# The double smeared null energy condition

Jackson R. Fliss<EMAIL_ADDRESS>ITFA,  Ben Freivogel<EMAIL_ADDRESS>ITFA,  GRAPPA,
Universiteit van Amsterdam, Science Park 904, 1098 XH Amsterdam, the
Netherlands  Eleni-Alexandra Kontou<EMAIL_ADDRESS>ITFA,  GRAPPA,
Universiteit van Amsterdam, Science Park 904, 1098 XH Amsterdam, the
Netherlands

###### Abstract

The null energy condition (NEC), an important assumption of the Penrose
singularity theorem, is violated by quantum fields. The natural generalization
of the NEC in quantum field theory, the renormalized null energy averaged over
a finite null segment, is known to be unbounded from below. Here, we propose
an alternative, the double smeared null energy condition (DSNEC), stating that
the null energy smeared over two null directions has a finite lower bound. We
rigorously derive DSNEC from general worldvolume bounds for free quantum
fields in Minkowski spacetime. Our method allows for future systematic
inclusion of curvature corrections. As a further application of the techniques
we develop, we prove additional lower bounds on the expectation values of
various operators such as conserved higher spin currents. DSNEC provides a
natural starting point for proving singularity theorems in semi-classical
gravity.

###### Contents

  1. I Introduction
  2. II Renormalization in Quantum field theory
  3. III Derivation of a general Minkowski bound
    1. III.1 Two-dimensional smeared null-energy
    2. III.2 Double smeared null energy in higher dimensions
    3. III.3 The smeared $\phi\phi$ correlator
    4. III.4 Smeared higher-spin currents
  4. IV Worldvolume QNEI
    1. IV.1 A general quantum null energy inequality
    2. IV.2 Timelike smearing
    3. IV.3 Double null smearing
  5. V Boost optimization of the bound
    1. V.1 $4$-dimensions
    2. V.2 General dimensions
    3. V.3 Even dimensions
  6. VI Discussion
  7. A On the validity of the Assumption 1
  8. B SNEC from DSNEC

##  I Introduction

The Null Energy Condition (NEC) is obeyed by all sensible classical theories,
but even the most familiar quantum field theories can violate this condition.
This violation suggests the possible construction of exotic geometries such as
traversable wormholes and bouncing cosmologies in semi-classical gravity.
Therefore it is interesting and relevant to investigate (i) the extent of NEC
violation that is possible in quantum field theory and (ii) what this implies
for physically realizable geometries in semi-classical gravity.

In this paper, we report progress on the first of these fronts. In particular
we prove bounds on ‘smeared’ null energy: the null components of the stress
tensor, averaged over a spacetime region. Our main result can be stated
roughly as follows. Define the operator $T_{--}^{smear}$ by averaging over a
distance $\delta_{+}$ in the $x^{+}$ direction and $\delta_{-}$ in the $x^{-}$
direction. We will sometimes refer to this operator the double (null) smeared
null energy, or “DSNE”. In dimensions higher than 2 this operator is smeared
over only a subspace of the entire spacetime.

In the simple context of free scalar quantum fields in Minkowski spacetime, we
prove that (schematically)

$\expectationvalue{T_{--}^{smear}}\geq-{{\mathcal{N}}_{2}[\gamma]\over(\delta^{+})^{n/2-1}(\delta^{-})^{n/2+1}}$
(1)

where $n$ is the spacetime dimension and $\mathcal{N}_{2}$ is a dimensionless
parameter depending on the number of scalar fields and the details of how the
operator is smeared. For massless fields, $\mathcal{N}_{2}$ is simply
proportional to the number of fields; however, for massive fields
$\mathcal{N}_{2}$ depends on the smearing lengths through the dimensionless
combination of the mass and the smearing lengths,
$\gamma=\delta^{+}\delta^{-}m^{2}$. For small $\gamma$ and smooth smearing
functions, $\mathcal{N}_{2}$ is an $O(1)$ factor times the number of fields.
However, in Fliss:2021gdz  it was shown that for large masses and in a class
of squeezed states $\mathcal{N}_{2}$ becomes exponentially small in $\gamma$.
Here we show, for general states in free theories, that $\mathcal{N}_{2}\to 0$
as $\gamma\to\infty.$ The precise form of our bound is given in equation (49),
and the connection to the schematic form above is demonstrated in (51) and
(52).

The universal, power-law dependence on the smearing lengths $\delta^{\pm}$
follows from symmetry arguments, namely the transformation of $T_{--}$ under
boosts, as well as the overall engineering dimension of the operator. The
nontrivial result is that this operator is indeed bounded from below. This
result was first suggested and coined the “Double Smeared Energy Condition” or
“DSNEC” in Fliss:2021gdz  but was not proven there.

Our result is closely related to the Smeared Null Energy Condition (SNEC)
Freivogel:2018gxj  which constrains the null energy, averaged over a portion
of a single null geodesic. The SNEC, however, does not have a finite field
theory limit as its lower bound diverges when the UV cutoff of the theory goes
to zero. Here, we will see that smearing in the perpendicular null direction
(“fattening” the geodesic slightly) leads to an operator that is bounded below
in quantum field theory. Additionally, we show that the Averaged Null Energy
Condition (ANEC) can be derived from DNEC at the appropriate limit for
$\delta_{-}\to\infty$.

Along the way we develop technology for constructing lower bounds on a wide
class of smeared operators, at least in the context of free scalars on
Minkowski spacetime. As a “test drive” of this technology we also prove
smeared bounds on $\phi^{2}$ expectation values as well as on expectation
values of higher-spin currents, $\mathcal{J}_{--\ldots-}$. For the latter we
arrive at a lower bound morally similar to (1)

$\expectationvalue{\mathcal{J}_{--\ldots-}^{smear}}\geq-{{\mathcal{N}}_{\mathfrak{s}}\over(\delta^{+})^{n/2-1}(\delta^{-})^{n/2-1+\mathfrak{s}}}$
(2)

with dependence on the smearing lengths, $\delta^{\pm}$, fixed by engineering
dimension and the transformation of $\mathcal{J}_{--\ldots-}$ under boosts.
Here $\mathfrak{s}\in 2\mathbb{N}$ is the spin. This bound implies the
“higher-spin ANEC” Hartman:2016lgu , as we will show later in the paper.

In regards to the relevance of these bounds in semi-classical gravity it is
necessary for us to move away from Minkowski space. While we do not attempt to
implement the DSNEC in a full semi-classical setting, as a first step we will
rephrase our bound in the context of an absolute inequality that does not make
use of a reference state (such as the Minkowski vacuum). In particular we show
that a general world-volume inequality proven by Fewster and Smith
Fewster:2007rh  implies the DSNEC for massless fields in Minkowski space. This
world-volume inequality provides a blueprint for the future incorporation of
curvature effects.

All of the above bounds will make use of a fixed momentum space reference
frame. As a final result of this paper we will show how to use this ambiguity
to our advantage by varying the bound over choices of reference frame. To be
specific about the scope of this optimization, we vary over boosts acting on
the domain of positive frequencies. The result of this optimization is a lower
bound with restored Lorentz covariance and with unexpected, non-linear,
dependence on the smearing functions, seen in equation (126). We will show
that in even dimensions we can cast this bound in a simple (though still non-
linear) form in position space (V.3).

A brief summary of the organization of this paper is as follows. Below we
remark on previous work in the realm of null energy bounds in quantum field
theory and will also fix our conventions. In section II we discuss
renormalization schemes in quantum field theory, including normal ordering (in
the context of Minkowski space) and Hadamard renormalization. In section III
we discuss bounding smeared operators in Minkowski space; in this section we
will derive the precise form of the DSNEC (section III.2), bounds on
$\phi^{2}$ (section III.3), and on higher-spin currents (section III.4).
Afterwards, in section IV, we will recast the DSNEC in the context of an
absolute quantum energy inequality. After a brief introduction of the relevant
technology (section IV.1) we will rederive the massless DSNEC and discuss
massive corrections to the bound (section IV.3). Following that, in section V
we perform the optimization over boosted domains and discuss the form of the
lower bound we find. Lastly, in the discussion, section VI, we will discuss
our results in the context of field theory and in semi-classical gravity and
what open questions remain at this stage.

Relation to previous work.

Ford Ford:1978qya  was the first to introduce bounds on the averaged
renormalized energy density and flux of quantum fields now known as Quantum
Energy Inequalities (QEIs). Since then, there has been much progress proving
QEIs for a variety of fields on flat and curved spacetimes (see Kontou:2020bta
and Fewster2017QEIs  for recent reviews). Most of these results are for
averages over timelike curves. Here, we focus on progress on bounds over null
geodesics.

  * •

ANEC: The averaged null energy condition (ANEC) states that the integral of
the null energy (classical or quantum) over an entire null (achronal) geodesic
is non-negative

$\int_{-\infty}^{\infty}d\lambda\,\langle T_{--}\rangle\geq 0\,.$ (3)

The ANEC has been proven for flat Fewster:2006uf  and curved spacetimes
Kontou:2015yha  for free fields using QEIs and in Minkowski spacetime for
interacting fields using general quantum information bounds Faulkner:2016mzt
and causality Hartman:2016lgu . It also follows from the quantum null energy
condition (QNEC) Bousso:2015wca , discussed below, and using holography
Kelly:2014mra , just to mention a few results. In section III.2 we show that
the ANEC follows from the DSNEC. There are no known counterexamples to self-
consistent achronal ANEC in the semi-classical regime.

  * •

Null QEIs: The first null QEIs bounds were obtained in two spacetime
dimensions, starting with Flanagan Flanagan:2002bd  for free fields in flat
and curved spacetimes. Fewster and Hollands Fewster:2004nj  proved a null QEI
for classes of interacting conformal field theories (CFTs), a result recently
generalized to curved spacetimes Freivogel:2020hiz .

In four spacetime dimensions the situation is very different. Fewster and
Roman Fewster:2002ne  showed using an explicit counterexample that finite
lower bounds of null QEIs do not exist. In their work they used a sequence of
vacuum–plus–two–particle states. As the three-momenta of the excited modes
become more and more parallel to the spatial part of the null vector tangent
to the geodesic, the bound diverges to negative infinity.

To circumvent that problem, Freivogel and Krommydas Freivogel:2018gxj
suggested the SNEC

$\int^{+\infty}_{-\infty}d\lambda g^{2}(\lambda)\langle
T_{--}\rangle\geq-\frac{4B}{G_{N}}\int^{+\infty}_{-\infty}d\lambda\left(g^{\prime}(\lambda)\right)^{2}\,,$
(4)

where $g(\lambda)$ is a differentiable ‘smearing function’ that controls the
region where the null energy is averaged, $B$ is an unknown dimensionless
constant and $G_{N}$ is the Newton constant. When gravity is coupled to such
theories, the renormalized $G_{N}$ to 1-loop order is
$G_{N}\sim\ell^{n-2}_{UV}/N$ where $\ell_{UV}$ is the UV cutoff of the theory
and $N$ the number of fields. The presence of the UV cutoff ensures that the
bound remains finite in cases such as the Fewster-Roman counterexample. The
SNEC has been proven for free fields in Minkowski spacetime Fliss:2021gdz  but
such a proof cannot easily be generalized for interacting fields and
spacetimes with curvature. Additionally the bound diverges when the UV cutoff
is taken to zero. We comment on the relationship between DSNEC and SNEC in
Appendix B.

  * •

QNEC: The Quantum Null Energy Condition (QNEC) Bousso:2015mna  is an extension
of the NEC to a local lower bound on the null stress tensor valid in generic
quantum field theories in Minkowski space-time Bousso:2015wca ;
Balakrishnan:2017bjg ; Malik:2019dpg . The QNEC bounds a state’s null energy
at a point by the second variation of the entanglement entropy of the state
reduced on a portion of a null hypersurface with respect to infinitesimal
null-deformations of its entangling surface:

$\langle T_{--}\rangle\geq\frac{1}{2\pi a}S_{ent}^{\prime\prime}$ (5)

where $a$ is the induced area element on the entangling surface at the given
point. In this sense, the QNEC is a state dependent bound of an entirely
different character than discussed in section II since the right-hand side
cannot be written as the expectation value of an operator.

  * •

Singularity theorems: The Penrose singularity theorem Penrose:1964wq  proves
null geodesic incompleteness using as an assumption the NEC thus it is
inapplicable in semiclassical gravity. Efforts to weaken the energy condition
required in the theorem started with the works of Tipler Tipler:1978zz  and
Borde Borde:1987qr . Fewster and Galloway Fewster:2010gm  and more recently
Fewster and Kontou Fewster:2019bjg  proved singularity theorems with
conditions inspired by QEIs. The first semiclassical singularity theorem for
timelike geodesic incompleteness was recently proven Fewster:2021mmz  and the
required initial contraction estimated for cosmological spacetimes. An
analogous theorem for null geodesic incompleteness was proven using SNEC as an
assumption Freivogel:2020hiz . While we do not prove a singularity theorem in
this work, the derivation of DSNEC is partly inspired by the need to have a
semiclassical replacement of the NEC as an assumption to singularity theorems.

  * •

QFC: A different approach is the Quantum Focusing Conjecture (QFC)
Bousso:2015mna  which provides an elegant proposal for how to generalize
singularity theorems to semi-classical gravity by promoting the classical
expansion to a quantum expansion. It depends both on null variations of the
geometric area element and the outer entanglement entropy. Thus it has a
state-dependence of a similar character to the QNEC and in fact the QNEC
follows as a consequence of the QFC. This poses the following issue with the
QFC: the initial condition needed to prove a singularity theorem is that the
non-positive quantum expansion on some surface. However, it is not clear that
the quantum expansion is an observable quantity (again, since the entanglement
entropy cannot be written as the expectation value of an operator). Our
approach is complementary to the QFC and appropriate for proving singularity
theorems from purely geometric aspects of the initial surface.

Conventions

Unless otherwise specified, we work in $n$ spacetime dimensions, assume
$\hbar=c=1$ and use metric signature $(+,-,\ldots,-)$. While we will make
statements involving general metrics, $g_{\mu\nu}$, we will perform concrete
calculations primarily in Minkowski space. When considering null subspaces we
will denote, w.l.o.g., null coordinates111Note importantly a discrepancy in
integration measures $dtdx^{1}=\frac{1}{2}dx^{+}dx^{-}$. In the interest of
comparison to previous results and to be clear on this front, we will always
denote integrations with respect to null-coordinates by $d^{2}x^{\pm}$.
Similarly integrations in momentum space will follow a similar notation:
$d^{2}k_{\pm}:=dk_{+}dk_{-}=\frac{1}{2}dk_{0}dk_{1}$. $x^{\pm}=t\pm x^{1}$ and
transverse coordinates, $\vec{y}=(x^{2},\ldots,x^{n})$:

$ds^{2}=dt^{2}-\sum_{i=1}^{n}(dx^{i})^{2}=dx^{+}dx^{-}-\sum_{a=2}^{n}(dy^{a})^{2}.$
(6)

Null derivatives will be denoted as
$\partial_{\pm}:=\frac{1}{2}\left(\partial_{t}\pm\partial_{1}\right)$. In
momentum space this implies the following notation
$k_{\pm}:=\frac{1}{2}(k_{0}\pm k_{1})$; the inner product with coordinates
remains unchanged,
$k_{\mu}x^{\mu}=k_{0}\,t+k_{i}x^{i}=k_{+}x^{+}+k_{-}x^{-}+k_{a}y^{a}$.

