Averaged null energy condition from causality
Received: May
Averaged null energy condition from causality
Thomas Hartman 0
Sandipan Kundu 0
Amirhossein Tajdini 0
Ithaca 0
U.S.A. 0
0 Department of Physics, Cornell University
Unitary, Lorentzinvariant quantum crocausality: commutators vanish at spacelike separation. For interacting theories in more than two dimensions, we show that this implies that the averaged null energy, R duTuu, must be nonnegative. This nonlocal operator appears in the operator product expansion of local operators in the lightcone limit, and therefore contributes to npoint functions. We derive a sum rule that isolates this contribution and is manifestly positive. The argument also applies to certain higher spin operators other than the stress tensor, generating an in  nite family of new constraints of the form R duXuuu u for the coupling constants of spinning operators in conformal eld theory, which include as special cases (but are generally stronger than) the existing constraints from the lightcone bootstrap, deep inelastic scattering, conformal collider methods, and relative entropy. We also comment on the relation to the recent derivation of the averaged null energy condition from relative entropy, and suggest a more general connection between causality and informationtheoretic inequalities in QFT.
Conformal Field Theory; Field Theories in Higher Dimensions

3.1
3.2
4.1
4.2
4.3
4.4
4.5
4.6
5.1
5.2
5.3
1 Introduction
2 Derivation of the ANEC
3 Average null energy in the lightcone OPE
Lightcone OPE
Contribution to correlators
3.3 Scalar example
4 Sum rule for average null energy
Analyticity in position space
Rindler positivity
Bound on the real part
Factorization
Sum rule
Nonconformal QFTs
5 HofmanMaldacena bounds
Conformal collider redux
Relation to scattering with smeared insertions
Relation to the shockwave kinematics
6 New constraints on higher spin operators
6.1 Es in the lightcone OPE
6.2
Sum rule and positivity
6.3 Comparison to other constraints and spin114 example
A Rotation of threepoint functions
B Normalized three point function for hJ XJ i
C Free scalars
1
Introduction
The average null energy condition (ANEC) states that
Z 1
0 ;
{ 1 {
inequality plays a central role in many of the classic theorems of general relativity [1{4].
Matter violating the ANEC, if it existed, could be used to build time machines [5, 6]
and violate the second law of thermodynamics [7]. And, unlike most of the other energy
conditions discussed in relativity (dominant, strong, weak, null, etc.), the ANEC has no
known counterexamples in consistent quantum
eld theories (assuming also that the null
geodesic is achronal [8]).
Though often discussed in the gravitational setting, the ANEC is a statement about
QFT that is nontrivial even in Minkowski spacetime without gravity. In this context,
the rst general argument for the ANEC in QFT was found just recently by Faulkner,
Leigh, Parrikar, and Wang [9]. (Earlier derivations [10{14], were restricted to free or
superrenormalizable theories, or to two dimensions.) The crucial tool in the derivation of
Faulkner et al. is monotonicity of relative entropy. Assuming all of the relevant quantities
are well de ned in the continuum limit, the argument applies to a large (and perhaps dense)
set of states in any unitary, Lorentzinvariant QFT.
Separately, the ANEC for a special class of states in conformal eld theory was derived
recently using techniques from the conformal bootstrap developed in [15, 16]. These special
cases of the ANEC, known as the HofmanMaldacena conformal collider bounds [17], were
derived in [18, 19]. The derivation relied on causality of the CFT, in the microscopic sense
that commutators must vanish outside the lightcone, applied to the 4point correlator
h [T; T ] i where T is the stress tensor and
is a scalar. However, it was not clear from the
derivation why the bootstrap agreed with the ANEC as applied by Hofman and Maldacena,
or whether there was a more general connection between causality and the ANEC in QFT.
In this paper, we simplify and extend the causality argument and show that it implies
the ANEC more generally. We conclude that any unitary, Lorentzinvariant QFT with
an interacting conformal xed point in the UV must obey the ANEC, in agreement with
the informationtheoretic derivation of Faulkner et al. The argument assumes no higher
spin symmetries at the UV
xed point, so it requires d > 2 spacetime dimensions and
does not immediately apply to free (or asymptotically free) theories. A byproduct of the
analysis is a sum rule for the integrated null energy in terms of a manifestly positive 4point
function. Furthermore, we argue that the ANEC is just one of an in nite class of positivity
constraints of the form
Z
duXuu u
0
(1.2)
where X is an evenspin operator on the leading Regge trajectory (normalized
appropriately)  i.e., it is the lowestdimension operator of spin s
2. This implies new constraints
on 3point couplings in CFT; we work out the example of spin1/spin1/spin4 couplings.
Another interesting corollary is that, like the stress tensor, the minimaldimension
operator of each even spin must couple with the same sign to all other operators in the theory.
(There may be exceptions under certain conditions; see section 6 for a discussion of the
subtleties.) In analogy with the HofmanMaldacena conditions on stress tensor couplings,
we conjecture that (1.2) evaluated in a momentum basis is optimal, meaning that the
{ 2 {
HJEP07(21)6
resulting constraints on 3point couplings can be saturated in consistent theories. This
remains to be proven.1
The connection between causality and null energy is well known in the gravitational
context (see for example [20, 21] and the references above) and in AdS/CFT (see for
example [22, 23]). In a gravitational theory, null energy can backreact on the geometry in
a way that leads to superluminal propagation in a curved background. Our approach is
quite di erent, since we work entirely in quantum
eld theory, without gravity, and invoke
microcausality rather than superluminal propagation in curved spacetime. On the other
hand, in holographic theories, the derivation of the ANEC in [22] only relies on physics close
to the boundary, so it is natural to guess that it can be rephrased as a general derivation
The F theorem, which governs the renormalization group in three dimensions [29{32], was
derived from strong subadditivity of entanglement entropy but has resisted any attempt at
a derivation using more traditional tools. On the other hand, its higherdimensional cousin,
the atheorem in four dimensions, was derived by invoking a causality constraint [33] (and
in this case, attempts to construct an entanglement proof have been unsuccessful). So
causality and entanglement constraints both tie deeply to properties of the
renormalization group, albeit in di erent spacetime dimensions. Another tantalizing hint is that in
holographic theories, RG monotonicity theorems in general dimensions are equivalent to
causality in the emergent radial direction [29, 30].
These clues suggest that the two types of Lorentzian constraints  from causality
and from quantum information  are two windows on the same phenomena in quantum
eld theory. It would be very interesting to explore this further. For instance, perhaps
the F theorem can be understood from causality; after all, a holographic violation of the
1In free eld theory our methods do not apply directly, but a simple mode calculation in an appendix
presented here has several advantages over previous bootstrap methods in [16, 18, 19].
