w_{k}\;,\\\ [a_{k},D_{k^{\prime}}]&=z\leavevmode\nobreak\
D_{k^{\prime}}\leavevmode\nobreak\ \delta(k-k^{\prime})\leavevmode\nobreak\
w_{k}\;,\\\
[D^{\dagger}_{k^{\prime}},a^{\dagger}_{k}]&=z^{*}\leavevmode\nobreak\
D^{\dagger}_{k^{\prime}}\leavevmode\nobreak\
\delta(k-k^{\prime})\leavevmode\nobreak\ w_{k}\;.\\\ \end{split}$ (6.24)

After performing similar calculations to the above two cases, we obtain the
following expressions for the expectation value of the stress tensor and its
fluctuations:

$\begin{split}\langle T^{s}_{++}\rangle_{coh}=2\leavevmode\nobreak\
w^{2}_{k}\leavevmode\nobreak\ s^{2}\left[1-\cos
2(\gamma-x^{+}_{0}\leavevmode\nobreak\ w_{k})\leavevmode\nobreak\
e^{-4\tau^{2}w^{2}_{k}}\right]\;,\end{split}$ (6.25)

$\begin{split}\langle
T^{s}_{++}T^{s}_{++}\rangle_{coh}&=\frac{0.042}{\tau^{4}}+2\leavevmode\nobreak\
w^{4}_{k}\leavevmode\nobreak\ s^{4}\leavevmode\nobreak\
e^{-8\tau^{2}w^{2}_{k}}+2\leavevmode\nobreak\ w^{4}_{k}\leavevmode\nobreak\
s^{4}\leavevmode\nobreak\
e^{-8\tau^{2}w^{2}_{k}}\cos(4\gamma-x^{+}_{0}w_{k})\\\
&+4S^{4}w^{4}_{k}-8S^{4}w^{4}_{k}\cos
2(\gamma-x^{+}_{0}w_{k})e^{-4\tau^{2}w^{2}_{k}}-\\\
&-\frac{2\leavevmode\nobreak\ w^{2}_{k}\leavevmode\nobreak\
s^{2}\leavevmode\nobreak\ e^{-2\tau^{2}w^{2}_{k}}}{\tau^{2}}\cos
2(\gamma-x^{+}_{0}w_{k})+2s^{2}\frac{w^{2}_{k}}{\tau^{2}}e^{-2\tau^{2}w^{2}_{k}}+\\\
&+\frac{2w^{3}_{k}\leavevmode\nobreak\ s^{2}\leavevmode\nobreak\
e^{-2\tau^{2}w^{2}_{k}}}{\tau}\leavevmode\nobreak\
\sqrt[]{\frac{\pi}{2}}\leavevmode\nobreak\
\left(1+\mbox{Erf}\left(\sqrt[]{2}\leavevmode\nobreak\
w_{k}\leavevmode\nobreak\
\tau\right)-\mbox{Erfc}\left(\sqrt[]{2}\leavevmode\nobreak\
w_{k}\leavevmode\nobreak\ \tau\right)\right)\;.\\\ \end{split}$ (6.26)

It is worth noticing that the quantity $\langle T^{s}_{++}\rangle_{coh}$ is
always positive (6.25), and thus the semi-classical coherent states do not
violate the quantum version of NEC. All that is left, is to compute the
measure of the stress tensor fluctuations, and show that it is indeed $<<1$.

$\begin{split}\frac{\sigma^{2}_{coh}}{\langle
T^{s}_{++}\rangle^{2}_{coh}}&=\frac{\langle
T^{s}_{++}T^{s}_{++}\rangle_{coh}-\langle T^{s}_{++}\rangle^{2}_{coh}}{\langle
T^{s}_{++}\rangle^{2}_{coh}}=\\\
&=\Big{(}\frac{0.042}{\tau^{4}}+2\leavevmode\nobreak\
w^{4}_{k}\leavevmode\nobreak\ s^{4}\leavevmode\nobreak\
e^{-8\tau^{2}w^{2}_{k}}+2\leavevmode\nobreak\ w^{4}_{k}\leavevmode\nobreak\
s^{4}\leavevmode\nobreak\ e^{-8\tau^{2}w^{2}_{k}}\cos
4(\gamma-x^{+}_{0}w_{k})\\\ &+4s^{4}w^{4}_{k}-8s^{4}w^{4}_{k}\cos
2(\gamma-x^{+}_{0}w_{k})e^{-4\tau^{2}w^{2}_{k}}-\\\
&-\frac{2\leavevmode\nobreak\ w^{2}_{k}\leavevmode\nobreak\
s^{2}\leavevmode\nobreak\ e^{-2\tau^{2}w^{2}_{k}}}{\tau^{2}}\cos
2(\gamma-x^{+}_{0}w_{k})+2s^{2}\frac{w^{2}_{k}}{\tau^{2}}e^{-2\tau^{2}w^{2}_{k}}+\\\
&+\frac{2w^{3}_{k}\leavevmode\nobreak\ s^{2}\leavevmode\nobreak\
e^{-2\tau^{2}w^{2}_{k}}}{\tau}\leavevmode\nobreak\
\sqrt[]{\frac{\pi}{2}}\leavevmode\nobreak\
\left(1+\mbox{Erf}\left(\sqrt[]{2}\leavevmode\nobreak\
w_{k}\leavevmode\nobreak\
\tau\right)-\mbox{Erfc}\left(\sqrt[]{2}\leavevmode\nobreak\
w_{k}\leavevmode\nobreak\ \tau\right)\right)-\\\ &-4\leavevmode\nobreak\
w^{4}_{k}\leavevmode\nobreak\ s^{4}\left[1-\cos
2(\gamma-x^{+}_{0}\leavevmode\nobreak\ w_{k})\leavevmode\nobreak\
e^{-4\tau^{2}w^{2}_{k}}\right]^{2}\Big{)}/\langle
T^{s}_{++}\rangle^{2}_{coh}\;.\\\ \end{split}$ (6.27)