For the Fourier transform we use the following convention

$\tilde{f}(k)=\int_{\mathbb{R}^{n}}d^{n}xf(x)e^{ikx}=\int
dtdx^{1}\,d^{n-2}\vec{y}\,f(x)e^{ikx}\,.$ (7)

Finally, for future reference, we define $f_{\sqrt{\text{G}}}$, the square-
root of a normalized Gaussian with unit variance:

$f_{\sqrt{\text{G}}}(s):=\frac{1}{\left(2\pi\right)^{1/4}}e^{-s^{2}/4}\,.$ (8)

##  II Renormalization in Quantum field theory

We consider the massive minimally coupled classical scalar field $\phi$ with
field equation

$(\Box_{g}+m^{2})\phi=0\,,$ (9)

where $m$ has dimensions of inverse length. The Lagrangian is

$L[\phi]=\frac{1}{2}\left((\nabla\phi)^{2}-m^{2}\phi^{2}\right)\,.$ (10)

Varying the action with respect to the metric gives the stress-energy tensor

$T_{\mu\nu}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}g^{\lambda\rho}\nabla_{\lambda}\phi\nabla_{\rho}\phi+\frac{1}{2}m^{2}g_{\mu\nu}\,.$
(11)

After quantization, our main object of interest is the two point function,

$W_{\psi}(x,x^{\prime})\equiv\langle\phi(x)\phi(x^{\prime})\rangle_{\psi},$
(12)

where $\psi$ is a quantum state of interest. The class of states we consider
in this paper are the Hadamard states KhavkineMoretti-aqft  whose two point-
functions have well-known singularity structures.

We renormalize the stress-energy tensor following the axioms of Hollands and
Wald Hollands:2001nf ; Hollands:2004yh . First let’s define the point-split
stress-energy operator

$\mathbb{T}^{\text{split}}_{\mu\nu^{\prime}}(x,x^{\prime})=\nabla^{(x)}_{\mu}\otimes\nabla^{(x^{\prime})}_{\nu^{\prime}}-\frac{1}{2}g_{\mu\nu^{\prime}}(x,x^{\prime})g^{\lambda\rho^{\prime}}(x,x^{\prime})\nabla^{(x)}_{\lambda}\otimes\nabla^{(x^{\prime})}_{\rho^{\prime}}+\frac{1}{2}m^{2}g_{\mu\nu^{\prime}}(x,x^{\prime})\mathbb{1}\otimes\mathbb{1}\,,$
(13)

where
$g_{\mu\nu^{\prime}}(x,x^{\prime})=g_{\mu\rho}(x)\,{g^{\rho}}_{\nu^{\prime}}(x,x^{\prime})$
is the parallel propagator implementing parallel transport of vectors along
the unique geodesic connecting $x$ and $x^{\prime}$ (we assume the points are
close enough to be in a geodesic convex neighborhood). That is, if $V$ is a
tangent vector on $x^{\prime}$, then the vector at $x$ after parallel
transport along the geodesic is given by

$V^{\mu}(x)={g^{\mu}}_{\nu^{\prime}}(x,x^{\prime})V^{\nu^{\prime}}(x^{\prime})\,.$
(14)

which defines ${g^{\mu}}_{\nu^{\prime}}$. Note that in the coincidence limit

$\lim_{x\to
x^{\prime}}{g^{\mu}}_{\nu^{\prime}}(x,x^{\prime})={g^{\mu}}_{\nu}(x^{\prime})=\delta^{\mu}_{\nu}\,.$
(15)

Then we can define

$\langle T^{\text{fin}}_{\mu\nu}\rangle_{\psi}(x)=\lim_{x^{\prime}\to
x}{g_{\nu}}^{\nu^{\prime}}(x,x^{\prime})\mathbb{T}^{\text{split}}_{\mu\nu^{\prime}}\circ(W_{\psi}-H_{(k)})(x,x^{\prime})\,,$
(16)

where $H_{(k)}$ are terms up to order $k$ of the Hadamard parametrix, a bi-
distribution that encodes the singularity structure of the two-point function
of Hadamard states, expressed as an infinite series. We will discuss this
parametrix in more detail in section IV. We have $W_{\psi}-H_{(k)}\in
C^{2}(X)$ for $k$ large enough, and an ultra regular domain $X$ Fewster:2007rh
. Then
$\mathbb{T}^{\text{split}}_{\mu\nu^{\prime}}\circ(W-H_{(k)})(x,x^{\prime})$ is
defined and continuous along coincident points. As a bit of short-hand for the
future, we will denote the coincident limit of a generic bi-distribution,
$\mathcal{B}(x,x^{\prime})$, as

$\left[\\!\\!\left[\mathcal{B}\right]\\!\\!\right](x^{\prime})=\lim_{x\to
x^{\prime}}\mathcal{B}(x,x^{\prime}).$ (17)

By $\langle T^{\text{ren}}_{\mu\nu}\rangle$ we denote the expectation value of
the renormalized stress-energy tensor following the axioms of Hollands:2001nf
; Hollands:2004yh . The difference of $\langle T^{\text{ren}}_{\mu\nu}\rangle$
between two Hadamard states $\psi$ and $\psi_{0}$ is smooth at the coincident
limit $x\to x^{\prime}$

$\langle T^{\text{ren}}_{\mu\nu}\rangle_{\psi}-\langle
T^{\text{ren}}_{\mu\nu}\rangle_{\psi_{0}}=\left[\\!\\!\left[{g_{\nu}}^{\nu^{\prime}}\,\mathbb{T}^{\text{split}}_{\mu\nu^{\prime}}\circ(W_{\psi}-W_{\psi_{0}})\right]\\!\\!\right]\,.$
(18)

Then any remaining finite renormalization must take the form of a state-
independent conserved local curvature term $C_{\mu\nu}$ that vanishes in
Minkowski space, and the finite renormalized expectation value of the quantum
stress energy is given by

$\langle T^{\text{ren}}_{\mu\nu}\rangle_{\psi}(x)=\langle
T^{\text{fin}}_{\mu\nu}\rangle_{\psi}-Q(x)g_{\mu\nu}(x)+C_{\mu\nu}(x)\,,$ (19)

where $Q$ is a term introduced by Wald Wald:1978pj  to preserve the
conservation of the stress-energy tensor.

In Minkowski space we have a distinguished state, the Minkowski vacuum,
annihilated by the generators of the Poincaré group. We will always denote
this state by $\Omega$. This defines a canonical renormalization scheme via
subtraction by the Minkowski vacuum, i.e. normal ordering. We will denote it
by ${:}\;\;\;{:}$ as is customary

$\langle{:}T_{\mu\nu}{:}\rangle_{\psi}:=\left[\\!\\!\left[{g_{\nu}}^{\nu^{\prime}}\,\mathbb{T}^{\text{split}}_{\mu\nu^{\prime}}\circ(W_{\psi}-W_{\Omega})\right]\\!\\!\right]\,.$
(20)

We will extend this definition to a general operator statement:

${:}\mathcal{O}{:}\equiv\mathcal{O}-\langle\mathcal{O}\rangle_{\Omega}.$ (21)

The Hadamard series coincides with the singularity structure of the Minkowski
vacuum and so in Minkowski space

$\langle{:}T_{\mu\nu}{:}\rangle_{\psi}=\langle T^{\text{fin}}\rangle_{\psi}.$
(22)

When it is clear by context that we are working in Minkowski space (for
example in the next section) we will drop the ${:}\;\;\;{:}$ from $T_{\mu\nu}$
with it being clear the renormalization scheme being used.

Because of the subtraction of divergences, operators that are classically
positive can acquire negative quantum expectation values after
renormalization. It is the goal of this paper to diagnose the magnitude of
this negative expectation value in the form of a lower bound, or a quantum
inequality. The most general form of a quantum inequality that bounds
$\mathcal{O}$ is

$\langle\mathcal{O}(f)\rangle_{\Psi}\geq-\langle\mathcal{Q}(f)\rangle_{\Psi}\,,$
(23)

where $f$ is a non-negative smearing function on spacetime. In general the
operator $\mathcal{Q}(f)$ could be an unbounded operator.

We call difference QEIs the ones where we bound the smooth difference between
the expectation values of two Hadamard states

$\langle\mathcal{O}(f)\rangle_{\Psi}-\langle\mathcal{O}(f)\rangle_{\Psi_{0}}\geq-\langle\mathcal{Q}_{\Psi_{0}}(f)\rangle_{\Psi}\,.$
(24)

In that case the bound can depend on both the reference state $\Psi_{0}$ and
the state of interest $\Psi$. If instead we renormalize using the Hadamard
parametrix the QEI is called absolute. If the reference state is the massless
Minkowski vacuum the two kinds coincide. If the bound depends on $\Psi$ then
it is a state dependent bound. The bounds of interest in this paper are state
independent and take the form

$\langle\mathcal{O}(f)\rangle_{\Psi}-\langle\mathcal{O}(f)\rangle_{\Psi_{0}}\geq-\mathcal{Q}_{\Psi_{0}}(f)\,.$
(25)

##  III Derivation of a general Minkowski bound

In this section we derive a quantum energy inequality for Minkowski spacetime
over a general domain. We want to bound smeared quantities of the form

$A_{\mathcal{O}\mathcal{O}}\equiv\int_{\Sigma_{p}}d^{p}x\,g(x)^{2}\langle:\mathcal{O}(x)^{2}:\rangle_{\psi}$
(26)

where $\Sigma_{p}$ is a $p$-dimensional time-like subspace of
$\mathbb{R}^{1,n-1}$. We will only consider time-like subspaces in this paper
as it is expected that if $\Sigma_{p}$ is a space-like subspace then the right
hand side of any prospective bound will diverge leaving the bound trivial. We
will additionally assume from here on that $\Sigma_{p}$ is flat and
translationally invariant such that fields admit a (partial) Fourier transform
along $\Sigma_{p}$ and denote the space of these momenta as
$\tilde{\Sigma}_{p}$. Using this partial Fourier transform we can point-split
the operators:

$A_{\mathcal{O}\mathcal{O}}=\int_{\tilde{\Sigma}_{p}}\frac{d^{p}\xi}{(2\pi)^{p}}\int_{\Sigma_{p}}d^{p}xd^{p}x^{\prime}\;e^{i\xi\cdot(x-x^{\prime})}g(x)g(x^{\prime})\left(\langle\mathcal{O}(x)\mathcal{O}(x^{\prime})\rangle_{\psi}-\langle\mathcal{O}(x)\mathcal{O}(x^{\prime})\rangle_{\Omega}\right)\,.$
(27)

Now we make the main assumption

Assumption 1: _The commutator of $\mathcal{O}$ with itself is a c-number:_

$[\mathcal{O}(t,\vec{x}),\mathcal{O}(0,\vec{0})]\propto\mathbb{1}$ (28)

_where $\mathbb{1}$ is the identity operator on the Hilbert space._

This assumption is certainly satisfied when $\mathcal{O}$ is a free field or
derivative there-of. More generally Assumption 1 is very constraining and as
explained in Appendix A, it is likely that this assumption is only satisfied
by generalized free fields, i.e. operators whose higher-point functions can be
evaluated via Wick contractions222We thank Tarek Anous and Mert Besken for a
discussion on this point. We pause to note that while for most of this paper
we will focus on free theories, the construction in this section and the bound
(36) are valid e.g. for interacting theories with some large $N$ parameter
that suppresses non-Wick terms in $\mathcal{O}$ higher-point functions by
powers of $1/N$, in the $N\rightarrow\infty$ limit.

Under Assumption 1 the integrand is symmetric under $x\leftrightarrow
x^{\prime}$ and we can then restrict the $k$ integration to a half-
space333That is, under the parity map ${\bf P}:\vec{k}\rightarrow-\vec{k}$ on
$\tilde{\Sigma}_{p}$, $D$ is such that $\tilde{\Sigma}_{p}=D\sqcup{\bf
P}(D)$., $D\subset\tilde{\Sigma}_{p}$. Importantly, the first term of (27) is
a positive-definite regardless of the choice of domain as it can be written as
the inner product

$\int_{D}\frac{d^{p}\xi}{(2\pi)^{p}}\langle\mathcal{O}_{g}(\xi)\psi|\mathcal{O}_{g}(\xi)\psi\rangle\,,\qquad\qquad|\mathcal{O}_{g}(\xi)\psi\rangle:=\int
d^{p}xe^{i\xi x}g(x)\mathcal{O}(x)|\psi\rangle.$ (29)

Thus if we are interested in bounding $A_{\mathcal{O}}$ from below, it
suffices to focus on the second term

$A_{\mathcal{O}\mathcal{O}}\geq-\mathcal{Q}_{\mathcal{O}\mathcal{O}}[D],\ \ \
\ \ \mathcal{Q}_{\mathcal{O}\mathcal{O}}[D]\equiv
2\int_{D}\frac{d^{p}\xi}{(2\pi)^{p}}\int_{\Sigma_{p}}d^{p}xd^{p}x^{\prime}\;g(x)g(x^{\prime})e^{i\xi(x-x^{\prime})}\langle\mathcal{O}(x)\mathcal{O}(x^{\prime})\rangle_{\Omega}\,.$
(30)

It is worth clarifying the role of $D$ in (30): the restriction of momentum
integration to $D$ prohibits the $\xi$ integral from reproducing the delta
function $\delta^{p}(x-x^{\prime})$; this changes the somewhat fictitious
point splitting in (27) to an effective point splitting, softening the contact
divergences in (30) and allowing us to write non-trivial lower bounds, as we
will soon see. Within the regime of validity of our main assumption, (30) is
otherwise fairly general and valid for arbitrary dimensions, masses, etc. If
one knows the vacuum 2-point function in position space, one can pick a domain
$D$ and just integrate.

Let us pause to note some useful choices of domains. The first is what we call
the canonical domain; it is given by the half-space of positive frequencies in
$\tilde{\Sigma}_{p}$:

$D_{0}:=\\{\xi\in\tilde{\Sigma}_{p}\,|\,\xi_{0}\geq 0\\},$ (31)

for which the bounds take the general form

$\mathcal{Q}_{\mathcal{O}\mathcal{O}}[D_{0}]=2\int_{\Sigma_{p}}d^{p}xd^{p}x^{\prime}\;g(x)g(x^{\prime}){i\delta^{p-1}(\vec{x}-\vec{x}^{\prime})\over
t-t^{\prime}+i\epsilon}\langle\mathcal{O}(x)\mathcal{O}(x^{\prime})\rangle_{\Omega}\,.$
(32)

When $p\geq 2$, we also have a family of bounds obtained by boosting $D_{0}$
in (w.l.o.g) the $(\xi^{0},\xi^{1})$ plane by a parameter $\eta\in\mathbb{R}$:

$D_{\eta}:=\\{\xi\in\tilde{\Sigma}_{p}\,|\,\xi_{\eta}:=e^{\eta}\xi_{+}+e^{-\eta}\xi_{-}\geq
0\\}.$ (33)

It is clear that when $p\geq 2$ that $D_{0}$ is a special case of $D_{\eta}$
with $\eta=0$.