First, it makes manifest the connection between causality constraints and integrated null
energy. Second, it produces optimal constraints (for example the full HofmanMaldacena
bounds on hT T T i) without the need to decompose the correlator into a sum over composite
operators in the dual channel. This decomposition, accomplished in [19], was technically
challenging for spinning probes, and becomes much more unweildy with increasing spins
(say, for hT T T T i). The simpli cation here comes from the fact that the new approach
allows for smeared operator insertions, and these can be used to naturally project out an
optimal set of positive quantities. Finally, the new method produces stronger constraints
on the 3point functions of nonconserved spinning operators. On the other hand, this
approach does not give us the solution of the crossing equation in the lightcone limit or
the anomalous dimensions of highspin composite operators as in [19].
Outline.
The main argument is given rst, in section 2. The essential new ingredient
that it relies on is the fact that the null energy operator appears in the lightcone OPE; this
is derived in section 3. For readers already familiar with the chaos bound [15] and/or earlier
causality constraints [16], sections 2{3 give a complete derivation of R duTuu
0. The sum
rule is derived in section 4, where we also review the methods of [15, 16]. In section 5, we
show how to smear operators to produce directly the conformal collider bounds in the new
approach  this section is in a sense super uous because conformal collider bounds follow
from the ANEC, but it is useful to see directly how the two methods compare. In all cases
we are aware of, this particular smearing produces the optimal set of constraints on CFT
3point couplings. Finally, in section 6, we generalize the argument to the ANEC in any
dimension d > 2, as well as to an in nite class of higher spin operators X.
2
Derivation of the ANEC
In this section we outline the main argument. Various intermediate steps are elaborated
upon in later sections. Our conventions for points x 2 R1;d 1 are
x = (t; y; ~x) or (u; v; ~x);
u = t
y;
v = t + y :
(2.1)
In expressions where some arguments are dropped, those coordinates are set to zero.
is
always a real scalar primary operator. Hermitian conjugates written as Oy(x) act only on
we set d = 4 in the rst few sections, leaving the general case (d > 2) for section 6.
De ne the average null energy operator
E =
Z 1
1
duTuu(u; v = 0; ~x = 0) :
(2.2)
{ 4 {
ψ
O
O
ψ
null energy integrated over the red line, which in the limit of large u takes the form hO R duTuuOi.
This is then related to an expectation value by a Euclidean rotation.
The goal is to show that this is positive in any state, h jE j i
conformal eld theories. In CFT, it is su cient to show that
0.
We rst discuss
hOy(t = i ; y = 0) E O(t =
i ; y = 0)i
0
states can be created this way.2
for an arbitrary local operator O (not necessarily primary). The insertion of O in imaginary
time creates a state on the t = 0 plane so that this 3point function can be interpreted as an
expectation value hOjE jOi. And, in a conformal eld theory, a dense subset of normalizable
In section 3, we show that the nonlocal operator E makes a universal contribution
to correlation functions in the lightcone limit. The key observation is that the operator
product expansion of two scalars in the lightcone limit can be recast as
in [36, 37]) including the contributions of all minimaltwist operators, (@u)nTuu for n
We have assumed that the theory is interacting, so there are no conserved currents of spin
> 2 [38], and that there are no very lowdimensions scalars in this OPE. (The second
2On the sphere, these states are complete by the stateoperator correspondence. On the plane, therefore,
any state the consists of local operators smeared over some
nite region can be created this way, and by
the ReehSchlieder theorem, such states are dense in the Hilbert space [34, 35].
{ 5 {
(2.3)
(2.4)
(2.5)
HJEP07(21)6
assumption is not necessary for the derivation of the sign constraints since we can project
onto stress tensor exchange; see section 6.)
Now consider the normalized 4point function
G = hO(y = ) (u; v) ( u; v)O(y = )
i ;
hO(y = )O(y = )ih (u; v) ( u; v)i
in the regime (2.5), as illustrated in
gure 1. O denotes the Rindler re ection of the
operator O. For scalars, O(t; y; ~x)
Oy( t; y; ~x); see section 4.2 for the action on spinning operators. The OPE (2.4) gives
HJEP07(21)6
G = 1 +
N
T vu2hO(y = ) E O(y = )
i
with N
= hO(y = )O(y =
)i > 0. The correction term, hO(y = ) E O(y =
)i, is
computed by a residue of the null line integral, and is purely imaginary.
Although the correction in (2.7) is small, it is growing with u. Corrections of this
form were studied by Maldacena, Shenker, and Stanford in [15], and in the CFT context
in [16, 18], where it was shown that if such a term appears, it must have a negative
imaginary part.3 Therefore
ihO(y = ) E O(y = )
i
This conclusion relies on a number of analyticity and positivity conditions that the
correlator must satisfy; we check these conditions and review the argument in detail below. It
can be understood as a causality constraint. If the correction has the wrong sign, then it
requires the correlator to have singularities in a disallowed regime, and these singularities
lead to nonvanishing commutators at spacelike separation.
This is not yet (2.3), since in one case the operators are inserted in Minkowski space
and in the other case o set in imaginary time. In fact, these are equivalent, by acting with
a rotation R that rotates by 2 in the Euclidean y plane (with
= it):
ihO(y = ) E O(y = )i = h(R O)y(t = i ) E R O(t =
(The null contour de ning E is also trivially rotated in relating these two expressions.) The
ANEC, in an arbitrary state in CFT, then immediately follows from (2.8).
For comparison to previous work, we note that the arguments in [15, 16, 18] were
phrased in terms of conformal cross ratios, and it was important that the correlator was
evaluated on the `2nd sheet', i.e. after a particular analytic continuation in the cross ratios.
The current approach is equivalent. The analytic continuation is entirely captured by the
choice of contour that de nes E in the formulas above, implicit in the way we have ordered
the operators, as we will discuss in detail in section 3.
3The chaos bound of MSS refers to largeN theories, and the interest in [15] was in a di erent regime
of the correlator (the Regge regime). Here, as in [16], we are applying similar methods to the lightcone
regime of a smallN theory. In the Regge/chaos regime, the small parameter that controls the OPE is
1=N , whereas here it is the expansion in v as we approach the lightcone. These limits do not, in general,
commute, even in large N theories, so the physics of the two classes of constraints is di erent.
{ 6 {
(2.6)
(2.7)
(2.8)
but
intricate.
It was previously noted in [40] that the ANEC would follow if E appears as the leading
term in a re ectionpositive OPE. This is similar in some ways to our argument, but the
details are di erent. The lightcone OPE invoked in the rst step actually produces not hE i
1 + ihE i, and because of the crucial factor of i, the argument for positivity is more
Conformal invariance was used several times in the derivation above, but under mild
assumptions the conclusions apply also to nonconformal QFTs with an interacting
xed
point in the UV. The approach to the lightcone is controlled by the UV
xed point, so
if the
xed point is an interacting CFT, then the OPE formula (2.4) still applies. (This
would not be true if the UV
xed point were free, since then an in nite tower of higher
spin currents would contribute to the lightcone OPE at leading order.) One might worry
that in the limit u
1, some pairs of operators in the 4point function are at large timelike
separation, so perhaps there are signi cant infrared e ects. We do not have a complete
argument that it is impossible, so leave this as an open question. A similar OPE argument
was used in [39] to derive Bousso's covariant entropy bound, and it was argued that such
e ects should be absent  the same arguments apply here, so we consider this a mild
assumption. See section 4.6 for further discussion.
