On the positivity of light-ray operators
B. W. Freivogel1 and H. Stoffels2
1
arXiv:2501.08386v1 [hep-th] 14 Jan 2025
ITFA and GRAPPA, University of Amsterdam, 904 Science Park, 1098 XH
Amsterdam, the Netherlands
2
School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh
Campus, EH14 4AS Edinburgh, Scotland
January 16, 2025
Abstract
´
We consider light-ray operators L2n = dx+ (x+ )2n T++ , where x+ is a null coordinate
and n a positive integer, in QFT in Minkowski spacetime in arbitrary dimensions. These
operators are generalizations of the average null energy operator, which is positive. We give
a proof that the light-ray operators are positive in a non-minimally coupled but otherwise
free scalar field theory, and we present various arguments that show that L2 is positive
semi-definite in two-dimensional conformal field theories. However, we are also able to
construct reasonable states which contradict these results by exploiting an infrared loophole
in our proof. To resolve the resulting tension, we conjecture that the light-ray operators are
positive in a more restrictive set of states. These states satisfy stronger conditions than the
Hadamard condition, and have the interpretation of states that can be physically prepared.
Our proposal is nontrivial even in two-dimensional CFT.
1
Introduction
All sensible classical theories obey the Null Energy Condition (NEC) [1, 2], which states that
for any null vector la , the stress-energy tensor Tab must obey Tab la lb ≥ 0. The NEC plays a key
role in constraining possible solutions of general relativity: exotic spacetimes such as traversable
wormholes [3], bouncing cosmologies [4], and black hole mimickers [5] are all forbidden by the
NEC.
In contrast, every quantum field theory (QFT) allows for violations of the NEC [6], allowing
some of these exotic solutions to exist. For example, one can construct a traversable wormhole
within the Standard Model of particle physics coupled to general relativity [7]. In order to
understand what types of exotic solutions are possible, it is necessary to determine to what
extent QFTs can violate the NEC.
In this paper, we are motivated by these gravitational questions but work in the simplest
setting: a QFT in Minkowski spacetime, without gravity. Two important ways to implement
the NEC in this context have been proposed. One is the Quantum Null Energy Condition
(QNEC) [8, 9], which bounds hTab la lb iψ from below using the entanglement entropy. The other
one is the Averaged Null Energy Condition (ANEC) [10, 11], which states that the NEC holds
on average along a null geodesic with tangent vector la , i.e.
ˆ ∞
hT++ iψ dx+ ≥ 0 ,
(1)
−∞
where we assumed, without loss of generality, that x+ = x0 + x1 is an affine parameter along the
null geodesic, and |ψi is any state of the field. Our goal is to determine whether any other energy
1
conditions are generally true, and to that end we expand on our work in [12] by considering the
following light-ray operators (first introduced in [13] and further studied in e.g. [14, 15]):
ˆ ∞
(x+ )s T++ dx+ ,
(2)
Ls =
−∞
where x+ is a null coordinate along the light-ray and we will further restrict to s ∈ 2N0 . A
particularly well-motivated constraint on these light-ray operators is suggested by the Conformally invariant Averaged Null Energy Condition (CANEC) [16, 17]. When taken at face value
and applied to a null geodesic embedded in the Minkowski spacetime lightcone of (without loss
of generality) the origin, this condition would claim that
ˆ ∞
d
x+ hT++ iψ dx+ ≥ 0 ,
(3)
−∞
for any state |ψi and spacetime dimension d. For even d, (3) can be seen to constrain Ld .
If these light-ray operators are positive1 , this would give an interesting set of additional
constraints on negative null energy in QFT. We expect that this would be useful in the gravity applications mentioned above, among other reasons. In this paper, we present intriguing
evidence for the positivity of these operators in a particular class of states.
We begin in section 2 by presenting a proof from [12] in free scalar field theory that many
light-ray operators (dependent on the non-minimal coupling) are indeed positive; however, we
immediately follow this with a calculation which finds counterexamples to the positivity of every
L2n except the ANEC operator L0 . We repeat these steps for a conformal field theory (CFT)
in section 3, where we present various arguments for the positivity of L2 followed by an explicit
counterexample. Finally, we discuss our results in more detail in section 4.
Clearly, our counterexamples violate some assumptions in the proofs, so in section 4 we
attempt to determine for what class of states the light-ray operators are positive. We establish
the following data points:
• States in free field theory which are superpositions of the vacuum and two particles with
sufficiently weak support at vanishing null momentum (p+ → 0) satisfy positivity of L2n .
To determine whether this support is sufficiently weak, considering L2(n+1) might suffice:
we suggest that hL2n iψ ≥ 0 if |hL2(n+1) iψ | < ∞ and ξ ≤ (2n − 1)/(8n), where ξ is the
non-minimal coupling.
• States in two-dimensional CFT prepared by acting with the stress tensor in a compact
region of Euclidean time satisfy positivity of L2 .
• States in two-dimensional CFT prepared by acting with sources in a compact region
of Lorentzian spacetime have finite expectation values of L2n . If the relation between
positivity of L2n and finiteness of L2(n+1) conjectured for the free field continues to hold
for the two-dimensional CFT, then the finiteness of all L2n implies their positivity.
Collecting these data points, we propose the following conjecture: Light-ray operators are positive semi-definite in physically preparable states. By ‘physically preparable states’ we mean
states that approach the vacuum sufficiently fast in the far IR and the far UV.
Note that our proposal is nontrivial even in two-dimensional CFT, and therefore could be
falsified or modified in the near future. In addition, the criterion of ‘physically preparable’
states will need to be further developed and quantified.
1
We often use simply ‘positive’ to mean positive semi-definite.
2
Future directions and relation to other approaches. Our results may be relevant to the
programme initiated in [18], which studied constraints on the central charges of CFTs as they
arise from the ANEC. If light-ray operators besides the ANEC operator are indeed positive, they
could be used in a very similar way to further bound the central charges of CFTs. Furthermore,
our results are relevant to the energy flow operators defined in [19] as
ˆ ∞
1
ν
d−2
Eω (n) =
lim
r
dt e−iωt(nν n̄ ) Tρσ (rn + tn̄)n̄ρ n̄σ ,
(4)
µ
nµ n̄ r→∞
−∞
where nµ = (1, n) and n̄µ are a null vectors, with nµ n̄µ 6= 0. These operators measure the
energy flux in a direction n reaching a detector at lightlike infinity if the temporal resolution
of the detector is characterised by ω [19]. Our light-ray operators can be realised as derivatives
with respect to ω of Eω (n) at ω = 0, and together form its real part. Furthermore, there may
be a connection to the higher-spin ANEC [11, 20] (although we do not explore this connection
in the present paper): in a free, minimally coupled scalar field, our light-ray operators are the
momentum space analogues of the spinning operators constrained by the higher-spin ANEC.
Finally, there is a connection between our work and [21], which showed that regions with negative
energy density cannot be localised arbitrarily far away from regions with positive energy density.
Similarly, positivity of L2n (for n ≥ 1) indicates that on a null geodesic, negative null energy
densities must be flanked by positive null energy densities.
2
Free field theory: proof and counterexamples
In this section, we start our investigation of the light-ray operators (2) by considering them in
a toy model: the non-minimally coupled but otherwise free real scalar field φ in d spacetime
dimensions. The classical action of this theory in a general background geometry is
ˆ
√ 1
(5)
dd x −g ∇a φ∇a φ + (m2 + ξR)φ2 ,
S=−
2
where m is the mass of the field, ξ the non-minimal coupling to the Ricci scalar R, g the
determinant of the metric tensor and ∇a the covariant derivative. A useful feature of this
d−2
theory, besides its mathematical simplicity, is that S is Weyl invariant if m = 0 and ξ = 4(d−1)
,
meaning that upon quantisation φ is either a simple QFT or a simple CFT, depending on the
values of m and ξ.
We first give a proof that there are many positive light-ray operators in this theory, and
then present a class of states which provide explicit counterexamples to this positivity result.
2.1
Proof of positivity in a free scalar field
The classical energy-momentum tensor associated with the action in (5) follows by varying S
with respect to the metric, leading to [22]
Tab = ∇a φ∇b φ −
1
1
∇c φ∇c φ + m2 φ2 gab + ξ Rab − gab R + gab ∇c ∇c − ∇a ∇b φ2 .
2
2
(6)
The equations of motion for φ can be found by varying S with respect to φ, leading to
∇a ∇a − m2 − ξR φ = 0 .
(7)
We now specialise to Minkowski spacetime, which has Rab = 0, and choose (without loss of
generality) the null geodesic to be the curve along which all spacetime coordinates vanish except
3
x+ = x0 + x1 , which acts as the affine parameter. If la is the tangent vector to this geodesic,
dφ
a b
then la ∇a φ = dx
+ and T++ = Tab l l can be written as
a b
T++ = Tab l l =
dφ
dx+
2
−ξ
d2 φ2
.
dx+ 2
(8)
Furthermore, in Minkowski spacetime solutions to (7) can be decomposed in plane waves
exp (±ipµ xµ ) with p2 = −m2 . To canonically quantise the field, one defines the components of
the four-momentum as pµ = (ωp , p) such that pµ xµ = −ωp x0 + p · x and then promotes the
coefficients in the plane wave decomposition of φ to creation and annihilation operators [23]:
φ̂(x) =
ˆ
1 dd−1 p
ip·x
† −ip·x
p
â
e
+
â
e
,
p
p
(2π)d−1 2ωp
(9)
We are interested in Tab la lb , which by inspection of (8) means that the two-point function is an
important quantity. Along the null geodesic, we have pµ xµ = p+ x+ with p+ = − 21 (ωp − p1 ),
and hence the (vacuum subtracted) two-point function on the null geodesic evaluates to
: φ̂(x+ )φ̂(y + ) : = φ̂(x+ )φ̂(y + ) − h0| φ̂(x+ )φ̂(y + ) |0i
ˆ
dd−1 p dd−1 k
1
−i(p+ x+ −k+ y + ) †
i(p+ x+ +k+ y + )
p
=
e
â
â
+
e
â
â
p
k
k
p
(2π)d−1 (2π)d−1 4ωp ωk
+ h.c. , (10)
where ‘h.c.’ denotes the hermitian conjugate and |0i is the Minkowski vacuum. To simplify
our subsequent calculations, we change integration variables from p1 to p+ using dp+ /p+ =
−dp1 /ωp , leading to
1
: φ̂(x )φ̂(y ) : =
2
+
+
ˆ 0
dp+ dk+ †
+
+
+
+
Âp Âk e−i(p+ x −k+ y ) + Âp Âk ei(p+ x +k+ y ) + h.c.
−∞ 2π 2π
(11)
Here, p and k are functions of p+ and k+ respectively, and we define the shorthand Âp as
1
Âp =
p+
ˆ
dd−2 p⊥ √
ωp âp .
(2π)d−2
(12)
We construct T++ from (11) by applying the appropriate differential operator (as read off from
(8)) and taking the coincidence limit y + → x+ ; this leads to
d2
d d
+
+
:
φ̂(x
)
φ̂(y
)
:
−ξ
: φ̂(x+ )φ̂(x+ ) :
lim
+
+
+
2
+
+
dx
dy
dx
y →x
ˆ
1 0 dp+ dk+ h
+
=
p+ k+ + (p+ − k+ )2 ξ e−i(p+ −k+ )x †p Âk
2 −∞ 2π 2π
T++ (x+ ) =
+
i
− p+ k+ − (p+ + k+ )2 ξ ei(p+ +k+ )x Âp Âk + h.c. (13)
We emphasise that this expression is only valid on this particular null geodesic.
Having constructed T++ along the null geodesic, we turn to the light-ray operators. Upon
substitution of (13) into (2) for s = 2n, we encounter x+ -integrals of the form
ˆ ∞
ˆ ∞
1
+
+
+ 2n −iαx+
2n
=
dx (x ) e
∂α
(14)
dx+ e−iαx = 2π(−1)n δ(2n) (α) .
2n
(−i)
−∞
−∞
4
The superscript on the delta function denotes the 2nth derivative with respect to the argument
of the delta function. Thus, substituting (13) into (2) for s = 2n and applying (14) for α =
±p+ − k+ gives
ˆ
h
(−1)n 0
dp+ dk+ p+ k+ + (p+ − k+ )2 ξ †p Âk δ(2n) (p+ − k+ )
L2n =
4π
−∞
i
− p+ k+ − (p+ + k+ )2 ξ Âp Âk δ(2n) (p+ + k+ ) + h.c. (15)
Note that the argument of the second delta function, p+ + k+ , only vanishes when p+ = k+ = 0
(since p+ , k+ ≤ 0 they cannot cancel each other). For a massive field, p+ = 0 corresponds to an
infinite momentum p1 , so if we restrict to states that don’t excite modes of infinite momentum,
expectation values of Âp Âk in a massive theory are unsupported whenever δ(2n) (p+ + k+ ) is
supported. This allows us to ignore the contribution of the second delta function in (15) to any
expectation value of L2n .
For m = 0, the situation is more complex, although the same conclusion should hold. To
see why, consider the expression for p+ in terms of p:
p+ = −
1
1 q 2
m + (p⊥ )2 + (p1 )2 − p1 .
ω p − p1 = −
2
2
(16)
It can be seen from this expression that p⊥ 6= 0⊥ fulfils the same role as a non-zero mass in
ensuring that p+ < 0 for finite p1 (we still restrict to states which do not excite modes with
infinite momentum). One may have p+ = 0 at finite p1 for p⊥ = 0⊥ , but this is a single
point in the d − 2 dimensional space over which the integral in (12) is taken. Therefore, if we
additionally restrict to states for which the expectation value of âp âk is a sufficiently smooth and
finite function of p and k, then the contribution of p⊥ = 0⊥ to the integral in (12) vanishes.
Thus, like for the massive theory, the second delta function in (15) does not contribute to
expectation values of L2n in a massless theory. Please note that this argument fails when d = 2,
since there is no p⊥ in that case.
Put together, we assume that we are working in d ≥ 3 or with a massive field, so that the
second term in (15) can be ignored and we can focus on evaluating the contribution of the first
term. This term naturally splits into a term which is independent of ξ and one which isn’t;
we refer to these terms as the minimal and non-minimal coupling terms respectively. Let us
consider them individually, and define u = p+ + k+ and v = p+ − k+ ; the minimal coupling
term then becomes
ˆ u
ˆ
(−1)n 0
min
L2n =
dv p+ k+ †p Âk ∂v2n δ(v) + h.c.
du
8π
−u
−∞
ˆ u
ˆ
(−1)n+1 0
=
dv ∂v p+ k+ †p Âk ∂v2n−1 δ(v)
du
8π
−u
−∞
ˆ
h
iu
(−1)n 0
+ h.c. (17)
du p+ k+ †p Âk ∂v2n−1 δ(v)
+
−u
8π
−∞
The step from the first to the second line is an integration by parts, for which we need to take
the derivative of creation and annihilation operators. This is uncommon, but we interpret it
as a formal operation which acquires its meaning inside an expectation value. To be explicit,
consider the following state:
|hi =
ˆ
†
dd−1 kj âkj 

