two cases is connected to the fluctuation
of the operator we are interested in, and more accurately to its variance. The
phenomenon which gives rise to this “in between” expectation value is quantum
interference. This phenomenon is a central feature of quantum theories and
deserves a more detailed discussion to which we turn to in the chapter of
semiclassical gravity (ch. 6). For now, let us define the form of a naive
Quantum Null Energy Condition, whose violations we will discuss in this
chapter:

$\langle\psi\rvert T_{\mu\nu}\lvert\psi\rangle K^{\mu}K^{\nu}\geq 0.$ (4.1)

The quantities involved in the above equation are the expectation value of
$\hat{T}_{\mu\nu}$ for some state $\psi$ and a null vector $K^{\mu}$. It is a
well-known fact that the expectation value of $\hat{T}_{\mu\nu}$ in QFT is a
divergent, state-dependent object. Thus, one should regulate
$\hat{T}_{\mu\nu}$ before computing $\langle\psi\rvert
T_{\mu\nu}\lvert\psi\rangle$ and contracting it with the null vector.

###  4.2 1+1 CFT

For the following discussion, we suggest that the reader unfamiliar with
Conformal Field Theories(CFT’s) takes a look at the introductory [9] notes
or/and the book [10]. We will introduce the tools that we shall use in our
calculations shortly, but first some linguistics. By the word operator in CFT,
we mean a one or more fields, e.g. $\phi$ and $\phi\phi$, or some more
complicated expression like $T_{\mu\nu}$. Moreover, since there is an
operator-state correspondence, we may even refer to a whole state as a field
and vice versa. The reader will also encounter the terms primary and first
descendant. For the purposes of this chapter, he/she should think of them as
nothing more than a ground state and the first excited state of the theory.
1+1 CFT gives as the remarkable technology to construct a theory (Lagrangian)
independent expression for the $\hat{T}_{\mu\nu}$ in terms of the Virasoro
Generators, i.e. the generators of the conformal algebra.

Although the calculation we perform here is simple, the author of this thesis
does not know of any piece of literature which includes it. We have chosen to
do our calculations in the flat 1+1-D case for a non-interacting CFT since the
message we are trying to convey does not require further complexification.
Nevertheless, it is possible to generalize our results to higher dimensional
CFT’s, and curved backgrounds.

We start by picking a state that we believe will exhibit (Q)NEC violating
behavior. This state is a superposition of a primary field $O_{2}\lvert
0\rangle$ of conformal weight $h_{2}$ and its first descendant $O_{1}\lvert
0\rangle$. As we already discussed, the descendant of a primary field/state is
created by the action: $O_{1}\lvert 0\rangle=L_{-1}O_{2}\lvert 0\rangle$. The
route we choose for our calculation is the following:

1) We compute the expectation value of the holomorphic part of
$\hat{T}_{\mu\nu}$ on the Euclidean plane.

2) We conformally transform it to a Euclidean cylinder.

3) We Wick rotate to the Lorentzian signature, to obtain the final expression
for $\hat{T}_{\mu\nu}$.

As we mentioned, the holomorphic and anti-holomorphic part of $T_{\mu\nu}$ on
the circle can be written in terms of Virasoro generators, a fact which
greatly simplifies our calculations. The expressions are:

$T_{zz}=\sum_{n=-\infty}^{\infty}L_{n}z^{-n-2}\;,$ (4.2)

$\widetilde{T}_{\widetilde{z}\widetilde{z}}=\sum_{n=-\infty}^{\infty}\widetilde{L}_{n}\widetilde{z}^{-n-2}\;.$
(4.3)

where the terms with tildes in the expression are the anti-holomorphic ones.
For the computation we also need the Virasoro commutation relations:

$[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c}{12}(m^{3}-m)\delta_{m+n,0}\;.$ (4.4)

The state $O_{2}\lvert 0\rangle=\lvert O_{2}\rangle$ is normalized, i.e.
$\langle O_{2}\lvert O_{2}\rangle=1$, but the first excited state $\langle
O_{1}\lvert O_{1}\rangle=2h_{2}$ is not. We choose to normalize the overall
superposed state in the following way:

$N^{2}(\langle O_{1}\rvert-\langle O_{2}\rvert)(\lvert O_{1}\rangle-\lvert
O_{2}\rangle)=1\implies N=\frac{1}{(h_{2}+1)^{1/2}}\;.$ (4.5)

We should note here that although we have chosen to define the superposed
state with a minus relative sign between the primary and the descendant, a
positive relative sign would give an almost identical result. We can now
proceed with our computation of the $\langle\psi\rvert T_{zz}|\psi\rangle$:

$(\langle O_{1}\rvert-\langle O_{2}\rvert)T_{zz}(\lvert O_{1}\rangle-\lvert
O_{2}\rangle)=N^{2}\left[\left(3h_{2}+2h^{2}_{2}\right)z^{-2}-2h_{2}z^{-1}-2h_{2}z^{-3}\right]\;,$
(4.6)

$\left(\langle O_{1}\rvert-\langle
O_{2}\rvert\right)\widetilde{T}_{\widetilde{z}\widetilde{z}}\left(\lvert
O_{1}\rangle-\lvert
O_{2}\rangle\right)=N^{2}\left[\left(3\widetilde{h}_{2}+2\widetilde{h}^{2}_{2}\right)\widetilde{z}^{-2}-2\widetilde{h}_{2}\widetilde{z}^{-1}-2\widetilde{h}_{2}\widetilde{z}^{-3}\right]\;.$
(4.7)

Notice the generality of this result, a common feature of calculations in 1+1
CFT. We have not chosen a specific conformal weight yet, thus the above result
holds for any primary. All one has to do is to pick his/her favorite theory
plug in the conformal weight of his/her chosen primary and obtain $\langle
O_{1}\rvert-\langle O_{2}\rvert)T_{zz}(\lvert O_{1}\rangle-\lvert
O_{2}\rangle)$. However, we do not stop our calculation here, since we would
like the quantities described by the stress-energy tensor to be physical
observables and in Euclidean coordinates that is not always the case. To go
from the stress-energy tensor on the Euclidean plane to the one on the
conformally equivalent Euclidean cylinder, one has to perform a conformal
transformation. The stress-energy tensor transforms in the following way:

$\langle
T^{\prime}_{z^{\prime}z^{\prime}}(z^{\prime})\rangle=\left(\frac{\partial
z^{\prime}}{\partial
z}\right)^{-2}\left[T_{zz}(z)-\frac{c}{12}S(z^{\prime},z)\right].$ (4.8)

Where $z^{\prime}$ is the cylinder and $z$ the plane coordinates and
$S(z^{\prime},z)$ is the Schwartzian derivative. We are going to compute only
the holomorphic part, since the calculation for the anti-holomorphic part is
exactly the same. After computing the Schwartzian and the other derivatives,
we obtain:

$\langle T_{cyl}(z^{\prime})\rangle=-z^{2}\langle
T_{plane}(z)\rangle+\left\langle\frac{c}{24}\right\rangle.$ (4.9)

Plugging in the $T_{zz}$ from (4.6), we get the holomorphic part of the
stress-energy tensor on the cylinder:

$\langle
T_{z^{\prime}z^{\prime}}\rangle=N^{2}\left[\left(3h_{2}+2h^{2}_{2}\right)-2h_{2}z-2h_{2}z^{-1}\right]+\left\langle\frac{c}{24}\right\rangle.$
(4.10)

To obtain the desired Lorentzian component $T_{\tau\tau}$, we perform a Wick
rotation:

$\begin{split}\langle T_{\tau\tau}\rangle&=-\langle
T_{z^{\prime}z^{\prime}}\rangle-\langle\widetilde{T}_{\widetilde{z}^{\prime}\widetilde{z}^{\prime}}\rangle\\\
&=N^{2}\Big{[}\left(3h_{2}+2h^{2}_{2}+\mbox{anti.-hol.}\right)-\left(4h_{2}\cos(\sigma-\tau)+\mbox{anti.-hol.}\right)-\\\
&-\left\langle\frac{c}{24}\right\rangle-\left\langle\frac{c}{24}2h_{2}\right\rangle\Big{]}.\end{split}$
(4.11)

Similarly, we can compute the $\langle T_{\sigma\sigma}\rangle$ component. Our
result (4.11) allows us to calculate the expectation value of the energy
momentum tensor for the superposed state $N(\langle O_{1}\rvert-\langle
O_{2}\rvert)$ in any 1+1 CFT for any primary of conformal weight $h_{2}$. It
is easy to see that the maximum negative value of the $\langle
T_{\tau\tau}\rangle$ is reached when we are on the light-cone of Lorentzian
cylinder. For that value we plot $\langle T_{++}\rangle$ against $h_{2}$, in
other words, we see the flow of the expectation value of the SET in the space
of primaries:

Figure 7: $\langle T_{++}\rangle$ plotted against the conformal weight of
$\mathcal{O}_{2}$.

Another interesting plot is the one that describes how the $\langle
T_{++}\rangle$ component of the stress tensor evolves with $\sigma$ and $\tau$
for a specific primary. Our choice is the vertex operator of the free scalar
theory, whose conformal weight is $h_{2}\sim k^{2}$ where $k$ is the momentum.
From the above we see that for $h_{2}\geq 0.5$, the state is not NEC
violating, so we choose the intermediate value $h_{2}=0.2$:

Figure 8: $\langle T_{++}\rangle$, when the conformal weight of
$\mathcal{O}_{2}$ is $h_{2}=0.2$, plotted against $\sigma-\tau$ ($x_{-}$ in
light-cone coordinates).

Comment: An important feature of our result is that the negativity of the
expectation value emerges from an oscillatory term. If we wish to compute the
overall energy, we should integrate this quantity over the whole space, which
is the general procedure one follows to obtain a conserved charge (Energy)
from its conserved current ($T_{\mu\nu}$). The integration over the
oscillatory term will clearly yield zero, hence the result we will obtain will
be strictly positive. The locality of the effect is a hint that the NEC
violating behavior of these states is connected to interference.

###  4.3 Vacuum plus 2 particles

In this section, we turn to the free massless scalar field theory on 4-D
Minkowski spacetime. The state considered is a superposition of the vacuum
plus two modes with specified momentum k. The calculation of the expectation
value of the stress tensor in this state was first produced by L.H.Ford and
M.J. Pfenning [12]. A calculation we followed, with some slight differences
that we point out along the way.

####  4.3.1 Framework

We start by performing the usual quantization procedure. The action is:

$S=\int_{-\infty}^{\infty}(-g)^{1/2}d^{4}x\left(\frac{1}{2}\partial_{\lambda}\phi\partial^{\lambda}\phi\right)\;.$
(4.12)

It is customary to define the stress-energy tensor in this procedure as:

$T_{\mu\nu}=-2\frac{\delta\mathcal{S}_{matter}}{\sqrt[]{-g}\leavevmode\nobreak\
\delta g_{\mu\nu}}\;.$ (4.13)

Thus, for the Minkowski spacetime we get:

$T_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}\eta_{\mu\nu}\partial_{\lambda}\phi\partial^{\lambda}\phi\;.$
(4.14)

One difference between the Ford-Pfenning [12] calculation and ours is that we
will not consider the part proportional to the metric. This is simply because
at the end of the day we will contract the expectation value of the full
stress tensor twice with a null vector, thus the second term on the r.h.s. of
(4.14) will vanish.