3
Average null energy in the lightcone OPE
In this section, we will derive the universal contribution of the null energy operator E ,
de ned in (2.2), to npoint correlation functions in (3 + 1) dimensions. The general case
(d 6= 4 and/or spin > 2) is in section 6.
3.1
Consider a scalar primary . In general, two nearby operators can be replaced by their
operator product expansion,
(x2) (x1) =
X C
i
i
(x1
x2)Oi
(x2) :
(3.1)
In the limit that x2
x1 becomes null, the OPE is organized as an expansion in twist,
i `i, where
is scaling dimension and ` is spin. For now we will assume that the stress
tensor T
is the unique operator of minimal twist. (This assumption is not necessary
for the ANEC, as long as the theory is interacting and the stress tensor is the only
spin2 conserved current; see section 6.) Then the leading contributions to the OPE in the
lightcone limit v ! 0 are
(u; v) ( u; v) = h (u; v) ( u; v)i 1 + vu3 X cn(u@u)nTuu(0) +
#
;
(3.2)
with corrections suppressed by powers of v. We have inserted the operators symmetrically
in the uvplane with ~x = 0, and expanded about the midpoint. Other descendants of T
are subleading because they must come with powers of v in order to contract indices.
"
{ 7 {
1
n=0
The constants cn can be determined by plugging (3.2) into the 3point function h
and comparing to the known answer, which can be found in [41]. However it is more
elucidating to rewrite the lightcone OPE as an integral, rather than a sum. In fact (3.2) is
exactly equivalent to
h (u; v) ( u; v)i
T i, and designing the kernel to reproduce the known answer. Alternatively, we
can expand the integrand as Tuu(u0) = Tuu(0) + u0@uTuu(0) +
, do the integral, and check
that (3.3) reproduces (3.2) with the correct cn's. The OPE coe cient c
T is xed by the
In (3.3), the lightcone OPE is expressed as an integral of Tuu over the null ray
connecting the two
's. It is an operator equation, meaning it can be used inside correlation
functions, though we must be careful about convergence (or, equivalently, coincident point
singularities).4
In the limit u ! 1, the integration kernel is trivial, so the lightcone OPE produces
the integrated null energy operator:
where T = 1c0T 2 . This equation holds in the limit where we rst take v ! 0, then u ! 1,
assuming that all other operator insertions are con ned to some nite region. Corrections
are subleading in 1=u or v.
3.2
Contribution to correlators
Now consider the correlator
h (x1) (x2)O(x3)
O(xn)i :
(O may have spin; its indices are suppressed.)
If all points are spacelike separated, then the
OPE is convergent. If, on the other
hand, for instance x1
x3 is spacelike but x2
x3 is timelike, then the full
OPE may
diverge. Still, we expect that any
nite number of terms in the lightcone OPE produce
a reliable asymptotic expansion in the limit v2 ! v1. This is argued in detail for 4point
functions in section 4.5 of [16] by comparing to a di erent, convergent OPE channel. (More
heuristically, it is reasonable to trust a divergent expansion in v as v ! 0 so long as
subsequent terms are highly suppressed, just as in ordinary perturbation theory.) For npoint
functions, a similar argument holds. The conclusion is that (3.4) can be used inside
arbitrary correlation functions, as long as we take the limits with all other quantities held xed.
4See [42] for recent progress in writing general OPEs by integrating over causal diamonds. The OPE (3.2)
can also be derived using shadow operators, as described in appendix B of that paper. It is interesting to
note the similarity to the formula for the vacuum modular Hamiltonian of an interval in 1+1 dimensions,
H
integral expression for the null OPE of twist line operators.
x2=L2)Ttt, and also to the recent derivation of the Bousso bound [39], which relied on an
{ 8 {
(3.4)
(3.5)
ψ
O(x3) O(x4)
O(x5) O(x6)
ψ
1
ju1j
Z 1
1
{ 9 {
but are widely separated in u, the leading nonidentity term in the
over the red null line.
OPE is R duTuu, integrated
Operators are ordered by the standard prescription (reviewed in detail in [16]): to
compute
hO1(x1)O2(x2)
On(xn)i ;
shift ti ! ti i i with 1 > 2 >
> n, and de ne the correlator by analytic continuation
from the Euclidean. In the domain with a xed imaginary time ordering, the function is
analytic, and sending i ! 0, it produces the Lorentzian correlator with operators ordered
as written in (3.6). When we apply the lightcone OPE (3.4), this translates into a choice of
contour for the uintegral. For concreteness, set n = 6 and suppose the O's are all at t = ~x =
0, with two O's in each Rindler wedge, as in gure 2. (Generalizing to arbitrary Minkowski
insertions x3;:::;n 2 R1;d 1 with xi2 > 0 is straightforward.) Suppressing coordinates set to
zero, rst consider the ordering
h (u1; v1) ( u1; v1)O(y3)O(y4)O(y5)O(y6)i
with v1 < 0 < u1. In the limit jv1j
1, the OPE gives the leading terms
h (u1; v1) ( u1; v1)ih 1 + T v1u12
duTuu(u; v = 0) O(y3)O(y4)O(y5)O(y6)i :
The integral has singularities at
u = y3; : : : ; y6. The i prescription says that to compute
(3.6)
(3.7)
(3.8)
HJEP07(21)6
the correlator ordered as in (3.7), the contour in the complex uplane goes below the poles:
−y6
−y5
−y4
−y3
as (or faster than) hTuu(u)T
(x)i
vanishes, only that ituhas no terms
For example, the timeordered correlator
As juj ! 1, applying the OPE to all the O's implies that the integrand falls o the same
u 6, so we can deform the contour and the integral
vanishes. (This does not mean that the stress tensor contribution to the lightcone OPE
v1u12. The rst nonzero contribution is actually
v1=u31, using (3.3).) Other orderings are obtained by deforming the contour across poles.
h (u1; v1)O(y3)O(y4)O(y5)O(y6) ( u1; v1)i
y
y
y
y
is again computed by (3.8), but now integrating on the following contour:
−y6
−y5
−y4
−y3
The integral is equal to the sum of residues at u =
y5; y6 or at u =
y4; y3, so this
ordering does have terms
3.3
Scalar example
v1u12.
In the language of [15, 16], the trivial contour (3.9) is the 1st sheet (or Euclidean) correlator,
while the nontrivial contour (3.11) produces the correlator on the 2nd sheet.