q
h (k1 , . . . , kN )
|0i ,
(2π)d−1 2ωkj
j∈N
Y


(18)
with N = {1, . . . , N }, the set of positive integers up to and including a positive integer N , and h
some sufficiently smooth function which vanishes sufficiently fast at large momenta. This state
5
includes the possibility of exciting a single mode multiple times, since we are allowed to make
h sharply peaked when ki − kj approaches 0 for any pair i, j. Very general states can therefore
be constructed as superpositions of the vacuum and states like |hi, with different choices of h
(and accordingly different N ). We then define ∂pn+ Âp as the operator which acts on |hi as
∂pn+ Âp |hi =
ˆ
N
dd−2 p⊥ X
(2π)d−2 i=1
ˆ
∂pn+
h (k1 , . . . , ki = p, . . . , kN )
√
2 p+
!
†
dd−1 kj âkj 

q
|0i .
(2π)d−1 2ωkj
j∈N \i
Y


(19)
The rationale behind this definition is that this is what one would get by acting on |hi with Âp
and then differentiating n times. With this construction, we have converted the derivatives of
an operator to the familiar derivatives of functions, allowing us to define e.g.
(21)
∂p+ Âp Âf (p+ ) = (∂p+ Âp )Âf (p+ ) + Âp (∂p+ Âf (p+ ) )
(22)
∂p+ Âf (p+ ) = ∂p+ f ∂f Âf (p+ )
∂p+ αÂp + β Âf (p+ ) = α∂p+ Âp + β∂p+ Âf (p+ )
∂p+ f (p+ )Âp = (∂p+ f )Âp + f ∂p+ Âp
(20)
(23)
for constant α, β ∈ C and a differentiable function f .
Having defined the derivative of an operator, we now consider the boundary terms in (17).
They can be treated using the arguments we used to ignore the second delta function in (15):
the derivative of the delta function is only supported when its argument (here ±u) vanishes,
which does not happen at finite momenta for the massive field. For the massless field, a non-zero
p⊥ takes on the role of the mass while the integral in (12) prevents the modes with p⊥ = 0⊥
and finite p1 from contributing to the expectation value. Furthermore, we restrict to states
in which modes of infinite momentum are not excited, so that the expectation value of †p Âk
vanishes sufficiently fast when p+ + k+ → 0. Putting this together with our earlier assumption
that either m 6= 0 or d ≥ 3, we recognise that the boundary term in (17) can be discarded.
If we furthermore assume that the expectation value of any derivatives of †p Âk also vanishes
when p+ , k+ → 0, we can repeat this another 2n − 1 times, to obtain
ˆ u
ˆ
(−1)n 0
2n
†
dv
δ(v)∂
du
Lmin
=
p
k
Â
Â
+ + p k + h.c.
v
2n
8π
−u
−∞
!
ˆ
2n
δ(p+ − k+ ) X
(−1)n 0
2n
j 2n−j j
†
dp+ dk+
=
(−1)
∂
∂
p
k
Â
Â
+ h.c.
+
+
k
p
p
k
+
+
j
4π
22n
−∞
j=0
!
ˆ 0
2n
(−1)n X
2n
j
(−1)
dp+ ∂p2n−j
(p+ †p )∂pj+ (p+ Âp ) + h.c. ,
(24)
= 2n+2
+
j
2
π j=0
−∞
where the step from the first to the second line used ∂v = 21 (∂p+ − ∂k+ ). Since we have restricted
to states and models in which modes with either p+ → −∞ or p+ → 0 are sufficiently suppressed,
we can integrate by parts in this expression while discarding the boundary terms until the 2n
derivatives are distributed equally between p+ †p and p+ Âp , leading to
Lmin
2n =
2n
(−1)n X
22n+2 π j=0
2n
j
!
(−1)n
ˆ 0
−∞
dp+ ∂pn+ (p+ †p )∂pn+ (p+ Âp ) + h.c.
(25)
The sum of binomial coefficients can be evaluated using the following identity:
2n
X
j=0
2n
j
!
6
= 22n .
(26)
Expanding the derivatives in (25), we therefore arrive at the following expression:
1
4π
ˆ 0
†p
dp+ p+ ∂pn+ †p + n∂pn−1
+
1
=
4π
ˆ 0
†p ∂pn−1
Âp + h.c. ,
dp+ (p+ )2 ∂pn+ †p ∂pn+ Âp + n(n − 1)∂pn−1
+
+
Lmin
2n =
−∞
−∞
p+ ∂pn+ Âp + n∂pn−1
Âp + h.c.
+
i
h
(27)
where we combined the cross terms with the quadratic terms via integration by parts. Note
that the derivatives in the second term of (27) are not sensible for n = 0; since the n-dependent
prefactor of this term vanishes in this case, we don’t consider this a problem.
Next, we consider the non-minimal coupling term. Using the same u and v as in (17) and
using the same arguments to discard the boundary terms that arise in the integration by parts,
we find that we can express this term as
ˆ u
ˆ
(−1)n ξ 0
Lnon−min
=
dv v 2 †p Âk ∂v2n δ(v) + h.c.
du
2n
8π
−u
−∞
ˆ u
ˆ
(−1)n 2n(2n − 1)ξ 0
dv δ(v)∂v2n−2 †p Âk + h.c. ,
du
(28)
=
8π
−u
−∞
where we kept only those terms in the expansion of ∂v2n (v 2 †p Âk ) that have precisely two derivatives acting on v 2 ; after all, more derivatives would annihilate v 2 , while fewer would lead to
surviving factors of v that vanish upon evaluation of the integral over v. Note that for n = 0,
the derivatives in the final expression do not make sense; however, since the prefactor vanishes
in this case, we do not consider that to be a problem.
By using that ∂v = 12 (∂p+ − ∂k+ ), (28) can be written in terms of p+ and k+ again, after
which the delta function can be used to evaluate the integral over k+ :
X
(−1)n 2n(2n − 1)ξ 2n−2
Lnon−min
=
2n
2n
2 π
j=0
!
2n − 2
j
j
(−1)
ˆ 0
−∞
dp+ ∂p2n−2−j
†p ∂pj+ Âp + h.c.
+
(29)
For the non-minimal coupling term, the last step is to apply the procedure that led to (25), i.e.
integrate by parts to distribute the 2n − 2 derivatives equally between †p and Â, upon which
we can apply the identity (26) to evaluate the sum over binomial coefficients. The outcome is
Lnon−min
= −
2n
ˆ 0
2n(2n − 1)ξ
4π
−∞
dp+ ∂pn−1
†p ∂pn−1
Âp + h.c.
+
+
(30)
The overall minus sign arises due to the fact that we now have 2(n − 1) derivatives to distribute,
instead of the 2n derivatives in the minimal coupling term. Combining (27) and (30) we obtain
the following expression for the light-ray operators:
ˆ 0
i
dp+ h
† n−1
.
(31)
(p+ )2 ∂pn+ †p ∂pn+ Âp + n (n − 1 − 2ξ(2n − 1)) ∂pn−1
Â
∂
Â
L2n =
p
p
p
+
+
−∞ 2π
For n = 0, the second term in the square brackets vanishes because of the prefactor. We can
then observe that the first term is a manifestly positive operator (multiplying an operator with
its hermitian conjugate, so that any expectation value would be the norm of a state), meaning
that L0 is positive. This is consistent with our expectations, since L0 is the ANEC operator.
To make further progress for n ≥ 1, we eliminate the explicit dependence on p+ by defining
B̂(p) =
r
−
p+ n−1
∂
Âp ,
µ p+
7
(32)
with µ > 0 some arbitrary fixed energy scale. If we then change variables to q with p+ = −µe−q
and denote B̂q = B̂ (p(q)), we find that
ˆ ∞
i
µe−q dq h
∂q (eq/2 B̂q† )∂q (eq/2 B̂q ) + n (n − 1 − 2ξ(2n − 1)) eq B̂q† B̂q
2π
ˆ−∞
∞
1
1
µdq
†
†
†
+ n(n − 1 − 2ξ(2n − 1)) B̂q B̂q ,
∂q B̂q ∂q B̂q + ∂q (B̂q B̂q ) +
=
2
4
−∞ 2π
L2n =
(33)
where we expanded ∂q (eq/2 B̂q† )∂q (eq/2 B̂q ) and combined the cross terms into the full derivative
∂q (B̂q† B̂q ); we can ignore this term since it is a boundary term which vanishes due to our
assumption that no modes with infinite momentum have been excited.
We finish our calculation by noting that the remaining operators in (33) form a linear
combination of operators multiplied by their respective hermitian conjugates, meaning that any
expectation value would be a sum of (positive definite) norms of states. We can choose states
in which the expectation value of B̂q† B̂q is either oscillating rapidly or almost constant as a
function of q, leading to the dominance of respectively the first or the last term in (33); hence,
L2n will be positive if and only if the term in curly brackets is non-negative. This condition is
trivially satisfied for n = 0, reproducing the ANEC [24]; for n ≥ 1, we can formulate it in two
equivalent ways, namely as a bound on ξ given n or as a bound on n given ξ:
ξ≤
2n − 1
8n
2n ≥
1
,
1 − 4ξ
(34)
where the form on the right assumes ξ < 41 ; inspection of the form on the left reveals that there
is no positive L2n with n ≥ 0 if ξ ≥ 14 .
Two values of ξ are of particular interest: minimal coupling ξ = 0 and conformal coupling