To analyze the above expression, we need to understand the role of $s$. It is
easy to understand from (6.23) that $s$ is connected to the number of single
mode excitations of our state, and hence to obtain a coherent state $s$ should
be big. For $s=0$, one gets back the vacuum and the more $s$ grows, the more
coherent the state becomes. Even without taking this into account, the
exponential damping terms may seem enough make these fluctuations small.
Nevertheless, one should also check the interesting limit where these damping
terms reduce to $\mathcal{O}(1)$ factors; namely the limit
$w_{k}\rightarrow\frac{1}{\tau}$. Remembering that $s$ is big, and noticing
that it appears in the denominator of (6.27) as $\mathcal{O}(s^{4})$ we keep
only terms of $\mathcal{O}(s^{4})$ in the numerator, after having taken the
limit of interest. After taking into account the above considerations, we
obtain the following form for the fluctuations measure in a coherent state:

$\begin{split}\frac{\sigma^{2}_{coh}}{\langle
T^{s}_{++}\rangle^{2}_{coh}}&=\Big{(}2\leavevmode\nobreak\
w^{4}_{k}\leavevmode\nobreak\ s^{4}\leavevmode\nobreak\
e^{-8}+2\leavevmode\nobreak\ w^{4}_{k}\leavevmode\nobreak\
s^{4}\leavevmode\nobreak\ e^{-8}\cos 4(\gamma-x^{+}_{0}w_{k})\\\
&+4s^{4}w^{4}_{k}-8s^{4}w^{4}_{k}\cos 2(\gamma-x^{+}_{0}w_{k})e^{-4}-\\\
&-4\leavevmode\nobreak\ w^{4}_{k}\leavevmode\nobreak\ s^{4}\left[1-\cos
2(\gamma-x^{+}_{0}\leavevmode\nobreak\ w_{k})\leavevmode\nobreak\
e^{-4}\right]^{2}\Big{)}/\langle T^{s}_{++}\rangle^{2}_{coh}=\\\
&=0(<<1)\;.\\\ \end{split}$ (6.28)

Hence, even in this extreme limit, the fluctuations are negligible. The above
result provides us with evidence to support our claim that for semi-classical
states the fluctuations of the stress tensor are small, and subsequently also
support our conjecture (sec. 6.2).

####  6.3.4 Proposed Bound

As we have discussed extensively in this chapter (ch. 6), the existence of
states that violate a quantum null energy condition is connected to the
features of the probability distribution of the stress energy tensor’s
eigenvalues in these states. To first order, this reduces to the study of the
said distribution’s variance $\sigma^{2}$. In this spirit, we propose a
quantum-corrected QFT bound which takes into account the relation between the
violations of NEC in some state and $\sigma^{2}$. It is a semi-local bound,
which we believe should hold for all, not just semi-classical, states.

Bound:

$\boxed{\langle T^{s}_{kk}\rangle_{\psi}+\sigma_{\psi}\geq 0}$ (6.29)

Here $\sigma$ is the standard deviation and $\langle T^{s}_{kk}\rangle$ is the
usual expectation value of the null-null component of the smeared stress
tensor, in some state $\psi$. It is important to note that our bound holds not
only for the Gaussian function but for any smearing function satisfying the
characteristics described in sec. 6.3. We have not proven the above bound,
thus it still remains a proposal. Nevertheless, we have yet to see a class of
states which violates it.

Just as in the case of our conjecture, to provide evidence that our bound
holds, we present our results for the vacuum, 0+2 particles state (sec. 4.3)
and the coherent state

For the vacuum, since $\langle T^{s}_{kk}\rangle_{0}=0$, the bound reduced to
the following simple formula:

$\sigma_{0}=\leavevmode\nobreak\ \sqrt[]{\frac{0.042}{\tau^{4}}}\geq 0.$
(6.30)

Next, we present the example of the
$\lvert\psi\rangle=\frac{\sqrt[]{3}}{2}\rvert 0_{k}\rangle+\frac{1}{2}\rvert
2_{k}\rangle$ state. Again here we consider the 1+1-D case, where
$T^{s}_{kk}\equiv T^{s}_{++}$. Remembering our computation in (6.21), we
notice that the contribution of the $\langle T^{s}_{++}\rangle^{2}_{0+2}$ term
is of sub-leading order in $w_{k}\leavevmode\nobreak\ L$, in fact it is of
$\mathcal{O}(1)$. Thus, $\sigma$ is of $\mathcal{O}(w_{k}\leavevmode\nobreak\
L)$ and $\langle T^{s}_{++}\rangle_{0+2}$ is again of $\mathcal{O}(1)$ and our
bound is once again satisfied trivially.