As a simple application, let us reproduce the well known hypergeometric formula for
the fourpoint conformal block in the lightcone limit. Consider the fourpoint function of
identical scalars,
Gscalar(z; z) = h (0) (z; z) (
1
) (
1
)i :
(3.12)
With these kinematics, the cross ratios are simply the lightcone coordinates of the second
insertion. For z; z 2 (0; 1), all points are spacelike. Plugging in the lightcone OPE (3.3)
gives an integral that is easily recognized as a hypergeometric function, and we reproduce
the well known formula for the stress tensor lightcone block (see [36, 37]):
30z
z2
This is regular as z ! 0. But after going to the second sheet, i.e. sending log(1
log(1 z) 2 i, the behavior near the origin is
iz=z2. This growing term, with the correct
coe cient, is what is captured by the approximation where we replace the full lightcone
OPE by just the null energy operator, as in (3.4). So what we have shown is that these
growing terms, responsible for all the results in [16, 18, 19], are precisely the contributions
of the null energy operator E .
4
Sum rule for average null energy
Now we will ll in the details of the ANEC derivation in section 2. Most of this discussion
is a review of [15] and [16], some of it from a di erent perspective. First, we will collect
some facts about positionspace correlation functions, which hold in any unitary,
Lorentzinvariant QFT. Then we put them together to derive the sum rule and positivity condition.
4.1
Analyticity in position space
The point of view taken throughout the paper is that a QFT can be de ned by its Euclidean
npoint correlation functions [34, 35]. These are singlevalued, permutation invariant
functions of x1;:::;n 2 Rd, i.e. there are no branch cuts in Euclidean signature. This ensures that
in the Lorentzian theory, noncoincident local elds at t = 0 commute with each other.
The Euclidean correlators can be analytically continued to complex xi. However, there
are branch points when one operator hits the lightcone of another. (See [34, 35] for details
or section 3 of [16] for a review.) When we encounter one of these branch points, we must
choose whether to go around it by deforming t ! t + i or t ! t
i , and this selects
whether the two operators are timeordered or antitimeordered. Thus the i prescription
to compute a Minkowski correlator ordered as
h 1(x1) 2(x2) 3(x3)
i
Im t1 < Im t2 < Im t3 <
:
Im x1 C Im x2 C Im x3 C
;
is to gives each ti a small imaginary part, with
The resulting function is analytic as long as we maintain (4.2), so once we've speci ed the
i prescription, the analytic continuation from Euclidean is unambiguous.
In fact, the domain of analyticity of the npoint correlator G(xi), viewed as a function
on (a subdomain of a cover of) n copies of complexi ed Minkowski space, is larger than
indicated by (4.2). It is also analytic on the domain de ned by the covariant version of (4.2):
where x C y means `x is in the past lightcone of y'.5 (Actually, it is analytic on an even
larger domain, but (4.3) is all we need.6)
is derived in [43]7 and also in [15].
De ne the Rindler re ection
Correlation functions in Minkowski space restricted to the left and right Rindler wedges
obey a positivity property analogous to re ection positivity in Euclidean signature. This
x = (t; y; ~x) = ( t ; y ; ~x) :
(4.4)
HJEP07(21)6
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
The transverse coordinate ~x is taken to be real. For real (t; y), this re ects a point in the
right Rindler wedge to the left Rindler wedge. Acting on a spinning operator O,
O
(t; y; ~x) = (
1
)P Oy
( t ; y ; ~x)
where P is the number of tindices plus yindices. (The Hermitian conjugate on the
righthand side acts only on the operator, not the coordinates.) This operation is CPT together
with a rotation by
around the yaxis: O = J OJ with J = U (R(y; ))CP T [43].
We will insert points in a complexi ed version of the left and right Rindler wedges.
The complexi ed right wedge is de ned as
RC =
(u; v; ~x) : uv < 0; arg v 2
;
and the complexi ed left wedge is
LC = RC =
(u; v; ~x) : uv < 0; arg v 2
3
;
The positivity condition for 2npoint correlators is
hO1(x1) O2(x2)
On(xn)O1(x1)O2(x2)
On(xn)i > 0 ;
for x1;:::;n 2 RC and
Im t1
Im t2
Im tn :
5We de ne the imaginary part of a complexi ed point by the convention that a point in Minkowski space
R1;d 1 has xi = Re xi. Thus the real and imaginary parts each live in a copy of Minkowski space, not
Euclidean space.
6The full domain is described as follows [34, 35]. First act on the domain (4.3) by all possible complex
Lorentz transformations; this de nes the extended tube. Then, permute the n points and a repeat this
procedure, to de ne the permuted extended tube. In the theory of multiple complex variables, the domain
of analyticity cannot be an arbitrary shape  once we know a function is analytic on some domain, it
must actually be analytic on a (generally larger) domain called the envelope of holomorphy. The domain of
analyticity of the correlator G(xi) is the envelope of holomorphy of the union of permuted extended tubes.
7What we call Rindler positivity is not, however, quite the same as `wedge re ection positivity' referred
to in the title of the paper [43]. The di erence is discussed in section 3 of [43].
Note that for product operators, the order does not reverse under re ection:
O1O2 = O1 O2 :
For real insertions, Im ti = 0, the operators in (4.8) are ordered as written, which we will
refer to as `positive ordering'. They are not time ordered. For complex insertions, the
correlator is de ned by analytic continuation from Euclidean within the domain (4.2). The
re ected operators have
which explains why they must be ordered as in (4.8).
support in a single complexi ed Rindler wedge. That is,
Positivity applies also to smeared operators, and products of smeared operators, with
Im
t
1
Im
t
2
Im
tn ;
(4.11)
where the smearing functions f have support in some localized region of RC , and the
operator ordering in
is the same as in .
These positivity conditions hold in any unitary, Lorentzinvariant QFT [15, 43]. We
will not repeat the derivation, but an intuitive way to understand this is as follows. To be
concrete, consider a case of particular interest for the ANEC and the HofmanMaldacena
constraints that will be used below:
0 =
(t0; y0)
Z 1
0
d
Z 1
0
dy
Z
d
d 2~xf ( ; y; ~x)O(t =
with t0; y0 > 0, and assume f is nonzero in some nite region.
and O may be timelike
separated, in the sense that Re (x0
x) 2 R1;d 1 lies in the forward lightcone for points of
the integral. We want to understand why h 0 0i > 0. First, we can evolve
(t0; y0) back
to the t = 0 slice; it becomes nonlocal, but with support only in the right Rindler wedge.
The same can be done in
0, evolving ( t0; y0) forward to the t = 0 slice. Then in the
yplane, the 4point function h 0 0i is smeared over the regions shown in gure 3. These
insertions are symmetric under y !
y together with complex conjugation; therefore,
reection positivity of the Euclidean theory guarantees that this correlator is positive.
Keeping track of Lorentz indices on O leads to the same conclusion. For a more precise derivation
we refer to [43] for real insertions, and [15] for insertions in the complexi ed wedge RC .
To recap, although h
i does not look like a norm in the theory quantized on the
= 0 plane  since
j0i is not normalizable, and
6
=
y  it is a norm in the theory
quantized on the y = 0 plane. These two di erent quantizations correspond to two di erent
ways of analytically continuing a Euclidean theory to Minkowski space as shown in gure 4.