d−2
. For ξ = 0, (34) leads us to conclude that L2n is positive for all integer n ≥ 0,
ξ = 4(d−1)
whereas only those L2n with 2n ≥ d − 1 are positive for a conformally coupled theory. For the
free scalar CFT in a spacetime with even d, we therefore see that Ld , whose expectation values
are constrained by the CANEC on the Minkowski lightcone, is indeed positive.
2.2
Counterexamples in a free theory
In the previous section, a proof was given for the positivity of a range of light-ray operators.
However, for this calculation to work, we had to restrict to a particular set of states. The
central theme in this restriction is the behaviour of expectation values for large and small p+ :
the calculation in section 2.1 only holds if the expectation values of Âp Âk , †p Âk , and their
derivatives vanish sufficiently quickly as p+ , k+ → 0 and p+ , k+ → −∞. To illustrate these
restrictions more explicitly, we will construct a class of states in which L2n is not positive. We
do this by explicitly calculating the expectation value of T++ and demonstrating that it can be
made to have either sign by an appropriate choice of parameters.
2.2.1
Choice of state and consistency conditions
We consider states |ψi = c1 |0i + c2 |2i, where c1 , c2 ∈ C with |c1 |2 + |c2 |2 = 1, and |2i is a
normalised two-particle state constructed as
ˆ
dd−1 p dd−1 k p+ k+
1
⊥
⊥
† †
|2i = √
(35)
√ √ f (p )f (k )h(p+ , k+ )âp âk |0i ,
(2π)d−1 (2π)d−1 ωp ωk
2
where f : Rd−2 → C and h : R2 → C are (sufficiently smooth) smearing functions, and we
assume that h(p+ , k+ ) = h(k+ , p+ ). The normalisation of |2i can be fixed by choosing f and h
8
such that
A≡
ˆ
2
dd−2 p⊥
f (p⊥ ) =
d−2
(2π)
ˆ 0
dp+ dk+
p+ k+ |h(p+ , k+ )|2
−∞ 2π 2π
!−1/2
,
(36)
where we have defined the quantity A for future use. Similarly, we define B as the integral of f :
ˆ d−2 ⊥
d p
f (p⊥ ) .
(37)
B=
(2π)d−2
We guarantee that both A and B are finite by taking f ∈ L2 to be Lebesgue integrable.
To demonstrate that |ψi = c1 |0i + c2 |2i allows for expectation values of L2n with either
sign, we first consider the expectation value of T++ as written in (13). This operator naturally
splits into a diagonal component (proportional to â†p âk and its hermitian conjugate) and an
off-diagonal component (proportional to âp âk and its hermitian conjugate), which we denote
diag
cross respectively. Due to the vacuum subtraction, we find that
by T++
and T++
diag
diag
hT++
iψ = |c2 |2 h2| T++
|2i
ˆ 0
dp+ dk+ dq+ 2
2
= − |c2 | A|B|
p+ k+ + (p+ − k+ )2 ξ q+
−∞ 2π 2π 2π
+
× h∗ (p+ , q+ )h(q+ , k+ )e−i(p+ −k+ )x + c.c. , (38)
where ‘c.c.’ denotes the complex conjugate. For the off-diagonal terms, we find instead that
cross
cross
hT++
iψ = c∗1 c2 h0| T++
|2i + c.c.
ˆ 0
2
∗
c c2 B
dp+ dk+ +
p+ k+ − (p+ + k+ )2 ξ h(p+ , k+ )ei(p+ +k+ )x + c.c.
= − 1√
2
−∞ 2π 2π
(39)
To make the number of minus signs more manifest, we change p+ , k+ , q+ → −p+ , −k+ , −q+ and
define h(p+ , k+ ) = h(−p+ , −k+ ), to obtain
ˆ ∞
dp+ dk+ dq+ diag
2
2
p+ k+ + (p+ − k+ )2 ξ q+
hT++ iψ = |c2 | A|B|
2π 2π 2π
0
+
× h∗ (p+ , q+ )h(q+ , k+ )ei(p+ −k+ )x + c.c. (40)
ˆ
c∗1 c2 B 2 ∞ dp+ dk+ +
− √
p+ k+ − (p+ + k+ )2 ξ h(p+ , k+ )e−i(p+ +k+ )x + c.c. (41)
2π 2π
2
0
At this point, let us introduce a specific choice of h (recall that from now on, the light-ray
momenta are taken to be non-negative):
cross
hT++
iψ =
h(p+ , k+ ) =
a
pa+ k+
e−l(p+ +k+ ) ,
(p+ + k+ )b
(42)
where a, b ≥ 0 and l ∈ R>0 is a length scale. We leave f unspecified up to the condition (36); the
possible values of a and b are restricted by various consistency conditions. Firstly, we demand
that A is both finite and non-zero, which constrains a and b based on (36). This can be seen
by explicitly calculating the right-hand side of (36), for which we define the following function
of three parameters u, v, and γ, assuming u, v ≥ 0 and Re(γ) > 0:
ˆ ∞
xu y u −(x+y)γ
e
dx dy
Iu,v (γ) =
(x + y)v
0
ˆ ∞
1
=
dx dy dα αv−1 xu y u e−(x+y)(γ+α)
Γ(v) 0
ˆ ∞
dα αv−1
Γ(u + 1) 2
=
Γ(v)
(γ + α)u+1
0
1
Γ2 (u + 1)Γ(2u + 2 − v)
.
(43)
=
Γ(2u + 2)
γ 2u−v+2
9
In the first two steps, we made use of the following identity:
ˆ ∞
1
1
=
dα αz−1 e−αx ;
xz
Γ(z) 0
the final step to (43) was to apply the following definition of the Euler beta function:
ˆ ∞
αz1 −1 dα
Γ(z1 )Γ(z2 )
B(z1 , z2 ) =
.
=
z
+z
1
2
(1 + α)
Γ(z1 + z2 )
0
(44)
(45)
The integral representation is convergent for Re(z1 ), Re(z2 ) > 0 or, in terms of the parameters
present in (43), v > 0 and 2u + 2 − v > 0. This condition
qis relevant because, according to (36),
(42) corresponds to a normalised state when A = 2π/ I2a+1,2b (2l); this is finite since a ≥ 0
and l > 0, and as mentioned it is non-zero if b > 0 and 2a + 2 − b > 0.
The second consistency condition we impose is that |2i should be a Hadamard state, which
means that the (non-vacuum-subtracted) two-point function h2| φ̂(x)φ̂(y) |2i must have the divergence structure of h0| φ̂(x)φ̂(y) |0i as x → y [25, 26]. To see whether this is the case, it suffices
to check whether the normal-ordered two-point function h2| : φ̂(x)φ̂(x) : |2i is finite; if it is, then
the divergence of h2| φ̂(x)φ̂(x) |2i is the same as that in h0| φ̂(x)φ̂(x) |0i. Without restricting to
points on the null geodesic, the expectation value of (10) in the state |2i from (35) is given by
h2| : φ̂(x)φ̂(y) : |2i =
ˆ
2
2
dd−1 p dd−1 k dd−1 q p+ k+ q+
f ∗ (p⊥ )f (k⊥ ) f (q⊥ )
d−1
d−1
d−1
(2π)
(2π)
(2π)
ωp ωk ωq
× h∗ (p+ , q+ )h(q+ , k+ )ei(p·x−k·y) + c.c.
(46)
Clearly, if we restrict to positive f and take h as in (42), the integrand consists of manifestly
positive functions multiplied by a phase ei(p·x−k·y). Then, the expectation value achieves its
maximum value when x = y = 0 (in this case, it is the integral of a positive function):
h2| : φ̂(0)φ̂(0) : |2i =
ˆ
2
2
dd−1 p dd−1 k dd−1 q p+ k+ q+
f ∗ (p⊥ )f (k⊥ ) f (q⊥ )
d−1
d−1
d−1
(2π)
(2π)
(2π)
ωp ωk ωq
× h∗ (p+ , q+ )h(q+ , k+ ) + c.c.
ˆ ∞
dp+ dk+ dq+ ∗
h (p+ , q+ )h(q+ , k+ )q+ + c.c.
2π 2π 2π
0
ˆ ∞
a q 2a+1 e−l(p+ +k+ +2q+ )
dp+ dk+ dq+ pa+ k+
2
+
+ c.c.
= A|B|
2π
2π
2π
(p
+
q+ )b (q+ + k+ )b
+
0
= A|B|2
(47)
The real exponential ensures that the integral does not diverge due to the behaviour of the
integrand as p+ → ∞; we can ensure that the integral does not diverge due to the small-p+
behaviour of the integrand by demanding that 4a + 4 − 2b > 0, i.e. 2a + 2 − b > 0. Thus, if |2i is
normalisable, then the divergent part of h2| φ̂(x)φ̂(x) |2i is identical to that of h0| φ̂(x)φ̂(x) |0i,
meaning that |2i is a Hadamard state.
Our third and final consistency condition is that |2i should have a finite energy. The classical
expression for the Hamiltonian can be found using (6), namely as [23]
ˆ
ˆ
1 2
i
2 2
i 2
d−1
00
3
φ̇ + ∂i φ∂ φ + m φ − ξ∂i ∂ φ
,
(48)
H = d xT =
d x
2
where the dot denotes a derivative with respect to x0 = t, ∂i with i = 1, . . . , d − 1 are the spatial
derivatives, and we made use of the fact that we are working in Minkowski spacetime to set
Rab = 0 and to replace covariant derivatives by normal partial derivatives. Note that the final,
ξ-dependent term only contributes a boundary term which we set to zero as usual. Hence, the
10
classical Hamiltonian is the Hamiltonian of the minimally coupled free scalar field, regardless
of ξ; quantising this expression and normal ordering it leads to the well-known result
ˆ d−2 ⊥ ˆ ∞
ˆ
d p
dp+ ωp2 †
dd−1 p
†
ω
â
â
=
â âp .
(49)
Ĥ =
p
p
p
(2π)d−1
(2π)d−2 0 2π p+ p
The expectation value of Ĥ in |2i as defined by (35) is readily calculated to be
ˆ d−2 ⊥ d−2 ⊥ ˆ ∞
2
2
d p d q
dp+ dq+
h2| Ĥ |2i = 2
q+ p+ ωp f (p⊥ ) f (q⊥ ) |h(p+ , q+ )|2
d−2
d−2
(2π)
(2π)
2π 2π
0
(50)
