A New Spin on Causality Constraints

0.49 MB·

by  Thomas Hartman

A New Spin on Causality Constraints
Thomas Hartman, Sachin Jain, and Sandipan Kundu
Department of Physics, Cornell University, Ithaca, New York
[email protected], [email protected], [email protected]
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J Abstract

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2 Causality in a shockwave state is related to the analytic properties of a four-

] point correlation function. Extending recent results for scalar probes, we show
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t that this constrains the couplings of the stress tensor to light spinning operators
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in conformal field theory, and interpret these constraints in terms of the inter-
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action with null energy. For spin-1 and spin-2 conserved currents in four dimen-
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sions, theresultinginequalitiesareasubsetoftheHofman-Maldacenaconditions
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v for positive energy deposition. It is well known that energy conditions in holo-
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Contents
1 Introduction 2
1.1 Energy condition from causality . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Currents and stress tensors in d = 4 . . . . . . . . . . . . . . . . . . . . 3
1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Correlators across a shockwave 5
2.1 J and T in d = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 General probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Reflection Positivity for Spinning Correlators 13
3.1 Review of reflection positivity . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Positive expansions for spinning correlators . . . . . . . . . . . . . . . . 16
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Derivation of causality constraints 19
4.1 The sum rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Positivity of shockwave corrections . . . . . . . . . . . . . . . . . . . . 20
5 Connection to the lightcone bootstrap 21
A Review of spinning correlators 24
A.1 Embedding space notation . . . . . . . . . . . . . . . . . . . . . . . . . 24
A.2 Building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
A.3 2-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
A.4 3-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
A.5 4-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A.6 Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
B 3-point functions of conserved currents 27
B.1 JJT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
B.2 TTT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
C Conformal blocks for stress tensor exchange 30
1
1 Introduction
Werecentlyshowedthatcausalityimposesinequalitiesoncouplingconstantsinstrongly
interacting conformal field theory in d > 2 dimensions [1]. The constraints for scalar
correlators, derived by conformal bootstrap methods, were nontrivial only in special
cases, but we argued that similar logic would impose nontrivial constraints on the cou-
plings of spinning operators. In this paper we confirm this, deriving a set of causality
constraints on the couplings of spinning operators that hold in any unitary CFT in
d > 2 dimensions (up to a caveat below). Like the causality constraints of [2], or the
results of the computational conformal bootstrap [3,4], these are infrared constraints
imposed by ultraviolet consistency.
1.1 Energy condition from causality
We study the 4-point function (cid:104)ψOOψ(cid:105), where ψ is a scalar and O is a symmetric
traceless tensor operator of spin (cid:96). The spinning operator O is viewed as a ‘probe’ in
the background produced by ψ. Causality requires
(cid:2) (cid:3)
(cid:104)ψ(x ) O (x ) , O (x ) ψ(x )(cid:105) = 0 (1.1)
1 µ1...µ(cid:96) 2 ν1...ν(cid:96) 3 4
for spacelike separated probes, (x − x )2 > 0. We show that vanishing of this com-
2 3
mutator imposes a constraint on the coupling to null energy, in a limit with lightlike
operator insertions:
(cid:104)ε·O(0)ε∗·O(y+,y−)T (∞yˆ)(cid:105)
−− < 0 as y+ → 0 , (1.2)
(cid:104)ε·O(0)ε∗·O(y+,y−)(cid:105)
with y± = y ±t > 0. This particular 3-point coupling appears because it determines
the leading behavior of the conformal block for stress tensor exchange, OO → T → ψψ,