In general a choice of domain $D$ breaks any subgroup,
$H_{\tilde{\Sigma}_{p}}\subset SO(1,n-1)$, of Lorentz invariance originally
possessed by $\tilde{\Sigma}_{p}$ and so, unsurprisingly, (30) will depend on
a fixed reference frame. However since (30) applies, at least in principle,
for any domain, $D$, we are free to optimize over choices of $D$,

$A_{\mathcal{O}\mathcal{O}}\geq-\min_{D}\,\mathcal{Q}_{\mathcal{O}\mathcal{O}}[D]$
(34)

and we expect this minimization to restore covariance under
$H_{\tilde{\Sigma}_{p}}$444The argument is the following: any $D_{min}$ found
through a variational principle will be stationary under infinitesimal changes
of frame. This includes infinitesimal $H_{\tilde{\Sigma}_{p}}$
transformations.. In practice, however, this is somewhat unwieldy
minimization; in this paper we will content ourselves with varying over the
much smaller family of boosted domains, $D_{\eta}$, as this provides a
controlled one-parameter minimization over $\eta\in\mathbb{R}$. The
minimization over $D_{\eta}$ will likely not provide the tightest bound, but
will restore covariance under boosts in the $(k_{0},k_{1})$ plane. In general
this will not restore the full $H_{\tilde{\Sigma}_{p}}$ covariance (since this
subgroup could consist of boosts in multiple directions plus rotations in the
internal space) for the two-dimensional time-like domains we primarily
consider in this paper, this boost minimization will restore covariance.

Generally, the position space correlators are not simple objects to work with
(e.g. in massive theories). It will be convenient for us to express the above
entirely in the Fourier space, $\tilde{\Sigma}_{p}$. Translational invariance
of the vacuum implies

$\langle\tilde{\mathcal{O}}(k)\tilde{\mathcal{O}}(k^{\prime})\rangle_{\Omega}:=(2\pi)^{p}\delta^{p}(k+k^{\prime})G_{\mathcal{O}\mathcal{O}}(k^{\prime})$
(35)

for some $G_{\mathcal{O}\mathcal{O}}$. We then obtain

$\boxed{\mathcal{Q}_{\mathcal{O}\mathcal{O}}[D]=2\int_{\tilde{\Sigma}_{p}}\frac{d^{p}k}{(2\pi)^{p}}|\tilde{g}(k)|^{2}\int_{D}\frac{d^{p}\xi}{(2\pi)^{p}}\;G_{\mathcal{O}\mathcal{O}}(k-\xi)}$
(36)

Note the difference in integration regions between $\xi$ and $k$. We have
written this formula in a convenient and general form; below we apply it to
some specific situations.

###  III.1 Two-dimensional smeared null-energy

To begin let’s apply the bound (36) to the null-energy of a two-dimensional
massive scalar smeared over spacetime (i.e. we will take
$\Sigma_{2}=\mathbb{R}^{1,1}$). Indeed, $T_{--}$ can be written in the form
$:\mathcal{O}\mathcal{O}:$,

$T_{--}(x^{+},x^{-})=:\partial_{-}\phi\partial_{-}\phi:(x^{+},x^{-})$ (37)

and so identifying

$G_{\partial_{-}\phi\partial_{-}\phi}(k)=\frac{1}{8}k_{-}\,\Theta(k_{-})\,(2\pi)\delta\left(k_{+}-\frac{m^{2}}{4k_{-}}\right)$
(38)

we have

$\mathcal{Q}_{T_{--}}[D]=2\int_{D}\frac{d^{2}\xi_{\pm}}{(2\pi)^{2}}\int_{0}^{\infty}\frac{d\zeta_{-}}{(2\pi)}\zeta_{-}\,|\tilde{g}(k_{\pm})|^{2}\Big{|}_{k_{-}=\xi_{-}+\zeta_{-},\;\;k_{+}=\xi_{+}+\frac{m^{2}}{4\zeta_{-}}}$
(39)

More specifically, we can investigate $Q_{T_{++}}[D_{\eta}]$ for the boosted
domains, (33):

$\mathcal{Q}_{T_{--}}[D_{\eta}]=2\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}\int_{0}^{\infty}\frac{d\zeta_{-}}{2\pi}|\tilde{g}(k_{\pm})|^{2}\,\zeta_{-}\,\Theta\left(k_{\eta}-e^{-\eta}\zeta_{-}-e^{\eta}\frac{m^{2}}{4\zeta_{-}}\right)$
(40)

where $k_{\eta}=e^{\eta}k_{+}+e^{-\eta}k_{-}$. Doing the linear $\zeta_{-}$
integral between the endpoints of Heaviside domain,
$\frac{1}{2}e^{\eta}\left(k_{\eta}\pm\sqrt{k_{\eta}^{2}-m^{2}}\right),$ we
find

$\mathcal{Q}_{T_{--}}[D_{\eta}]=\frac{e^{2\eta}}{2\pi}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}\,|\tilde{g}(k_{\pm})|^{2}\,k_{\eta}\sqrt{k_{\eta}^{2}-m^{2}}\Theta(k_{\eta}-m)$
(41)

leading to a bound555This bound differs from an apparent factor of 4 from that
appearing in the appendix of Fliss:2021gdz  stemming from a difference in
normalization of the Fourier transform here (equation (7)) and in
Fliss:2021gdz :  $\tilde{g}_{here}(k):=\int d^{2}xe^{ikx}g(x)=\frac{1}{2}\int
d^{2}x^{\pm}\,e^{ikx}g(x)\equiv\frac{1}{2}\tilde{g}_{there}(k).$ (42) This
factor of 4 follows all comparisons to results in Fliss:2021gdz .

$\boxed{\int d^{2}x^{\pm}\,g(x^{\pm})^{2}\langle
T_{--}(x^{\pm})\rangle^{(2d)}_{\psi}\geq-\min_{\eta\in\mathbb{R}}\frac{e^{2\eta}}{\pi}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}\,|\tilde{g}(k_{\pm})|^{2}\,k_{\eta}\sqrt{k_{\eta}^{2}-m^{2}}\,\Theta(k_{\eta}-m)}$
(43)

###  III.2 Double smeared null energy in higher dimensions

We now look to apply (36) to the null-energy smeared in two null directions in
dimensions $n\geq 3$ in what was coined the DSNE in Fliss:2021gdz . Since we
are interested smearing in $x^{\pm}$ the relevant domain, $\Sigma_{2}$, is the
$(x^{+},x^{-})$ plane defined by the level-set
$\Sigma_{2}=\\{(x^{+},x^{-};y^{a}=0)|a=2,\ldots,n-1\\}$. The “partial Fourier
transform” $\tilde{\Sigma}_{p}$ is then spanned by two momenta, $k^{\pm}$.

To be specific we will continue to work with a free massive scalar field. The
relevant momentum space correlator is

$\langle\partial_{-}\phi(k_{+},k_{-},\vec{y}=0)\partial_{-}\phi(k^{\prime}_{+},k^{\prime}_{-},\vec{y}=0)\rangle_{\Omega}=(2\pi)^{2}\delta^{2}(k+k^{\prime})G_{T_{--}}(k^{\prime})$
(44)

with

$G_{T_{--}}(k)=\frac{V_{n-3}}{2(2\pi)^{n-3}}\,k_{-}^{2}(4k_{+}k_{-}-m^{2})^{n-4\over
2}\Theta\left(4k_{+}k_{-}-m^{2}\right)\Theta(k_{-})$ (45)

where $V_{n-3}=(2\pi^{\frac{n-2}{2}})/\Gamma\left(\frac{n-2}{2}\right)$ is the
volume of the angular $S^{n-3}$. The bound on the stress tensor is, for
general mass and dimension,

$\int d^{2}x^{\pm}g(x^{\pm})^{2}\langle
T_{--}(x^{\pm},\vec{y}=0)\rangle_{\psi}\geq-\min_{D}\;\mathcal{Q}_{T_{--}}[D]$
(46)

where we will write

$\mathcal{Q}_{T_{--}}[D]=\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{\pm})|^{2}h_{D}(k_{\pm})$
(47)

so that the function $h_{D}(k)$ encodes the choice of reference frame.
Processing $h_{D}$ a bit, for the boosted domain, $D_{\eta}$,

$\displaystyle h_{D_{\eta}}(k_{\pm})=$
$\displaystyle\frac{8V_{n-3}}{(2\pi)^{n-3}}\int\frac{d^{2}\zeta_{\pm}}{(2\pi)^{2}}\zeta_{-}^{2}(4\zeta_{+}\zeta_{-}-m^{2})^{\frac{n-4}{2}}\Theta(4\zeta_{+}\zeta_{-}-m^{2})\Theta(\zeta_{-})\Theta(k_{\eta}-\zeta_{\eta})$
$\displaystyle=$
$\displaystyle\frac{e^{2\eta}}{(4\pi)^{\frac{n-1}{2}}}\frac{1}{\Gamma\left(\frac{n+1}{2}\right)}\,k_{\eta}(k_{\eta}^{2}-m^{2})^{\frac{n-1}{2}}\Theta(k_{\eta}-m)$
(48)

where $k_{\eta}=e^{\eta}k_{+}+e^{-\eta}k_{-}$ and
$\zeta_{\pm}=k_{\pm}-\xi_{\pm}$. This provides a “first principles” derivation
of the bound suggested in Fliss:2021gdz  by investigating $T_{++}$ expectation
values in squeezed states:

$\boxed{\int_{\Sigma_{2}}d^{2}x^{\pm}g(x^{\pm})^{2}\langle
T_{--}(x^{\pm},\vec{y}=0)\rangle_{\psi}\geq-\min_{\eta\in\mathbb{R}}c_{T_{--}}^{(n)}e^{2\eta}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{\pm})|^{2}\,k_{\eta}(k_{\eta}^{2}-m^{2})^{\frac{n-1}{2}}\Theta(k_{\eta}-m)}$
(49)

with

$c_{T_{--}}^{(n)}=\frac{1}{(4\pi)^{\frac{n-1}{2}}\Gamma\left(\frac{n+1}{2}\right)}\,.$
(50)

To make the matching to Fliss:2021gdz  more explicit, let us call
$g(x^{+},x^{-})=\frac{1}{\sqrt{\delta^{+}\delta^{-}}}\mathcal{F}(x^{+}/\delta^{+},x^{-}/\delta^{-})$,
where $\mathcal{F}(s^{+},s^{-})$ is a function of dimensionless variables
dropping off quickly for $|s^{\pm}|\gg 1$ and normalized to $\int
d^{2}s^{\pm}\mathcal{F}(s^{+},s^{-})^{2}=1$. Calling
$e^{\tilde{\eta}}\equiv\sqrt{\frac{\delta^{-}}{\delta^{+}}}e^{\eta}$ and
$\rho_{\pm}\equiv\delta^{\pm}k_{\pm}$, and denoting
$\rho_{\tilde{\eta}}=e^{\tilde{\eta}}\rho_{+}+e^{-\tilde{\eta}}\rho_{-}$, the
universal power-law dependence on $\delta^{+}$ and $\delta^{-}$ can be scaled
out of the integral:

$\int_{\Sigma_{2}}\frac{d^{2}x^{\pm}}{\delta^{+}\delta^{-}}\mathcal{F}(x^{+}/\delta^{+},x^{-}/\delta^{-})^{2}\langle
T_{--}(x^{\pm},\vec{y}=0)\rangle_{\psi}\geq-\frac{\mathcal{N}_{2,n}[\gamma]}{(\delta^{+})^{\frac{n-2}{2}}(\delta^{-})^{\frac{n+2}{2}}}$
(51)

where

$\mathcal{N}_{2,n}:=\min_{\tilde{\eta}\in\mathbb{R}}c_{T_{--}}^{(n)}e^{2\tilde{\eta}}\int\frac{d^{2}\rho_{\pm}}{(2\pi)^{2}}|\tilde{\mathcal{F}}(\rho_{+},\rho_{-})|^{2}\,\rho_{\tilde{\eta}}(\rho_{\tilde{\eta}}^{2}-\gamma^{2})^{\frac{n-1}{2}}\Theta(\rho_{\tilde{\eta}}-\gamma)$
(52)

is now a dimensionless parameter depending on the (dimensionless) Fourier-
transformed smearing function, $\tilde{\mathcal{F}}(\rho_{\pm}):=\int
d^{2}s\,e^{i\rho_{\pm}s^{\pm}}\mathcal{F}(s^{\pm})$ and the dimensionless
combination of the mass and smearing lengths,
$\gamma^{2}:=\delta^{+}\delta^{-}m^{2}$. This is precisely the form of the
bound proposed in the introduction, (1), and what was referred to as the DSNEC
in Fliss:2021gdz . Note that in Fliss:2021gdz  $\tilde{\eta}$ was implicitly
set to zero however we have left it as tuneable degree of freedom in our bound
above. This suggests, following the discussion at the beginning of this
section, a further optimization over $\tilde{\eta}$. We will do this in
section V.

Equation (52) shows that the prefactor $\mathcal{N}_{2,n}\to 0$ as the mass
becomes large compared to the smearing lengths. To see this, note that as
$\gamma\to\infty$, the theta function restricts the integral on the right side
of (52) to very large $\rho_{\tilde{\eta}}$. (For this discussion, we can just
pick any value of the boost $\tilde{\eta}$.) Since the smearing function
$\tilde{\mathcal{F}}(\rho_{\pm})$ must fall off at large dimensionless momenta
$\rho$, the integrand becomes small in the region of integration, so the
entire expression approaches zero as $\gamma\to\infty$.

ANEC

Having derived the DSNEC we now take a brief opportunity to show that it
implies the ANEC. We want to take the limit $\delta^{+}\to 0$ and
$\delta^{-}\to\infty$ while holding $\delta^{+}\delta^{-}\equiv\alpha^{2}$
fixed. To recover the ANEC limit we require that the smearing function
satisfies

$\lim_{x^{+}\to
0}\lim_{x^{-}\to\infty}\frac{1}{\delta^{+}}\mathcal{F}(x^{+}/\delta^{+},x^{-}/\delta^{-})^{2}=A\delta(x^{+}-\beta)\,,$
(53)

where $A$ and $\beta$ are real numbers. An example of such a function that
satisfies (53) is the Gaussian,
$\mathcal{F}(s^{+},s^{-})=f_{\sqrt{G}}(s^{+})f_{\sqrt{G}}(s^{-})$. Then
Eq.(51) becomes

$\int_{-\infty}^{\infty}dx^{-}\mathcal{\langle}T_{--}(x^{+}=\beta,x^{-},\vec{y}=0)\rangle_{\psi}\geq-\lim_{\delta^{+}\to
0}\frac{\mathcal{N}_{2,n}[\gamma]}{A\alpha^{n}}\delta^{+}=0\,.$ (54)

We note that $\mathcal{N}_{2,n}$ remains fixed in this limit.