The integral over q⊥ evaluates to A by definition, but the evaluation of the integral over p⊥ is
1
hindered by the presence
of a factor of ωp . However, from the definition 2p+ = ωp − p we can
find that ωp p+ = p2+ + (p⊥ )2 + m2 /4. Thus, the energy in |2i can be written as
" ˆ
1
h2| Ĥ |2i = 2A
4
2
dd−2 p⊥ ⊥ 2
2
(p
)
+
m
f (p⊥ )
d−2
(2π)
+A
ˆ ∞
0
dp+ dq+
q+ |h(p+ , q+ )|2
2π 2π
ˆ ∞
dp+ dq+ 2
p q+ |h(p+ , q+ )|2
2π 2π +
0
.
(51)
We do not choose a specific f , but we will restrict to f for which the remaining integral over p⊥ is
convergent. We can analyse the integrals over p+ and q+ by substituting (42) and using the same
arguments as we did to conclude that h0| : φ(0)φ(0) : |0i is finite: the real exponential ensures
that the high-p+ behaviour of the integrand does not lead to a divergence of the integral, while
the small-p+ behaviour of the integrand yields no divergences if a and b satisfy some conditions.
For (51), these conditions are 4a+ 3 > 2b and 4a+ 5 > 2b for the first and second set of integrals
respectively. While the latter is implied by the earlier constraint 2a + 2 − b > 0, the former is
stricter. Thus, from now on we assume that 4a + 3 − 2b > 0 as well as a ≥ 0 and b > 0 in (42);
this ensures that |2i is normalisable and Hadamard, and that it has a finite energy.
2.2.2
Computation of the counterexample
Now that we have defined a particular class of states, we can compute the expectation value
of L2n explicitly; we start by calculating hT++ iψ . Using (43), the off-diagonal contribution to
hT++ iψ can be calculated quite straightforwardly. Substituting (42) into (41), we observe that
ˆ
a
c∗1 c2 B 2 ∞ dp+ dk+ p+ k+ − (p+ + k+ )2 ξ pa+ k+
+
cross
hT++ iψ = − √
e−(p+ +k+ )(l+ix ) + c.c.
b
2π 2π
(p+ + k+ )
2
0
2
∗
c c2 B
= − 1 2 √ Ia+1,b (l + ix+ ) − ξIa,b−2 (l + ix+ + c.c.
4π 2
2
Γ (a + 1)Γ(2a + 4 − b)
c∗ c2 B 2 a + 1
= − 1 2√
−ξ
+ c.c. ,
(52)
Γ(2a + 2)(l + ix+ )2a+4−b
4π 2 4a + 6
where the convergence of Ia+1,b and Ia,b−2 is guaranteed by the condition 4a + 3 > 2b. We
continue by computing the contribution to hT++ iψ from diagonal terms. This starts similarly
to the computation for the off-diagonal terms, namely by substituting (42) into (40):
ˆ ∞
2 )ξ pa k a q 2a+1
dp+ dk+ dq+ (1 − 2ξ)p+ k+ + (p2+ + k+
diag
2
2
+ + +
hT++ iψ = |c2 | A|B|
2π 2π 2π
(p+ + q+ )b (q+ + k+ )b
0
+
+
× e−(l−ix )p+ e−(l+ix )k+ e−2lq+ + c.c.
=
|c2 |2 A|B|2 +
+
+
(1
−
2ξ)J
(l
−
ix
)
+
ξJ
(l
−
ix
)
+
ξJ
(l
−
ix
)
a+1,a+1
a+2,a
a,a+2
8π 3
+ c.c. , (53)
11
where we have defined the following function, again assuming Re(γ) > 0:
ˆ ∞
xp y q z 2a+1
∗
∗
dx dy dz
Jp,q (γ) =
e−γx e−γ y e−(γ+γ )z
b (z + y)b
(x
+
z)
0
ˆ ∞
1
∗
∗
dx dy dz dα dβ αb−1 β b−1 xp y q z 2a+1 e−(γ+α)x e−(γ +β)y e−(γ +α+γ+β)z
= 2
Γ (b) 0
ˆ ∞
Γ(2a + 2)
αb−1 Γ(p + 1) β b−1 Γ(q + 1)
1
,
(54)
dα dβ
= 2
Γ (b) 0
(γ + α)p+1 (γ ∗ + β)q+1 (γ + α + γ ∗ + β)2a+2
where we again made use of (44). The remaining integrals are difficult to compute, but since
diag
iψ over x+ we are mainly interested in its behaviour when
we eventually wish to integrate hT++
+
x → ±∞. To find this behaviour, we first remark that for 4a + 3 > b and γ = l − ix+ with
l > 0, the relevant integrals converge. This can be seen by noting that the integrand has no
singularities for finite α and β, and then changing to polar coordinates r, θ (with r 2 = α2 + β 2 )
to observe that the integrand (together with the Jacobian) decays as 1/r 5−2b+p+q+2a ; since we
are interested in cases where p + q = 2a + 2, this is sufficiently fast to guarantee convergence.
We then consider Jp,q (l − ix+ ) for x+ → ∞ (the case x+ → −∞ is completely analogous).
To characterise the behaviour of Jp,q in this limit, we make all quantities in the integral dimensionless using x+ as a length scale: we define α′ = α/x+ , β ′ = β/x+ , δ = l/x+ , and a constant
prefactor Cp,q = Γ(p + 1)Γ(q + 1)Γ(2a + 2)/ Γ2 (b)(−i)p iq to obtain
ˆ ∞
dα′ dβ ′ α′b−1 β ′b−1
Cp,q
+
.
Jp,q (l − ix ) = + 2a+4−2b+p+q
(x )
(1 + iδ + iα′ )p+1 (1 − iδ − iβ ′ )q+1 (2δ + α′ + β ′ )2a+2
0
(55)
We now wish to isolate the contribution to the integral which dominates the limit δ → 0. First,
we can recognise that since δ ≪ 1, we can Taylor expand (1+ iδ + iα′ )−p−1 and (1− iδ − β ′ )−q−1 :
#
"ˆ
∞
dα′ dβ ′ α′b−1 β ′b−1
Cp,q
+
+ O(δ)
Jp,q (l − ix ) = + 2a+4−2b+p+q
′ p+1 (1 − iβ ′ )q+1 (2δ + α′ + β ′ )2a+2
(x )
0 (1 + iα )
Cp,q
≡ + 2a+4−2b+p+q [Kp,q (δ) + O(δ)] .
(56)
(x )
To find out how Kp,q scales with δ for δ ≪ 1, we choose some ε with 2δ ≪ ε ≪ 1 and split
the integral Kp,q in four parts, namely 0 ≤ α′ , β ′ ≤ ε; 0 ≤ α′ ≤ ε < β ′ ; 0 ≤ β ′ ≤ ε < α′ ; and
ε < α′ , β ′ . In the latter three parts, we note that α′ + β ′ ≫ 2δ, so that (2δ + α′ + β ′ )−2a−2 can
also be Taylor expanded, leaving a contribution that is independent of δ and O(δ) terms. In the
first part (with 0 ≤ α′ , β ′ ≤ ε), we can Taylor expand (1 + iα′ )−p−1 and (1 − iβ ′ )−q−1 , yielding
ˆ ε
(1 + O(α′ )) (1 + O(β ′ )) α′b−1 β ′b−1 dα′ dβ ′
+
O
δ0 .
(57)
Kp,q (δ) =
′ + β ′ )2a+2
(2δ
+
α
0
Note that factors of α′ and β ′ in the numerator effectively suppress the integral with factors of
ε, so the dominant contribution is given by the term with as few of them as possible:
ˆ ε ′b−1 ′b−1 ′ ′
α
β
dα dβ
Kp,q (δ) = (1 + O(ε))
+
O
δ0 .
(58)
′
′ 2a+2
0 (2δ + α + β )
Let us consider positive integer values of b. We can then integrate by parts in the β ′ integral:
"
#ε
ˆ ε
1
β ′b−1
′b−1
α
Kp,q (δ) = (1 + O(ε))
−
2a + 1 (2δ + α′ + β ′ )2a+1 0
0
!
ˆ ε
b−1
β ′b−2 dβ ′
′
+
dα
+
O
δ0 . (59)
2a + 1 0 (2δ + α′ + β ′ )2a+1
12
Observe now that the boundaryterm at β ′ = ε can once again be Taylor expanded in δ, leading
to a contribution which is O δ0 . The boundary term at β ′ = 0 either vanishes (for b > 1) or it
represents the relevant value of the integral (for b = 1). In the latter case, we can immediately
perform the α′ integral as well:
ˆ
1
dα′
1 + O(ε)
1 + O(ε) ε
0
+
O
δ
=
+
O
δ0 ,
(60)
Kp,q (δ) =
2a + 1 0 (2δ + α′ )2a+1
(2a + 1)2a (2δ)2a
assuming that a > 0. In the case that a = 0 we have
ˆ ε
dα′
0
+
O
δ
Kp,q (δ) = (1 + O(ε))
=
−
(1
+
O(ε))
ln
δ
+
O
δ0 .
′
0 2δ + α
(61)
Plugging these expressions back into (56) shows that for b = 1, a > 0, Jp,q (l − ix+ ) decays as
1/(x+ )p+q+2 when x+ → ∞, while Jp,q depends on x+ as ln x+ /l /(x+ )p+q+2 in the same limit
for a = 0, b = 1.
For the case of integer b ≥ 2, we return to (59) and discard the boundary term at β ′ = 0.
This can be repeated b − 2 times (which is possible since 4a + 3 > 2b implies 2a + 2 > b), to find
ˆ
(1 + O(ε))Γ(b)Γ(b − 1) ε
α′b−1 dα′
Kp,q (δ) =
+
O
δ0 .
(62)
′ 2a+2−b
Γ(2a + 2)
0 (2δ + α )
We would like to perform the integral over α′ as well, but there is no guaranteed ordering of the
exponents 2a + 2 − b and b − 1 while it is precisely this ordering which determines the relevant
small α-behaviour of the integrand. We therefore consider three cases separately: 2a + 2 > 2b,
2a + 2 = 2b, or 2a + 2 < 2b.
If 2a + 2 > 2b, we continue integrating by parts and discarding the boundary terms to find
Kp,q (δ) =
1 + O(ε)
Γ2 (b)Γ2 (b − 1)
+
O
δ0 .
Γ(2a + 2)Γ(2a + 2 − b) (2δ)2a+2−2b
(63)
When 2a + 2 = 2b, we can still integrate by parts b − 1 times before anything is different:
ˆ
(1 + O(ε)) Γ(b)Γ(b − 1) ε dα′
(1 + O(ε)) Γ2 (b)
Kp,q (δ) =
ln
δ
+
O
δ0 .
=
−
(64)
′
Γ(2b)
2δ
+
α
(b
−
1)Γ(2b)
0
In the case that 2a + 2 < 2b, we observe that the integral is convergent in the limit δ →
0, meaning that the O δ0 -terms actually dominate the expansion in δ. Thus, we conclude
that for any value of a ≥ 0 and
b ∈ Z≥1 , we have that asymptotically, Kp,q scales as ∼
ja,b (x+ )(x+ )2a+2−2b + O (l/x+ )0 , with
+
ja,b (x ) =