The scalar $\phi(x)$ we considered is a function of spacetime and must be a
solution to the classical Equations Of Motion (EOM), obtained by variating the
above action w.r.t. it. It is easy to see that the EOM is simply the Klein-
Gordon equation:

$\partial_{\mu}\partial^{\mu}\phi(x)=0.$ (4.15)

A solution to this equation is the plane wave, thus we Fourier expand
$\phi(x)$ in terms of a complete set of plane wave mode function:

$f_{k}(x)=\left(2wL^{3}\right)^{-(1/2)}e^{i\textbf{k x }-iwt}.$ (4.16)

By $x$ we denote spacetime coordinates, by bold $\textbf{k},\textbf{x}$ space
coordinates and by $t$ the usual Minkowski time. The Fourier expansion of
$\phi(x)$ in terms of annihilation-creation operators and mode functions is:

$\phi(x)=\sum_{k}{}[a_{k}f_{k}(x)+a^{\dagger}_{k}f^{*}_{k}(x)].$ (4.17)

The next step is to require that $\hat{a},\hat{a}^{\dagger}$’s obey the usual
commutation relations:

$\begin{split}[\hat{a}_{k},\hat{a}^{\dagger}_{k^{\prime}}]&=\delta_{kk^{\prime}},\\\
[\hat{a}_{k},\hat{a}_{k^{\prime}}]&=0,\\\
[\hat{a}^{\dagger}_{k},\hat{a}^{\dagger}_{k^{\prime}}]&=0.\end{split}$ (4.18)

We also define the vacuum as the state annihilated by annihilation operators
$\hat{a}_{k}$ and the first excited state as the state created by the action
of one $\hat{a}^{\dagger}_{k}$ on the vacuum. In formulas:

$\begin{split}\hat{a}_{k}\lvert 0\rangle&=0,\\\ \hat{a}^{\dagger}_{k}\lvert
0\rangle&=\lvert 1_{k}\rangle.\end{split}$ (4.19)

It is also useful to define the action of $\hat{a},\hat{a}^{\dagger}$’s on a
state of the Fock space that includes $n$ modes with $k_{th}$ momentum, i.e.:

$\begin{split}\hat{a}^{\dagger}_{k}\lvert
n_{k}\rangle&=\sqrt{n+1}\lvert{n+1}_{k}\rangle,\\\ \hat{a}_{k}\lvert
n_{k}\rangle&=\sqrt{n}\lvert{n-1}_{k}\rangle.\end{split}$ (4.20)

The field $\phi$ gets promoted to an operator $\hat{\phi}$ with again the
usual commutation relations and we have thus quantized our theory. All that is
left to do, is plug $\hat{\phi}$ in the formula for the stress tensor (4.14),
which is then also promoted to an operator, and compute its expectation value
in the state of interest. We will compute the expectation value of each
component of $\hat{T}_{\mu\nu}$ separately so that the negative energy density
contribution becomes manifest from the start. From now on we drop the hats for
simplicity, bearing in mind that all $T_{\mu\nu}$ and
$a_{k},a^{\dagger}_{k}$’s are operators:

$\begin{split}T_{tt}&=\sum_{k}{}\sum_{k^{\prime}}{}\left[(-iw)a_{k}f_{k}+(+iw)a^{\dagger}_{k}f^{*}_{k}\right]\left[(-iw^{\prime})a_{k^{\prime}}f_{k^{\prime}}+(iw^{\prime})a^{\dagger}_{k^{\prime}}f^{*}_{k^{\prime}}\right]\\\
&=\sum_{k}{}\sum_{k^{\prime}}{}ww^{\prime}\left[-a_{k}a_{k^{\prime}}f_{k}f_{k^{\prime}}+a_{k}a^{\dagger}_{k^{\prime}}f_{k}f^{*}_{k^{\prime}}+a^{\dagger}_{k}a_{k^{\prime}}f_{k^{\prime}}f^{*}_{k}-a^{\dagger}_{k}a^{\dagger}_{k^{\prime}}f^{*}_{k^{\prime}}f^{*}_{k^{\prime}}\right],\end{split}$
(4.21)

where we have used the fact that $\partial_{t}\phi=-iw\phi$ and
$\partial_{t}\phi^{\dagger}=iw\phi$. There is one more obstacle one needs to
surpass before obtaining the final result, and that is the divergent Vacuum
Expectation Value (VEV) of the stress-energy tensor. That is:

$\langle 0\rvert T_{tt}\lvert 0\rangle=\frac{1}{2L^{3}}\sum_{k}{}w=\infty.$
(4.22)

In QFT, the set of techniques used to treat such infinities, is called
renormalization. The renormalization scheme we follow is normal ordering and
subtraction of the VEV of the stress energy tensor. To justify this
subtraction of the VEV one should remember that in QFT, we only measure energy
difference, and not the energies themselves. Thus, it is common practice to
compute the renormalized energy densities whose values are taken relative to
the infinity background energy. A quantity which involves annihilation and
creation operators is said to be normal ordered when all creation operators
are on the left side of all annihilators. After normal ordering taking into
account the commutation relations (4.18), (4.21) becomes:

$\begin{split}T_{tt}=&\sum_{k}{}\sum_{k^{\prime}}{}ww^{\prime}\left[-a_{k}a_{k^{\prime}}f_{k}f_{k^{\prime}}+a^{\dagger}_{k^{\prime}}a_{k}f_{k}f^{*}_{k^{\prime}}+a^{\dagger}_{k}a_{k^{\prime}}f_{k^{\prime}}f^{*}_{k}-a^{\dagger}_{k}a^{\dagger}_{k^{\prime}}f^{*}_{k^{\prime}}f^{*}_{k^{\prime}}\right]+\\\
&+\frac{1}{2L^{3}}\sum_{k}{}w,\end{split}$ (4.23)

where the last term is exactly the divergent VEV. The fact that the vacuum
term arises as the divergence of the $T_{tt}$ gives additional justification
to our renormalization scheme. Hence, the normal ordered finite energy density
component of SET is given by the operator statement:

$:T_{tt}:=T_{tt}-\langle 0\rvert T_{tt}\lvert 0\rangle.$ (4.24)

Using the operator on the l.h.s. of (4.24) we can obtain non-divergent
expectation values for different physical states.

####  4.3.2 Calculation

As we mentioned above, a state that will give rise to a negative energy
density is a certain superposition of the vacuum plus two excited modes of the
same momentum. For our calculations we choose a normalization which gives the
state the following form:

$\lvert\psi\rangle=\frac{\sqrt{3}}{2}\lvert 0_{k}\rangle+\frac{1}{2}\lvert
2_{k}\rangle.$ (4.25)

Now we are ready to perform the calculation of the expectation value of the
operator $:T_{tt}:$ in the above state. Using (4.24) and (4.23):

$\langle\psi\rvert:T_{tt}:\lvert\psi\rangle=\frac{\sqrt{3}}{4}\langle
0_{k}\rvert:T_{tt}:\lvert 2_{k}\rangle+\frac{\sqrt{3}}{4}\langle
2_{k}\rvert:T_{tt}:\lvert 0_{k}\rangle+\frac{1}{4}\langle
2_{k}\rvert:T_{tt}:\lvert 2_{k}\rangle.$ (4.26)

The cross-terms after some trivial algebra yield:

$\begin{split}\frac{\sqrt{3}}{4}\langle 0_{k}\rvert:T_{tt}:\lvert
2_{k}\rangle+\frac{\sqrt{3}}{4}\langle 2_{k}\rvert:T_{tt}:\lvert
0_{k}\rangle&=w^{2}\left(-\frac{\sqrt{6}}{4}\right)(f_{k})^{2}+w^{2}\left(-\frac{\sqrt{6}}{4}\right)(f^{*}_{k})^{2}\\\
&=-\frac{w}{2L^{3}}\frac{\sqrt{6}}{2}\cos(\textbf{kx}-wt).\end{split}$ (4.27)

And the diagonal term:

$\frac{1}{4}\langle 2_{k}\rvert:T_{tt}:\lvert
2_{k}\rangle=\frac{w}{2L^{3}}\;.$ (4.28)

Thus, we obtain the renormalized expectation value of the energy density of
the vacuum plus two particles state in the massless free scalar theory:

$\langle\psi\rvert:T_{tt}:\lvert\psi\rangle=\frac{w}{2L^{3}}\left[1-\frac{\sqrt{6}}{2}\cos(\textbf{kx}-wt)\right].$
(4.29)

We notice again that the negative contribution comes from an oscillating term.

###  4.4 Casimir

In 1948, the Dutch physicist Hendrik Brugt Gerhard Casimir showed that the
vacuum energy density for the electromagnetic field between two perfectly
conducting plates, is negative relative to the value of the Minkowski vacuum
[13]. A heuristic explanation is that certain modes are missing in the
spacetime region between the plates. Namely, modes with a wavelength longer
than the distance $L$ between the plates simply do not “fit”. Therefore,
compared to the Minkowski vacuum which contains all modes, the energy density
is necessarily lower.

The counterparts of this electromagnetic effect have been computed by several
authors [6], [8] for the case of the scalar field. In this section, we quote
their results for the cases of minimal and conformal coupling for the
interaction between gravity and a massive scalar field, in 4-D.

The expectation value of the full stress energy tensor between the plates, for
a conformal coupling to gravity:

$\langle 0\lvert T_{\mu\nu}\rvert
0\rangle_{conformal}=\frac{\pi^{2}}{1440L^{4}}\begin{bmatrix}-1&0&0&0\\\\[3.00003pt]
0&1&0&0\\\\[3.00003pt] 0&0&1&0\\\\[3.00003pt] 0&0&0&-3\\\\[3.00003pt]
\end{bmatrix},$ (4.30)

and for a minimal coupling:

$\langle 0\lvert T_{\mu\nu}\rvert 0\rangle_{minimal}=\langle 0\lvert
T_{\mu\nu}\rvert
0\rangle_{conformal}+\frac{\pi^{2}}{48L^{4}}\frac{(3-2\sin^{2}(\frac{\pi
z}{L}))}{\sin^{4}(\frac{\pi z}{L})}\begin{bmatrix}-1&0&0&0\\\\[3.00003pt]
0&1&0&0\\\\[3.00003pt] 0&0&1&0\\\\[3.00003pt] 0&0&0&0\\\\[3.00003pt]
\end{bmatrix},$ (4.31)

where L is the distance between the plates in the z-axis. To explicitly see
that these are indeed NEC violating stress tensors, one can pick a null
vector, e.g:

$K^{\mu}=\begin{pmatrix}-1\\\ 0\\\ 0\\\ 1\end{pmatrix},$ (4.32)

and compute the contaction of interest:

$\begin{split}\langle T_{\mu\nu}\rangle_{conformal}K^{\mu}K^{\nu}&=\langle
T_{xx}\rangle_{conformal}+\langle T_{zz}\rangle_{conformal}\\\
&=-\frac{\pi^{2}}{360L^{4}}.\end{split}$ (4.33)

Similarly to the case of the Ellis wormhole (3.5), NEC is violated again in
only one direction, namely z. Several authors, e.g. [14], have tried to find a
stress tensor for the Casimir effect inside a wormhole. In this case, the
effect arises due to the topology of spacetime itself, and more precisely the
hole which contains the throat. The region in the neighborhood of the throat
cannot contain modes with a diameter bigger than that of the throat. The
$\langle\hat{T}_{\mu\nu}\rangle_{casimir}$, even for a very long and thin
throat is zero for null rays directed exactly parallel to the throat.