###  III.3 The smeared $\phi\phi$ correlator

Though the main focus of this paper is on smeared null-energy, we comment on
the generality of (30) by applying to two additional situations in this
section and in section III.4. To start, we can posit a bound on the smeared
correlator of the $n$-dimensional massive scalar smeared over two null
dimensions, $\Sigma_{2}$:

$\int
d^{2}x^{\pm}\,g(x^{\pm})^{2}\,\langle\phi^{2}(x^{+},x^{-},\vec{y}=0)\rangle_{\psi}\geq-\min_{D}\mathcal{Q}_{\phi\phi}[D]$
(55)

We start with the “partial Fourier transform” of the vacuum correlator

$\langle\phi(k_{+},k_{-},\vec{y}=0)\phi(k_{+}^{\prime},k_{-}^{\prime},\vec{y}=0)\rangle_{\Omega}=(2\pi)^{2}\delta^{2}(k+k^{\prime})G_{\phi\phi}(k^{\prime})$
(56)

with

$G_{\phi\phi}(k)=\frac{V_{n-3}}{2(2\pi)^{n-3}}(4k_{+}k_{-}-m^{2})^{n-4\over
2}\Theta\left(4k_{+}k_{-}-m^{2}\right)\Theta(k_{+})\,.$ (57)

Writing

$\mathcal{Q}_{\phi\phi}[D]=\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{\pm})|^{2}h_{D}(k_{\pm})$
(58)

then by similar techniques to the previous section, for the boosted domains,
$D_{\eta}$ we find

$h_{D_{\eta}}(k_{\pm})=\frac{4}{(4\pi)^{n/2}}\Theta(k_{\eta}-m)k_{\eta}^{n-2}\int_{m^{2}/k_{\eta}^{2}}^{1}dw\,\left(w-\frac{m^{2}}{k_{\eta}^{2}}\right)^{\frac{n-4}{2}}\log\left(\frac{1+\sqrt{1-w}}{1-\sqrt{1-w}}\right)\,,$
(59)

where we changed variables to $w=4\zeta_{+}\zeta_{-}/k_{\eta}$. When the field
is massless, the dependence on $k_{\eta}$ is completely power-law and the
integral can be evaluated when exactly for $n\geq 3$ (when $n=2$ the integral
diverges leaving the bound trivial). The end result is

$\boxed{\int_{\Sigma_{2}}d^{2}x^{\pm}\,g(x^{\pm})^{2}\langle\phi^{2}(x^{\pm},\vec{y}=0)\rangle_{\psi}\Big{|}_{m^{2}=0}\geq-\min_{\eta\in\mathbb{R}}c_{\phi\phi}^{(n)}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{\pm})|^{2}\,k_{\eta}^{n-2}\Theta(k_{\eta})}$
(60)

with

$c_{\phi\phi}^{(n)}=\frac{4}{(4\pi)^{\frac{n-1}{2}}}\left(\frac{n-4}{n-2}\right)\frac{\Gamma\left(\frac{n-4}{2}\right)}{\Gamma\left(\frac{n-2}{2}\right)\Gamma\left(\frac{n-1}{2}\right)}.$
(61)

and $k_{\eta}=e^{\eta}k_{+}+e^{-\eta}k_{-}$. To our knowledge, neither this
bound (or its massive counterpart, (59)) have appeared explicitly in the
literature.

###  III.4 Smeared higher-spin currents

As a final example, we use (30) to derive a similar double-smeared bound on
null higher-spin currents. To be specific, we will focus on the massless
scalar (again in $n$ dimensions). As is well-known, there are a tower of even-
spin conserved currents with null-components given by

$\mathcal{J}_{--\ldots-}=:\partial_{-}^{\mathfrak{s}/2}\phi\,\partial_{-}^{\mathfrak{s}/2}\phi:\qquad\qquad\mathfrak{s}\;\;\text{even}$
(62)

up to total derivative. Given the above recipe, bounding
$\mathcal{J}_{--\ldots-}$ only amounts a modification in the kernel $G$:

$G_{\mathcal{J}_{--\ldots-}}(k)=\frac{V_{n-3}}{2(2\pi)^{n-3}}\,k_{-}^{\mathfrak{s}}\,(4k_{+}k_{-})^{\frac{n-4}{2}}\Theta(k_{+})\Theta(k_{-}).$
(63)

By similar mathematics as above, the associated $h_{D_{\eta}}(k)$ can be
evaluated as

$\displaystyle h_{D_{\eta}}(k)=$ $\displaystyle\frac{4}{(4\pi)^{n/2}}\int
dq_{-}du\,q_{-}^{\mathfrak{s}-1}u^{\frac{n-4}{2}}\Theta(\zeta_{-})\Theta(u)\Theta\left(k_{\eta}-e^{-\eta}\,\zeta_{-}-e^{\eta}\frac{u}{4\zeta_{-}}\right)$
$\displaystyle=$
$\displaystyle\frac{e^{\mathfrak{s}\eta}}{2^{\mathfrak{s}-4}(4\pi)^{n/2}\mathfrak{s}}\left(\sum_{\ell\;\text{odd}}^{\mathfrak{s}-1}\binom{\mathfrak{s}}{\ell}\frac{\Gamma\left(\frac{\ell+2}{2}\right)}{\Gamma\left(\frac{\ell+n}{2}\right)}\right)k_{\eta}^{\mathfrak{s}+n-2}\Theta(k_{\eta})$
(64)

leading to a bound of the following form

$\boxed{\int_{\Sigma_{2}}d^{2}x^{\pm}\,g(x^{\pm})^{2}\,\langle{\mathcal{J}}_{--\ldots-}(x^{\pm},\vec{y}_{\perp}=0)\rangle_{\psi}\geq-\min_{\eta\in\mathbb{R}}c^{(n)}_{\mathfrak{s}}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k)|^{2}\,e^{\mathfrak{s}\eta}\,k_{\eta}^{\mathfrak{s}+n-2}\Theta(k_{\eta})}$
(65)

with

$c^{(n)}_{\mathfrak{s}}=\frac{1}{2^{\mathfrak{s}-4}(4\pi)^{n/2}\,\mathfrak{s}}\sum_{k\;\text{odd}}^{\mathfrak{s}-1}\binom{\mathfrak{s}}{k}\frac{\Gamma\left(\frac{k+2}{2}\right)}{\Gamma\left(\frac{k+n}{2}\right)}.$
(66)

and again, $k_{\eta}=e^{\eta}k_{+}+e^{-\eta}k_{-}$.

HSANEC

We can put (65) in a similar form to that of the DSNEC by again defining the
smearing lengths explictly in our smearing function,
$g(x^{+},x^{-})=\frac{1}{\sqrt{\delta^{+}\delta^{-}}}\mathcal{F}(x^{+}/\delta^{+},x^{-}/\delta^{-})$,
for some dimensionless smooth function $\mathcal{F}(s^{+},s^{-})$ dropping off
quickly for $|s^{\pm}|\gg 1$. Rescaling our boost parameter
$e^{\tilde{\eta}}:=\sqrt{\frac{\delta^{-}}{\delta^{+}}}e^{\eta}$ and
integration variable $\rho_{\pm}=\delta^{\pm}k_{\pm}$ the bound takes the
schematic form

$\int\frac{d^{2}x^{\pm}}{\delta^{+}\delta^{-}}\mathcal{F}(x^{+}/\delta^{+},x^{-}/\delta^{-})^{2}\langle\mathcal{J}_{--\ldots-}(x^{\pm},\vec{y}_{\perp}=0)\rangle_{\psi}\geq-\frac{\mathcal{N}_{\mathfrak{s},n}}{(\delta^{+})^{\frac{n-2}{2}}(\delta^{-})^{\frac{n-2}{2}+s}}$
(67)

where $\mathcal{N}_{\mathfrak{s},n}$ is an O(1) dimensionless factor depending
on the details of the smearing function:

$\mathcal{N}_{\mathfrak{s},n}=\min_{\tilde{\eta}\in\mathbb{R}}c_{\mathfrak{s}}^{(n)}\int\frac{d^{2}\rho_{\pm}}{(2\pi)^{2}}\,e^{s\tilde{\eta}}|\tilde{\mathcal{F}}(\rho_{\pm})|^{2}\rho_{\tilde{\eta}}^{s+n-2}\Theta(\rho_{\tilde{\eta}}).$
(68)

Now once again we can let
$\mathcal{F}(s^{+},s^{-})=f_{\sqrt{\text{G}}}(s^{+})f_{\sqrt{\text{G}}}(s^{-})$
factorize where $f_{\sqrt{\text{G}}}$ is the square-root of the normalized
Gaussian with unit variance, (8). We then multiply both sides of the bound by
$\delta^{-}$ and take the limit $\delta^{+}\rightarrow 0$ while holding
$\alpha\equiv\delta^{+}\delta^{-}$ fixed. In this limit we recover the HSANEC
proposed by Hartman:2016lgu

$\int
dx^{-}\langle\mathcal{J}_{--\ldots-}(x^{+}=0,x^{-},\vec{y}_{\perp}=0)\rangle_{\psi}\geq-\lim_{\delta^{+}\rightarrow
0}\frac{\sqrt{2\pi}\mathcal{N}_{\mathfrak{s},n}}{\alpha^{\frac{n-2}{2}+s-1}}\,(\delta^{+})^{\mathfrak{s}-1}=0\,.$
(69)

##  IV Worldvolume QNEI

Having explained the method for deriving the DSNEC (among other bounds) as
difference inequalities in Minkowski space, we will show in this section how
this bound is implied by an existing absolute QEI Fewster:2007rh  averaged
over a spacetime worldvolume. This QEI is valid for general curved spacetimes,
however here we focus on Minkowski space and will only comment on curvature
effects in the discussion, section VI. First, we will describe the QEI in
question. Then, we use it to obtain a familiar timelike averaged bound first
derived by Fewster and Roman Fewster:2002ne  in 4 dimensions as a pedagogical
example. We will then proceed to apply the QEI to the DSNE, confirming the
bounds described in Sec. III.2.

###  IV.1 A general quantum null energy inequality

We start by stating the general form of the QEI of Ref. Fewster:2007rh :

$\displaystyle\int_{\Sigma}d\text{vol}(x)g^{2}(x)\left[\\!\\!\left[\mathcal{D}\otimes\mathcal{D}(W_{\Psi}-H_{(k)})\right]\\!\\!\right]\geq$
$\displaystyle\qquad\qquad-2\int_{\mathcal{D}}\frac{d^{n}\xi}{(2\pi)^{n}}\left[|h_{\kappa}|^{1/4}g_{\kappa}\otimes|h_{\kappa}|^{1/4}g_{\kappa}\vartheta_{\kappa}^{*}(\mathcal{D}\otimes\mathcal{D}\tilde{H}_{(k)})\right]^{\wedge}(-\xi,\xi)\,,$
(70)

where $\mathcal{D}$ is partial differential operator of order at most one with
smooth real-valued coefficients. For convenience, we have introduced a
notation $[\cdot]^{\wedge}(\xi_{1},\xi_{2})$ for a bi-Fourier transform in two
arguments $x$ and $x^{\prime}$, i.e.

$[B]^{\wedge}(\xi_{1},\xi_{2}):=\int
d^{n}x\,d^{n}x^{\prime}\,e^{i\xi_{1}x+\xi_{2}x^{\prime}}B(x,x^{\prime})\,.$
(71)

The Hadamard bi-distribution $H$, expressed as an infinite series in even
dimensions, is given by Decanini:2005eg

$H(x,x^{\prime})=\frac{\Gamma\left(\frac{n-2}{2}\right)}{4\pi^{n/2}}\left\\{\frac{U(x,x^{\prime})}{\sigma_{+}(x,x^{\prime})^{n/2-1}}+V(x,x^{\prime})\ln{\left[\frac{\sigma_{+}(x,x^{\prime})}{\ell^{2}}\right]}+W(x,x^{\prime})\right\\}\,,$
(72)

where $\ell$ is an arbitrary length scale, and for odd dimensions

$H(x,x^{\prime})=\frac{\Gamma\left(\frac{n-2}{2}\right)}{4\pi^{n/2}}\left\\{\frac{U(x,x^{\prime})}{\sigma_{+}(x,x^{\prime})^{n/2-1}}+W(x,x^{\prime})\right\\}\,.$
(73)

The bi-distributions $U(x,x^{\prime})$ and $V(x,x^{\prime})$ are regular in
the coincidence limit and can be expressed as power-series in $\sigma$

$U(x,x^{\prime})=\sum_{\ell=0}^{\infty}U_{\ell}(x,x^{\prime})\sigma(x,x^{\prime})^{\ell}\,,\qquad\qquad
V(x,x^{\prime})=\sum_{\ell=0}^{\infty}V_{\ell}(x,x^{\prime})\,\sigma(x,x^{\prime})^{\ell}\,,$
(74)

with symmetric coefficients calculated uniquely by requiring that $H$ obeys
the field equation (9) at each order with the appropriate boundary conditions
Decanini:2005eg . In contrast,

$W(x,x^{\prime})=\sum_{\ell=0}^{\infty}W_{\ell}(x,x^{\prime})\,\sigma(x,x^{\prime})^{\ell}\,,$
(75)

are not uniquely specified as $W_{0}(x)$ is undetermined. This coefficient
depends on the state of the quantum field and once it is fixed the
$W_{\ell}$’s can also be determined using the recursion relations derived from
the field equation.

We will specify the order of the Hadamard series using the following
convention: by $H_{k}$ we denote the term of order $\sigma^{k}$ 666By
convention terms of the form $\log{\sigma}$ are order zero. while by $H_{(k)}$
all the terms up to order $k$. In Ref. Fewster:2007rh , it was required that
$k=\max\\{n+3,5\\}$ for the QEI of Eq. (IV.1). However, in Ref. Kontou:2014tha
it was shown that only $k=2$ are needed for a first order differential
operator.

We define

$\tilde{H}(x,x^{\prime})=\frac{1}{2}[H(x,x^{\prime})+H(x^{\prime},x)+iE(x,x^{\prime})]\,,$
(76)

where $E(x,x^{\prime})$ is the antisymmetric part of the two-point function.

The function $\sigma$ is the squared invariant length of the geodesic between
$x$ and $x^{\prime}$, negative for timelike separation. In flat space

$\sigma(x,x^{\prime})=-\eta_{\mu\nu}(x-x^{\prime})^{\mu}(x-x^{\prime})^{\nu}\,,$
(77)

where $\eta_{\mu\nu}$ is the Minkowski metric. By $F(\sigma_{+})$, for some
distribution $F$, we mean the distributional limit

$F(\sigma_{+})=\lim_{\epsilon\to 0^{+}}F(\sigma_{\epsilon})\,,$ (78)

where

$\sigma_{\epsilon}(x,x^{\prime})=\sigma(x,x^{\prime})+2i\epsilon(t(x)-t(x^{\prime}))+\epsilon^{2}\,.$
(79)

Following Ref. Fewster:2007rh  and Fewster:2018pey  we define a _small
sampling domain_. A small sampling domain $\Sigma$ is defined to be an open
subset of $(\mathcal{M},g)$ that (i) is contained in a globally hyperbolic
convex normal neighbourhood of $M$, (ii) may be covered by a single
_hyperbolic coordinate chart_ $\\{x^{\mu}\\}$, which requires that
$\partial/\partial x^{0}$ is future pointing and timelike and that there
exists a constant $c>0$ such that

$c|u_{0}|\geq\sqrt{\sum_{j=0}^{3}u_{j}^{2}}$ (80)

holds for the components of every causal covector, $u$, at each point of
$\Sigma$. That statement means that the coordinate speed of light is bounded.
Now we may express the hyperbolic chart $\\{x^{\mu}\\}$ by a map $\kappa$
where $\Sigma\to\mathbb{R}^{n}$,
$\kappa(p)=(x^{0}(p),x^{1}(p),\ldots,x^{n-1}(p))$. Any function $g$ on
$\Sigma$ determines a function $g_{\kappa}=g\circ\kappa^{-1}$ on
$\Sigma_{\kappa}=\kappa(\Sigma)$. In particular, the inclusion map
$\iota:\Sigma\to\mathcal{M}$ induces a smooth map
$\iota_{\kappa}:\Sigma_{\kappa}\to\mathcal{M}$. We have
$\vartheta:\Sigma\times\Sigma\to\mathcal{M}\times\mathcal{M}$ the map
$\vartheta(x,x^{\prime})=(\iota\otimes\iota)(x,x^{\prime})$. Here
$h=\iota^{*}g$ is a Lorentzian metric on $\Sigma$ and $h_{\kappa}$ is the
determinant of the matrix $\kappa^{*}h$. Then the bundle $\mathcal{N}^{+}$ of
non-zero future pointing null covectors on $(M,g)$ pulls back under
$\iota_{\kappa}$ so that

$\iota^{*}_{\kappa}\mathcal{N}^{+}\subset\Sigma_{\kappa}\times D\,,$ (81)

where $D\subset\mathbb{R}^{n}$ is the set of all $u_{a}$ that satisfy Eq.
(80). As in Minkowski space there is some freedom in choosing $D$. One example
of appropriate $D$ is $D_{0}$ which is the set of all $u_{a}$ with $u_{0}>0$
so it is a proper subset of the upper half space
$\mathbb{R}^{+}\times\mathbb{R}^{n-1}$.