1
ln x+ /l


0
a>b−1
a=b−1
a<b−1
.
(65)
If we plug this behaviour back into (56), we find that Jp,q asymptotically depends on x+ as
Jp,q (l − ix+ ) ∼
ja,b (x+ )
ja,b (x+ )
C
C
+
=
+ + 4a+6−2b ,
+
p+q+2
+
p+q+2a+4−2b
+
2a+4
(x )
(x )
(x )
(x )
(66)
where C is some x+ -independent number and the second expression takes into account the fact
that we are interested in the case p + q = 2a + 2, according
to (53).
´
cross + T diag i . We begin
We are now in a position to calculate hL2n iψ = dx+ (x+ )2n hT++
++ ψ
with the cross term contribution, which using (52) reads
!
ˆ ∞
ˆ
c∗1 c2 B 2 (x+ )2n
c1 c∗2 (B ∗ )2 (x+ )2n
+
+
+ 2n
cross
dx
dx (x ) hT++ iψ = Da,b,ξ
+
,
(67)
(l + ix+ )2a+4−b
(l − ix+ )2a+4−b
−∞
R
13
where we have denoted all real numerical prefactors in (52) by Da,b.ξ . (67) can be evaluated
using the residue theorem if we choose a and b such that 2a + 4 − b = 2n + 1 (so that we have a
first order pole) and c1 , c2 , and f such that c∗1 c2 B 2 ∈ R (so that the integrand in (67) vanishes
sufficiently quickly as x+ → ∞). Importantly, for this choice of parameters we find that the
residue at either pole is finite and non-zero, meaning that the cross term contribution to hL2n iψ
is finite and non-zero (up to the vanishing of Da,b,ξ , to which we return momentarily). Also
note that the choice 2n + 1 = 2a + 4 − b is only possible for n > 43 .
For the contribution of the diagonal terms, we have not found a closed-form expression,
diag
but we have argued that hT++
iψ is non-singular for x+ ∈ R, and (66) gives the asymptotic
diag
iψ asymptotically scales as
behaviour. In particular, for 2a + 4 − b = 2n + 1, (x+ )2n hT++
ja,b (x+ )
C
+ + 2n .
(68)
+
b+1
(x )
(x )
´
diag
iψ converges for e.g. a = n−1
This decay is sufficiently fast to guarantee that dx+ (x+ )2n hT++
and b = 1 if n ≥ 1.
cross i
Thus, that there are choices of the parameters a, b, n, c1 , and c2 for which both hT++
ψ
diag
and hT++ iψ make finite and non-zero contributions to hL2n iψ . However, observe from (52)
and (53) that the former is linear in c2 , while the latter is proportional to |c2 |2 . Thus, by
cross i dominate the value of
taking |c2 | ≪ 1, we can make any non-zero contribution from hT++
ψ
hL2n iψ , and by choosing an appropriate sign for c2 we can make this value negative, providing
an explicit counterexample to the conclusion of section 2.1.
A potential issue with this counterexample is that the non-minimal coupling ξ may be such
that the factor Da,b,ξ vanishes in (67). From (52), we see that this happens whenever
diag
iψ ∼
(x+ )2n hT++
a+1
.
(69)
4a + 6
This form suggests a method to circumvent this issue, as ξ is a parameter of the theory, but
a is a parameter of the state under consideration. Thus, having chosen a = n − 1 for n ≥ 1,
if we do encounter a theory with ξ = n/(4n + 2) we simply choose a different state (e.g. with
a = n − 21 and b = 2 for n ≥ 2, or a = 12 , b = 2 for n = 1) for which Da,b,ξ does not vanish.
We therefore have counterexamples to the positivity of all L2n with n ≥ 1. The states |2i
from (35) with (42) do not provide a counterexample for n = 0, since the requirement that they
should have a finite energy imposes the constraint 2n = 2a + 3 − b > 23 . This is to be expected,
since the free scalar field obeys the ANEC [24].
Finally, we consider how the counterexample presented in this section relates to the proof
given in section 2.1. We do this by considering the expectation value of (15) in a state |ψi =
c1 |0i + c2 |2i with (35), (42), and 2n + 3 = 2a + 4 − b. We focus in particular on the off-diagonal
part, proportional to Âp Âk and its hermitian conjugate; evaluating hÂp Âk iψ leads to
"
#
ˆ
a+1
a
pa+1
ξpa+ k+
(−1)n+1 c∗1 c2 B 2 ∞
+ k+
cross
√
dp+ dk+
hL2n iψ =
−
e−l(p+ +k+ ) δ(2n) (p+ + k+ )
b
b−2
(p
+
k
)
(p
+
k
)
2π 2
+
+
+
+
0
ξ=
+ c.c.
(70)
To evaluate these integrals, we define p̄ = p+ + k+ and δp = p+ − k+ , so that p+ k+ = 14 (p̄2 − δp2 )
and hence
(−1)n c∗1 c2 B 2
√
−
4a+1 π 2
ˆ ∞
ˆ p̄
"
#
(p̄2 − δp2 )a+1 ξ(p̄2 − δp2 )a −lp̄ (2n)
dδp
dp̄
−
e δ
(p̄) + c.c.
4p̄b
p̄b−2
−p̄
0
"ˆ
#
ˆ
1
(−1)n c∗1 c2 B 2 ∞
2
2 a 2a+3−b −lp̄ (2n)
√
= −
dp̄
du (1 − u − 4ξ)(1 − u ) p̄
e δ
(p̄) + c.c.
4a+2 π 2 0
−1
hLcross
2n iψ =
= −
(−1)n c∗1 c2 B 2 F h 2n 2a+3−b −lp̄ i
√
∂p̄ p̄
e
+ c.c. ,
p̄=0
4a+2 π 2
14
(71)
where we defined F in the final line as the integral in square brackets in the second line, and the
integral over p̄ could be performed using integration by parts while ignoring boundary terms
because the polynomial factor ensured that p̄ = 0 does not contribute and the exponential term
suppressed any contribution from p̄ → ∞. We now recall that we chose 2a+3−b = 2n, meaning
that there is one term from the remaining derivative which survives at p̄ = 0:
hLcross
2n iψ = −
(−1)n c∗1 c2 B 2 F
√
Γ(2a + 4 − b) + c.c. .
4a+2 π 2
(72)
Clearly, this does not need to vanish, and it is now clear that this is because h does not vanish
sufficiently fast as p+ , k+ → 0 (note that 2a + 3 − b > 2n, corresponding to a more rapidly
decaying h, would suffice to obtain hLcross
2n iψ = 0).
3
CFT examples and counterexamples
We would like to know whether the above results are an artifact of working in free theories, or
if they are generic. To address this, we work in another framework where things are somewhat
tractable: CFTs.
For simplicity, we will work in two-dimensional CFT; we follow the conventions of [27]. In
this context, it is useful to write the light-ray operators L2n from (2) in terms of the Virasoro
operators Ln . To this end, we first Wick rotate by setting x2 = ix0 , and then define complex
coordinates z = ix1 + x2 and z̄ = −ix1 + x2 so that x+ = −iz and x− = −iz̄; as usual,
tracelessness of the energy-momentum tensor implies that Tz z̄ = Tz̄z = 0 while its conservation
implies that Tzz = Tzz (z) is holomorphic. We define T (z) = −2πTzz so that T++ = −Tzz =
T (z)/(2π) and
ˆ
ˆ
(−1)n
+
+ 2n
+
dz z 2n T (z) .
(73)
dx (x ) T++ (x ) =
L2n =
2πi iR
R
We can now make a Möbius transformation w = −(z + 1)/(z − 1) which sends the imaginary
axis to the (positively oriented) unit circle |w|2 = 1 and allows us to express L2n as
˛
˛
dz 2n dw 2 ′
(w − 1)2n ′
(−1)n
(−1)n
dw
z
dw
T (w) ,
(74)
T (w) =
L2n =
2πi |w|=1
dw
dz
4πi |w|=1
(w + 1)2n−2
where T ′ = (dz/dw)2 T is the energy-momentum tensor in the w-coordinate. Closing the contour
is only unambiguous for two integer values of n, namely n = 0 or n = 1; for higher n, the contour
crosses a pole at w = −1. Using (74), the ANEC operator L0 can be expressed as
˛
dw
1
1
(w + 1)2 T ′ (w) = L0 + (L1 + L−1 ) ,
(75)
L0 =
2 |w|=1 2πi
2
¸
where the Virasoro operators Lj are defined as Lj = (2πi)−1 dw wj+1 T ′ (w). We can give a
very similar expression for the first even light-ray operator L2 :
˛
1
dw
1
L2 = −
(w − 1)2 T ′ (w) = L0 − (L1 + L−1 ) .
(76)
2 |w|=1 2πi
2
Expressing L2n with n ≥ 2 is more complicated, primarily because the contour crosses a pole at
w = −1 for these n. There are two ways to confront this issue: we deform the contour to include
this pole, or we deform it to exclude it (equivalently, one can use an iǫ prescription to move
the pole into or out of the contour). Since a Möbius transformation can map the interior of a
circle to its exterior and vice versa, these approaches should be equivalent, i.e. we should have
T ′ (w = −1) = 0 in any correlator if we include the pole in the contour. This is not unreasonable
15
since w = −1 corresponds to x+ → ±∞, so T ′ (−1) = 0 expresses the idea that fields should
decay sufficiently fast as we approach the conformal boundary of spacetime.
Assuming that we deform the contour to exclude the pole at w = −1, the light-ray operators
with n ≥ 2 can be written as
!
!
˛
2n X
∞
X
dw (w − 1)2n ′
(−1)n
2n
2 − 2n
k
T (w) =
(−1)
L2n =
Lk+l−1 , (77)
2n−2
k
l
2
|w|=1 2πi (w + 1)
k=1 l=0
where we have Taylor expanded (w + 1)−2n+2 , and a binomial coefficient with a negative upper
entry should be interpreted in terms of gamma functions. For simplicity and since the positivity
of L0 is well-established, we will restrict our attention to L2 .
3.1
Proofs that
´
(x+ )2 T++ dx+ ≥ 0 in two-dimensional CFT
We now consider the second simplest light-ray operator in two-dimensional CFT, L2 . There is
a very quick argument that this operator is positive (see e.g. [14]). The ANEC operator L0 is
known to be positive, and an inversion y = 1/x maps L2 to the ANEC operator:
ˆ
ˆ
+ + 2
dy + Ty+ y+ = L0 .
(78)
dx (x ) Tx+ x+ =
L2 =
R
R
Inversion is part of the global conformal group, so there is no correction to the classical transformation of the stress tensor. The only potential subtlety with this argument relates to the
points x+ = 0 and x+ → ±∞. The region of integration x+ ∈ (−∞, 0) ∪ (0, ∞) maps to the
identical region of y integration, but x+ = ±∞ map to y + = 0 and vice versa. So one might
worry that states with interesting behaviour near these special points could possibly lead to a
subtlety in the equivalence between L0 and L2 .
There is a second simple argument that L2 is positive, suggested to us by Jan de Boer. As
shown in (75) and (76), L0 and L2 can be expressed very similarly in terms of the Virasoro operators L0 , L−1 , and L1 . Since the ANEC (i.e. the positivity of L0 ) can be proven algebraically
by writing an arbitrary state as a superposition of primaries and descendants, it is tempting to
perform a similar calculation for L2 .
It is most convenient to use a representation of the global conformal group rather than the
larger Virasoro algebra, and write a general state as
|ψi =
∞
XX
∆ n=0
c∆n Ln−1 |∆i ,
(79)
where the c∆n are free complex constants (chosen to normalise |ψi). We aim to calculate
∞
1 X X
′
hL2 iψ =
c∗ ′ ′ c∆n ∆′ Ln1 (2L0 − L1 − L−1 )Ln−1 |∆i .
2 ∆,∆′ n,n′ =0 ∆ n
(80)
Different primaries do not mix in this transition amplitude, so we can set ∆ = ∆′ . The only
non-zero terms are
∞
1XX
∗
h∆| (2Ln1 L0 Ln−1 c∗∆n c∆n − Ln+1
L−1 Ln−1 c∗∆n+1 c∆n − Ln1 L1 Ln+1
1
−1 c∆n c∆n+1 ) |∆i .
2 ∆ n=0
(81)
Since the different primaries do not mix, we see that L2 is positive if and only if its expectation
value is non-negative in any state which is a superposition of descendants of a single primary.
Let us therefore fix ∆ and suppress the label indicating the primary in the c-coefficients, giving
hL2 iψ =
hL2 iψ =
∞
1X
n+1
h∆| 2c∗n cn Ln1 L0 Ln−1 − c∗n+1 cn Ln+1
− c∗n cn+1 Ln+1
Ln+1
−1 L1
1
−1 |∆i .
2 n=0
16
(82)
Let us also redefine the cn to absorb the normalization,
q
and use that
cn h∆| Ln1 Ln−1 |∆i ≡ bn ,
(83)
L0 Ln−1 |∆i = (∆ + n)Ln−1 |∆i ,
(84)
so that
v
u