However, this should have been expected even from studying the simple Ellis
wormhole. In sec. 3.1 we showed that the direction in which the violation
should occur is the l direction, i.e. the direction parallel to the “walls”
which contain the throat. Here, we showed that the violation occurs in the
direction normal to the plates, which in the wormhole case is the direction
vertical to the walls. Thus the NEC violating directions do not match.

Our simple suggestion, if one insists to use a Casimir stress tensor, is to
place two conducting plates in the direction vertical to the “walls”. This
will give rise to a stress tensor, which at least violates NEC in the right
direction.

###  4.5 Perturbed vacuum states

An interesting, general class of states are the states which are close to the
vacuum. A way to obtain these states is:

$e^{i\int J(x)d^{d}x\Phi(x)}\lvert 0\rangle,$ (4.34)

where $J(x)$ is a small source777$J(x)$ here is a real source, and not the
purely mathematical entity one uses when he wants to obtain n-point functions
from the path integral., and $\Phi(x)$ a free massless scalar. This a nice way
to write such states, since one can expand in powers of $J$ and pick up more
and more contributions to the expectation value of the stress tensor in these
states. One of the reasons why these states are interesting is that we the
freedom to choose function $J$ to be any function we want.

Below, we compute the expectation value of the stress tensor for this state,
in 1+1-D light-cone coordinates in Minkowski spacetime. However, our result
can be generalized to higher dimensions. The stress tensor for the free
massless scalar in light-cone coordinates has the components:

$\begin{split}T_{++}&=\partial_{+}\Phi(z)\partial_{+}\Phi(z),\\\
T_{--}&=\partial_{-}\Phi(z)\partial_{-}\Phi(z).\end{split}$ (4.35)

where we have chosen to work in the coincident limit. From now on, we use the
notation $\Phi(z)=\Phi_{z}$ for aesthetic reasons. Next, we compute the
expectation value of $T_{++}$ up to order $J^{2}$, using the expansion:

$\begin{split}\langle T_{++}\rangle_{per.}&=\langle 0\rvert e^{-i\int
J_{x}d^{2}x\Phi_{x}}T_{++}e^{i\int J_{x}d^{2}x\Phi_{x}}\lvert 0\rangle\\\
&=\langle 0\rvert\left(1-i\int d^{2}xJ_{x}\Phi_{x}-\frac{1}{2}\int
d^{2}xd^{2}wJ_{x}J_{w}\Phi_{x}\Phi_{w}\right)\cdot\left(\partial_{+}\Phi_{z}\partial_{+}\Phi_{z}\right)\cdot\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \cdot\left(1+i\int d^{2}wJ_{w}\Phi_{w}-\frac{1}{2}\int
d^{2}xd^{2}wJ_{x}J_{w}\Phi_{x}\Phi_{w}\right)\lvert 0\rangle.\end{split}$
(4.36)

  1. 1.

Order $J^{0}$: one obtains the usual VEV, $\langle 0\lvert T_{++}\rvert
0\rangle$ which has already been extensively studied.

  2. 2.

Order $J^{1}$: These are 3-point functions, and it is a well-known fact that
for a free scalar field all odd n-point functions vanish. A heuristic way to
understand this is by considering the Feynman diagram for a 3-point function;
this is just the 3-point vertex. Since there are no interactions, there is no
mechanism through which a “particle” can decay in two or two “particles” can
combine in one. A more rigorous way to see this is to turn to the path
integral formalism and obtain the n-point functions using the random source-
functional derivative trick. He/she will then arrive at the same conclusion.

  3. 3.

Order $J^{2}$: These terms do contribute to (4.36) and we shall devote the
rest of this section in calculating this contribution.

There are three terms of order $J^{2}$. Our first step is to deal with their
cumbersome 4-point functions, by invoking the Wick’s Theorem:

$\begin{split}A&=\langle\Phi_{x}\partial_{+}\Phi_{z}\partial_{+}\Phi_{z}\Phi_{w}\rangle_{0}\\\
&=2\cdot\langle\Phi_{x}\partial_{+}\Phi_{z}\rangle_{0}\cdot\langle\partial_{+}\Phi_{z}\Phi_{w}\rangle_{0}\;,\end{split}$
(4.37)

$\begin{split}B&=-\frac{1}{2}\langle\Phi_{x}\Phi_{w}\partial_{+}\Phi_{z}\partial_{+}\Phi_{z}\rangle_{0}\\\
&=-\langle\Phi_{x}\partial_{+}\Phi_{z}\rangle_{0}\cdot\langle\Phi_{w}\partial_{+}\Phi_{z}\rangle_{0}\;,\end{split}$
(4.38)

$\begin{split}C&=-\frac{1}{2}\langle\partial_{+}\Phi_{z}\partial_{+}\Phi_{z}\Phi_{x}\Phi_{w}\rangle_{0}\\\
&=-\langle\partial_{+}\Phi_{z}\Phi_{x}\rangle_{0}\cdot\langle\partial_{+}\Phi_{z}\Phi_{w}\rangle_{0}\;.\end{split}$
(4.39)

By simply adding A,B and C one obtains:

$\begin{split}A+B+C&=\langle\partial_{+}\Phi_{z}\Phi_{w}\rangle_{0}\cdot\left(\langle\Phi_{x}\partial_{+}\Phi_{z}\rangle_{0}-\langle\partial_{+}\Phi_{z}\Phi_{x}\rangle_{0}\right)+\\\
&+\langle\Phi_{x}\partial_{+}\Phi_{z}\rangle_{0}\cdot\left(\langle\partial_{+}\Phi_{z}\Phi_{w}\rangle_{0}-\langle\Phi_{w}\partial_{+}\Phi_{z}\rangle_{0}\right).\end{split}$
(4.40)

Noticing that all operators are Hermitian, using the identity $\langle
AB\rangle=\langle BA\rangle^{*}$ and $\langle AB\rangle-\langle
AB\rangle^{*}=2i\cdot\mbox{Im}\langle AB\rangle$ for (4.40) , we arrive to:

$\begin{split}A+B+C&=2i\cdot\langle\partial_{+}\Phi_{z}\Phi_{w}\rangle_{0}\cdot\mbox{Im}\langle\Phi_{x}\partial_{+}\Phi_{z}\rangle_{0}+\\\
&+2i\cdot\langle\Phi_{x}\partial_{+}\Phi_{z}\rangle_{0}\cdot\mbox{Im}\langle\partial_{+}\Phi_{z}\Phi_{w}\rangle_{0}\;.\end{split}$
(4.41)

The Feynman propagator in two dimensions reads:

$\langle\Phi_{x}\Phi_{z}\rangle_{0}=-\frac{1}{4\pi}\ln\frac{1}{(z_{+}-x_{+})(z_{-}-x_{-})-i\epsilon}\;.$
(4.42)

Remembering that all the derivatives are actually
$\partial_{+}=\frac{\partial}{\partial z^{+}}$, we use (4.42) to compute all
the relevant quantities in (4.41):

$\langle\Phi_{x}\partial_{+}\Phi_{z}\rangle_{0}=\frac{\partial}{\partial
z^{+}}\langle\Phi_{x}\Phi_{z}\rangle_{0}=\frac{1}{4\pi}\frac{1}{(z_{+}-x_{+})-i\epsilon}\;,$
(4.43)

$\mbox{Im}\langle\Phi_{x}\partial_{+}\Phi_{z}\rangle_{0}=\frac{\epsilon}{4\pi}\frac{1}{(z_{+}-x_{+})^{2}-\epsilon^{2}}\;,$
(4.44)

$\langle\partial_{+}\Phi_{z}\Phi_{w}\rangle_{0}=-\frac{\partial}{\partial
z^{+}}\langle\Phi_{z}\Phi_{w}\rangle_{0}=-\frac{1}{4\pi}\frac{1}{(w_{+}-z_{+})-i\epsilon}\;,$
(4.45)

$\mbox{Im}\langle\partial_{+}\Phi_{z}\Phi_{w}\rangle_{0}=\frac{\epsilon}{4\pi}\frac{1}{(w_{+}-z_{+})^{2}-\epsilon^{2}}\;.$
(4.46)

In the epsilon prescription we have used, there is always an implicit limit of
$\epsilon\rightarrow 0$. Taking this limit we see that (4.44) and (4.46) are
nothing more than an alternative definition for the delta function:

$\delta(x)=\frac{1}{\pi}\lim_{\epsilon\to
0}\frac{\epsilon}{x^{2}+\epsilon^{2}}\;.$ (4.47)

After having brought the 4-point functions to the form we wanted, computing
the integrals for the order $J^{2}$ terms becomes an easy task:

$\begin{split}\mathcal{O}(J^{2}):&-\frac{2i}{16\pi}\int dx_{+}dx_{-}\int
dw_{+}dw_{-}J(x_{+},x_{-})J(w_{+},w_{-})\frac{1}{(w_{+}-z_{+})-i\epsilon}\delta(z_{+}-x_{+})-\\\
&-\frac{2i}{16\pi}\int dx_{+}dx_{-}\int
dw_{+}dw_{-}J(x_{+},x_{-})J(w_{+},w_{-})\frac{1}{(z_{+}-x_{+})-i\epsilon}\delta(w_{+}-z_{+})=\\\
&=-\frac{2i}{16\pi}\int dx_{-}\int
dw_{+}dw_{-}J(c(x_{+}),x_{-})J(w_{+},w_{-})\frac{1}{(w_{+}-z_{+})-i\epsilon}-\\\
&-\frac{2i}{16\pi}\int dx_{+}dx_{-}\int
dw_{-}J(x_{+},x_{-})J(c^{\prime}(w_{+}),w_{-})\frac{1}{(z_{+}-x_{+})-i\epsilon}\;.\end{split}$
(4.48)

Relabeling $x_{-}\rightarrow w_{-}$ the last to integrals combine, and we
obtain:

$\begin{split}\mathcal{O}(J^{2})&=-\frac{i}{8\pi}\int dx_{+}dx_{-}\int
dw_{-}J(x_{+},x_{-})J(z_{+},w_{-})\left(\frac{1}{(z_{+}-x_{+})-i\epsilon}-\frac{1}{(z_{+}-x_{+})+i\epsilon}\right)=\\\
&=-\frac{i}{8\pi}\int dx_{+}dx_{-}\int
dw_{-}J(x_{+},x_{-})J(z_{+},w_{-})\frac{2i\epsilon}{(z_{+}-x_{+})^{2}+\epsilon^{2}}=\\\
&=\frac{1}{4\pi}\int dx_{+}dx_{-}\int
dw_{-}J(x_{+},x_{-})J(z_{+},w_{-})\delta(x_{+}-z_{+})=\\\
&=\frac{1}{4\pi}\left(\int dx_{-}J(z_{+},x_{-})\right)^{2}\;.\end{split}$
(4.49)

As we mentioned, J is a general function which describes the form of the
source distribution. Finally, the result for the entire expectation value of
the stress tensor up to second order in J is:

$\langle T_{++}\rangle_{per.}=\langle
T_{++}\rangle_{0}+\frac{1}{4\pi}\left(\int dx_{-}J(z_{+},x_{-})\right)^{2}.$
(4.50)

which hold for a general class of functions J. This result shows that there is
a general class of states for which the quantity of interest is strictly
positive. We take this as evidence that the spectrum of NEC violating states
in QFT’s is smaller than one would generally expect.

