JHEP07(2017)066
Published for SISSA by Springer
Received: May 22, 2017
Accepted: June 28, 2017
Published: July 14, 2017
Averaged null energy condition from causality
Thomas Hartman, Sandipan Kundu and Amirhossein Tajdini
Department of Physics, Cornell University,
Ithaca, NY, U.S.A.
E-mail: hartman@cornell.edu, kundu@cornell.edu, at734@cornell.edu
Abstract: Unitary, Lorentz-invariant quantum field theories in flat spacetime obey mi-
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
duT
uu
,
must be non-negative. This non-local operator appears in the operator product expansion
of local operators in the lightcone limit, and therefore contributes to n-point 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 infi-
nite family of new constraints of the form
R
duX
uuu···u
≥ 0. These lead to new inequalities
for the coupling constants of spinning operators in conformal field 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 condi-
tion from relative entropy, and suggest a more general connection between causality and
information-theoretic inequalities in QFT.
Keywords: Conformal Field Theory, Field Theories in Higher Dimensions
ArXiv ePrint: 1610.05308
Open Access,
c
The Authors.
Article funded by SCOAP
3
.
https://doi.org/10.1007/JHEP07(2017)066
JHEP07(2017)066
Contents
1 Introduction 1
2 Derivation of the ANEC 4
3 Average null energy in the lightcone OPE 7
3.1 Lightcone OPE 7
3.2 Contribution to correlators 8
3.3 Scalar example 10
4 Sum rule for average null energy 11
4.1 Analyticity in position space 11
4.2 Rindler positivity 12
4.3 Bound on the real part 14
4.4 Factorization 15
4.5 Sum rule 15
4.6 Non-conformal QFTs 17
5 Hofman-Maldacena bounds 17
5.1 Conformal collider redux 17
5.2 Relation to scattering with smeared insertions 18
5.3 Relation to the shockwave kinematics 20
6 New constraints on higher spin operators 20
6.1 E
s
in the lightcone OPE 20
6.2 Sum rule and positivity 21
6.3 Comparison to other constraints and spin-1-1-4 example 22
A Rotation of three-point functions 23
B Normalized three point function for hJ XJ i 24
C Free scalars 25
1 Introduction
The average null energy condition (ANEC) states that
Z
∞
−∞
dλ hT
αβ
iu
α
u
β
≥ 0 , (1.1)
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JHEP07(2017)066
where the integral is over a complete null geodesic, and u is the tangent null vector. This
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 field 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 first 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 defined in the continuum limit, the argument applies to a large (and perhaps dense)
set of states in any unitary, Lorentz-invariant QFT.
Separately, the ANEC for a special class of states in conformal field theory was derived
recently using techniques from the conformal bootstrap developed in [15, 16]. These special
cases of the ANEC, known as the Hofman-Maldacena 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 4-point correlator
hφ[T, T ]φi where T is the stress tensor and φ 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, Lorentz-invariant QFT with
an interacting conformal fixed point in the UV must obey the ANEC, in agreement with
the information-theoretic derivation of Faulkner et al. The argument assumes no higher
spin symmetries at the UV fixed 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 4-point
function. Furthermore, we argue that the ANEC is just one of an infinite class of positivity
constraints of the form
Z
duX
uu···u
≥ 0 (1.2)
where X is an even-spin operator on the leading Regge trajectory (normalized appropri-
ately) — i.e., it is the lowest-dimension operator of spin s ≥ 2. This implies new constraints
on 3-point couplings in CFT; we work out the example of spin-1/spin-1/spin-4 couplings.
Another interesting corollary is that, like the stress tensor, the minimal-dimension opera-
tor 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 Hofman-Maldacena conditions on stress tensor couplings,
we conjecture that (1.2) evaluated in a momentum basis is optimal, meaning that the
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JHEP07(2017)066
resulting constraints on 3-point 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 different, since we work entirely in quantum field 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
using the OPE.
Causality vs. quantum information. Our derivation bears no obvious resemblance
to the relative entropy derivation of Faulkner et al., except that both seem to rely on
Lorentzian signature. (Our starting point is Euclidean, and we do not make any assump-
tions about the QFT beyond the usual Euclidean axioms, but we do analytically continue
to Lorentzian.) It is intriguing that causality and information-theoretic inequalities lead
to overlapping constraints in this context.
There are significant hints that this connection between entanglement and causality
is more general. This is certainly true in 2d CFT; see for example [24]. It is also clear
in general relativity; for example, both the second law and strong subadditivity of the
holographic entanglement entropy require the NEC [7, 25, 26]. But there are also hints in
higher-dimensional QFT for a deeper connection between entanglement and causality con-
straints. Recent work on the quantum null energy condition [14, 27, 28] is one example, and
they are also linked by c-theorems for renormalization group flows in various dimensions.
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 higher-dimensional cousin,
the a-theorem 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 renormaliza-
tion group, albeit in different 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
field 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
1
In free field theory our methods do not apply directly, but a simple mode calculation in an appendix
demonstrates that the inequality (1.2) holds also for free scalars. This appears to have escaped notice. It
may have interesting implications for coupling quantum fields to stringy background geometry, just as the
ANEC plays an important role in constraining physical spacetime backgrounds. The operators X generalize
the stress tensor to the full leading Regge trajectory of the closed string. A first step would be to confirm
that (1.2) holds for other types of free fields.
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JHEP07(2017)066
F -theorem would very likely violate causality, too. It also suggests that the higher-spin
causality constraints (1.2) on the leading Regge trajectory could have an information-
theoretic origin, presumably involving non-geometric deformations of the operator algebra.
Comparison to previous methods. Both conceptually and technically, the argument
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 Hofman-Maldacena
bounds on hTT 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 TT T i). The simplification 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 3-point functions of non-conserved 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 high-spin composite operators as in [19].
Outline. The main argument is given first, 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
duT
uu
≥ 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 superfluous 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
3-point couplings. Finally, in section 6, we generalize the argument to the ANEC in any
dimension d > 2, as well as to an infinite 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 ∈ R
1,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. ψ is
always a real scalar primary operator. Hermitian conjugates written as O
†
(x) act only on
the operator, not the coordinates, so [O(t, . . . )]
†
= O
†
(t
∗
, . . . ). To simplify the formulas
we set d = 4 in the first few sections, leaving the general case (d > 2) for section 6.
Define the average null energy operator
E =
Z
∞
−∞
duT
uu
(u, v = 0, ~x = 0) . (2.2)
– 4 –