∞
u h∆| Ln+1 Ln+1 |∆i
1X
1
−1
2(∆ + n)|bn |2 − (b∗ bn+1 + b∗ bn )t
 .
hL2 iψ =
n
n+1
2 n=0
h∆| Ln1 Ln−1 |∆i
(85)
Now we use
n n
h∆| Ln+1
Ln+1
1
−1 |∆i = (n + 1)(2∆ + n) h∆| L1 L−1 |∆i ,
(86)
which can be derived from the global conformal algebra, so that
∞
q
1X
2(∆ + n)|bn |2 − (b∗n bn+1 + b∗n+1 bn ) (n + 1)(2∆ + n) .
hL2 iψ =
2 n=0
(87)
To see that L2 is positive, we find it helpful to rewrite this as a matrix equation: collecting the
bn into a vector ~b, we have
hL2 iψ = ~b† M~b
(88)
with
√
2∆
0
0
2∆
−
p
 √
2(∆
+
1)
−
2(2∆
+
1)
− 2∆
p 0
p


− 2(2∆ + 1)
2(∆ + 2)
− 3(2∆ + 2)
M = 0
p
 0
0
−
3(2∆
+
2)
2(∆ + 3)

..
..
..
..
.
.
.
.


...
. . .


. . .
 .
. . .

..
.
(89)
Showing that L2 is positive is equivalent to showing that M is a positive matrix, in that all
eigenvalues are non-negative. This can be accomplished by writing

√
2∆ √ −1
0
0
...
√
 0
2∆ + 1 √ − 2
0
. . .


√


†
 .

0
0
2∆
+
2
−
3
.
.
.
(90)
M = S S with S = 
√


 0
0
0
2∆
+
3
.
.
.