##  5 Local and Non-Local Constraints

In the first three sections of this chapter, we present a selection of the
most relevant constraints on exotic matter that currently exist in literature.
We start by describing a local condition (sec. 5.1), then discuss a global
condition (sec. 5.2) and finally turn to a “semi-local” one (sec. 5.3). We
have noticed that there are some general characteristics shared by members of
each of the above categories. Local conditions, like NEC, are imposed at each
and every point of spacetime. However, they lack restrictive power. The global
ones are the most restrictive, but since they are imposed on an entire
geodesic cannot be used to prove theorems for which point-wise conditions are
necessary. Finally, the semi-classical conditions sit somewhere in the middle.
They have good restrictive power, and are good candidates for proving more
theorems than the global ones. However, they have their own issues, which we
shall discuss at the end of sec. 5.3.

###  5.1 QNEC

One of the most exciting, recent developments in the field of Energy
Conditions is the Quantum Null Energy Condition(QNEC), which was introduced by
R.Bousso, A.Wall et al. [68]. The goal of [68] was to unify the Covariant
Bousso bound [70] with the classical focusing theorem (th. 2.1), using a
universal inequality, introduced by the same authors, called the Quantum
Focusing Conjecture(QFC). The QFC and how QNEC arises from it will be
discussed in more detail in sec. 5.4 .

An important feature of QNEC is that it is a statement independent of gravity;
QNEC is a pure QFT statement. As such, it admits a proof within QFT, which was
given by the same authors in [69]. Another interesting property of QNEC is
that, much like the classical NEC, it is a local condition, i.e. it is imposed
on spacetime points. However, it does not hold for every point in spacetime
for all spacetimes, but only for points that lie on stationary null
surfaces888A null surface with expansion $\theta=0$ everywhere on it..
However, in Minkowski space, any point and null vector on it lie on such a
surface, e.g a Rindler Horizon.

Although QNEC has a clear meaning as an energy condition, the proof and even
the interpretation of the quantities involved, is rather cumbersome. Hence,
before we present the statement itself, we shall provide a detailed
introduction, following closely the work in [69].

We start by picking a Cauchy Surface $M$. A Cauchy surface is a spacelike
slice, $t=constant$, which contains all information about our theory. Then, we
place a QFT on this surface. Since the proof holds for spacetime dimensions
$D\geq 2$, we work with a co-dimension one ($D-1$) surface. Next, we define a
smooth co-dimension two surface $\Sigma$, which divides $M$ in two parts.

Since a QFT leaves on $M$, $\Sigma$ is now an entangling surface; the field
theory degrees of freedom on one side of $M$ are entangled with the degrees of
freedom which live on the other side. Hence, one can define $S_{out}$, which
denotes the entanglement entropy of the fields residing in one side of
$\Sigma$. Next we shoot a co-dimension one, Null Hupersurface $N$, orthogonal
to $\Sigma$. We pick a point $p$ which lies on $\Sigma$, and a null vector
$k^{\mu}$. The vector $k^{\mu}$ is a member of a vector field $k^{\mu}(y)$
defined in the neighborhood of $p$, which is orthogonal to $\Sigma$ and
tangent to $N$. Here, $y$ is a co-ordinate on $\Sigma$, called the “transverse
direction”, which labels all possible $p$’s.

The vector $k^{\mu}(p)$ generates a null geodesic on $N$, parametrized by an
affine parameter $\lambda(p)$. Thus, the vector field which consists of all
$k^{\mu}(y)$ in the neighborhood of $p$, defines a family of geodesics,
parametrized by affine parameters $\lambda(y)$. By deforming $\Sigma$ along
$N$ in the neighborhood of any $y$, one can obtain a family of surfaces which
are now functionals of the affine parameter $\Sigma[\lambda(y)]$.

Figure 9:  (Taken from [69])Assuming the 4-D case and omitting one spatial
dimension, one obtains the above figure. The null hypersurface $N$ is in
green, in yellow we see the “outside” half of the Cauchy surface M and the
black line is co-dim 2 surface $\Sigma$. The deformation of $\Sigma$, and
hence the variation, is performed along a “pencil” of infinitesimal area
$\mathcal{A}$, which contains point $p$. After deforming $\Sigma$ once more in
the neighborhood of $p$, we obtain the second variation of the entanglement
entropy of the fields living on half of M:
$\frac{\delta^{2}S_{out}}{\delta\lambda(p)\delta\lambda(p)}$.

Finally, we can define a family of entanglement entropies
$S_{out}[\lambda(y)]$ for all Cauchy splitting $\Sigma[\lambda(y)]$, which are
the relevant quantities for the QNEC statement.

$\langle
T_{kk}(p)\rangle_{\psi}\geq\frac{\hbar}{2\pi\mathcal{A}}\frac{\delta^{2}S_{out}}{\delta\lambda(p)\delta\lambda(p)}\;,$
(5.1)

where $T_{kk}(p)$ is the double contraction of the stress tensor of the QFT at
point $p$ with the null vector $k^{\mu}$ normal to $\Sigma$ at $p$. The r.h.s.
is the diagonal term of the second functional derivative of
$S_{out}[\lambda(y)]$. One of the reasons why QNEC is defined in such a
complicated way is that $S_{out}[\lambda(y)]$ is not a local quantity. Thus,
if one wants to involve such a quantity in strictly local bound like QNEC,
he/she must find a way to distil the local information it contains. This is
done, by writing the entropy as a functional of $\lambda(y)$ and taking its
derivative at coincident points.

###  5.2 ANEC

Seeing our failure to generalize a local NEC in QFT, an option is to consider
non-local conditions in hope that most, or at least some, of the results based
on NEC can be salvaged. Historically the first to suggest such a condition was
Tipler in 1977 [27]. Tipler considered the average over one of these point-
wise conditions, namely WEC (sec. 2.1). The averaging was taken over the whole
worldline of the observer, and the resulting bound is known as the Averaged
Weak Energy Condition. However, we are interested in NEC, so we shall discuss
AWEC’s counterpart, the Averaged Null Energy Condition(ANEC):

$\int^{\infty}_{-\infty}\langle T_{\mu\nu}K^{\mu}K^{\nu}\rangle d\lambda\geq
0.$ (5.2)

The statement is the following: the integral over the entirety of a null
geodesic with affine parameter $\lambda$, of the projected $T_{\mu\nu}$ on the
null vector $K^{\mu}$ tangent to the geodesic, must always be greater or equal
than zero. A list of classical GR results that can be restored by imposing
ANEC can be found in [28].

An interesting aspect of ANEC is that it, much like QNEC, can be derived in
QFT. In 1991, Wald and Yurtsever [32] proved that ANEC holds in two and four-
dimensional Minkowski spacetime, for any Hadamard state along an achronal null
geodesic. There are other proofs since then, like the one from Ford and Roman
[56], who proved the above using an entirely different method, or the one from
Eleni-Alexandra Kontou and Ken D. Olum [55]. In [55] the above results are
additionally generalized for spacetimes with small curvature. In a nutshell,
ANEC holds for free fields in Minkowski spacetime but can be violated in some
curved spacetimes.

One may think that a traversable wormhole spacetime clearly violates ANEC, and
if we accept ANEC as our new NEC for QFT such solutions should be considered
unphysical. But there is a way out, even for wormholes as simple as the ones
we considered in sec. 3.1 and the reason is that ANEC is defined as the
integral over the “whole” null geodesic. Therefore, one could imagine placing
the NEC violating matter in some local region of spacetime so that the
flaring-out condition (sec. 3.2) is satisfied and start throwing smaller and
smaller pebbles of “normal” matter on his way to infinity. We would like our
pebbles to be smaller and sparser as we reach infinity so that the asymptotic
flatness condition is satisfied and our metric does not change drastically.
This can be done in such a way, that the NEC violating energy the light ray
“sees” is compensated by the NEC satisfying pebbles, resulting in geodesics
that do not violate ANEC.

###  5.3 Quantum Inequalities

A set of semi-local conditions in QFT are the Quantum Inequalities(QI’s),
originally introduced by Ford in 1978 [57]. The quantum inequalities are a bit
more local than the averaged energy conditions since they involve integrals
over only a specific interval of interest and not the whole spacetime. They
are essentially restrictions on the magnitude/time duration on the negative
energy density (or flux) in some time interval. The form of these restrictions
is the following:

$\langle\hat{\rho}\rangle=\int^{\infty}_{-\infty}\langle\rho\rangle
f(\tau)d\tau\geq-\frac{C\hbar}{c^{3}\tau^{4}_{0}}\;.$ (5.3)

Here $\rho$ is the renormalized expectation value of the energy density of a
free field in an arbitrary Hadamard state, measured by an inertial observer in
his rest frame. $\tau$ is the proper time measured by the observer and
$f(\tau)$ is a sampling function (a smooth peaked function of width
$\tau_{0}$, which integrates to 1). An example of such a function is the
Lorentzian function:

$f(\tau)=\frac{\tau_{0}}{\pi}\frac{1}{\tau^{2}+\tau^{2}_{0}}\;.$ (5.4)

The above QI is the QI for the energy density and it states that the amount of
the smeared negative energy density $\hat{\rho}$ is inversely proportional to
the fourth power of the duration of the interval $\tau^{4}_{0}$. Simply put,
the more negative energy you see, the shorter it can exist. It is interesting
to notice that these bounds resemble the uncertainty principle, although it
has not been assumed in their derivation.

However, the sampling procedure does not come risk-free. If the probability
distribution function of the eigenvalues of the operator that we sample is
some strange function (not Gaussian, Lorentzian etc.) at least in the
neighborhood of the expectation value, the sampled function may inherit
features from the sampling function.

These constraints have been derived for free fields, namely: massless &
massive free scalar, E/M field, massive spin-1 field and Dirac field. They
have been derived for arbitrary smooth sampling functions in 2-D & 4-D
Minkowski and a number of curved spacetimes.

Going back to our favorite traversable wormhole example we notice that from
the bounds discussed so far, QI’s place the strongest restriction in their
existence. If one wishes to have a sufficiently big, macroscopic traversable
wormhole so that he can travel through it comfortably, he would in principle
need a huge amount of negative energy. This, in turn, implies that the
wormhole would remain open for a ridiculously small amount of proper time.
This is certainly bad news for sci-fi travelers. However, to their relief,
there is still a way out. There exist wormhole solutions introduced by
M.Visser et al. [58], which require arbitrary small energy condition
violations.

Furthermore, in this thesis we have defined a traversable wormhole as a
wormhole spacetime through which a light signal can travel back and forth.
Hence, for our purposes, neither the wormhole nor the period of its existence
needs to be especially big.