$\begin{split}\sigma_{0+2}+\langle
T^{s}_{++}\rangle_{0+2}\approx\sigma_{0+2}\geq 0.\end{split}$ (6.31)

Finally, again in the relevant limit $w_{k}\rightarrow\frac{1}{\sigma}$, since
the variance for our class of coherent states is zero, the only contribution
to the bound comes from the, always positive, stress tensor:

$\begin{split}\sigma_{coh}+\langle
T^{s}_{++}\rangle_{coh}=2\leavevmode\nobreak\ w^{2}_{k}\leavevmode\nobreak\
s^{2}\left[1-\cos 2(\gamma-x^{+}_{0}\leavevmode\nobreak\
w_{k})\leavevmode\nobreak\ e^{-4\tau^{2}w^{2}_{k}}\right]\geq 0.\end{split}$
(6.32)

##  7 Conclusions

The simple quantum version of NEC is violated in QFT (sec. 4.1), but as we saw
in sec. 4.5 it is harder to violate than expected. In many cases, the NEC
violating states (sec. 4.2, sec. 4.3, squeezed vacuum states etc.) are
superpositions, whose interference takes the form of an oscillatory term
responsible for the violation. However, not all representatives of these
classes violate NEC. Whether the oscillatory term leads to violations of NEC
or not, depends on the normalization of the state; an example where this
becomes apparent is the 0+2 particles states, for which exactly half of the
phase space covered by these states gives rise to NEC violations and the other
half does not [12]. An interesting feature of these states is that they
satisfy ANEC, making them a good candidate for exotic matter sources, in
contrast to states (sec. 4.4, moving mirrors, Hawking radiation etc.) which do
not contain such oscillatory terms and usually lead to more severe violations.

A local version of NEC in QFT is most likely not very useful. It is
interesting that a state dependent, NEC-like, local bound can be proven for
certain QFT’s. However, the bound itself is not strict enough, and cannot be
“tuned” to provide us with information about the non-local scale at which the
NEC violating contribution become negligible. As we have discussed, the more
local the bound, the greater the amount of exotic matter it allows for. We
strongly believe that for a NEC-like bound to make sense in QFT, it should be
semi-local. Hence, we have proposed a bound for a smeared version of the
stress energy tensor, which is (much like QI’s) semi-local. In addition, this
result is unique since it is the only bound that takes into account
characteristic features of the probability distribution of the stress tensor.
Moreover, it has a very simple interpretation in terms of our conjecture:
since the variance of the stress tensor for NEC violating states is large, the
negative contribution due to $\langle T^{s}_{kk}\rangle_{\psi}$ should be
compensated by the standard deviation. However, this raises the question of
whether or not these contributions get overcompensated, making our bound too
strong. Hence before worrying about proving this proposed bound, we must check
which one of the existing bounds is the sharpest one.

So far, there exists no bound in QFT that allows us to generalize all theorems
of GR and exclude Wormholes and other exotic spacetimes. QNEC (sec. 5.1) is a
local QFT bound that does not restrict the amount by which NEC is violated
enough to do the above. The semi-local QI’s are doing a much better job at
that, but there exist solutions in literature which manage to avoid them [58].
A similar statement can be made about ANEC, which is indeed the most
restrictive bound we have, but its power is severely weakened by its strictly
global nature. The only way we know of to salvage everything, at least at the
semi-classical level, is for our conjecture (sec. 6.2) to be true.

However, we should not fail to mention that one can think of a natural
counter-example to this conjecture, which is the Hawking radiation. Two
essential points of the black hole evaporation process are that it takes place
in the semi-classical regime and that NEC is violated. Nevertheless, in [77]
Wu and Ford, using their quantity $\Delta$, calculated that the fluctuations
of the Hawking flux to be big. Whether or not the Hawking radiation
contradicts our conjecture is an interesting question that we would like to
think about in the future.

It is true that some constructs that seem to shape our classical reality are
not inherited by the quantum one (e.g. determinism). In the end, semi-
classical gravity has one foot in the classical world and one in the quantum,
and it may very well be that not everything we like in GR will also work in
semi-classical gravity.

Acknowledgments I am grateful for conversations with Vassilis Anagiannis,
Mikola Schlottke, Isaías Roldán, Vittorio Ricciardulli, Yaroslav Shustrov, and
Erik van Heusden. I would like to thank Diego Hofman and Jan Pieter van der
Schaar for their advise and for stimulating discussions. Finally, I would like
to thank my supervisor Ben Freivogel whose intuition and deep thought guided
me throughout this project.

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