..
..
..
..
..
.
.
.
.
.
Matrices of the form M = S † S are positive, so this establishes that the expectation value is
non-negative in all states of the form (79). Note that a nearly identical argument works for the
ANEC operator, which differs from L2 only by the sign of the off-diagonal terms in (81).
A faster argument is as follows: L2 is related to the ANEC operator L0 by the replacements
L±1 → −L±1 . This change of sign leaves the global conformal algebra (which is the relevant
algebra as we consider global descendants of quasi-primary states) invariant, and hence L2 must
be positive if the ANEC holds, which we assume2 . This argument shows that as long as the
mapping from the plane to the cylinder does not have any subtleties, L2 must be positive.
2
We thank Jan de Boer for this elegant argument.
17
3.2
(Counter-)example of the positivity of L2
To see how the arguments from the previous section play out in more explicit detail, let us
consider the expectation value of L2 in several concrete states. We find that it is possible to
construct a reasonable-looking state in any two-dimensional CFT which violates the positivity
of L2 , although we are not able to violate the ANEC.
A commonly considered state is the one that is obtained by acting with a primary operator
Oh,h̄ , where (h, h̄) denotes its conformal weights, at negative Euclidean time −it (t ≥ 0):
|ψi =
ˆ ∞
0
dt f (t)Oh,h̄ (−it, 0) |0i ,
(91)
with f a state preparation function; preparing states with Euclidean time evolution combined
with operator insertions is standard in CFT. The expectation value of the stress tensor follows
from the T OO three-point function, which is universal and leads to
ˆ ∞
h
dt dt′ f ∗ (t)f (t′ )
(92)
hψ|ψi hT++ (x+ )iψ =
2π 0 (t + t′ )2(h+h̄−1) (x+ − it)2 (x+ + it′ )2
ˆ ∞
dt dt′ f ∗ (t)f (t′ )
hψ|ψi =
,
(93)
(t + t′ )2(h+h̄)
0
where we remind the reader that z = ix+ . (92) is in general quite cumbersome to compute, so
let us instead focus on a particular f , e.g. f (t) = α(2t0 )h+h̄ δ(t − t0 ) with t0 > 0 and α ∈ C
constants. This choice leads to
hT++ (x+ )iψ =
2ht20
π (x+ )2 + t20
2 ,
(94)
which is manifestly positive and decays as (x+ )−4 as x+ → ±∞. Thus, hL2 iψ is finite and
positive given (91) with f (t) = (2t0 )h+h̄ δ(t − t0 ).
Another commonly considered state is obtained by inserting the energy-momentum tensor
(of which every two-dimensional CFT has one) at negative Euclidean time. In particular, we
consider its superposition with the vacuum (we also could have taken a superposition of (91)
with the vacuum, but since the T O two-point function vanishes, (92) would remain unchanged):
ˆ
2π ∞
dt f (t)T++ (−it, 0) |0i ,
(95)
|ψi = |0i + √
c 0
where c is the central charge and f , as before, is the state preparation function. We now have
ˆ
2π ∞
+
hψ|ψi hT++ (x )iψ = √
dt f (t)hT++ (x+ )T++ (−it)i + c.c.
c 0
ˆ
4π 2 ∞
+
dt dt′ f ∗ (t′ )f (t)hT++ (it′ )T++ (x+ )T++ (−it)i (96)
c 0
ˆ
4π 2 ∞
dt dt′ f ∗ (t′ )f (t) T++ (it′ )T++ (−it) ,
(97)
hψ|ψi = 1 +
c 0
where the quantities on the right side are all vacuum expectation values. Using the universal
two- and three-point functions of the energy-momentum tensor in CFT, this becomes
√ ˆ ∞
ˆ ∞
c
1
dt f (t)
dt dt′ f ∗ (t′ )f (t)
hψ|ψi hT++ (x+ )iψ =
+
c.c.
+
(98)
4π 0 (x+ + it)4
2π 0 (t + t′ )2 (x+ + it)2 (x+ − it′ )2
ˆ
1 ∞ dt dt′ f ∗ (t′ )f (t)
,
(99)
hψ|ψi = 1 +
2 0
(t + t′ )4
18
This formula depends on the CFT under consideration only through the central charge c. As
before, we could consider f (t) = α(2t0 )2 δ(t − t0 ) with t0 > 0 and α ∈ C constants; using the
residue theorem, we find that the terms in (98) that are linear in f do not contribute to hL2 iψ
in this case, leaving only the positive contribution of the final term of (98).
However, it is clear that the sign of the first terms in (98) can be tuned by choosing the
phase of f appropriately. We can therefore choose a simple state preparation function to find a
counterexample to the positivity of L2 , e.g.
f (t) = αΘ(t − t0 ) ,
(100)
with t0 > 0 and α ∈ C constants. Here the Θ-function is taken to be 1 for positive argument
and 0 for negative argument. We discuss whether this state should be allowed in section 4.
Evaluating the integrals, we have
√
ˆ
dt dt′
|α|2 ∞
α c
+
+
c.c.
+
(101)
hψ|ψi hT++ (x )iψ =
12π(t0 − ix+ )3
2π t0 (t + t′ )2 (x+ + it)2 (x+ − it′ )2
ˆ
|α|2 ∞ dt dt′
hψ|ψi = 1 +
.
(102)
2 t0 (t + t′ )4
The α2 -term in the first line is inconvenient to integrate, so for simplicity we work at small
α. We still need to check that this term gives a finite contribution to L2 . The contribution to
hψ|ψi hL2 iψ of this term is
|α|2
2π
ˆ ∞
dx+ (x+ )2
−∞
ˆ ∞
t0
dt dt′
.
(t + t′ )2 (x+ + it)2 (x+ − it′ )2
(103)
The integral converges absolutely, so we can interchange the order of integration and do the x+
integral by contours, yielding
ˆ
ˆ ∞ ′
ˆ ∞
tt dtdt′
dt dt′
|α|2 ∞
2
+ + 2
∼
|α|
.
(104)
dx (x )
′ 2 +
2 +
′ 2
′ 5
2π −∞
t0 (t + t )
t0 (t + t ) (x + it) (x − it )
The remaining integral is finite:
ˆ ∞
t0
tt′ dtdt′
1
∼
.
(t + t′ )5
t0
(105)
Similarly, we find that the α2 -term in the norm of the state is finite. Having established this,
we work to first order in α, finding
√
α c
+ c.c. + O α2 ;
(106)
hψ|ψi hT++ (x+ )iψ =
+
3
12π(t0 − ix )
this allows us to evaluate the expectation value of L2 , namely
√ ˆ
α c ∞ dx+ (x+ )2
+ c.c. + O α2 .
hψ|ψi hL2 iψ =
+
3
12π −∞ (t0 − ix )
(107)
This integral diverges due to the large-x+ behaviour of the integrand unless we choose α ∈ R;
then, the integral converges and we obtain
√
α c
hψ|ψi hL2 iψ = −
+ O α2 .
(108)
6
Since this term is linear in α, it can be chosen to have either sign. Therefore, this simple state
gives a negative expectation value for the L2 operator.
19
4
Discussion and conclusions
In the preceding sections, we have given both proofs and counterexamples for the hypothesis that
the light-ray operators L2n from (2) are positive operators, in the context of a non-minimally
coupled free scalar field and general two-dimensional CFTs. The counterexamples served to
illustrate the properties that characterise the set of states for which L2n is positive. In this
section, we will discuss the relationship between the various proofs and their counterexamples
in more detail, as well as some implications of the positivity of the light-ray operators.
4.1
The free scalar field
The most important assumption about the class of states to which the proof in section 2.1
applies, is that modes with vanishing or divergent lightcone momentum p+ are sufficiently suppressed. This assumption allowed us to discard the off-diagonal term in (15) and subsequently
ignore any boundary terms when integrating by parts. We also argued that it is reasonable:
for massive fields, it corresponds to the assumption that modes with infinite momentum are
sufficiently suppressed. For massless fields in d ≥ 3 spacetime dimensions, it is equivalent to the
statement that expectation values of creation and annihilation operators should be sufficiently
smooth (implying that no mode has a divergent occupation number) as well as sufficiently
suppressed at high momenta.
Despite this apparently reasonable condition, section 2.2 identified a class of states which
satisfy several acceptability criteria (as Hadamard states with finite energy and norm) but still
yield an indefinite expectation value of L2n . As shown at the end of section 2.2, this happens
because these states have insufficiently suppressed support at vanishing lightcone momentum:
as p+ + k+ → 0, h from (42) vanishes polynomially, so one of its derivatives does not vanish at
p+ = k+ = 0. To further emphasise this point, note from (67) that choosing a and b such that
cross from contributing to hL
hL2n iψ is indefinite prevents T++
2m iψ if m < n. Instead, hL2m iψ is
diag
determined by T++
with a sign that cannot be tuned, recovering the result of section 2.1. The
reason for this is suggested by (71): lower light-ray operators are less demanding with regard
to the rate at which modes of vanishing lightcone momentum should be suppressed.
cross to hL
On the other hand, the contribution of T++
2M iψ with M > n is divergent (as can
+
2M
+2
be seen in (67) by expanding (x )
around il), meaning that L2M is not well-defined as
an operator in this particular state. Combining this fact with our observations on L2m with
m < n, it seems that the counterexamples from section 2.2 are quite special: states which are
counterexamples to the positivity of L2n are not counterexamples to the positivity of L2m with
m < n, while hL2M iψ with M > n diverges in these states. If this is not an artefact of our
class of states and our specific theory, then we could give an alternative characterisation of the
combination of states and light-ray operators for which the computation in section 2.1 holds:
namely, that hL2n iψ ≥ 0 when n satisfies (34) and hL2(n+1) iψ < ∞.
Furthermore, let us mention two more physical ramifications of the proof and its counterexamples in section 2. First, one of our motivations to study light-ray operators was that the
CANEC, applied to the lightcone in Minkowski spacetime, would imply that Ld is positive for
holographic CFTs for even d ≥ 4 [16, 17] (for d = 2, [17] reproduces the ANEC). However, 2.2
provides counterexamples to such a statement for the conformally coupled scalar field. This
implies at least one of three things: the strong coupling which characterises holographic CFTs
renders Ld positive, the CANEC only applies to a restricted set of states, or the CANEC does
not apply to all congruences in all spacetime geometries. Since [16, 17] only gave a holographic
proof of the CANEC in a limited class of geometries, it stands to reason that at least the latter
possibility is true; assessing the former two will require further investigation.
Our second point is concerned with the Rindler vacuum. This state has a negative null
energy density everywhere except at the origin, and thus forms another counterexample to the
20
positivity of L2n ; this implies that it cannot satisfy the assumptions we made in section 2.1.
This raises the question of how one should think about the states for which L2n is indefinite:
are they unphysical, or are they still relevant to the theory? In the former case, it would be
necessary to explain why an observer accelerating at a constant rate could not access their
natural vacuum, and in the latter case one would need to explain what (if anything) physically
distinguishes states for which L2n is positive from those with indefinite L2n .
A way to circumvent the problem posed by the Rindler vacuum would be to argue that it is
not realistic due to its divergent energy density at a single point. However, we could consider a
state which approximates the Rindles vacuum away from the origin and behaves more smoothly
near it. Such a smoothed Rindler vacuum could still be a counterexample to the positivity of
L2n , since L2n suppresses the (positive) contributions from the region near the origin. On the
other hand, smoothing out the divergence near x+ = 0 might change the null energy density
at large x+ in such a way that L2n stays positive. In any case, the physical interpretation of
states with indefinite L2n remains an interesting avenue of further research.
4.2
d = 2 conformal field theory
Having concluded our treatment of the free, non-minimally coupled scalar field, we went on to
consider light-ray operators in a d = 2 CFT in section 3. This is an important complement to
the free theories from section 2, since they correspond to massless fields in two dimensions (the
only context in which the free field computation does not apply) and allow for interactions. We
focused our efforts on L2 , since it is closely related to the ANEC operator and, as shown in
(76), it has a very simple expression in terms of the Virasoro operators.
We then proved the positivity of L2 in three ways. First, we mapped it to the ANEC
operator with an inversion, which assumed that expectation values of T++ vanish sufficiently fast
as x+ → ±∞; this is similar to our assumptions for the free field. The other proofs expressed
L2 in terms of Virasoro operators, which are well-defined in the theory on the (Lorentzian)
cylinder or in radial quantisation on the plane. Positivity of L2 in the representation of the
global conformal group then followed by explicitly calculating its expectation value and by
relating it to the ANEC operator via a change of sign for L±1 .
Nevertheless, in section 3.2 we have found a set of states that allow for negative expectation
values of L2 , meaning that they must somehow violate the assumptions of our proofs. This is
most straightforwardly interpreted for our algebraic proofs, which assumed that we work with
states that are well-defined on the cylinder (i.e. states from the representation of the global
conformal group). Apparently, the state (95) with (100) is not a good state on the cylinder.
However, from the point of view of the theory on the plane quantized on equal-time surfaces
(e.g. t = 0) rather than in radial quantization, it is not clear that there is any problem with
this state. The state is normalizable and has finite energy. Because it is prepared by evolving
in Euclidean time, with operator insertions away from zero Euclidean time, the short distance
behaviour is well-behaved, meaning that the state satisfies the Hadamard condition. From
the point of view of the theory on the plane, there does not seem to be any justification for
prohibiting this class of states.
A natural question, then, is whether there is a simple definition of the class of states for
which L2 is positive. We can give some data points with regards to this question. Consider
again our state |ψi defined in (95). In this state, we have
hψ|ψi hL2 iψ =