####  5.3.1 ER=EPR and the Lost Wormhole

One cannot help but notice, that an interesting question is raised in the
above discussion. If the exotic matter can indeed only exist for a certain
amount of time, what happens to the wormhole when this negative stress-energy
vanishes? Clearly, the above question requires a detailed analysis of the
dynamics of the problem. Instead, we give a heuristic description using
intuitions coming from information theory and the ER=EPR proposal.

To justify our way of thinking we first discuss the dynamical Schwarzschild
wormhole in the ER=EPR context. In the past few years, it has become a slogan
of quantum information theorists that entanglement is connectedness. Following
this logic, one should realize that although the proposal of [59] is somewhat
“fringy” at first sight, it has some reasonable intuitive origins. The
proposal in its original form is the claim that pairs of maximally entangled
black holes are connected via ER (Einstein-Rosen) bridges. In this picture,
the entangled black holes are treated as a complex EPR (Einstein-Podolsky-
Rosen) pair. For our purposes, a stronger version of the proposal also
introduced in [59] is in order: every pair of entangled particles is connected
via an ER bridge. Assuming the above, we proceed to a heuristic description of
the interior of the Schwarzschild spacetime.

As we discussed in sec. 3.1, following the evolution of spacelike slices (fig.
5) of the Kruskal diagram (fig. 4) one can see a non-traversable wormhole
forming and then disappearing when the slices start “touching” the
singularity. This means that for some period of time, the universes I and III
are connected and then are disconnected again. This connection signals the
existence of entanglement between the degrees of freedom residing on universes
I and III since entanglement is connectedness and vice versa. In the ER=EPR
picture, things are even clearer. The existence of an ER bridge means that
there are entangled pairs, whose one component leaves in I and the other in
III.

Another factor which plays an important role to our description of this
scenario is the breaking of the entanglement. According to the AMPS firewall
proposal [60], when entanglement is broken, a huge amount of energy is
released. Regardless if one trusts the proposal or not, there is a simple
lesson to take home. When entanglement is broken, energy is indeed released.
To get some intuition as to why this is true, think about the following
example. Take the vacuum state on a Cauchy slice with an entangling surface
dividing it in two. Now assume that for some reason the entanglement between a
single pair of field degrees of freedom is broken. Since the vacuum is defined
as the state of lower energy, the state you obtained will be by necessity of
higher energy.

Having acquired the required pieces, we present our picture starting from the
point where axes T and R meet in fig. 4. As time passes, I and III become more
and more disconnected and hence, more and more energy is released. When
finally I and III are completely separate, a singularity has formed, i.e. the
energy released from the breaking of the entanglement has become the mass of
the black hole. Clearly, there is no simple observable that can allow one to
test if this process takes place or not, since the interior is protected the
whole time by the existence of the event horizon.

Next, we turn to our traversable wormhole case. It is clear from our
definition (def. 3.1) that the solutions we are interested in cannot contain
event horizons, so everything now is out in the open. In our discussion, the
initial system is that of an Ellis wormhole, for simplicity. According to QI’s
the negative energy supporting the wormhole can exist for a certain amount of
time. After that it gets annihilated via some, still not discussed, physical
mechanism. More precisely, according to the Quantum Interest Conjecture [61] ,
it gets compensated by a positive energy flux.

Using the same logic, we propose another heuristic and even more ambitious
interpretation of what happens after the vanishing of the exotic matter
supporting an Ellis wormhole. After the negative energy is removed, the
wormhole snaps and the outcome depends highly on the amount of positive
energy, which shall compensate for it. As for the ER=EPR interpretation, the
entanglement between the degrees of freedom in the two asymptotic regions is
broken. So maybe, this is where the energy (or at least part of it) came from;
the broken entanglement. Another way to phrase it is to say that the energy
was hidden in the form of entanglement. Again, one could in principle compute
the energy released by the entanglement between the two regions and check if
it corresponds to the energy predicted by [61]. However, only if the energy
due to the broken entanglement is bigger, this calculation can be used to
disprove our heuristic picture, since it may be only part of the overall
positive compensating flux.

Finally, we should mention that throughout our discussion we carefully left an
important question unanswered. Namely: what is entangled? This is a deep
question whose answer we cannot hope to justify within this simple and
seemingly unrelated work. However, our educated guess is: Qubits of a
microscopic theory from which spacetime emerges. For some time, the existence
of an underlined structure of spacetime has been a subject of many popular
projects ([62], [63]). Recently, the idea that both gravity and spacetime
arise from the entanglement between degrees of freedom of a microscopic
theory, has been used to reproduce even observational data ([64]). In fact, in
[64], the idea that a mass corresponds to the decrease of entanglement entropy
has been a central argument.

Even though the above discussion was somewhat hand-waving, we argued that the
connection between energy, entanglement and connectedness is a deep one. It is
not only at the center of quantum information theory research, but it has
proven to be a useful tool in our search for pieces of the puzzle that is
quantum gravity.

###  5.4 QFC

The Quantum Focusing Conjecture (QFC), as the name suggests, is a conjectured
generalization of the Classical Focusing (sec. 2.2.2). The statement of the
conjecture is that the Quantum Expansion $:=\Theta$ is non-increasing:

$0\geq\Theta^{\prime}\;.$ (5.5)

As in the case of QNEC (sec. 5.1), the meaning of this statement can be
explained in a few words, but the derivation and interpretation of the
quantities involved are rather involved. In the first part of this section we
shall assume the responsibility of summarizing the above. For a more detailed
treatment, we refer the reader to the original paper [68].

The setup is the same as that of sec. 5.1 and $S^{\prime\prime}_{out}$ is
again the second functional derivative of the entanglement entropy of the
fields sitting on the one-half of the Cauchy surface $M$.
$\theta^{\prime}=\frac{d\theta[\lambda(y)]}{d\lambda(y)}$ is the rate of the
expansion we defined earlier in 2.2, now seen also as a functional derivative.

The motivation for QFC came from studying the generalized entropy, a quantity
introduced by Bekenstein in [74], which is defined as the sum of all Black
Hole horizons in Planck units and the entropy of the matter fields outside
them.

$S_{gen}\equiv\frac{A}{4G_{N}\hbar}+S_{out}\;.$ (5.6)

The original purpose of Bekenstein was to find an entropy that can be used to
prevent violations of the Generalized Second Law of thermodynamics. A famous
example of such a violation occurs in the evaporation process of a Black hole,
in which the area of the Horizon decreases due to an ingoing NEC violating
flux. Computing the entanglement entropy of the radiated quanta
$S^{\prime\prime}_{out}$, one can see that the entropy loss due to the area
decrease is overcompensated.

One of the most remarkable features of $S_{gen}$ is that, although it involves
the UV-divergent von Neumann entropy $S_{out}$, it is itself finite and cut-
off independent. In [68], a mechanism through which both the leading and
subleading divergences of $S_{out}$999The entanglement entropy on the one side
of a sharp boundary such as $\Sigma$, is highly divergent due to short
distance entanglement between field degrees of freedom very close to the
boundary. are canceled by the renormalization of $G_{N}$ in
$\frac{A}{4G_{N}\hbar}$ is presented. In short, these divergences are absorbed
into counterterms of the gravitational action.

To obtain the quantum expansion from the $S_{gen}$, we look a fig. 9. Using
the same logic as in QNEC, we cast the quantities involved in (5.6) in the
form of functionals:

$S_{gen}[\Sigma(\lambda(y))]\equiv\frac{A[\Sigma(\lambda(y))]}{4G_{N}\hbar}+S_{out}[\Sigma(\lambda(y))].$
(5.7)

Then $\Theta$ is defined as the first functional derivative of $S_{gen}$:

$\Theta[\Sigma(y);y_{1}]\equiv\frac{4G\hbar}{\sqrt[]{g(y_{1})}}\frac{\delta
S_{gen}}{\delta\Sigma(y_{1})}\equiv\lim_{{\mathcal{A}}\to
0}\frac{4G\hbar}{\mathcal{A}}\frac{\delta S_{gen}}{\delta\Sigma(y_{1})}\;.$
(5.8)

Where $\sqrt[]{g(y_{1})}$, is the finite area element of the metric restricted
on $\sigma$. Finally, the QFC inequality reads:

$\frac{\delta}{\delta\Sigma(y_{2})}\Theta[\Sigma(y);y_{1}]\leq 0\;.$ (5.9)

In words, $\Theta$ cannot increase at point $y_{1}$, if the slice of N defined
by $\Sigma(y)$ is infinitesimally deformed along the generator $y_{2}$ of N in
the same direction.

Since the only important quantity here is $S_{gen}$, the regime of validity of
the conjecture, coincides with the regime of validity of $S_{gen}$.
Originally, $S_{gen}$ was defined for causal horizons, but following [71] [72]
[73] the writers assumed that $S_{gen}$ can be used to describe any co-
dimension 2, Cauchy splitting surface $\Sigma$. Moreover, the surface need not
be connected, nor compact.

After having formally defined QFC, it is interesting to see how one can arrive
at the QNEC bound (sec. 5.1) from it. Here it is useful to define the
classical infinitesimal expansion as:

$\theta\equiv\lim_{\textbf{A}\to
0}\frac{1}{\mathcal{A}}\frac{d\mathcal{A}}{d\lambda}\;,$ (5.10)

where $\mathcal{A}$ is an infinitesimal area element which separates
infinitesimal neighboring null geodesics. Now combining (5.10) and the
definitions of $\Theta$ (5.9) and $S_{gen}$ (5.7):

$\Theta=\theta+\frac{4G\hbar}{\mathcal{A}}S^{\prime}_{out}\;.$ (5.11)

Above and from now the limit is implicit but always there. By taking another
variation at the same point, i.e. looking only at the diagonal part of QFC,
one has:

$\begin{split}0\geq\Theta^{\prime}&=\theta^{\prime}+\frac{4G\hbar}{\mathcal{A}}(S_{out}^{\prime\prime}-S_{out}^{\prime}\theta)\\\
&=-\frac{1}{2}\theta^{2}-\sigma^{2}+\omega^{2}-8\pi G_{N}\langle
T_{kk}\rangle+\frac{4G\hbar}{\mathcal{A}}(S_{out}^{\prime\prime}-S_{out}^{\prime}\theta)\;.\end{split}$
(5.12)

If one chooses a congruence for which $\theta=\sigma=\omega=0$, then the above
statement reduces to QNEC:

$\langle
T_{kk}\rangle\geq\frac{\hbar}{2\pi\mathcal{A}}S^{\prime\prime}_{out}\;.$
(5.13)

Moreover, if one assumes that the classical limit corresponds to simply taking
$\hbar\rightarrow 0$, then one obtains the classical focusing theorem from
(5.12):

$\theta^{\prime}\leq 0\;,$ (5.14)

and from (5.13), an expression claimed to be the classical NEC of the form:

$\langle T_{kk}\rangle\leq 0\;.$ (5.15)

However, we believe that going from the quantum to the classical case is more
subtle than taking $\hbar\rightarrow 0$. In addition, (as we will discuss
extensively in ch.6) $\langle T_{kk}\rangle$ is not always well defined
classically.