In conformal eld theory, the positivity properties discussed here follow from the fact
that the conformal block expansion has positive coe cients (in the appropriate channel), as
f ∗O
ψ
Z
f O
orange plane, parameterized by ( ; y; ~x)) is the same in both pictures, but the de nition of states
and corresponding notion of Minkowski spacetime (vertical blue planes) is di erent in the two cases.
On the left, the continuation to Lorentzian is
! it, states of the theory are de ned on the plane
= 0, and y is a space direction. On the right, the continuation to Lorentzian is y ! it0, states
are de ned at y = 0, and
= y0 is a space direction. The two theories are identical, since they
are determined by the same set of Euclidean correlators, but the map of observables and matrix
elements from one description to the other is nontrivial.
described in [16, 18]. We have chosen a di erent but conformally related kinematics in the
present paper because (i) it makes the positivity conditions more manifest, (ii) positivity
in the new kinematics does not require conformal invariance, and (iii) it allows us to smear
operators while easily maintaining positivity properties needed to derive the constraints.
4.3
Bound on the real part
With operators inserted symmetrically across the Rindler horizon, the positiveordered
correlator is real and positive. The timeordered correlator is generally complex, but it
inherits from (4.8) bounds on its magnitude and real part. This was derived in [15] by
interpreting the correlator as a Rindler trace and applying CauchySchwarz inequalities.
The positivity condition for CFT shockwaves derived in [16, 18] using the decomposition
into conformal partial waves can also be restated in this way.
Here we repeat the CauchySchwarz derivation, but in Minkowski language. Let A, B
be operators (possibly nonlocal) with support in the right wedge RC . The positiveordered
correlator de nes a positive inner product (A; B)
inequality applies,
hABi. Therefore the CauchySchwarz
In the derivation of the ANEC in section 2 we considered
jhABij2
hAAihBBi :
Ganec = hO (u; v) ( u; v)Oi :
(There O was local, but for the present purposes it can also be smeared.) Applying
CauchySchwarz with A = O ( u; v), B =
( u; v)O,
Re Ganec
jGanecj
hO
O ih O
Oi
1=2
where
( u; v);
(u; v). Note that both of the correlators on the righthand
side are positiveordered.
4.4
Factorization
In the limit u ! 1 (with everything else held xed or v
1=u), positiveordered correlators
factorize into products of twopoint functions. In a CFT, this follows from the OPE. In this
limit, we can replace O by a local operator, and the conformal cross ratios of the 4point
function hO
O i are z; z
0. Positive ordering means that we do not cross any branch
cuts to reach this regime [15, 16], so the correlator can be computed by the usual Euclidean
OPE and is dominated by the identity term. Thus
hO
O i
h O
Oi
hOOih
i
and (4.17) becomes
Re Ganec
jGanecj
hOOih (u; v) ( u; v)i + " :
The correction term " on the right is necessary because the positiveordered correlator does
not exactly factorize. It has corrections from subleading operator exchange. But " is
suppressed by positive powers of v and 1=u, so it can be neglected everywhere in the derivation.
This argument relied on conformal invariance, but factorization is expected to hold
in a general interacting theory (possibly excluding integrable theories). This is motivated
on physical grounds in [15] by relating it to thermalization of nite temperature Rindler
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
energy, hOjE jOi. Let
Now we use the method of [16] to derive a manifestly positive sum rule for averaged null
v =
;
u = 1= ;
(4.20)
with 0 <
1 and consider the function G de ned in (2.6) as a function of complex ,
with all other coordinates xed. The function G( ) obeys two important properties:
(i) G( ) is analytic on the lowerhalf
plane in a region around
from (4.3) with complexi ed points labeled as x1 = ( u; v), x2 = (y = ), x3 =
(y =
), x4 = (u; v).
(ii) For real
with j j < 1,
Re G
1 + " ;
where the correction " is suppressed by positive powers of both
and . This follows
directly from (4.19).
of a halfdisk, just below
Equipped with these facts, the sum rule is derived by integrating G( ) over the boundary
The radius R of the semicircle does not matter, as long as it is in the regime (2.5), i.e.,
R
1. The integral over a closed contour vanishes, Re d (1 G( )) = 0 :
We split this into the contributiuon from the semicircle and from the real line, then use the
OPE (2.7) to evaluate the correlator on the semicircle and do the integral. The result is
ψ
σ
I
ihO(y = ) E O(y = ) =
d Re (1
G( )) :
N
Z R
T
R
The righthand side is positive by property (ii) above. Using (2.9), the sum rule can be
written as a manifestly positive integral for the expectation value of average null energy:
hOjE jOi =
1
Z R
T
R
Re
1
G
u =
; v =
1
where jOi
the Euclidean
yplane).
p1N O(t =
i )j0i (and if O is not a scalar, the operator is rotated by =2 in
Note that two distinct positivity conditions came into play. First, there was Rindler
positivity, property (ii). Applied to the correlator G in the lightcone regime, this would
imply Re hO(y = )E O(y = )
i
0. However, this 3point function is purely imaginary,
so this constraint is trivial. Rindler positivity is nontrivial only near the origin of the
plane (the Reggelike limit), where the OPE is not dominated by the lowdimension (or
lowtwist) operators. Second, there is the positivity condition on the imaginary part of
the OPE correction, coming from the sum rule. It is this second, less direct consequence
of re ection positivity that leads to the ANEC. This is similar to the use of dispersion
relations for scattering amplitudes in momentum space  the optical theorem gives a
positivity condition on the forward amplitude, and sum rules relate this positive quantity
to the amplitude in other regimes (see for example [44]).
(4.21)
(4.22)
(4.23)
(4.24)
(4.25)
As mentioned in section 2, we expect the main conclusions and the sum rule to hold even in
nonconformal QFTs as long as there is an interacting UV
xed point. This is essentially
because it is a UV argument. The applicability of the lightcone OPE was discussed at the
end of section 2. Conformal symmetry was used again when we invoked the stateoperator
correspondence to claim that any normalizable state can be created by inserting a local
i . This made the derivation simpler, since we could restrict to local
operator insertions O(y = ; t = 0; ~x = 0). But in fact this restriction was not necessary. The
ingredients that go into the sum rule  positivity, etc  still apply if O is a nonlocal
operator de ned by smearing local operators (and their products) over a complexi ed version
of the Rindler wedge. In any QFT a dense set of states can be created in this way [34, 35].
5
HofmanMaldacena bounds
The conformal collider bounds of Hofman and Maldacena [17] are constraints on CFT
3point functions that come from imposing
v!1
lim v2h" O(P )y
Z 1
1
duTuu(u; v)" O(P )i
0 :
(5.1)
Here O(P ) is a wave packet with timelike momentum P = !t^, created by inserting a
spinning operator near the origin:
Z
" O(P ) =
dtdydd 2~x e (t2+y2+~x2)=D2 e i!t"
O
(t; y; ~x) ;
!D
The positivity condition (5.1) was an assumption in [17], motivated by the fact that this
computes the energy measured in a faraway calorimeter if we prepare a CFT in the state
" O(P )j0i. It leads to constraints on the 3coupling constants that appear in hOT Oi.