√ ˆ
ˆ ∞
ˆ ∞
2
dx+ (x+ )2  c ∞ dt f (t)
dt f (t)e−αt 
, (109)
αdα
+ c.c. +
2π
2 0 (x+ + it)4
(x+ + it)2
−∞
0
0
ˆ ∞
where we applied (44) to the last term of (98) to make its positivity manifest. In fact, observe
that the t-integral in this last term is the Laplace transform of f (t)/(x+ + it)2 ; by Lerch’s
theorem [28] this vanishes if and only if f is a null function, i.e. its integral vanishes on any
21
subset of its domain. Thus, negative contributions to hL2 iψ from the terms linear in the state
preparation function f are always at least partially compensated by a positive contribution
from the last term in (109). Now, suppose that f is sufficiently well-behaved that the order
of integration can be interchanged in the terms linear in f . In that case, we can perform the
x+ -integral first, obtaining
ˆ ∞
dx+ (x+ )2
=0;
(110)
+
4
−∞ (x + it)
so as long as the order of integration can be interchanged, the terms linear in f give zero
contribution, and L2 is positive. The condition for changing the order of integration is given by
the Fubini theorem, which states that we can change the order of integration if
ˆ ∞
ˆ ∞
(x+ )2 f (t)
+
<∞.
(111)
dx
dt
(x+ + it)4
−∞
0
This will be the case as long as f (t) vanishes sufficiently fast as t → 0 and as t → ∞. The
state preparation function (100) violates the latter condition, and indeed, it provides a simple
counterexample to the positivity of L2 .
Extrapolating from these data points, one might guess that L2 is positive for any state
prepared by acting with operators that are compactly supported away from t = 0 in the lower
half Euclidean plane. This is consistent with the proof for the positivity of L2 : any such state is
well-defined on the cylinder and therefore satisfies the assumptions of section 3.1. Additionally,
we find that the state created by a single operator insertion in (91) has hL2 iψ ≥ 0 as well if
f is compactly supported away from t = 0. After all, if f is compactly supported, (92) shows
that hT++ iψ decays as (x+ )−4 for x+ → ±∞; if f is not supported at t = 0, this in turn implies
that the x+ - and t-integrals involved in hL2 iψ converge absolutely and can be exchanged. The
x+ -integral can then be evaluated by closing the contour around e.g. the pole at x+ = it (the
decay as x+ → ∞ is fast enough to allow this), leading to
ˆ ∞
ˆ ∞
ˆ ∞
2
dα α2(h+h̄)
dt dt′ f ∗ (t)f (t′ )t′ t
= 2h
dt f (t)e−αt t , (112)
hψ|ψi hL2 iψ = 2h
Γ(2h + 2h̄ + 1) 0
(t + t′ )2(h+h̄)+1
0
0
which is manifestly positive. As before, Lerch’s theorem implies that the t-integral, which is the
Laplace transform of f (t)t, vanishes if and only if f (t)t is at most a null function. In practice,
this means that (112) only saturates hL2 iψ ≥ 0 as (91) approaches the vacuum.
Having studied L2 in some detail, we also offer some comments on the other light-ray
operators L2n in two-dimensional CFTs. Consider for example hL2n iψ for n ≥ 2 and the
counterexample state |ψi from section 3.2 (i.e. (95) with (100)); from (107) one can immediately
read off that even at first order in α, hL2n iψ diverges for n ≥ 2. This is consistent with the
pattern we noted for the counterexamples to the positivity of L2n in free field theory: for |ψi
from (95), L0 is positive, L2 is indefinite but finite, and L2n with n ≥ 2 is divergent.
This divergence is not unique to the |ψi from (95): because light-ray operators involve
improper integrals with positive powers of x+ , we expect there to be many seemingly sensible
states where the expectation value of the light-ray operator diverges. A simple example is (91),
a state created with a single compactly supported primary operator insertion; as noted before,
(92) then implies that hT++ iψ ∼ (x+ )−4 asymptotically, which means that L0 and L2 are the
only finite light-ray operators in this state.
One way to counteract this divergence is to combine insertions of Oh,h̄ with insertions of the
energy-momentum tensor. For example, to construct a state in which hL4 iψ is finite we could
consider a superposition of (91) and (95), choosing the state preparation functions proportional
to delta functions:
|ψi = |0i +
2πα1 (2t1 )2
√
T++ (−it1 , 0) |0i + α2 (2t2 )h+h̄ Oh,h̄ (−it2 , 0) |0i
c
22
(113)
with α1 , α2 ∈ C and t1 , t2 > 0 constants. Because the T O- and T T O-correlators vanish, the
expectation value T++ (x+ ) ψ is straightforwardly computed:
√
2|α1 |2 t21
α1 t21 c
2h|α2 |2 t22
+
c.c.
+
hψ|ψi hT++ (x )iψ =
+
.
π(x+ + it1 )4
π|x+ + it1 |4 π|x+ + it2 |4
+
(114)
To have a finite hL4 iψ , we should now demand that hT++ iψ asymptotically decays faster than
(x+ )−5 . This condition requires α1 ∈ R, with either of
1√
c±
α1 = −
2
s
c
t2
− h|α2 |2
4
t1
2
.
(115)
With this relation in mind, we can compute hψ|ψi hL4 iψ explicitly using the residue theorem:
√
hψ|ψi hL4 iψ = 3h|α2 t2 |2 (t1 − t2 ) − 5 c α1 t31 .
(116)
By fixing α2 t2 and considering t2 ≫ t1 , the negative first term of this expression can be seen
to dominate. Therefore, (113) with (115) represents a finite counterexample to the positivity of
L4 . Note that, given (114), this state results in a positive hL2 iψ . Again, this is consistent with
our observation from free field theory that a finite hL2(n+1) i seemingly implies a positive hL2n i.
Nevertheless, the fact that even an innocuous-looking state like (91) leads to divergent
hL2n≥4 i may make it seem like only fine-tuned states have finite expectation values of the lightray operators. However, consider the set of states we can actually prepare in CFT. We are
restricted to act in Lorentzian time with unitary operators, so a natural set of states to consider
is the states obtained by acting with sources ji in Lorentzian time:
!
ˆ
X
2
jk (x)Ok (x) |0i .
(117)
|ψi = exp i d x
k
Here, the integral is over Lorentzian spacetime, and we have suppressed tensor indices. Suppose
now that we make the natural assumption that we can only turn on the sources in some finite
spacetime region, i.e. the jk (x) are compactly supported. Because T++ is holomorphic, we can
compute T++ (x+ ) on the line t = 0. Since operators at large x+ commute with all operators
in the state preparation, the Campbell identity implies that the set of states in (117) has
hT++ (x+ )iψ = 0
(118)
for sufficiently large |x+ |. Therefore, in the natural set of states (117), all light-ray operators
have finite expectation values. If the relation between the finiteness of L2(n+1) and the positivity
of L2n which we observed in free field theory holds for two-dimensional CFTs as well, then (118)
implies that all light-ray operators are positive in the set of states (117). We leave the question
of whether this is indeed the case to future work.
Acknowledgements
We thank Jan de Boer, Diego M. Hofman, Ken D. Olum, David Simmons-Duffin, and Anthony
J. Speranza for useful discussions. BF is partially supported by Heising-Simons Foundation
‘Observational Signatures of Quantum Gravity’ QuRIOS collaboration grant.
References
[1] R. V. Buniy, S. D. H. Hsu, and B. M. Murray. The null energy condition and instability.
Phys. Rev. D, 74, September 2006.
23
[2] S. Dubovsky, T. Grégoire, A. Nicolis, and R. Rattazzi. Null energy condition and superluminal propagation. Journal of High Energy Physics, 2006(03):025, March 2006.
[3] D. Hochberg and M. Visser. Null Energy Condition in Dynamic Wormholes. Phys. Rev.
Lett., 81:746–749, jul 1998.
[4] D. Battefeld and P. Peter. A critical review of classical bouncing cosmologies. Physics
Reports, 571:1–66, 2015. A critical review of classical bouncing cosmologies.
[5] R. Penrose. Gravitational Collapse and Space-Time Singularities. Phys. Rev. Lett., 14:57–
59, January 1965.
[6] H. Epstein, V. Glaser, and A. Jaffe. Nonpositivity of the energy density in quantized field
theories. Il Nuovo Cimento, 36(3):1016–1022, April 1965.
[7] J. Maldacena, A. Milekhin, and F. Popov. Traversable wormholes in four dimensions.
Classical and Quantum Gravity, 40(15):155016, jul 2023.
[8] R. Bousso, Z. Fisher, S. Leichenauer, and A. C. Wall. Quantum focusing conjecture. Phys.
Rev. D, 93:064044, March 2016.
[9] S. Balakrishnan, T. Faulkner, Z. U. Khandker, and H. Wang. A general proof of the
quantum null energy condition. Journal of High Energy Physics, 2019(9), September 2019.
[10] T. Faulkner, R. G. Leigh, O. Parrikar, and H. Wang. Modular Hamiltonians for deformed half-spaces and the averaged null energy condition. Journal of High Energy Physics,
2016(9), September 2016.
[11] T. Hartman, S. Kundu, and A. Tajdini. Averaged null energy condition from causality.
Journal of High Energy Physics, 2017(7), July 2017.
[12] H. Stoffels. Positive thinking: A search for energy conditions, November 2024.
[13] H. Casini, E. Testé, and G. Torroba. Modular Hamiltonians on the null plane and the
Markov property of the vacuum state. Journal Of Physics A: Mathematical And Theoretical, 50(36):364001, August 2017.
[14] A. Belin, D. M. Hofman, G. Mathys, and M. T. Walters. On the stress tensor light-ray
operator algebra. Journal of High Energy Physics, 2021(5), May 2021.
[15] M. Besken, J. de Boer, and G. Mathys. On local and integrated stress-tensor commutators.
Journal of High Energy Physics, 2021(7), July 2021.
[16] N. Iizuka, A. Ishibashi, and K. Maeda. Conformally invariant averaged null energy condition from AdS/CFT. Journal of High Energy Physics, 2020(3), March 2020.
[17] N. Iizuka, A. Ishibashi, and K. Maeda. The averaged null energy conditions in even dimensional curved spacetimes from AdS/CFT duality. Journal of High Energy Physics,
2020(10), October 2020.
[18] D. M. Hofman and J. Maldacena. Conformal collider physics: energy and charge correlations. Journal of High Energy Physics, 2008(05):012, May 2008.
[19] G. P. Korchemsky and A. Zhiboedov. On the light-ray algebra in conformal field theories.
Journal of High Energy Physics, 2022(2), February 2022.
[20] P. Kravchuk and D. Simmons-Duffin. Light-ray operators in conformal field theory. Journal
of High Energy Physics, 2018(11), November 2018.
24
[21] D. D. Blanco and H. Casini. Localization of Negative Energy and the Bekenstein Bound.
Phys. Rev. Lett., 111:221601, November 2013.
[22] D. G. Figueroa, A. Florio, T. Opferkuch, and B. A. Stefanek. Lattice simulations of nonminimally coupled scalar fields in the Jordan frame. SciPost Phys., 15:077, September
2023.
[23] M. E. Peskin and D. V. Schroeder. An Introduction to Quantum Field Theory. CRC Press,
2019.
[24] G. Klinkhammer. Averaged energy conditions for free scalar fields in flat spacetime. Phys.
Rev. D, 43:2542–2548, April 1991.
[25] B. S. Kay and R. M. Wald. Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. Physics
Reports, 207(2):49–136, August 1991.
[26] C. J. Fewster and R. Verch. The necessity of the Hadamard condition. Classical and
Quantum Gravity, 30(23):235027, November 2013.
[27] P. di Francesco, P. Mathieu, and D. Sénéchal. Conformal Field Theory. Springer Science
& Business Media, January 1997.
[28] M. Lerch. Sur un point de la théorie des fonctions génératrices d’Abel. Acta Mathematica,
27:339 – 351, 1903.
25