##  6 Semi-Classical Gravity

In the current chapter, we discuss Semi-Classical Gravity and its relevance to
the violations of NEC in QFT. Semi-classical gravity is an approximation which
is used when one wants to describe a theory for which matter is quantized, but
gravity (spacetime) is not. There are many versions of this approximation, but
most of them have the following equation at their core:

$G_{\mu\nu}=8\pi G_{N}\langle T_{\mu\nu}\rangle_{\psi}\;.$ (6.1)

In this equation, the expectation value of the quantized matter fields in some
state $\lvert\psi\rangle$ is used as a source of a classical fixed background
spacetime. Now, this is not a very precise statement since, as we know from
GR, there are more than one metrics that can correspond to the same stress
tensor101010An example is the vacuum $T_{\mu\nu}=0$, where solutions to the
Einstein equations are Schwarzschild, Minkowski, etc.. Moreover, if one tries
to find one of these solutions, he would have the usual, cumbersome task of
solving some very difficult partial, non-linear differential equations. In
fact, this task is so difficult, that there is no good “general” mathematical
procedure of producing exact solutions to these equations. The ones we have
were mainly obtained by assuming a lot of symmetry (stationarity, spherical
symmetry etc.).

However, the Einstein equation can be solved quite easily following another
route. Instead of choosing a stress tensor and requesting that it solves these
equations, we can start with a metric of a Lorentzian manifold and compute the
corresponding stress tensor (ch. 1). Then, the task simplifies to finding a
“physical” theory for which the expectation value of the stress tensor in one
of its states, is the one predicted by the Einstein equations. Again this is
not an easy task, but it is in principle easier than the previous one. This
way of solving the Einstein equations solves both the problems of “uniqueness”
and difficulty, and it is the scheme people usually use to construct wormholes
or other exotic solutions.

That said, another reason for which one should be worried about the semi-
classical gravity equation, is that it involves a classical object
$G_{\mu\nu}$, and a quantum expectation value $\langle
T_{\mu\nu}\rangle_{\psi}$. However, in support of the validity of (6.1), one
can provide the following argument. There is a regime in which quantum effects
of matter are important but quantum effects of gravity are not. For example,
there clearly exists an energy scale in which quantum contributions to gravity
have not yet become important, but quantum contributions to matter fields have
become so important that they dominate the processes. If this were not true,
it would be impossible for us to make predictions for experiments at CERN
without considering quantum gravity corrections. In fact, Quantum
Electrodynamics (QED), our most precise quantum theory, is an example in which
quantum gravity corrections are not taken into account. Nevertheless, in the
next chapter (ch. 6.1) we shall see that things are more subtle and that semi-
classical gravity is not only sensitive to the energy scale.

So we have presented a short introduction to semi-classical gravity, and now
as promised we shall see its connection to NEC. In previous chapters, we noted
that a lot of work has been done towards generalizing GR theorems beyond GR
but also finding a way to source wormholes and other exotic spacetime
solutions. However, a very important question that has not been addressed in
the above literature is the following: in which regime do the above attempts
take place? And the much-expected answer is the regime of semi-classical
gravity. We do not yet hold enough pieces of the puzzle that is quantum
gravity, to be able to even determine its shape, hence it is not strange that
so much work has been done in the domain of semi-classical gravity. A
reasonable claim supporting this choice is that semi-classical gravity is some
limit of quantum gravity.

###  6.1 Regime of validity of semi-classical gravity

Semi-classical gravity, like any other approximation, holds in a specific
regime and as it should, breaks down if one tries to treat something that
cannot be treated this regime. The regime of validity of semi-classical
gravity is the set of conditions which we silently imposed when we defined it
in the previous chapter. These conditions were 1) that the background metric
is fixed and 2) that gravity is not quantized.

For reasons that will become clear in sec. 6.2, we focus on when semi-
classical gravity breaks down, or cannot be trusted. To our knowledge this
happens in the following cases:

  1. 1.

The semi-classical gravity approximation breaks down in high energy scales,
i.e. close to Planck scale, where the full quantum gravity treatment is
needed.

  2. 2.

The semi-classical gravity approximation breaks down when the fluctuation
induced to the metric by the fluctuations of the stress energy tensor, namely
the 2-point function $\langle T_{\mu\nu}T_{\sigma\rho}\rangle_{\psi}$, in the
state where the expectation value is taken, are large (sec. 6.2.1). (Even away
from Planck scale)

The first case is a well-known and studied break down of semi-classical
gravity, and therefore we will not discuss it in great detail. However, the
latter, which has received less attention, is central to our main topic.
Hence, we shall devote several of the upcoming sections to its analysis (sec.
6.2.1, sec. 6.2.2.

###  6.2 Conjecture

Putting everything discussed in the previous two chapters (ch. 6, sec. 6.1)
together, led us to the following question: Can semi-classical gravity be used
to treat the QFT states in which (4.1) is violated? We believe that the answer
to this question is negative. Following this intuition, we propose a simple,
no-go conjecture:

Conjecture: For states in which $\langle
T_{\mu\nu}K^{\mu}K^{\nu}\rangle_{\psi}<0$, the semiclassical
equation $G_{\mu\nu}=8\pi G_{N}\langle T_{\mu\nu}\rangle_{\psi}$ is not a
valid approximation.

where $\langle T_{\mu\nu}K^{\mu}K^{\nu}\rangle_{\psi}\geq 0$ is the naive
version of the Quantum Null Energy Condition we defined in sec. 4.1. If our
conjecture holds, it provides us with a way out of the stalemate the
violations of NEC in QFT put us in. The route is the following: if $\langle
T_{\mu\nu}K^{\mu}K^{\nu}\rangle_{\psi}\geq 0$ holds in semi-classical gravity,
then no W/H’s or other exotic spacetimes can exist and we are free to use
$\langle T_{\mu\nu}K^{\mu}K^{\nu}\rangle_{\psi}\geq 0$ to generalize all of
the GR theorems that used NEC as an assumption to the semi-classical gravity
regime. This exit exists, of course, only in the regime of semi-classical
gravity, but as we discussed this is the actual regime of interest. In the
following chapters, we shall see the ways in which states with $\langle
T_{\mu\nu}K^{\mu}K^{\nu}\rangle_{\psi}<0$, violate the conditions of semi-
classical gravity.

####  6.2.1 Fluctuations of the Stress Tensor

The first to connect the NEC violating states with large fluctuations of their
stress tensor were Cheng-I Kuo & L.H. Ford [5]. In their paper, they propose
the dimensionless quantity $\Delta$, as a measure of the fluctuations of the
stress tensor, where $\Delta$ is defined us:

$\Delta_{\mu\nu\rho\sigma}=\left\lvert\frac{\langle:T_{\mu\nu}(x)T_{\rho\sigma}(y):\rangle_{\psi}-\langle:T_{\mu\nu}(x):\rangle_{\psi}\langle:T_{\rho\sigma}(y):\rangle_{\psi}}{\langle:T_{\mu\nu}(x)T_{\rho\sigma}(y):\rangle_{\psi}}\right\rvert\;.$
(6.2)

Since computing all these components requires a rather long algebraic
calculation, they suggest that taking the coincident limit $x\rightarrow y$
for the $\mu=\nu=\rho=\sigma=0$ component is enough to prove their point.
Thus, the quantity they actually proceed to calculate is:

$\Delta(x)=\left\lvert\frac{\langle:T_{00}(x)T_{00}(x):\rangle_{\psi}-\langle:T_{00}(x):\rangle_{\psi}\langle:T_{00}(x):\rangle_{\psi}}{\langle:T_{00}(x)T_{00}(x):\rangle_{\psi}}\right\rvert\;.$
(6.3)

Here $\langle:T_{\rho\sigma}(x):\rangle_{\psi}$ is the expectation value of
the normal ordered stress tensor in some state $\lvert\psi\rangle$ and
$\langle:T_{\mu\nu}(x)T_{\rho\sigma}(y):\rangle_{\psi}$ is the normal ordered
2-point function. As we have explained in sec. 4.3, the normal ordering
procedure is a regularization which consists of putting all creation operators
to the left and subtracting the VEV of the operator of interest. The
fluctuations of the stress tensor are said to be small in the state of
interest, if $\Delta\ll 1$ and large if $\Delta\approx 1$. We must note here
that this condition is as strong as NEC since it is local in the same sense,
i.e. it is imposed on each and every point of spacetime.

Thereafter, they proceed to calculate $\Delta$ for coherent states and some
known NEC violating states: the Casimir vacuum, the vacuum plus two particles
and the squeezed vacuum states. For the coherent states, they find $\Delta=0$,
which was to be expected since it is common wisdom that coherent states
correspond to classical field excitations. Thus it would be highly unusual if
they were incompatible with semi-classical gravity. To our knowledge, there is
no well-behaved theory that contains coherent states which exhibit NEC
violating behavior. For all NEC violating states mentioned above, the value of
$\Delta$ is $\approx 1$ or greater.

However, there is room for doubt as to if this quantity is indeed enough to
quantify the amount by which the semi-classical gravity approximation is
violated. The first caveat we identify, is that the object
$\langle:T_{00}(x)T_{00}(x):\rangle_{\psi}$ is only part of
$\langle:T_{00}(x)::T_{00}(x):\rangle_{\psi}$, which is the actual 2-point
function at the coincident limit. Although
$\langle:T_{00}(x)::T_{00}(x):\rangle_{\psi}$ contains divergent
pieces111111In some very rare cases of states and for some specific component
of the stress tensors, the two are equivalent, e.g for the photon number
eigenstate:
$\langle:T_{00}(x)T_{00}(x):\rangle\approx\langle:T_{00}(x)::T_{00}(x):\rangle$
, one should find a way of extracting the information encoded in these pieces
rather than just subtracting them. In later work, Ford & Wu realized this
problem, but did not present a non-divergent, well-defined quantity that
measures the fluctuations $\langle:T_{00}(x)::T_{00}(x):\rangle_{\psi}$. In
sec. 6.3, we present a quantity which not only solves this problem but is in
principle more physical.

####  6.2.2 Very Quantum States

In sec. 4.1, we mentioned an example that makes using the expectation value of
an operator in a classical equation counter-intuitive. Hereon, we will call
such states very quantum states and we will argue against their compatibility
with the semi-classical approximation.

The operator of interest is again the stress energy tensor. In the probability
distribution of $T_{\mu\nu}$’s eigenvalues for very quantum states, one can
find eigenvalues “far” from each other which carry high probability weight.
Since the eigenvalues of the stress tensor are matter distributions in
spacetime, if they are far apart in the probability distribution functional,
they correspond to very different geometries. Quantitatively, one expects the
fluctuations of $T_{\mu\nu}$ to be big and to induce equally big metric
fluctuations, leading to a breakdown of the semi-classical approximation. In
fig. 10, we give a simplified example of such a state. The simplification
refers to the fact that the eigenvalues of $T_{\mu\nu}$ describe the matter
density at points, not areas. However, this last inconsistency can be restored
if one uses a “smeared” version of $T_{\mu\nu}$, as we shall see in sec. 6.3.