To conclude the introduction of this general QEI we should note that it is not
covariant in full generality because it depends on the coordinates used and
the choice of tetrad near $\Sigma$. However, Ref. Fewster:2007rh  following
methods of Fewster:2006kt  showed that covariance can be rescued if we
restrict the freedom to choose coordinates and the tetrad in a covariant
fashion.

Now we state a specific example of the QEI of Eq. (IV.1) where the
differential operator $\mathcal{D}=\ell^{\mu}\nabla_{\mu}$, where $\ell^{\mu}$
is a future pointing null vector, thus the quantity bounded is the null
energy. The bound has the form

$\displaystyle\int_{\mathcal{M}}d\text{vol}g^{2}(x)\langle
T^{\text{ren}}_{\mu\nu}\ell^{\mu}\ell^{\nu}\rangle_{\psi}\geq$
$\displaystyle\qquad-2\int_{D}\frac{d^{n}\xi}{(2\pi)^{n}}\left[|h_{\kappa}|^{1/4}g_{\kappa}\otimes|h_{\kappa}|^{1/4}g_{\kappa}\left((\ell^{\mu}\nabla_{\mu}\otimes\ell^{\nu^{\prime}}\nabla_{\nu^{\prime}}\tilde{H}_{(2)}\right)_{\kappa}\right]^{\wedge}(-\xi,\xi)$
$\displaystyle\qquad\qquad+\int_{\mathcal{M}}d\text{vol}g(x)^{2}\,C_{\mu\nu}\ell^{\mu}\ell^{\nu}\,.$
(82)

Massless fields

Let us first discuss the bound for massless fields in Minkowski space for
which Eq. (IV.1) becomes

$\int_{\mathcal{M}}d\text{vol}g(x)^{2}\langle
T^{\text{ren}}_{\mu\nu}\ell^{\mu}\ell^{\nu}\rangle_{\psi}\geq-2\int_{D}\frac{d^{n}\xi}{(2\pi)^{n}}\left[(g\ell^{\mu}\nabla_{\mu}\otimes
g\ell^{\nu^{\prime}}\nabla_{\nu^{\prime}})\tilde{H}_{-n/2+1}\right]^{\wedge}(-\xi,\xi)\,,$
(83)

where only the most singular term of (72)-(73) is relevant:

$\tilde{H}_{-n/2+1}(x,x^{\prime})=H_{-n/2+1}(x,x^{\prime})=\frac{\Gamma\left(\frac{n-2}{2}\right)}{4\pi^{n/2}}\frac{1}{\sigma_{+}(x,x^{\prime})^{n/2-1}}\,.$
(84)

Since we are in Minkowski space the timelike curve can be parametrized by $t$.
Then we can define $\Delta t=t-t^{\prime}$ and
$\Delta\vec{x}=\vec{x}-\vec{x}^{\prime}$

$H_{-n/2+1}(x,x^{\prime})=\frac{(-1)^{n/2-1}\Gamma\left(\frac{n-2}{2}\right)}{4\pi^{n/2}((\Delta
t-i\epsilon)^{2}-|\Delta\vec{x}|^{2})^{n/2-1}}\,.$ (85)

To proceed we pick the direction $(-)$ for the null vector $\ell^{\mu}$
defined by $x^{-}=t-x$ while $x^{+}=t+x$. Then

$\ell^{\mu}\partial_{\mu}=\partial_{-}=\frac{1}{2}(\partial_{t}-\partial_{x})\,,$
(86)

and

$H_{-n/2+1}(x,x^{\prime})=\frac{(-1)^{n/2-1}\Gamma\left(\frac{n-2}{2}\right)}{4\pi^{n/2}(\Delta
x^{-}\Delta x^{+}-2i\epsilon\Delta t-\Delta y^{2})^{n/2-1}}\,,$ (87)

where we remind the reader that $y$ denotes the transverse spatial dimensions.
Applying the derivatives gives

$(\ell^{\mu}\partial_{\mu})(\ell^{\nu^{\prime}}\partial_{\nu^{\prime}})\tilde{H}_{-n/2+1}(x,x^{\prime})=\frac{(-1)^{n/2}\Gamma\left(\frac{n+2}{2}\right)(\Delta
x^{+})^{2}}{4\pi^{n/2}(\Delta x^{-}\Delta x^{+}-2i\epsilon\Delta t-\Delta
y^{2})^{n/2+1}}\,.$ (88)

So we have

$\displaystyle\int_{\mathcal{M}}d\text{vol}g(x)^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq$ (89)
$\displaystyle-\frac{(-1)^{n/2}\Gamma\left(\frac{n+2}{2}\right)}{2\pi^{n/2}}\int_{D}\frac{d^{n}\xi}{(2\pi)^{n}}\left[g(x)g(x^{\prime})\frac{(\Delta
x^{+})^{2}}{(\Delta x^{-}\Delta x^{+}-2i\epsilon\Delta t-\Delta
y^{2})^{n/2+1}}\right]^{\wedge}\\!\\!(-\xi,\xi)\,.$

Massive fields

For massive fields there are additional singular terms in the Hadamard
parametrix that are relevant for the renormalization of the stress tensor. The
details of these terms are dimension dependent and are fixed requiring the
expansion (74) to satisfy the massive field equation at each order. Likewise
the number of terms relevant for renormalizing the stress tensor is dimension
dependent. In particular, following the standard Hadamard renormalization
prescription of subtracting only the singular terms (what one might regard as
a minimal subtraction scheme) only a finite number of coefficients $U_{\ell}$
and $V_{\ell}$ are relevant. We emphasize that in Minkowski space this is, in
principle, a different scheme than normal ordering: since the Minkowski two-
point function is typically a transcendental function of the mass, subtraction
by the vacuum expectation value is a subtraction in all orders of $m^{2}$.
Since these two schemes differ only by terms vanishing in the coincident limit
they yield the same local operator, $T^{\text{ren}}_{--}$, however for the
derivation of a lower bound we will find different results. To see this, in
section IV.3 we will construct the corresponding lower bound implied by (IV.1)
for the 4d massive scalar and discuss it in comparison to the Minkowski
difference inequality (49).

###  IV.2 Timelike smearing

As a brief check on the content of (89), let us reproduce the time-like null-
energy bound in 4d first derived in Fewster:2002ne ; this will also provide a
blue-print calculation for the double smeared quantities to follow. To
implement a worldline smearing we will take

$g(t,\vec{x})^{2}=g_{0}(t)^{2}\delta^{3}(\vec{x})\,.$ (90)

To take the “square-root” of the delta function, we will regard it is as the
limit of a sharply-peaked Gaussian, i.e.

$g(t,\vec{x})=\lim_{\sigma\rightarrow
0}g_{0}(t)\frac{1}{\sigma^{3/2}({2}\pi)^{3/4}}e^{-\frac{\vec{x}^{2}}{{4}\sigma^{2}}}\,,$
(91)

where its Fourier transform is given by

$\tilde{g}(\omega,\vec{k})=\lim_{\sigma\to
0}\tilde{g}_{0}(\omega)\sigma^{3/2}\pi^{3/4}2^{9/4}e^{-\sigma^{2}\vec{k}^{2}}\,.$
(92)

From Eq. (89) for $n=4$, The bound on the time-like smeared null-energy then
is

$\displaystyle\int dtg_{0}(t)^{2}\langle
T^{\text{ren}}_{--}(t)\rangle_{\psi}\geq-\lim_{\sigma\rightarrow
0}\frac{\sigma^{3}2^{9/2}}{\pi^{1/2}}\int_{D}\frac{d^{4}\xi}{(2\pi)^{4}}\int\frac{d\omega}{(2\pi)}\frac{d\omega^{\prime}}{(2\pi)}\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{d^{3}\vec{k}^{\prime}}{(2\pi)^{3}}\,$
$\displaystyle\quad\int
d^{4}x\,d^{4}x^{\prime}e^{-\sigma^{2}\vec{k}^{2}-\sigma^{2}\vec{k}^{\prime
2}}e^{i\omega
t+i\omega^{\prime}t^{\prime}}e^{i\vec{k}\vec{x}+i\vec{k}^{\prime}\vec{x}^{\prime}}e^{-i\xi\Delta
x}\tilde{g}_{0}(\omega)\tilde{g}_{0}(\omega^{\prime})\frac{(\Delta
x^{+})^{2}}{(\Delta x^{-}\Delta x^{+}-2i\epsilon\Delta t-\Delta
y^{2})^{3}}\,.$

We will shift the $x$ integral to $s\equiv\Delta x=x-x^{\prime}$. The
$x^{\prime}$ integral yields the delta functions
$(2\pi)^{4}\delta(\omega+\omega^{\prime})\delta^{3}(\vec{k}+\vec{k}^{\prime})$,
which we then collapse and perform the Gaussian integration over $\vec{k}$:

$\displaystyle\int dtg_{0}(t)^{2}\langle
T^{\text{ren}}_{--}(t)\rangle_{\psi}\geq-\lim_{\sigma\rightarrow 0}$
$\displaystyle\frac{1}{\pi^{2}}\int_{D}\frac{d^{4}\xi}{(2\pi)^{4}}\int\frac{d\omega}{(2\pi)}|\tilde{g}_{0}(\omega)|^{2}$
(94) $\displaystyle\qquad\qquad\times\int d^{4}s\,e^{i\omega s_{0}-i\xi
s}e^{-\frac{\vec{s}^{2}}{8\sigma^{2}}}\frac{(s^{+})^{2}}{(s^{-}s^{+}-2i\epsilon
s_{0}-s_{y}^{2})^{3}}.$

We now make a specific choice of smooth sampling domain, namely
$D_{0}=\\{\xi_{0}\geq 0\\}$. The integration over $\vec{\xi}$ then introduces
$(2\pi)^{3}\delta^{3}(\vec{s})$ which, upon collapsing, leaves the
$\sigma\rightarrow 0$ limit safe:

$\int dtg_{0}(t)^{2}\langle
T^{\text{ren}}_{--}(t)\rangle_{\psi}\geq-\frac{1}{\pi^{2}}\int_{0}^{\infty}\frac{d\xi_{0}}{2\pi}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}|\tilde{g}_{0}(\omega)|^{2}\int
ds_{0}\frac{1}{(s_{0}-i\epsilon)^{4}}e^{i(\omega-\xi_{0})s_{0}}\,.$ (95)

Using the general result

$\lim_{\epsilon\rightarrow 0}\int_{-\infty}^{\infty}ds\,\frac{e^{i\omega
s}}{(s-i\epsilon)^{p}}=\frac{2\pi
e^{ip\pi/2}}{\Gamma(p)}\,\omega^{p-1}\,\Theta(\omega),\qquad\qquad
p\in\mathbb{R}$ (96)

and writing $\zeta=\omega-\xi_{0}$ we have

$\int dtg_{0}(t)^{2}\langle
T^{\text{ren}}_{--}(t)\rangle_{\psi}\geq-\frac{1}{3\pi}\int_{0}^{\infty}\frac{d\omega}{2\pi}|\tilde{g}_{0}(\omega)|^{2}\int_{0}^{\omega}\frac{d\zeta}{2\pi}\zeta^{3}=-\frac{1}{24\pi^{2}}\int_{0}^{\infty}\frac{d\omega}{2\pi}|\tilde{g}_{0}(\omega)|^{2}\omega^{4}\,.$
(97)

Using $\widetilde{g^{\prime}_{0}}(\omega)=i\omega\tilde{g_{0}}(\omega)$ and
Parseval’s theorem

$\int_{0}^{\infty}\frac{d\omega}{2\pi}|\tilde{g}_{0}(\omega)|^{2}\omega^{4}=\int_{0}^{\infty}\frac{d\omega}{2\pi}|\widetilde{g^{\prime\prime}_{0}}(\omega)|^{2}=\frac{1}{2}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}|\widetilde{g^{\prime\prime}_{0}}(\omega)|^{2}=\frac{1}{4}\int_{-\infty}^{\infty}dtg^{\prime\prime}_{0}(t)^{2}\,,$
(98)

we arrive at a nice representation in position space, matching the result of
Fewster:2007rh

$\boxed{\int dtg_{0}(t)^{2}\langle
T^{\text{ren}}_{--}(t)\rangle_{\psi}\geq-\frac{(\ell_{-}\cdot\partial_{t})^{2}}{12\pi^{2}}\int_{-\infty}^{\infty}dtg_{0}^{\prime\prime}(t)^{2}.}$
(99)

where
$\ell_{-}\cdot\partial_{t}=\eta_{\mu\nu}(\partial_{-})^{\mu}(\partial_{t})^{\nu}=\frac{1}{2}$.

###  IV.3 Double null smearing

Now we want to apply the worldvolume QEI, (89), to the main object of
interest, the DSNE, in $n$ spacetime dimensions. To do so we write the
smearing function as

$g(x^{+},x^{-},\vec{y})^{2}=g(x^{+},x^{-})^{2}\delta^{n-2}(\vec{y})\,,$ (100)

and so

$g(x^{+},x^{-},\vec{y})=\lim_{\sigma\to
0}g(x^{+},x^{-})\frac{1}{\sigma^{(n-2)/2}(2\pi)^{(n-2)/4}}e^{-|\vec{y}|^{2}/4\sigma^{2}}\,.$
(101)

Then the bound of (89) can be written as

$\displaystyle\int d^{2}x^{\pm}g(x^{\pm})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}$ $\displaystyle\geq-\lim_{\sigma\to
0}\frac{\Gamma\left(\frac{n+2}{2}\right)2^{3n/2}\sigma^{n-2}}{(-1)^{n/2}\pi}\int_{D}\frac{d^{n}\xi}{(2\pi)^{n}}$
$\displaystyle\int\frac{d^{2}k_{\pm}d^{2}k_{\pm}^{\prime}}{(2\pi)^{4}}\tilde{g}(k_{+},k_{-})\tilde{g}(k_{+}^{\prime},k_{-}^{\prime})\int\frac{d^{n-2}\vec{k}_{y}d^{n-2}\vec{k^{\prime}}_{y}}{(2\pi)^{2(n-2)}}e^{-\sigma^{2}|\vec{k}_{y}|^{2}-\sigma^{2}|\vec{k^{\prime}}_{y}|^{2}}$
$\displaystyle\int
d^{n}xd^{n}x^{\prime}e^{ikx+ik^{\prime}x^{\prime}}e^{-i\xi(x-x^{\prime})}\frac{(\Delta
x^{+})^{2}}{(\Delta x^{-}\Delta x^{+}-2i\epsilon\Delta t-\Delta
y^{2})^{n/2+1}}\,.$

A calculation following a wholly similar logic as section IV.2 (we refer the
reader interested in following the details to look there) leads to

$\displaystyle\int d^{2}x^{\pm}g(x^{\pm})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-$
$\displaystyle\frac{2\Gamma\left(\frac{n+2}{2}\right)}{(-1)^{n/2}\pi^{n/2}}\int_{D}\frac{d^{2}\xi_{\pm}}{(2\pi)^{2}}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}$
(103) $\displaystyle\qquad\times\int
d^{2}s^{\pm}e^{i(k-\xi)_{\pm}s^{\pm}}\frac{1}{(s^{+}-i\epsilon)^{n/2-1}(s^{-}-i\epsilon)^{n/2+1}}\,.$

Utilizing (96) we arrive at

$\displaystyle\int d^{2}x^{\pm}g(x^{\pm})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}$ $\displaystyle\geq$
$\displaystyle-\frac{8}{\pi^{n/2-2}\Gamma\left(\frac{n-2}{2}\right)}\int_{D}\frac{d^{2}\xi_{\pm}}{(2\pi)^{2}}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}$
(104)
$\displaystyle\times(k_{+}-\xi_{+})^{n/2-2}\Theta(k_{+}-\xi_{+})(k_{-}-\xi_{-})^{n/2}\Theta(k_{-}-\xi_{-})\,.$

Up to now we have been fairly agnostic about the domain, $D$, beyond that it
satisfies the criteria of a small sampling domain. The freedom to choose this
domain is very much analogous to the choice of domain we encountered in the
difference inequalities in section III. In principle this freedom of a small
sampling domain is a parameter that can be optimized, however, much like in
section III we restrict our focus to boosted domains of the form
$D_{\eta}:=\\{\xi_{\eta}=e^{\eta}\xi_{+}+e^{-\eta}\xi_{-}\geq 0\\}$ and
optimizing over $\eta\in\mathbb{R}$.