Since we have shown that E is a positive operator, the inequality (5.1) follows from
the above analysis. But it is instructive to see how constraints in this particular state are
related to our discussion of Minkowski scattering and the ANEC sum rule. That is the
goal of this section. In particular, we will show exactly how to smear the probe operators
in the previous analyses [16, 18, 19] to produce the HofmanMaldacena inequalities. This
avoids the step of decomposing the correlator into the crossed channel, used in [19] in order
to improve upon the bounds derived in [18].
5.1
Conformal collider redux
First we will restate the HofmanMaldacena condition in a way that makes all of the
integrals trivial. We perform the Fourier transform over t rst. In the regime !D
1 it is
dominated by a saddlepoint at t =
2
i !D2. Therefore, instead of viewing this as a wavepacket
with frequency !, we can replace it by an operator inserted at a xed, imaginary value of t:
Z
" O(P )
dydd 2~x e (y2+~x2)=D2 "
O
(t =
with
> 0. Also, in this limit, we only need to integrate over the position of one of the O
insertions, since the other integral gives an overall factor. The remaining gaussian can be
dropped, and the nal d
1 integrals are done by residues.8
In (5.1), the state is created near the origin of Minkowski space, and the average null
energy is evaluated near future null in nity. For comparison to the rest of the paper, it
is more convenient to shift coordinates so that the null energy is integrated over a ray at
v = 0, and O is inserted near spatial in nity. That is, (5.1) is equivalent to
(5.4)
(5.5)
(5.6)
where E is integrated over v = 0 as in (2.2), and
j" O i
dy~dd 2~x " O(t =
i ; y = y~; ~x)j0i :
In (5.4), the wavepacket is implemented by the order of limits: rst we do the uintegral
(by residues) to compute E , then take
! 1, then perform the integrals over ~x; y~.
To recap, the HofmanMaldacena constraints are restated as:
Z
dy~dd 2
~x lim
!1
2h" O(t = i ; y = ; ~0) E " O(t =
i ; y = y~; ~x)i
0 :
This expression is a convenient way to compute the explicit constraints in terms of the
3point coupling constants.
5.2
Relation to scattering with smeared insertions
The state (5.5) is most naturally created by smearing an operator near spatial in nity, but
like any localized state in a CFT, it can also be created by inserting a single, nonprimary
i ; y = ~x = 0.9 It is straightforward to
nd the operator explicitly by
a series expansion of the wavepacket integral, R dydd 2~x O(t =
i ; y; ~x)e ((y
)2+~x2)=D2 .
Applying section 2 to this particular operator is one way to derive the HofmanMaldacena
inequalities directly from the causality of the 4point function. However, we would need
to be more careful about the order of limits in the series de ning the operator and various
other steps of the calculation, especially since we are expanding a wavepacket about a point
very far from its center.
This problem is avoided if we instead apply the causality argument directly to a
correlator with smeared operator insertions, corresponding to wavepackets o set far into
imaginary time. As explained around gure 4, the interpretation of the ANEC as an expectation
8Dropping the gaussian can lead to unphysical divergences in the remaining integrals, depending on the
dimensions of the operators. These are dealt with by dimensional regularization in the transverse directions,
d 92Ev!erdywhere in this section, the limit
2 + with
! 0 at the end [45].
1
juj
1
1 and
1. As we move the wavepacket further, the constraints obtained this way approach
arbitrarily close to the HofmanMaldacena type constraints. (And, in the version of the argument with
smeared insertions, we put a hard cuto
on the wavepacket at some distance where it is exponentially
suppressed, in order to avoid coincident point singularities.)
! 1 should be interpreted as large but nite , with jvj
value and the interpretation of the ANEC as it appears in the lightcone OPE di er by a
=2 rotation in the Euclidean
yplane. Therefore, after the rotation, in order to make
contact with section 2 we are led to study the correlator
where
GHM = hOHM (u; v) ( u; v)OHM i
hOHM OHM ih (u; v) ( u; v)i
!1
Z
OHM = lim
d d
d 2~x " O(t = i ; y = ; ~x)e ((
)2+~x2)=D2
The insertions are now symmetric under Rindler re ection, and the wavepacket is centered
HJEP07(21)6
at large imaginary time. We could of course map the centers to real points in Minkowski,
but the smearing procedure in this frame would be more complicated.
As in the discussion of the HofmanMaldacena calculation above, the wavepacket can
be implemented by instead taking
OHM =
OHM =
Z
Z
d d
d 2~x " O(t =
i + i ; y = ; ~x) ;
d 0dd 2~x0 " Oy(t =
i + i 0; y =
; ~x0) ;
where "
= (
1
)P (" ) and P is the number of tindices plus yindices; the integral in E
is done rst, then the limit
! 1, then the remaining integrals.
The leading correction to (5.7) in the lightcone limit comes, as usual, from the
integrated null energy:
G
hOHM E OHM i :
The operator ordering here and in (5.7) is subtle: we do not follow the usual prescription
of analytic continuation (4.2) which orders operators by imaginary time. If we did, then
the ucontour (in the integral de ning E ) would not circle any poles, and the constrained
term would vanish as in (3.9). Instead we de ne the correlator in (5.7) with the ucontour
as follows:
u
The black circles indicate the wavepacket insertions. This picture can be interpreted two
ways:
rst, it shows the path of analytic continuation that de nes GHM , ie the route
taken by the 's as we push (u; v) forward in time starting from the Euclidean correlator.
OHM
OHM
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
Second, it is the contour of integration used when E appears in the OPE.10 The correlator
de ned in this way has the same properties as Ganec in section 2, so the sum rule (4.25)
still applies; thus the correction (5.11) has a positive imaginary part, and this inequality is
identical to the conformal collider constraints.
Relation to the shockwave kinematics
In earlier work [16, 18, 19] a di erent kinematics was used for the fourpoint function in
order to derive causality constraints. There, the kinematics corresponded to expectation
values of OO in a shockwave state created by
inserted near the origin. Although the
conformal cross ratios are the same in the two setups, the advantage of the scattering
kinematics used here is that positivity conditions are now manifest, using Rindler re ections
as in [15]. This makes it easy to generalize the argument to nonprimary, smeared insertions,
which does not appear to be straightforward using the kinematics of [16, 18, 19].