Figure 10: Superposition: 50% Jupiter ($318M_{\oplus}$) and 50% Earth
($1M_{\oplus}$). Expectation value is: Planet X($159.5M_{\oplus}$)

####  6.2.3 The Probability distribution

At the end of the day, if one wishes to make precise statements about the
expectation value of the stress tensor in some state, he needs to study the
probability distribution of the eigenvalues of $\hat{T}_{\mu\nu}(x)$ in that
state. The first thing that one should notice, is that each one of these
eigenvalues corresponds to a different distribution of matter in spacetime.
This means that by assuming the semi-classical gravity equation, each
eigenvalue of $\hat{T}_{\mu\nu}(x)$ corresponds to a different
spacetime121212We have already discussed in ch. 6 that $T_{\mu\nu}(x)$ does
not correspond to a single solution of the Einstein or SCG equation. However
this fact does not affect our argument., making the expectation value a sum
over possible spacetimes multiplied by the probability they have to occur.

Our purpose here is to use the probability distribution function, to
distinguish between semi-classical and non-semi-classical states. The most
important quantity for this purpose is the variance $\sigma^{2}$. The variance
we shall consider is $\sigma^{2}=\langle
T_{kk}(x)T_{kk}(x)\rangle_{\psi}-\langle T_{kk}(x)\rangle_{\psi}\langle
T_{kk}(x)\rangle_{\psi}$, for some state $\psi$. Here, use
$T_{kk}=T_{\mu\nu}K^{\mu}K^{\nu}$ since we are interested in the null behavior
of $T_{\mu\nu}$ and $\sigma^{2}$ should be a scalar quantity. For very quantum
states, the fluctuations are always bigger than the mean, leading to a big
variance. On the other hand, for semi-classical states, the variance in most
cases is close to zero since they should have sharply peaked around the mean
probability distributions. Another useful feature of the variance is that it
throws away states whose existence leads to a probability distribution of
$T_{\mu\nu}$ which contains heavy tails. For such distributions, the variance
is big, since the peak of the probability distribution is “drugged way” from
the mean, due to the huge expectation values of the tail.

Taking all the above into account we define the following quantity as a
measure of the magnitude of fluctuations of $\langle T_{kk}\rangle_{\psi}$:

$\frac{\sigma^{2}}{\langle T_{kk}(x)\rangle^{2}_{\psi}}=\frac{\langle
T_{kk}(x)T_{kk}(x)\rangle_{\psi}-\langle T_{kk}(x)\rangle_{\psi}\langle
T_{kk}(x)\rangle_{\psi}}{\langle T_{kk}(x)\rangle^{2}_{\psi}}\;.$ (6.4)

When $\frac{\sigma^{2}}{\langle T_{kk}(x)\rangle^{2}_{\psi}}<<1$ state $\psi$
is semiclassical. When $\frac{\sigma^{2}}{\langle
T_{kk}(x)\rangle^{2}_{\psi}}\approx 1\leavevmode\nobreak\ \mbox{or}>1$, the
state $\psi$ is very quantum. In addition to its other features, this is a
scalar quantity and hence co-ordinate invariant. Although the quantity (6.4)
has the required form, it contains the divergent quantity $\langle
T_{kk}(x)T_{kk}(x)\rangle_{\psi}$. Thus, in the next chapter we fix this
problem in a way that we believe is the most natural and efficient.

###  6.3 Smearing

The most natural way to regulate the divergences in $\langle
T_{kk}(x)T_{kk}(x)\rangle_{\psi}$ is to define a smeared version of the stress
tensor. Here smearing means integrating the root of all divergences, operator
$T_{kk}(x)$, with some smearing function $f_{x_{0}}(x)$ over the whole
spacetime; this results in the smearing of the quantities that the stress
tensor describes over some finite region controlled by the width of the
smearing function $\tau$.

$T^{s}_{kk}(\textbf{x}_{0},\tau,t)=\int^{\infty}_{-\infty}f_{x_{0}}(x)T_{kk}d^{d}x.$
(6.5)

First let us discuss the characteristics the function should have. It should
be a smooth function, and it should integrate to one:
$\int^{\infty}_{\infty}f_{x_{0}}(x)dx=1$. A function that satisfies the above
and is easy to handle, is a Gaussian function of the form:

$f_{x_{0}}(x)=\left(\frac{1}{4\pi\tau^{2}}\right)^{\frac{d}{2}}e^{-\left(\frac{x_{0}-x}{2\tau}\right)^{2}}.$
(6.6)

Where $\textbf{x}_{0}$ is the vector which defines the position of the center
of the Gaussian in space, and $\tau$ determines its width. Smearing does not
only act as a natural cut-off to the divergences of $\langle
T_{kk}(x)\rangle_{\psi}$ and $\langle T_{kk}(x)T_{kk}(x)\rangle_{\psi}$, but
also makes the planet intuition more precise. Comparing matter densities
$T_{kk}(x)$ at points with planets does not make so much sense. However,
$T^{s}_{kk}$ describes the matter density in a region, which makes it a much
more suitable quantity for the description of stellar objects. In fact
$T^{s}_{kk}$ is in general more physical than the stress tensor at a point
since this is what one expects to measure in real life experiments.

####  6.3.1 The Smeared Stress Energy Tensor

In this section, we define a smeared version of the stress energy tensor and
calculate $\langle T_{++}(x)\rangle$ for the NEC violating state sec. 4.25 in
1+1-D in light-cone co-ordinates. For that purpose, we make use of the
smearing function (6.6), and similar techniques to the ones we used in sec.
4.3. One main difference here is that, instead of the circle on which we
worked on in sec. 4.3, we now work on the infinite plain. Thus we are required
to cast the sums of sec. 4.3 into integrals, change our commutation relations
accordingly and take care of some IR divergences which were previously
controlled by the size of the circle. First, using the dispersion relation
$w=\lvert k\rvert$ we see that the scalar modes decouple to right and left
movers. Since we are interested in computing $\langle T_{++}(x)\rangle$ we
only consider the right movers, namely:

$\phi(x_{+})=\int_{w=0}^{w=\infty}\frac{dw}{w}\left(a_{w}e^{-ix^{+}w}+a^{\dagger}_{w}e^{ix^{+}w}.\right)$
(6.7)

The commutation relations of the annihilation and creation operators take the
following form:

$\begin{split}[\hat{a}_{k},\hat{a}^{\dagger}_{k^{\prime}}]&=w_{k}\leavevmode\nobreak\
\delta(k-k^{\prime}),\\\ [\hat{a}_{k},\hat{a}_{k^{\prime}}]&=0,\\\
[\hat{a}^{\dagger}_{k},\hat{a}^{\dagger}_{k^{\prime}}]&=0.\end{split}$ (6.8)

where $\delta(k-k^{\prime})$ is the Dirac delta function. The stress energy
tensor is:

$\begin{split}T_{++}=\int_{w=0}^{w=\infty}dw\int_{w^{\prime}=0}^{w^{\prime}=\infty}dw^{\prime}(&-a_{w}a_{w^{\prime}}e^{-ix^{+}(w+w^{\prime})}+a^{\dagger}_{w}a_{w^{\prime}}e^{-ix^{+}(w^{\prime}-w)}+\\\
&+a^{\dagger}_{w^{\prime}}a_{w}e^{-ix^{+}(w-w^{\prime})}-a^{\dagger}_{w}a^{\dagger}_{w^{\prime}}e^{ix^{+}(w^{\prime}+w)}).\end{split}$
(6.9)

Here and from now on, we use the normal-ordered stress tensor, omitting the
colons (:…:). The Gaussian smearing function we use, since we are interested
in integrating along $x^{+}$-axis is:

$f_{x^{+}_{0}}(x^{+})=\left(\frac{1}{4\pi\tau^{2}}\right)^{\frac{1}{2}}e^{-\left(\frac{x^{+}_{0}-x^{+}}{2\tau}\right)^{2}}.$
(6.10)

The integral over $x^{+}$ with integrand $T_{++}$ and $f_{x^{+}_{0}}(x^{+})$
is simple Gaussian integral which yields:

$\begin{split}T^{s}_{++}(\textbf{x}^{+}_{0},\tau,x^{+})&=\int_{w=0}^{w=\infty}dw\int_{w^{\prime}=0}^{w^{\prime}=\infty}dw^{\prime}(-a_{w}a_{w^{\prime}}e^{-ix^{+}_{0}(w+w^{\prime})-\tau^{2}(w+w^{\prime})^{2}}+\\\
&+a^{\dagger}_{w}a_{w^{\prime}}e^{-ix^{+}_{0}(w^{\prime}-w)-\tau^{2}(w-w^{\prime})^{2}}+a^{\dagger}_{w^{\prime}}a_{w}e^{-ix^{+}_{0}(w-w^{\prime})-\tau^{2}(w-w^{\prime})^{2}}-\\\
&-a^{\dagger}_{w}a^{\dagger}_{w^{\prime}}e^{ix^{+}(w^{\prime}+w)-\tau^{2}(w+w^{\prime})^{2}}).\end{split}$
(6.11)

Hereafter, we take (6.11) as our definition for the smeared stress tensor
operator.

####  6.3.2 Fluctuations of Smeared Stress Tensor

The next object we need to compute in order to obtain the variance of the
stress tensor, are the fluctuations of the stress tensor. Much like the
original, not normal ordered $T_{++}$ the UV divergences of $T_{++}T_{++}$ are
due to its VEV. Thus, the vacuum state is the perfect candidate for testing
the validity of our claim that the $\langle
T^{s}_{++}T^{s}_{++}\rangle_{\psi}$ converges. Taking $\langle 0\rvert
T^{s}_{++}T^{s}_{++}\lvert 0\rangle$ one finds that only one term contributes,
namely:

$\begin{split}\langle 0\rvert T^{s}_{++}T^{s}_{++}\lvert
0\rangle&=\int_{w_{i}=0}^{w_{i}=\infty}\prod_{i=1}^{4}dw_{i}\cdot\langle
0\rvert a_{w_{1}}a_{w_{2}}a^{\dagger}_{w_{3}}a^{\dagger}_{w_{4}}\lvert
0\rangle\cdot\\\ &\cdot
e^{-ix^{+}_{0}(w_{1}+w_{2}-w_{3}-w_{4})-\tau^{2}[(w_{1}+w_{2})^{2}+(w_{3}+w_{4})^{2}]}\\\
&=\int_{w_{1}=0}^{w_{1}=\infty}\int_{w_{2}=0}^{w_{2}=\infty}dw_{1}dw_{2}\cdot
w_{1}w_{2}\cdot\\\ &\cdot 2e^{-2\tau^{2}(w_{1}+w_{2})^{2}}\\\
&=\frac{1}{\tau^{4}}\cdot 4.2\cdot 10^{-2}.\end{split}$ (6.12)

We see that (6.12) is indeed finite, and depends only on our cut-off. Now that
we have obtained a finite $\langle T^{s}_{++}T^{s}_{++}\rangle_{\psi}$ we
upgrade our measure of semi-classicality of states:

$\boxed{\frac{\sigma^{2}}{\langle
T^{s}_{kk}(x)\rangle^{2}_{\psi}}=\frac{\langle
T^{s}_{kk}(x)T^{s}_{kk}(x)\rangle_{\psi}-\langle
T^{s}_{kk}(x)\rangle^{2}_{\psi}}{\langle T^{s}_{kk}(x)\rangle^{2}_{\psi}}}$
(6.13)

and finally use it to test our conjecture.