Defining variables $\zeta_{\pm}=k_{\pm}-\xi_{\pm}$ (with the constraint
$k_{\eta}-\zeta_{\eta}\geq 0$ due to the domain $D_{\eta}$)

$\displaystyle\int d^{2}x^{\pm}g(x^{\pm})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}$ $\displaystyle\geq$
$\displaystyle-\frac{2}{\pi^{n/2}\Gamma\left(\frac{n-2}{2}\right)}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}$
$\displaystyle\int_{0}^{e^{\eta}k_{\eta}}d\zeta_{-}\int_{0}^{e^{-\eta}k_{\eta}-e^{-2\eta}\zeta_{-}}d\zeta_{+}(\zeta_{+})^{n/2-2}(\zeta_{-})^{n/2}\Theta(k_{\eta})$

The $\zeta_{\pm}$ integrals then give the final expression

$\displaystyle\boxed{\int d^{2}x^{\pm}g(x^{+},x^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-\frac{e^{2\eta}}{(4\pi)^{\frac{n-1}{2}}\Gamma\left(\frac{n+1}{2}\right)}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}\,k_{\eta}^{n}\,\Theta(k_{\eta}).}$

with $k_{\eta}=e^{\eta}k_{+}+e^{-\eta}k_{-}$. This is the same expression as
the one derived in Sec. III.2 (in the massless limit).

Including a mass

We can compute the mass corrections, in 4d, to the massless bound,(IV.3), by
using the expansion of the Hadamard parametrix. The relevant terms for the 4d
massive scalar in Minkowski spacetime are given by

$H_{(1)}(x,x^{\prime})=H_{-1}(x,x^{\prime})+H_{0}(x,x^{\prime})+H_{1}(x,x^{\prime})\,,$
(107)

where

$\displaystyle\tilde{H}_{-1}(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
H_{-1}(x,x^{\prime})=-\frac{1}{4\pi^{2}}\frac{1}{(\Delta x^{-}\Delta
x^{+}-2i\epsilon\Delta t-\Delta y^{2})}$
$\displaystyle\tilde{H}_{0}(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
H_{0}(x,x^{\prime})=-\frac{v_{0}}{4\pi^{2}}\ln{\left((\Delta x^{-}\Delta
x^{+}-2i\epsilon\Delta t-\Delta y^{2})/\ell^{2}\right)}$
$\displaystyle\tilde{H}_{1}(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
H_{1}(x,x^{\prime})=-\frac{v_{1}}{4\pi^{2}}(\Delta x^{-}\Delta x^{+}-\Delta
y^{2})\ln{\left((\Delta x^{-}\Delta x^{+}-2i\epsilon\Delta t-\Delta
y^{2})/\ell^{2}\right)}$

with coefficients777These coefficients differ slightly from Decanini:2005eg
due to a difference in definition in $\sigma(x,x^{\prime})$. Decanini:2005eg

$v_{0}=-\frac{1}{4}m^{2}\,,\qquad v_{1}=\frac{1}{32}m^{4}\,.$ (109)

Higher order terms vanish in the coincidence limit. Applying the derivatives
gives

$\displaystyle(\partial_{-})(\partial^{\prime}_{-})\tilde{H}_{(1)}(x,x^{\prime})$
$\displaystyle=$ $\displaystyle\frac{(\Delta x^{+})^{2}}{2\pi^{2}(\Delta
x^{-}\Delta x^{+}-2i\epsilon\Delta t-\Delta y^{2})^{3}}+\frac{m^{2}(\Delta
x^{+})^{2}}{16\pi^{2}(\Delta x^{-}\Delta x^{+}-2i\epsilon\Delta t-\Delta
y^{2})^{2}}$ (110) $\displaystyle\qquad\qquad+\frac{m^{4}(\Delta
x^{+})^{2}}{128\pi^{2}(\Delta x^{-}\Delta x^{+}-2i\epsilon\Delta t-\Delta
y^{2})}$

Tracking the terms through the calculation of the previous section we arrive,
intermediately, at

$\displaystyle\int d^{2}x^{\pm}g(x^{+},x^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-$
$\displaystyle\frac{4}{\pi^{2}}\int_{D_{\eta}}\frac{d^{2}\xi_{\pm}}{(2\pi)^{2}}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}\int
d^{2}s^{\pm}e^{i(k-\xi)_{\pm}s^{\pm}}$
$\displaystyle\quad\times\left\\{\frac{1}{(s^{+}-i\epsilon)(s^{-}-i\epsilon)^{3}}+\frac{m^{2}}{8}\frac{1}{(s^{-}-i\epsilon)^{2}}+\frac{m^{4}}{64}\frac{s^{+}}{(s^{-}-i\epsilon)}\right\\}$
$\displaystyle\geq-$ $\displaystyle
8\int\frac{d^{2}\zeta_{\pm}}{(2\pi)^{2}}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}\,\Theta(k_{\eta}-\zeta_{\eta})$
$\displaystyle\quad\times\left\\{\zeta_{-}^{2}\,\Theta(\zeta_{-})\Theta(\zeta_{+})-\frac{m^{2}}{4}\delta(\zeta_{+})\zeta_{-}\Theta(\zeta_{-})+\frac{m^{4}}{32}\delta^{\prime}(\zeta_{+})\Theta(\zeta_{-})\right\\}$

where in the second line we changed variables to
$\zeta_{\pm}=k_{\pm}-\xi_{\pm}$ and incorporated the boosted domain,
$D_{\eta}$ into an appropriate theta function. From here the $\zeta_{\pm}$
integrals are simple to do (noting that
$\frac{\partial}{\partial\zeta_{+}}\Theta(k_{\eta}-\zeta_{\eta})=-e^{\eta}\delta(k_{\eta}-\zeta_{\eta})$)

$\int d^{2}x^{\pm}g(x^{+},x^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-\frac{e^{2\eta}}{6\pi^{2}}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}\left(k_{\eta}^{4}-\frac{3}{2}m^{2}k_{\eta}^{2}+\frac{3}{8}m^{4}\right)\Theta(k_{\eta})\,.$
(112)

We note that this is a different lower bound for massive fields than what we
found by direct construction in Minkowski space, (49). As discussed at the end
of section IV.1, this is somewhat expected since Hadamard renormalization only
subtracts a finite number of singular terms while normal ordering subtracts an
expectation value containing all orders in a mass expansion. To check this
intuition we can compare (112) to a perturbative expansion of (49)

$\displaystyle\int d^{2}x^{\pm}g(x^{+},x^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-\frac{e^{2\eta}}{6\pi^{2}}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}\left\\{\left(k_{\eta}^{4}-\frac{3}{2}k_{\eta}^{2}\,m^{2}+\frac{3}{8}m^{4}\right)\Theta(k_{\eta})\right.$
$\displaystyle\qquad\qquad\qquad\left.+k_{\eta}^{4}\left(-m\delta(k_{\eta})+\frac{m^{2}}{2}\delta^{\prime}(k_{\eta})-\frac{m^{3}}{6}\delta^{\prime\prime}(k_{\eta})+\frac{m^{4}}{24}\delta^{\prime\prime\prime}(k_{\eta})\right)+\ldots\right\\}$
$\displaystyle\qquad\qquad\qquad\geq-\frac{e^{2\eta}}{6\pi^{2}}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}\left\\{\left(k_{\eta}^{4}-\frac{3}{2}k_{\eta}^{2}\,m^{2}+\frac{3}{8}m^{4}\right)\Theta(k_{\eta})+\ldots\right\\}$

where the second line, coming from expanding $\Theta(k_{\eta}-m)$, vanishes up
to order $m^{4}$. This expansion matches (112) up to the “$\ldots$” indicating
higher order mass terms. Note that while the coefficients of the mass terms
can come with either sign, the general expectation is that full difference
inequality (49) is a qualitatively stronger lower bound than (112). Indeed it
was shown in Fliss:2021gdz  that for Gaussian smearing functions of smearing
lengths $\delta^{\pm}$, that at large mass the integral in (49) can be
evaluated at saddle-point and is exponentially suppressed in
$m^{2}\delta^{+}\delta^{-}$ making the bound very tight. This is contrast to
(112), which gets weaker as the mass increases.

To illustrate that fact we write the bound as a function of
$\gamma:=(\delta^{+}\delta^{-}m^{2})^{1/2}$

$\int d^{2}x^{\pm}g(x^{\pm})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-\frac{c^{(4)}_{T_{--}}}{\delta^{+}(\delta^{-})^{3}}\tilde{Q}(\gamma)\,.$
(114)

and plot the two bounds as functions of $\gamma$. The two plots are shown in
figure 1.

Figure 1: In blue is the difference inequality, (49), which displays
exponential damping in $\gamma^{2}$. In orange is the absolute inequality,
(112), which increases (i.e. becomes weaker) for large masses. This plot was
made using square-root Gaussian smearing functions, (8),
$g(x^{+},x^{-})=\frac{1}{\sqrt{\delta^{+}\delta^{-}}}f_{\sqrt{\text{G}}}(x^{+}/\delta^{+})f_{\sqrt{\text{G}}}(x^{-}/\delta^{-})$
and choosing the boost parameter $e^{\eta}=\sqrt{\delta^{-}/\delta^{+}}$ to
scale out the dependence on the smearing parameters as outlined in the
paragraph above equation (51).

##  V Boost optimization of the bound

Having derived the DSNEC bounds in Sec. III.2 and Sec. IV.3 we now address the
issue of Lorentz covariance, namely that our boundary involves a explicit
reference frame. We remind the reader that is related to a freedom in the
choice of smooth sampling domain that we discussed earlier.

In this section we utilize this freedom (or at least a portion of it) optimize
the derived bound over Lorentz boosts of the domain. We show in principle how
this works in four spacetime dimensions where we derive an “boost-optimized”
bound. More generally, however, we show how a simple (albeit sub-optimal)
Lorentz covariant bound can be derived in general dimensions. For even
spacetime dimensions, this bound can be expressed directly in position space.

###  V.1 $4$-dimensions

We start with the expression Eq. (IV.3) in $n=4$ dimensions. Because the
integrand is even888Since $g$ is real in position space
$\tilde{g}(-k_{+},-k_{-})=\tilde{g}(k_{+},k_{-})^{\ast}$. about
$k_{\eta}\rightarrow-k_{\eta}$ we can extend $D_{\eta}$ to the entire
$\mathbb{R}^{2}$ plane at the expense of a factor of 1/2:

$\displaystyle\int d^{2}x^{\pm}g(x^{+},x^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq$
$\displaystyle-\frac{1}{12\pi^{2}}\int\frac{dk_{+}dk_{-}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}\bigg{(}\beta^{3}k_{+}^{4}+4\beta^{2}k_{+}^{3}k_{-}$
(115) $\displaystyle\qquad\qquad\qquad\qquad+6\beta
k_{+}^{2}k_{-}^{2}+4k_{+}k_{-}^{3}+\beta^{-1}k_{-}^{4}\bigg{)}\,,$

where $\beta\equiv e^{2\eta}$. We will proceed by assuming that the smearing
function factorizes as

$|\tilde{g}(k_{+},k_{-})|^{2}=|\tilde{g}_{+}(k_{+})|^{2}|\tilde{g}_{-}(k_{-})|^{2}$
(116)

normalized to

$\int\frac{dk_{\pm}}{2\pi}|\tilde{g}_{\pm}(k_{\pm})|^{2}=1.$ (117)

We will simplify the notation in what follows by defining moments

$\int\frac{dk_{\pm}}{2\pi}\,k_{\pm}^{n}\,|\tilde{g}_{\pm}(k_{\pm})|^{2}\equiv\langle
k_{\pm}^{n}\rangle\qquad\qquad n\in\mathbb{Z}\,.$ (118)

While hidden in this notation, it is important to keep in mind that $\langle
k_{\pm}^{n}\rangle$ is a functional of $\tilde{g}_{\pm}$, respectively.
Additionally note that the assumption that the smearing function factorizes
forces the odd moments to vanish. So we have

$\int d^{2}x^{\pm}g(x^{+},x^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-\frac{1}{12\pi^{2}}\bigg{(}\beta^{3}\langle
k_{+}^{4}\rangle+6\beta^{1}\langle k_{+}^{2}\rangle\langle
k_{-}^{2}\rangle+\beta^{-1}\langle
k_{-}^{4}\rangle\bigg{)}\equiv-\mathcal{Q}(\beta)\,.$ (119)

The minimizer of the bound, $\beta_{0}$, is a real positive solution to

$\mathcal{Q}^{\prime}(\beta_{0})=0\qquad\Rightarrow\qquad\beta^{4}_{0}\langle
k_{-}^{4}\rangle-6\beta^{2}_{0}\langle k_{+}^{2}\rangle\langle
k_{-}^{2}\rangle-3\langle k_{+}^{4}\rangle=0\,,$ (120)

and the boost-optimized bound is

$\mathcal{Q}(\beta_{0})=\frac{\sqrt{\langle
k_{-}^{4}\rangle}}{3\pi^{2}}\,\frac{\left(\langle k_{-}^{4}\rangle\langle
k_{+}^{4}\rangle+3\langle k_{-}^{2}\rangle\langle
k_{+}^{2}\rangle\left(3\langle k_{-}^{2}\rangle\langle
k_{+}^{2}\rangle+\sqrt{9\langle k_{-}^{2}\rangle^{2}\langle
k_{+}^{2}\rangle^{2}+3\langle k_{-}^{4}\rangle\langle
k_{+}^{4}\rangle}\right)\right)}{\left(3\langle k_{-}^{2}\rangle\langle
k_{+}^{2}\rangle+\sqrt{9\langle k_{-}^{2}\rangle^{2}\langle
k_{+}^{2}\rangle^{2}+3\langle k_{-}^{4}\rangle\langle
k_{+}^{4}\rangle}\right)^{3/2}}\,.$ (121)

Before moving on, let us remark on some features of the above bound. Firstly,
we emphasize that the optimization over boosts is only a one-parameter
characterization of the freedom in choosing a smooth sampling domain. Thus we
strongly suspect that (121) is not the truly optimal bound for $4d$ massless
scalars.