6
New constraints on higher spin operators
So far we have discussed constraints on the integrated stress tensor. As in many other
contexts (for example [16, 18, 19, 36, 44, 46], the positive sum rules for spin2 exchange
readily generalize to the exchange of higher spin operators. Let X be the lowestdimension
operator of spin s, where s > 2 is even. This operator is the dominant spins exchange in
the lightcone limit [36]. We will argue that
Es =
Z 1
1
duXuuu u(u; v = 0; ~x = 0)
(6.1)
is positive in any state. The resulting constraints agree in many cases with other
methods, but are generally stronger for nonconserved operators. It would be interesting to
check them in known conformal eld theories, for example by numerical bootstrap or other
methods. Although our OPE method does not apply to free theories, it is shown by direct
calculation in appendix C that (6.1) holds for free scalars.
The formulas in this section also hold for s = 2, so this generalizes the discussion in
the rest of the paper to the ANEC in any spacetime dimension d > 2.
6.1
Es in the lightcone OPE
First let us derive the contribution of the operator X and its descendants to the operator
product expansion of two scalars in the lightcone limit. Repeating the steps in section 3,
10An equivalent prescription is to de ne the correlator by rst expanding each wavepacket in a series
expansion around t = 0, then compute the usual timeordered correlator. Yet another equivalent prescription
is to
rst compute the timeordered correlator at
0 as we did in previous sections, then analytically
continue to nite .
we nd the lightcone OPE can be written as the integral
(u; v) ( u; v)jX = h (u; v) ( u; v)i
Z u
u
2 X c
X
p cX
2
X +s 1
X is the OPE coe cient, cX > 0 is the coe cient of the XX twopoint function,
X
s is the twist. (Conventions for the constants follow [47]). The notation
contribution of the operators (@u)nXuu u(0) for n
0. Other descendants are subleading
)jX means the contribution of the operator X, and the expression in (6.2) is the full
The OPE in the regime relevant to the ANEC is a divergent asymptotic series,
orga
, u = 1= . Let us focus on a particular power 1 s. Contributions of this
form come from operators with spin s0
s, so in the lightcone limit
contribution is the lowesttwist operator satisfying s0
! 0, the dominant
s. Denote by s the twist of the
spins operator with smallest dimension. It was argued in [36] that s is a monotonically
increasing, convex function of spin, with s
2(d
2).11 This guarantees that at the order
1= s 1, the leading contribution to the OPE indeed comes from X, which we de ned to
be the lowest dimension operator with spin s.
If
< d 2, an additional subtlety arises because there is an accumulation point in the
twist spectrum at
2
[36, 37]. It is unclear whether the asymptotic lightcone OPE can
be applied at orders in
beyond the accumulation point. Therefore in what follows we
assume the probe satis es
> X =2 (and that any other light scalars in the OPE that would
lead to accumulation points are absent). Note that this restriction was not necessary for the
ANEC, since it is already enforced by the unitarity bound when X is a conserved operator.
6.2
Sum rule and positivity
We can now simply repeat the argument of section 4, inserting a factor of s 2 into the sum
rule integral to project onto the spins contribution: H d (1
The other steps are identical, leading to
G) ! H d
s 2(1
G).12
X hO(y = ) Es O(y = )i =
N
X
s=2 Rli!m0 li!m0 Re
Z R
R
d
s 2(1
G( ))
(6.4)
11There, the argument held only above some unknown minimum spin, s
smin. An identical argument
can be made using the positionspace sum rules, following the same logic. In this case we know that the
sum rule is convergent for spin2, so this establishes monotonicity and convexity for s
12Here we have followed [19] to project onto a given spin. This method assumes that lower spin operators
have integer dimensions, to avoid additional nonanalytic contributions to the OPE which are subleading
in
but leading in , so can spoil the projection. However the method can be generalized by subtracting
these terms as well [48].
for any s
2 where
2
X +s . When s is an even integer, Rindler positivity
ensures that the right hand side of the above sum rule is nonnegative. Finally by acting
with a rotation R that rotates by 2 in the Euclidean y plane, we can generalize (2.9) for
arbitrary spinning operators:
hO(y = ) Es O(y = )i = is+1h(R O)y(t = i )Es(R O)(t =
This is derived in appendix A. Therefore, there is a constraint on the lowest dimensional
operator at each even spin:
(6.5)
(6.6)
(
1
) 2 c
s
X Es
0 :
Note that by taking X
X it is always possible to choose ( 1) 2 c
X > 0, and in
that case we have a positivity condition similar to the ANEC. However, once this choice
has been made for some coupling h
Xi we do not have the freedom to ip the sign for
a di erent probe, h 0 0Xi. This means that, like the stress tensor, the lowestdimension
operator of each spin must couple with the same sign to all possible probes.
In fact these conclusions apply to the tower of operators appearing in any given
OPE. In theories with global symmetries, di erent probes may lead to di erent in nite
s
families of constraints.
Comparison to other constraints and spin114 example
In many cases, this sum rule implies the same sign constraints on CFT 3point couplings
that have already been deduced by other methods:
For s = 2, the same results can be obtained from the ANEC using conformal
collider methods [17]. Therefore these constraints follow from monotonicity of relative
entropy [9].
For s > 2 with transverse polarizations "n = 0, where n is the null direction separating
the wavepacket insertion from the insertion of E , the results agree with deep inelastic
scattering [46]. In examples where the results are available, these also agree with the
lightcone bootstrap [18, 19]. If O is a conserved current, then we can always choose
a gauge where the polarizations are transverse.
For s = 2 and " n 6= 0  assuming O is not a conserved current  it was shown in [46]
that the ANEC is stronger than deep inelastic scattering and the lightcone bootstrap.
Therefore, analogously at higher spin, we expect the condition Es
0 to produce new
constraints, stronger than any of these other methods, when s > 2 and O is not conserved.
The simplest such case is s = 4 with O taken to be a spin1, nonconserved operator J
1. We will work out this example in detail and con rm that the of dimension
J > d
sign constraints are indeed new.
The most general 3point function hJ XJ i consistent with conformal symmetry is
written in appendix B, following [47]. It has four free numerical constants, 1;2;3;4. To derive
the constraints, we apply the HofmanMaldacena analysis to this 3point function. In
practice, this amounts to computing the integral (5.6), with O ! J and Tuu ! Xuuuu.
Requiring this to be positive gives a constraint of the form
"yA"
0 ;
where " = ("+; " ; ~"?) and A is a block diagonal matrix which depends on
X , J and the
i. The explicit formula can be found in (B.3). It follows that A must be a positive
semide nite matrix. Requiring the eigenvalues to be nonnegative gives quadratic inequalities
on the i; the explicit form is unilluminating, so we will not write it explicitly, but it is
easily found from (B.3).
Now let us compare to deep inelastic scattering [46]. Repeating their calculation for
the present example, we nd that the DIS constraints are
The constraints (6.8) are identical to the constraint (6.7) only if we set " = 0. The
constraints from the " 6
= 0 polarizations in (6.7) are stronger, so these are new  they
do not follow from any of the known methods based on conformal collider bounds, the
ANEC, relative entropy, DIS, or the lightcone bootstrap. The special role played by "
polarizations is analogous to the situation for the integrated null energy as described in [46].