####  6.3.3 Testing the Conjecture

In this section, we present our results for the fluctuations of three
representative classes of states. We first consider the fluctuation of the
very quantum vacuum, which we find to be very big in accordance with our
discussion. Next, we compute our measure of semi-classicality for the NEC
violating 0+2 particles state. We find that the quantity of interest (6.13) is
again $>>1$; a result which supports our conjecture (sec. 6.2). Finally, we
calculate (6.13) for a class of coherent states, which we find to be $<<1$.
Coherent states are semi-classical which, to the best of our knowledge, do not
violate NEC. Hence, this last result tests our definition of semi-classicality
and provides additional evidence that supports the validity of 6.2.

As promised, we first compute (6.13) and measure the quantumness of the
vacuum. Since the variance of the vacuum state is:

$\begin{split}\sigma^{2}_{0}=\langle 0\rvert T^{s}_{++}T^{s}_{++}\lvert
0\rangle-\langle 0\rvert T^{s}_{++}\lvert 0\rangle^{2}=\frac{1}{\tau^{4}}\cdot
4.2\cdot 10^{-2}.\end{split}$ (6.14)

We trivially obtain:

$\frac{\sigma^{2}_{0}}{\langle T^{s}_{kk}(x)\rangle^{2}_{0}}=\infty>>1.$
(6.15)

Thus our result agrees with the generally accepted idea that the vacuum is a
very quantum state. However, the vacuum does not violate NEC and thus the
above result, although interesting, does not validate or falsify our
conjecture. Hence, we turn to the simple 0+2-particles state we considered in
sec. 4.3. The only subtlety here is that if we naively consider the exact same
state as in sec. 4.3 some IR divergences will appear. That is because in the
“continuous limit” we consider for our calculation in the current chapter
(6.3), the state $\lvert 2_{k}\rangle$ is not properly normalized. Taking into
account the commutation relations (6.8), we choose the following
normalization:

$\lvert
2_{k}\rangle=\frac{a^{\dagger}_{k}a^{\dagger}_{k}}{\sqrt[]{2}\leavevmode\nobreak\
w_{k}\leavevmode\nobreak\ L}\lvert 0_{k}\rangle.$ (6.16)

Using the above, we compute the expectation value of our stress energy tensor.

$\begin{split}\langle
T^{s}_{++}\rangle_{0+2}=\frac{w_{k}}{L}\left(1-\sqrt[]{\frac{3}{2}}\cos(2x^{+}_{0}\cdot
w_{k})e^{-4\tau^{2}w^{2}_{k}}\right).\end{split}$ (6.17)

This result is similar to sec. 4.29, but here due to the smearing, the
negative contribution on the r.h.s of eq.(6.17) is exponentially damped. Since
the width $\tau$ controls the area of spacetime in which our source is
contained, eq.(6.17) has an interpretation familiar to us from QIs and ANEC.
The bigger the spacetime region we are considering, the smaller the NEC
violating flux.

To obtain the expression for the fluctuations of the stress tensor in the
0+2-particles state, one needs to go through lengthy wick contractions and
perform some simple Gaussian integrals. After following these algebraic steps,
one has:

$\begin{split}\langle
T^{s}_{++}T^{s}_{++}\rangle_{0+2}&=\frac{1}{L}\Big{(}\frac{w_{k}}{\tau^{2}}e^{-2\tau^{2}w^{2}_{k}}-\frac{w^{2}_{k}}{\tau}\leavevmode\nobreak\
\sqrt[]{\frac{\pi}{2}}\cdot\mbox{Erfc}(\sqrt[]{2}\tau w_{k})+\\\
&+\frac{w^{2}_{k}}{\tau}\cdot\sqrt[]{\frac{\pi}{2}}\cdot(1+\mbox{Erf}(\sqrt[]{2}\tau
w_{k})\big{)}-\\\ &-1.225\frac{w^{2}_{k}}{\tau}\cdot
e^{-2\tau^{2}w^{2}_{k}}\leavevmode\nobreak\ \cos(2x^{+}_{0}\cdot
w_{k})\Big{)}+\\\
&+\frac{w^{2}_{k}}{L^{2}}\Big{(}2+e^{-4\tau^{2}w^{2}_{k}}\Big{)}+\frac{0.0625}{\tau^{4}}\;,\end{split}$
(6.18)

and thus the variance is:

$\begin{split}\sigma^{2}_{0+2}&=\frac{1}{L}\Big{(}\frac{w_{k}}{\tau^{2}}e^{-2\tau^{2}w^{2}_{k}}-\frac{w^{2}_{k}}{\tau}\leavevmode\nobreak\
\sqrt[]{\frac{\pi}{2}}\cdot\mbox{Erfc}(\sqrt[]{2}\tau w_{k})+\\\
&+\frac{w^{2}_{k}}{\tau}\cdot\sqrt[]{\frac{\pi}{2}}\cdot(1+\mbox{Erf}(\sqrt[]{2}\tau
w_{k})\big{)}-\\\ &-1.225\frac{w^{2}_{k}}{\tau}\cdot
e^{-2\tau^{2}w^{2}_{k}}\leavevmode\nobreak\ \cos(2x^{+}_{0}\cdot
w_{k})\Big{)}+\\\
&+\frac{w^{2}_{k}}{L^{2}}\Big{(}2+e^{-4\tau^{2}w^{2}_{k}}\Big{)}+\frac{0.0625}{\tau^{4}}-\\\
&-\frac{w^{2}_{k}}{L^{2}}(1-\sqrt[]{\frac{3}{2}}\cos(2x^{+}_{0}\cdot
w_{k})e^{-4\tau^{2}w^{2}_{k}})^{2}.\end{split}$ (6.19)

Before computing the measure of semi-classicality for this state there are two
factors we should take into account. The first and most relevant one is that
$w_{k}L>>1$. Since we have taken the continuum limit, we secretly assumed that
the density of states is high enough to do so. Thus, $w_{k}$ must be
sufficiently big and since L the size of our space the dimensionless
$w_{k}L>>1$ must indeed be large. So, it is sensible to write the measure of
fluctuations in a way that clarifies which terms are relevant, and which are
not:

$\begin{split}\frac{\sigma^{2}_{0+2}}{\langle
T^{s}_{++}\rangle^{2}_{0+2}}&=\Big{(}\frac{w_{k}L}{\tau^{2}}e^{-2\tau^{2}w^{2}_{k}}-\frac{w^{2}_{k}L}{\tau}\leavevmode\nobreak\
\sqrt[]{\frac{\pi}{2}}\cdot\mbox{Erfc}(\sqrt[]{2}\tau w_{k})+\\\
&+\frac{w^{2}_{k}L}{\tau}\cdot\sqrt[]{\frac{\pi}{2}}\cdot(1+\mbox{Erf}(\sqrt[]{2}\tau
w_{k})\big{)}-\\\ &-1.225\frac{w^{2}_{k}L}{\tau}\cdot
e^{-2\tau^{2}w^{2}_{k}}\leavevmode\nobreak\ \cos(2x^{+}_{0}\cdot w_{k})+\\\
&+w^{2}_{k}\Big{(}2+e^{-4\tau^{2}w^{2}_{k}}\Big{)}+\frac{0.0625L^{2}}{\tau^{4}}-\\\
&-w^{2}_{k}(1-\sqrt[]{\frac{3}{2}}\cos(2x^{+}_{0}\cdot
w_{k})e^{-4\tau^{2}w^{2}_{k}})^{2}\Big{)}/\\\ &\leavevmode\nobreak\
\leavevmode\nobreak\ w^{2}_{k}(1-\sqrt[]{\frac{3}{2}}\cos(2x^{+}_{0}\cdot
w_{k})e^{-2\tau^{2}w^{2}_{k}})^{2}.\end{split}$ (6.20)

If one looks carefully (and/or treats this expression numerically), he will
notice that for all values of $\tau$ the fluctuations are indeed $>>1$.
Nevertheless, we take the interesting limit $w_{k}\rightarrow\frac{1}{\sigma}$
which gives a clear, analytic result:

$\begin{split}\frac{\sigma^{2}_{0+2}}{\langle
T^{s}_{++}\rangle^{2}_{0+2}}&=\Big{(}w_{k}L\leavevmode\nobreak\
e^{-2}-w_{k}L\leavevmode\nobreak\
\sqrt[]{\frac{\pi}{2}}\cdot\mbox{Erfc}(\sqrt[]{2})+\\\
&+w_{k}L\cdot\sqrt[]{\frac{\pi}{2}}\cdot(1+\mbox{Erf}(\sqrt[]{2})\big{)}-\\\
&-1.225\leavevmode\nobreak\ w_{k}L\cdot e^{-2}\leavevmode\nobreak\
\cos(2x^{+}_{0}\cdot w_{k})+\\\
&+\Big{(}2+e^{-4}\Big{)}+0.0625\leavevmode\nobreak\ w^{2}_{k}L^{2}-\\\
&-(1-\sqrt[]{\frac{3}{2}}\cos(2x^{+}_{0}\cdot w_{k})e^{-4})^{2}\Big{)}/\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\
(1-\sqrt[]{\frac{3}{2}}\cos(2x^{+}_{0}\cdot w_{k})e^{-2})^{2}\approx\\\
&\approx\frac{0.0625\leavevmode\nobreak\
w^{2}_{k}L^{2}}{(1-\sqrt[]{\frac{3}{2}}\cos(2x^{+}_{0}\cdot
w_{k})e^{-2})^{2}}+\mathcal{O}(w_{k}\leavevmode\nobreak\ L)>>1.\\\
\end{split}$ (6.21)

It crucial that we also present an example for which the measure of
fluctuations ((6.13)) becomes small ($>>1$). According to our conjecture, this
should be a semi-classical state, and as we have discussed the most natural
candidates are coherent states. To make our life easier we consider a state
which is a superposition of many particles of a single mode. Following Roy J.
Glauber [51] we use the displacement operator $D_{k}(z)$ acting on the vacuum
to create the above mention states.

$\lvert\psi\rangle_{coh}\equiv D_{k}(z)\lvert 0\rangle.$ (6.22)

Where

$D_{k}(z)\equiv e^{(z\leavevmode\nobreak\
a^{\dagger}_{k}-z^{*}\leavevmode\nobreak\ a_{k})}=e^{-\frac{\lvert
z\rvert^{2}}{2}w_{k}\delta(0)}\leavevmode\nobreak\ e^{z\leavevmode\nobreak\
a^{\dagger}_{k}}\leavevmode\nobreak\ e^{-z^{*}\leavevmode\nobreak\ a_{k}},$
(6.23)

and $z=s\leavevmode\nobreak\ e^{i\gamma}$. In (6.23) use made use of the
Baker–Campbell–Hausdorff formula in order compute the commutation relations we
need. Moreover, from this final form of $D_{k}(z)$ in (6.23), one can
immediately observe that the state we obtained is indeed a 1-mode excited
coherent state since the action of $e^{-z^{*}\leavevmode\nobreak\ a_{k}}$ on
the vacuum reduces to that of the identity.

The commutation relations we will need are the following:

$\begin{split}[a_{k},a^{\dagger}_{k^{\prime}}]&=\delta(k-k^{\prime})\leavevmode\nobreak\