Secondly, boost optimization has restored Lorentz covariance to (121): under
$(k_{+},k_{-})\rightarrow(\lambda k_{+},\lambda k_{-})$ in the integrals
defining the moments of $|\tilde{g}|^{2}$, $Q\rightarrow\lambda^{4}Q$
consistent with the engineering dimension of $T_{--}$ and under
$(k_{+},k_{-})\rightarrow(\lambda k_{+},\lambda^{-1}k_{-})$,
$Q\rightarrow\lambda^{-2}Q$ consistent with the weight of $T_{--}$ under
boosts. We will find this to also be true of the bounds we derive for general
dimensions.

Thirdly, unlike what is suggested prima facie by (IV.3), (121) is not linear
functional of the original smearing function $g^{2}$. This follows from
solving $Q^{\prime}(\beta_{0})=0$ in terms of the moments of
$|\tilde{g}|^{2}$, i.e. the optimizing boost parameter depends on the smearing
function. This is a common feature of the bounds we discuss for general
dimensions below.

Lastly, for general dimensions (and in particular odd dimensions) the
extension of $D_{\eta}\rightarrow\mathbb{R}^{1,1}$ is only valid for the
absolute value of $k_{\eta}^{n}$. As a consequence we will not be able to drop
odd moments of $|\tilde{g}|^{2}$. Additionally, solving the resulting
polynomial $\mathcal{Q}^{\prime}(\beta_{0})=0$ may not be analytically
possible in generic dimensions. As we will soon see we can circumvent these
difficulties and derive a generic expression at the expense of making the
bound slightly weaker.

###  V.2 General dimensions

We start by noting

$\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}\,k_{\eta}^{n}\,\Theta(k_{\eta})=\frac{1}{2}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}\,|k_{\eta}|^{n}\,.$
(122)

We will continue to assume that the smearing function factorizes as in Eq.
(116) and normalized as in Eq. (117), and we write

$\int d^{2}x^{\pm}g(x^{+},x^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-\frac{c_{T_{--}}^{(n)}}{2}\sum_{m=0}^{n}\left(\begin{array}[]{c}n\\\
m\end{array}\right)e^{(2+2m-n)\eta}\langle|k_{+}|^{m}\rangle\langle|k_{-}|^{n-m}\rangle$
(123)

where we recall the definition of $c_{T_{--}}^{(n)}$ in equation (50). Note
that we have made use of the triangle inequality after expanding
$|k_{\eta}|^{n}$ and so we have already weakened the bound. Next we implement
Hölder’s inequality on each term of the sum. The inequality is the following
finner1992generalization : given a probability measure $d\mu$ and two
measurable functions $f_{1}$ and $f_{2}$ then

$\int d\mu|f_{1}||f_{2}|\leq\left(\int
d\mu|f_{1}|^{p_{1}}\right)^{1/p_{1}}\left(\int
d\mu|f_{2}|^{p_{2}}\right)^{1/p_{2}},\quad p_{1,2}\geq 1,\qquad
1/p_{1}+1/p_{2}=1\,.$ (124)

For $\langle|k_{+}|^{m}\rangle$, for example, we use the measure
$d\mu=\frac{dk_{+}}{2\pi}|\tilde{g}_{+}|^{2}$, functions $f_{1}=|k_{+}|^{m}$,
$f_{2}=1$, and $p_{1}=n/m$, $p_{2}=\frac{n}{n-m}$. This doesn’t apply for the
$m=n$ term in the sum, however we don’t need to implement the inequality for
this term. We now have

$\int d^{2}x^{\pm}g(x^{+},x^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-\frac{c_{T_{--}}^{(n)}}{2}\sum_{m=0}^{n}\left(\begin{array}[]{c}n\\\
m\end{array}\right)e^{(2+2m-n)\eta}\langle|k_{+}|^{n}\rangle^{m/n}\langle|k_{-}|^{n}\rangle^{1-m/n}\,.$
(125)

Now defining
$\bar{\beta}:=e^{-\eta}\langle|k_{+}|^{n}\rangle^{-\frac{1}{2n}}\langle|k_{-}|^{n}\rangle^{\frac{1}{2n}}$
to arrive at

$\boxed{\int d^{2}x^{\pm}g(x^{+},x^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-\frac{c_{T_{--}}^{(n)}}{2}\,P_{n}(\bar{\beta})\,\langle|k_{+}|^{n}\rangle^{\frac{n-2}{2n}}\langle|k_{-}|^{n}\rangle^{\frac{n+2}{2n}}}$
(126)

where we recall the notation $\langle|k_{\pm}|^{p}\rangle:=\int
dk_{\pm}|\tilde{g}_{\pm}(k_{\pm})|^{2}|k_{\pm}|^{p}$. All residual dependence
on the boost parameter lies in

$P_{n}(\bar{\beta})=\sum_{m=0}^{n}\left(\begin{array}[]{c}n\\\
m\end{array}\right)\bar{\beta}^{n-2-2m}=\bar{\beta}^{-2}(\bar{\beta}^{-1}+\bar{\beta})^{n}\,,$
(127)

which a simple polynomial of $\bar{\beta}$ and whose optimal value is given by

$\bar{\beta}_{0}=\sqrt{\frac{n+2}{n-2}}\,,\qquad\qquad
P_{n}(\bar{\beta}_{0})=\frac{n-2}{n+2}\left(\sqrt{\frac{n-2}{n+2}}+\sqrt{\frac{n+2}{n-2}}\right)^{n}\,.$
(128)

Note that much like the $4d$ boost-optimal bound, (121), the general bound
(126) has restored covariance under boosts and rescalings of $(k_{+},k_{-})$
and so fixes the dependence on smearing lengths into the DSNEC form.

For instance, being explicit, we could choose the functions $g_{\pm}$ as

$g_{\pm}(x^{\pm})=\frac{1}{\sqrt{\delta^{\pm}}}f_{\sqrt{\text{G}}}(x^{\pm}/\delta^{\pm})$
(129)

where we recall that $f_{\sqrt{\text{G}}}$ is the square-root Gaussian, (8).
The moments are999We will choose the convention
$\tilde{g}_{\pm}:=\frac{1}{\sqrt{2}}\int
dx^{+}\,e^{ik_{\pm}x^{\pm}}g_{\pm}(x^{\pm})$ in accordance with the factor of
two encountered in footnote 5.

$\langle|k_{\pm}|^{m}\rangle=\frac{\Gamma\left(\frac{m+1}{2}\right)}{2^{\frac{m+1}{2}}\sqrt{2\pi}(\delta^{\pm})^{m}}\,.$
(130)

Then we see that Eq. (126) is precisely in the DSNEC form

$\int\frac{d^{2}x^{\pm}}{\delta^{+}\delta^{-}}f_{\sqrt{\text{G}}}(x^{+}/\delta^{+})^{2}f_{\sqrt{\text{G}}}(x^{-}/\delta^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-
A_{n}\frac{1}{(\delta^{+})^{n/2-1}(\delta^{-})^{n/2+1}}\,,$ (131)

where

$A_{n}=\frac{c_{T_{--}}^{(n)}P_{n}(\bar{\beta}_{0})\Gamma\left(\frac{n+1}{2}\right)}{2^{\frac{n+3}{2}}\sqrt{2\pi}}=\frac{1}{2(8\pi)^{n/2}}\frac{n-2}{n+2}\left(\sqrt{\frac{n-2}{n+2}}+\sqrt{\frac{n+2}{n-2}}\right)^{n}$
(132)

It is also helpful to compare this general bound to the “boost-optimal” bound,
(121) in four dimensions, again using square-root Gaussians as the smearing
functions. A simple calculation reveals that (121) and (126) imply,
respectively,

$\displaystyle\int\frac{d^{2}x^{\pm}}{\delta^{+}\delta^{-}}f_{\sqrt{\text{G}}}(x^{+}/\delta^{+})^{2}f_{\sqrt{\text{G}}}(x^{-}/\delta^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\gtrapprox$ $\displaystyle-\frac{2.18\times
10^{-3}}{(\delta^{+})^{3/2}(\delta^{-})^{5/2}}\,,\qquad\text{(Boost-
optimized)}$
$\displaystyle\int\frac{d^{2}x^{\pm}}{\delta^{+}\delta^{-}}f_{\sqrt{\text{G}}}(x^{+}/\delta^{+})^{2}f_{\sqrt{\text{G}}}(x^{-}/\delta^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\gtrapprox$ $\displaystyle-\frac{7.51\times
10^{-3}}{(\delta^{+})^{3/2}(\delta^{-})^{5/2}}\,.\qquad\text{(Equation
}\eqref{eqn:boundgen}\text{)}$ (133)

and so indeed (126) is a weaker bound.

###  V.3 Even dimensions

We finish this section of the paper by noting that while generally our bound
is most conveniently expressed in momentum space, in even dimensions the
integrals can be inverse Fourier transformed to integrals over local
quantities in position space (this is not true in odd dimensions because of
our bound makes use of the absolute value of the momenta). Indeed by noting

$\langle|k_{\pm}|^{n}\rangle_{\tilde{g}_{\pm}}=\frac{1}{2}\int\frac{dk_{\pm}}{2\pi}|\tilde{g}_{\pm}|^{2}\,k_{\pm}^{n}=\frac{1}{4}\int
dx^{\pm}\,(g^{(n/2)}(x^{\pm}))^{2}\,,$ (134)

where the superscript $(\cdot)^{(n/2)}$ indicates the $n/2$-th derivative, we
arrive at simple position space integrals

$\displaystyle\boxed{\int d^{2}x^{\pm}g(x^{\pm})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-\frac{c_{T_{--}}^{(n)}P_{n}(\bar{\beta}_{0})}{8}\left(\int
dx^{+}\\!(g^{(n/2)}_{+}(x^{+}))^{2}\right)^{\frac{n-2}{2n}}\\!\\!\left(\int
dx^{-}\\!(g^{(n/2)}_{-}(x^{-}))^{2}\right)^{\frac{n+2}{2n}}}\,.$

where we recall the definitions of the constant $c_{T_{--}}^{(n)}$ in equation
(50) and the coefficient $P_{n}(\bar{\beta}_{0})$ in equation (128). In four
dimensions, in particular, this is

$\int d^{2}x^{\pm}g(x^{+},x^{-})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\geq-\frac{16}{81\pi^{2}}\left(\int
dx^{+}(g_{+}^{\prime\prime}(x^{+}))^{2}\right)^{1/4}\left(\int
dx^{-}(g_{-}^{\prime\prime}(x^{-}))^{2}\right)^{3/4}\,.$ (136)

##  VI Discussion

In this work we investigated the double smeared null energy condition (DSNEC),
a proposed bound on the renormalized null energy smeared over two null
directions. For free fields in Minkowski space we derived this bound in two
separate ways. First, we derived the DSNEC as a quantum difference inequality
using the Minkowski vacuum as the reference state. We showed that this
derivation generalizes to bounds on a large set of operators including higher-
spin currents. Second, we showed that the DSNEC arises naturally from a
general absolute quantum worldvolume inequality. The formalism of this second
derivation allows for a straightforward generalization to curved spacetimes.
As both approaches require a fixed domain of momentum integration, we further
utilized this degree of freedom to optimize the bound over a set of boosted
domains. This results in a bound that restores Lorentz covariance and displays
an unexpected, non-linear dependence on the smearing function. Finally, we
showed how the averaged null energy condition (ANEC) and the smeared null
energy condition (SNEC) can be derived from DSNEC at the correct limit.

There are several interesting directions for future work. The most obvious is
to investigate the generalization of our bound to curved spacetimes. As
mentioned above, this is indeed a primary motivation for expressing the DSNEC
as an absolute inequality: as opposed to Minkowski space, there is no
preferred reference vacuum state in curved spaces. Renormalizing with respect
to the Hadamard parametrix provides a canonical way to derive the DSNEC while
allowing for curvature contributions. Generalizing the DSNEC to curved spaces
is also a chief concern for applications to semiclassical gravity, which we
will return to discuss shortly. Thus this is a direction of high interest and
importance.

Further probing the validity of the DSNEC, one can also speculate about its
application in generic quantum field theories in Minkowski spacetime. For
theories that are relevant perturbations away from free field theory, we
generally expect the DSNEC to hold following the argument given in
Fliss:2021gdz : at large momenta (compared to any inverse correlation lengths
of the theory) the divergences appearing in vacuum expectation values are
roughly given by those of the UV fixed point. Having shown that the DSNEC
holds for the free fixed point, it is reasonable to assume that the smeared
null energy is lower bounded in the above situations as well. The engineering
dimension of $T_{--}$ and covariance under boosts then fix the schematic form
of this bound. For strongly interacting theories, the question becomes more
subtle and a proof of the DSNEC will likely require formal CFT techniques. It
has been suggested that by looking at states prepared by stress-tensor
insertions Farnsworth:2015hum , that energy densities in 4d CFTs obey
worldvolume inequalities. It would be interesting to explore whether the same
states suggest the validity of the DSNEC and more generally, if the DSNEC can
be proven in CFTs. For non-conformal general interacting QFTs, the situation
is more difficult as no such QEIs have been derived. Their existence has been
established only for operators arising from the operator product expansion of
theories satisfying a microscopic phase space condition Bostelmann:2009efm .
However, it is not clear if these operators include components of the stress
energy tensor of interacting theories.

The other main direction for future work is to use this type of bound in order
to prove singularity theorems. Penrose showed, assuming the NEC, that trapped
surfaces lead to singularities Penrose:1964wq . These theorems are violated in
semi-classical gravity due to NEC violation. Thus, a result like our bound
gives the natural starting point (replacing the NEC) for proving semi-
classical singularity theorems. One main obstacle for the DSNEC as an
assumption is that singularity theorems require bounds on individual null
geodesics. In Appendix B we explore one “light-ray limit” of the DSNEC in
showing how to reproduce the SNEC. The SNEC has been used as an assumption to
a semiclassical singularity theorem for null geodesic incompleteness
Freivogel:2020hiz . However in the context of field theory alone, the SNEC
bound is not very useful as one must then make sense of the UV cutoff. An
alternative direction is using the DNEC along with “segment inequality”
theorems sprouse2000integral . Such theorems use worldvolume bounds on the
Ricci tensor to show that the length of any geodesic maximizing the distance
to a Cauchy surface is bounded, thus establishing singularity theorems.

###### Acknowledgements.

We thank Tarek Anous, Mert Besken, Chris Fewster and Dimitrios Krommydas for
helpful conversations. JRF is supported the ERC Starting Grant GenGeoHolo. BF
and E-AK are supported by the ERC Consolidator Grant QUANTIVIOL. This work is
part of the $\Delta$-ITP consortium, a program of the NWO that is funded by
the Dutch Ministry of Education, Culture and Science (OCW).