Acknowledgments
We are grateful to Nima AfkhamiJeddi, John Cardy, Ven Chandrasekaran, Netta
Engelhardt, Eanna Flanagan, Diego Hofman, Sachin Jain, Juan Maldacena, Samuel McCandlish,
Eric Perlmutter, David Poland, Andy Strominger, John Stout, and Aron Wall for useful
discussions. The work of TH and AT is supported by DOE grant DESC0014123, and the
work of SK is supported by NSF grant PHY1316222. We also thank the Galileo Galilei
Institute for Theoretical Physics and the organizers of the workshop Conformal Field
Theories and Renormalization Group Flows in Dimensions d > 2, as well as the Perimeter
Institute for Theoretical Physics and the organizers of the It from Qubit workshop, which
provided additional travel support and where some of this work was done.
A
Rotation of threepoint functions
In this appendix, we will derive equation (6.5). Let us start with the threepoint function
hO(y = ) Es O(y = )i, where O(y = ) is some arbitrary operator
O(y = ) =
X cn "n:On(y = ) ;
n
O(y = ) =
X cn "n:Ony(y =
n
) :
(A.1)
On's are (not necessarily a primary) operators with any spin and Es is de ned in
equation (6.1). " is de ned in the usual way " ::: = (
1
)P (" :::) , where P is the total
number of t and y indices. Let us now look at one of the terms of this three point
function: h"1:O1y( i ;
) Es "2:O2(i ; )i. Before performing the uintegral, this three point
function, in general has the following branch cut structure:
u
Now, using the integration contour shown in red in the above gure, one can show that
)
1
h"1:O1y( i ;
)Es "2:O2(i ; )i = ih"1:O1y(0;
duXuuu u(iu) "2:O2(0; )i : (A.3)
There is no ambiguity in the right hand side correlator and hence the i has been removed.
Let us now look at the three point function h"1:O1y(0;
)Xuuu u(iu) "2:O2(0; )i. This
threepoint function can be obtained by starting with some appropriate correlator in the
Euclidean space: (x0; x1; ~x) and performing an analytic continuation x0 = it; x1 = y. On the
other hand, we can start with the same Euclidean correlator and perform a di erent analytic
continuation: x1 = it; x0 = y to obtain a di erent Lorentzian correlator. Hence, these two
Lorentzian correlators should be related to each other. More explicitly, one can show that
h"1:O1y(y =
)Xuuu u(iu) "2:O2(y = )
i
= ish"~1:O1y(t = i )Xuuu u( u) "~2:O2(t =
i )i ;
where,
Therefore, nally we can write,
"~ ::: =
: : : " ::: ;
0 0
= B i 0 0C :
i 01
A
h"1:O1y(y =
)Es "2:O2(y = )i = is+1h"~1:O1y(t = i )Es "~2:O2(t =
Normalized three point function for hJ XJ i
Here we write the matrix A and two point function of states we used in the paper explicitly.
The three point function involving two same operators with spin one and another operator
(A.4)
(A.5)
(A.6)
HJEP07(21)6
In which conformal building blocks are expressed by 1
xjk
Hij =
2xij "j xij "i + xi2j "i "j
Vi
Vi;jk =
2 (xi2j xik "
xi2kxij "i)
For state de ned in 5.5 expectation value of spin 4 operator X is given by
c
X h"
J
j
duXuuuu(0; u)j" J i =
1
= (Vol)c
X 2 J (3 +
J
2 X 4 J 3 52 (2 J ) ( 3+2 X )
2X ) (3 +
2X ) (1 +
J +
2X )
fj"+j2(4 +
+ j" j2(4(4 + 2 J
X )(2 J +
X )( X + 2 J
2)( X + 2 J )(2 4
1)
X )(12
2X + 2 J (4 +
X )) 2+
+ 2(4 +
X )( 2
2 J +
X )((8 + 4 J
2 X ) 3 + ( X
2 J ) 4)
(96 + 16 J (4 +
J ) + 4 J (1 +
J ) X
+ 2("+"
+ " "+ )(2 J +
X )((16 + 4 X
2(3 + 2 J ) 2X +
2X + 2 J (2 +
3X ) 1)+
X )) 1
2(4 + 2 J
+ j"?j2(4 +
X )(2 +
X ) 2
4(4 +
d
1 and the convexity condition for the twist of a spin4 operator d 2
X
4
2) [36] (in d = 4), all gamma functions are positive.
The volume term in three point function is canceled out by the same factor in the two
point function, which is always positive:
h
J j" J i =
= (Vol)
3
23 2 J cJ ( 1 2
)
2 J 3 (1+
(B.2)
(B.3)
(B.4)
with spin 4 is given by
hJ (x1; "1)X(x2; "2)J (x3; "3)i =
1
(x212) 2X +2(x223) 2X +2(x231) J + 2X 1
+ 2V23(V3H12 +H23V1)+ 3V22H12H23 + 4V24H13g
(B.1)
f 1V1V3V24+
J ) f(j"+j2 +j" j2)(2 J 4) 2("+"
+" "+ )+j"?j2( J 1)g
C
Free scalars
In this appendix, we show that the inequality Es = R duXuuu u
scalar elds, with s
2 an even spin. For s = 2, this is the ANEC, proved for free scalars
in [10, 11]. The OPE methods in the body of the paper do not immediately apply, because
the expansion in twist has an in nite number of contributions already at leading order.
0 holds also for free
Instead we will follow the derivation of the ANEC in [10]. { 25 { The allnull components of the conserved, symmetric, traceless spins current in the theory of a free scalar takes the form
s
i=0
Xu u =
Es
Z
where ais are known coe cients. (The explicit formula is equation (4.99) in [49] but will
not be needed.) Therefore after integration by parts, the generalization of the averaged
null energy, up to an overall constant, is
the scalar eld operator
of (C.2) as
Z
d 1~ Z
k
The overall coe cient does not matter, since we only need to show that Es has a de nite sign
 if it is nonpositive, we can de ned X !
X to make it nonnegative. Classically, (C.2)
is obviously signde nite because the integrand is positive, but quantum mechanically this
is true only after doing the integral. To proceed, we use the standard mode expansion for
= R d
d 1~k[u~ka~k + h:c:] with u~
k
e ikx to write the integrand
d 1~k0 2( iku)s=2(iku0)s=2u~ku~k0 a~yk0 a~k
+ ( iku)s=2( iku0)s=2u~ka~ku~k0 a~k0 + (iku)s=2(iku0)s=2u~ka~yku~k0 a~yk0
The rst term is obviously a nonnegative operator, since it is R dd 1~ku~ka~k times its
Hermitian conjugate. The other terms can be negative (for example in squeezed states), but
disappear upon integrating over the null ray in (C.2), leaving only nonnegative
contributions. See [10] for details of these integrals, as well a careful demonstration that exchanging
the order of the uintegral and kintegrals is justi ed in states with a nite number of
particles and nite energy.
Open Access.
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