##  Appendix A On the validity of the Assumption 1

In this appendix we derive the constraints on what types of operators satisfy
Assumption 1 from section III. In line with the general philosophy that a QFT
is defined by its flow away from a conformal fixed point (and to be concrete),
let us spell out what Assumption 1 implies for primary operators in a CFT (if
Assumption 1 is satisfied by a primary then it is also satisfied by its
descendants by acting with derivatives). Thus the main object of interest is

$[\mathcal{O}_{\Delta}(t,\vec{x}),\mathcal{O}_{\Delta}(0)]$ (A.1)

where $\Delta$ is the conformal weight of $\mathcal{O}_{\Delta}$. For
simplicity we will focus scalar primaries although our main conclusions will
be unchanged for spinning operators. This can be evaluated with an appropriate
$i\varepsilon$ prescription and using the operator product expansion
(OPE)Besken:2020snx

$\displaystyle[\mathcal{O}_{\Delta}(t,\vec{x}),\mathcal{O}_{\Delta}(0)]=$
$\displaystyle\lim_{\varepsilon\rightarrow
0}\left\\{\mathcal{O}_{\Delta}(t-i\varepsilon,\vec{x})\mathcal{O}_{\Delta}(0)-\mathcal{O}_{\Delta}(t+i\varepsilon,\vec{x})\mathcal{O}_{\Delta}(0)\right\\}$
$\displaystyle=$ $\displaystyle\lim_{\varepsilon\rightarrow
0}\left.\sum_{\Delta^{\prime}}\mathcal{G}_{\Delta\Delta}^{\Delta^{\prime}}(\bar{t},\vec{x};\partial)\right|^{\bar{t}=t-i\varepsilon}_{\bar{t}=t+i\varepsilon}\mathcal{O}_{\Delta^{\prime}}(0)$
(A.2)

where the sum goes over all other primaries $\Delta^{\prime}$ and we intend
this as an operator statement true inside all Wightman functions with other
local operators; via the operator-state correspondence this statement holds in
a dense set of the Hilbert space. The identity, $\hat{1}$, (with
$\Delta^{\prime}=0$) always appears in the operator product expansion of
$\mathcal{O}_{\Delta}$ with itself. We are interested in what other primaries
can possibly contribute to this commutator. To isolate the contribution of a
primary $\mathcal{O}_{\Delta^{\prime}}$ we can consider the overlap of (A) in
the conformal vacuum, $\Omega_{c}$, with $\mathcal{O}_{\Delta^{\prime}}(y)$ in
the limit that $|y|\rightarrow\infty$. For instance a typical term is

$\lim_{|y|\rightarrow\infty}\langle\mathcal{O}_{\Delta^{\prime}}(y)\mathcal{O}_{\Delta}(x)\mathcal{O}_{\Delta}(0)\rangle_{\Omega_{c}}=\lim_{|y|\rightarrow\infty}\mathcal{G}_{\Delta\Delta}^{\Delta^{\prime}}(x;-\partial^{(y)})\frac{1}{|y|^{2\Delta^{\prime}}}$
(A.3)

where we have used the universal form of primary two-point functions. In the
$|y|\rightarrow\infty$ limit the contribution of descendants (coming from
acting by $\partial^{(y)}$) are subleading and so the leading contribution
comes the primary operator itself. We can also evaluate the left-hand side of
(A.3) using the universal form of conformal three-point function and find

$\mathcal{G}_{\Delta\Delta}^{\Delta^{\prime}}(x;0)=\lim_{|y|\rightarrow\infty}|y|^{2\Delta^{\prime}}\langle\mathcal{O}_{\Delta^{\prime}}(y)\mathcal{O}_{\Delta}(x)\mathcal{O}_{\Delta}(0)\rangle_{\Omega_{c}}=\lim_{|y|\rightarrow\infty}\frac{|y|^{2\Delta^{\prime}}c_{\Delta\Delta}^{\Delta^{\prime}}}{|y-x|^{\Delta^{\prime}}|x|^{2\Delta-\Delta^{\prime}}|y|^{\Delta^{\prime}}}=\frac{c_{\Delta\Delta}^{\Delta^{\prime}}}{|x|^{2\Delta-\Delta^{\prime}}}.$
(A.4)

Here $c_{\Delta\Delta}^{\Delta^{\prime}}$ are three-point coefficients and are
c-numbers specifying the CFT and its operator content. To take stock, the
contribution of primary operators to the commutator is then

$[\mathcal{O}_{\Delta}(t,\vec{x}),\mathcal{O}_{\Delta}(0)]\supset\sum_{\Delta^{\prime}}c_{\Delta\Delta}^{\Delta^{\prime}}\mathcal{O}_{\Delta^{\prime}}(0)\boldsymbol{f}_{\Delta-\frac{1}{2}\Delta^{\prime}}(t,\vec{x})$
(A.5)

with

$\boldsymbol{f}_{h}(t,\vec{x}):=\lim_{\varepsilon\rightarrow
0}\left\\{\frac{1}{(-(t-i\varepsilon)^{2}+\vec{x}^{2})^{h}}-\frac{1}{(-(t+i\varepsilon)^{2}+\vec{x}^{2})^{h}}\right\\}.$
(A.6)

Note that if a primary $\mathcal{O}_{\Delta^{\prime}}$ appears in the OPE
(i.e. $c_{\Delta\Delta}^{\Delta^{\prime}}\neq 0$), it is still possible to
vanish in the commutator as long as the associated
$\boldsymbol{f}_{\Delta-\Delta^{\prime}/2}$ vanishes as a distribution,

$\int dt\,\varphi(t)\boldsymbol{f}_{h}(t,\vec{x})=0\qquad\qquad\forall\,\text{
smooth }\varphi(t).$ (A.7)

This requires $h\in\mathbb{Z}_{\leq 0}$. For instance if $h\in\mathbb{Z}_{>0}$
then the poles in $\boldsymbol{f}_{h}$ can contribute to the integration
against a test function and so as a distribution Besken:2020snx

$\boldsymbol{f}_{h\in\mathbb{Z}_{>0}}(t,\vec{x})=\frac{2\pi
i}{\Gamma(h)}\left\\{(t-|\vec{x}|)^{-h}\partial_{t}^{h-1}\delta(t+|\vec{x}|)+(t+|\vec{x}|)^{-h}\partial_{t}^{h-1}\delta(t-|\vec{x}|)\right\\}.$
(A.8)

And if $h\notin\mathbb{Z}$, $\boldsymbol{f}_{h}$ possesses branch cuts
contributing to integrations against test functions leading to Besken:2020snx

$\boldsymbol{f}_{h\notin\mathbb{Z}}\propto\sin(\pi
h)(t^{2}-\vec{x}^{2})^{-h}\left\\{\Theta(t-|\vec{x}|)-\Theta(t+|\vec{x}|)\right\\}.$
(A.9)

Thus we are lead to conclude that as a necessary condition for only the
identity operator to appear in commutator
$[\mathcal{O}_{\Delta},\mathcal{O}_{\Delta}]$, the only primaries that can
appear in $\mathcal{O}_{\Delta}\mathcal{O}_{\Delta}$ OPE have conformal
dimensions

$\Delta^{\prime}=2\Delta+m\qquad\qquad m\in\mathbb{Z}_{\geq 0}.$ (A.10)

Since descendent operators have conformal dimensions differing from primaries
by positive integers this is also a sufficient condition. This is very
constraining of the operator spectrum of a CFT. We in fact already know one
set of primaries that naturally in appear in such an OPE which are the so-
called “double-trace” operators schematically of the form

$\mathcal{O}_{\Delta^{\prime}}(x)\;\sim\;:\mathcal{O}_{\Delta}(\partial^{2})^{\ell}\partial^{\mu_{1}}\ldots\partial^{\mu_{s}}\mathcal{O}_{\Delta}:(x)\qquad\qquad\Delta^{\prime}=2\Delta+2\ell+s.$
(A.11)

Excepting the possibility of multiple conformal modules possessing the same
conformal weight, we find that $\mathcal{O}_{\Delta}$ only has its double-
traces in its OPE which implies that higher-point functions follow from Wick
contractions, reminiscent of free fields. We make the passing remark that the
bounds we derive in III make no use of the particular form of commutator, only
that it is proportional to the identity and so we make no requirements on the
conformal dimension of $\mathcal{O}_{\Delta}$ itself. This allows for the
possibility for $\mathcal{O}_{\Delta}$ to be a generalized free field at some
interacting fixed point with some large parameter $N$ suppressing non-Wick
contractions by powers of $1/N$. Thus our construction in III provides lower
bounds on such primaries (and their descendants) to leading order in $1/N$.

Moving away from the fixed point we can ask how deformations of the CFT affect
the above analysis. For one, we expect that in order to preserve the equal-
time commutators, $[\mathcal{O}_{\Delta}(0,\vec{x}),\mathcal{O}_{\Delta}(0)]$,
in Assumption 1 irrelevant deformations should be prohibited as they might
contain derivative couplings (or perhaps might induce derivative couplings via
renormalization) that can alter the canonical structure. However, we also
require Assumption 1 to hold over all points in a causal domain and so we must
evolve $\mathcal{O}_{\Delta}(0,\vec{x})$ away from $t=0$ using the interacting
Hamiltonian. This will generically generate more operators unless the
deformation is Gaussian. Given these two arguments it seems that Assumption 1
will only hold for the simplest massive deformation of a generalized free
field fixed point: $m^{2}\mathcal{O}_{\Delta}\mathcal{O}_{\Delta}$.

##  Appendix B SNEC from DSNEC

It is clear that in a “light-ray” limit, say, by taking the
$\delta^{+}\rightarrow 0$, the right-hand side of the DSNEC diverges leaving a
trivial bound. This is expected on general grounds: the null-energy averaged
along a finite portion of a light-ray is unbounded from below in QFT
Fewster:2002ne . However, with the introduction of a UV cutoff of the theory,
the bound remains finite allowing the proof of SNEC Freivogel:2018gxj ;
Fliss:2021gdz . Here, we investigate the derivation of the “field theory”
version of SNEC from DNEC at the appropriate limit.

First we examine the schematic form of DNEC (51), imposing the following
cutoff: we take $\delta^{+}\to 0$ while
$\delta^{+}\delta^{-}\to\ell_{\text{UV}}^{2}$. Similarly to the way we derived
ANEC we require that the smearing function factorizes and
$\lim_{\delta^{+}0}f_{+}(x^{+}/\delta^{+})^{2}/\delta_{+}=\delta(x^{+}-\beta)$.
Then we have

$\int dx^{-}g_{-}(x^{-})^{2}\langle
T_{--}(x^{+}=\beta,x^{-})\rangle_{\Psi}\geq-\frac{\mathcal{N}_{2}}{\ell_{UV}^{n-2}(\delta^{-})^{2}}\,,$
(B.1)

consistent with a schematic form of the SNEC.

To investigate if the DNSEC implies a SNEC type bound with the same number of
derivatives on the smearing function we start from Eq. (IV.3). Picking the
$\eta\rightarrow-\infty$ limit of our boosted domains $D_{\eta}$ the equation
becomes 101010For the rest of this calculation we will ignore constant
prefactors of order one for simplicity.

$\int d^{2}x^{\pm}g(x^{\pm})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\gtrsim-\int
d^{2}k_{\pm}|\tilde{g}(k_{+},k_{-})|^{2}\int_{0}^{k_{-}}d\zeta_{-}\int_{0}^{\infty}d\zeta_{+}(\zeta_{+})^{n/2-2}(\zeta_{-})^{n/2}\Theta(k_{-})\,.$
(B.2)

Then the right-hand side is independent of $k_{+}$. One might now be tempted
to take the $\delta^{+}\rightarrow 0$ limit on both sides. However we have
only moved the divergence to a new place: the unbounded $\zeta_{+}$
integration. We cannot infinitely boost $D_{\eta}$ and expect a finite lower
bound (indeed in this limit $D_{\eta}$ fails to satisfy the criteria of a
small sampling domain). This is the point at which we implement the UV cutoff.
We place this cutoff covariantly on $\zeta_{\pm}$

$\zeta_{+}\zeta_{-}<\ell_{UV}^{-2}.$ (B.3)

In our first approach we additionally assume that momenta appearing in the
state are cutoff as $\zeta_{\pm}\leq\Lambda_{\pm}$ This is no longer a Lorentz
invariant cutoff as imposed in some versions of SNEC. However, this approach
will allow us to derive SNEC for momenta arbitrarily close to $\Lambda_{\pm}$.
Then the $\zeta_{+}$ integral is bounded and Eq. (B.2) becomes

$\int d^{2}x^{\pm}g(x^{\pm})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\gtrsim-\int
d^{2}k_{\pm}|\tilde{g}(k_{+},k_{-})|^{2}\Lambda_{+}^{n/2-1}k_{-}^{n/2+1}\,.$
(B.4)

We will assume that the support of the smearing function in momentum space,
$|\tilde{g}|^{2}$, only has support for momenta below these cutoffs,
$k_{\pm}\leq\Lambda_{\pm}$

$\int d^{2}x^{\pm}g(x^{\pm})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\gtrsim-\frac{1}{\ell_{UV}^{n-2}}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}\,k_{-}^{2}\gtrsim-\frac{1}{\ell_{UV}^{n-2}}\int
d^{2}x^{\pm}(\partial_{-}g(x^{\pm}))^{2}\,,$ (B.5)

which reproduces the SNEC bound.

In a different approach, we do not independently bound the $\zeta_{\pm}$
momenta but just implement the covariant cutoff of Eq. (B.3). On mass-shell,
$4\zeta_{+}\zeta_{-}=\vec{\zeta}_{\perp}^{2}+m^{2}$, so one can view this as a
cutoff on the transverse momenta accessible to the theory. This is the same
regime in which light-sheets admit a “pencil decomposition” and in which the
SNEC was proven in Fliss:2021gdz . Revisiting (IV.3) in the large $e^{-\eta}$
limit, the upper limit of the $\zeta_{+}$ integration is approximately
replaced with

$e^{-2\eta}(\zeta_{-}-k_{-})\leq\zeta_{+}-k_{+}<\frac{1}{\ell_{UV}^{2}\zeta_{-}}(1-\ell_{UV}^{2}k_{+}\zeta_{-})\,.$
(B.6)

If we additionally assume that the maximum momenta for which the smearing
function has support obeys $(k_{+})_{max}(k_{-})_{max}\ll\ell_{UV}^{-2}$, then
the second term of (B.6) is subleading (recall that $\zeta_{-}$ integral only
has support for $\zeta_{-}<k_{-}$). Implementing this back into (IV.3)

$\displaystyle\int d^{2}x^{\pm}g(x^{\pm})^{2}\langle
T^{\text{ren}}_{--}\rangle_{\psi}\gtrsim$
$\displaystyle-\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}\int_{0}^{k_{-}}d\zeta_{-}\int_{0}^{\frac{1}{\ell_{UV}^{2}\zeta_{-}}}d\zeta_{+}(\zeta_{+})^{n/2-2}(\zeta_{-})^{n/2}\Theta(k_{-})$
$\displaystyle\gtrsim$
$\displaystyle-\frac{1}{\ell_{UV}^{n-2}}\int\frac{d^{2}k_{\pm}}{(2\pi)^{2}}|\tilde{g}(k_{+},k_{-})|^{2}\,k_{-}^{2}$
$\displaystyle\gtrsim$ $\displaystyle-\frac{1}{\ell_{UV}^{n-2}}\int
d^{2}x^{\pm}(\partial_{-}g(x^{\pm}))^{2}$ (B.7)

again arriving at the SNEC. Note that in this covariant approach the
assumption that $(k_{+})_{max}(k_{-})_{max}\ll\ell_{UV}^{-2}$ places a strong
limitation on how finely one can probe the light-ray, depending on the UV
cutoff.

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