in the lightcone limit. Note the peculiar sign of the constraint — our sign conventions
are such that the usual null energy condition is (cid:104)T (cid:105) > 0. Of course this is not a
−−
contradiction, since (1.2) is not an expectation value.
Unlike the null energy condition, which is violated quantum mechanically, the en-
ergy condition (1.2) is derived from reflection positivity of 4-point functions in the
Euclidean theory and must hold in any unitary CFT in d > 2 obeying the usual ax-
ioms, with one notable caveat: We assume that no scalar fields with scaling dimension
2
in the range d−1 < ∆ ≤ d−2 appear in the OO OPE. This ensures that stress tensor
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exchange dominates the 4-point function (1.1) for null-separated probes.1
If (1.2) is violated, then the 4-point function (cid:104)ψOOψ(cid:105) has a singularity before the
Minkowski lightcone, and therefore a non-vanishing commutator at spacelike separa-
tion. In other words, the shockwave induces a time advance.
1.2 Currents and stress tensors in d = 4
The complete 3-point function (cid:104)OOT(cid:105) is fixed by conformal symmetry up to numer-
ical couplings, so (1.2) is a set of inequalities for these coupling constants. Different
polarizations lead to multiple, independent constraints. We will work out these con-
straints in detail for the case of spin-1 and spin-2 conserved probes in four spacetime
dimensions.
A convenient basis for 3-point tensor structures of conserved currents is provided
by free fields of various spins. (We assume parity.) The current 3-point functions in
any CFT can be expanded in this basis,
(cid:104)JJT(cid:105) = n (cid:104)JJT(cid:105) +n (cid:104)JJT(cid:105) (1.3)
s scalar f fermion
and
(cid:104)TTT(cid:105) = n˜ (cid:104)TTT(cid:105) +n˜ (cid:104)TTT(cid:105) +n˜ (cid:104)TTT(cid:105) , (1.4)
s scalar f fermion v vector
where (cid:104)OOT(cid:105) is proportional to the correlator in a theory of free scalars, fermions,
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or vectors, and n ,n ,n˜ ,n˜ ,n˜ are the coupling constants (see appendices for conven-
s f s f v
tions).
The conformal Ward identity relates one combination of couplings to the two-point
function, which must be positive in a unitary theory. With our normalizations, these
combinations are
n˜ 1 16(d−3)
s
c ∝ 4n˜ + > 0 , c ∝ n˜ + n˜ + n˜ > 0 . (1.5)
J f T f s v
d−2 2(d−1) d(d−2)
Hofman and Maldacena [6,7] (and others in d (cid:54)= 4 [8–10]) showed that in a theory
1InseveralinterestingCFTs,includingtheWilson-FisherandKlebanov-Wittenfixedpoints,there
isascalarinthisrangebutsymmetriespreventitfromappearinginthisOPE.See[5]foradiscussion
in the context of entanglement entropy (which has an enhanced divergence if such operators appear
in the TT OPE).
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satisfying an integrated null energy condition, the constraints are much stronger:
n ≥ 0, n ≥ 0 , n˜ ≥ 0, n˜ ≥ 0, n˜ ≥ 0 . (1.6)
s f s f v
In large-N holographic theories, identical constraints have been derived from causality
on the gravity side [7,8,11,12].
More generally, however, it is an open question whether these constraints hold in
every unitary CFT, and how they are related to causality.2 We will partially resolve
this. Taking various polarizations in (1.2) leads to
n ≥ 0 , n +8(d−1)n ≥ 0 (1.7)
s s f
and
d(d−2)2
n˜ ≥ 0, (d−2)2n˜ +2(d−1)3n˜ ≥ 0, n˜ +4dn˜ +128n˜ ≥ 0. (1.8)
s s f (d−1)3 s f v
Inthreedimensions, thereisnovectorstructure, soonlythefirsttwoinequalitiesapply.
In four dimensions, (1.8) implies that the anomaly coefficients a and c satisfy
13 a 31
≤ ≤ . (1.9)
54 c 18
1.3 Discussion
The constraints (1.7) and (1.8) are stronger than positivity of the 2-point function (1.5)
but weaker than (and implied by) the Hofman-Maldacena energy conditions. They are
necessary for causality, but we do not claim they are sufficient. Our derivation seems to
exploit only a limited form of reflection positivity, so a better organization of positivity
may lead to stronger constraints.
One approach to strengthen the causality constraints may be to combine the meth-
odsherewiththelightconebootstrap[16–26]. Causalityfixesthesignoftheanomalous
2A partial argument based on the OPE of non-local energy expectation values appears in [7].
A rather different argument, based on finite-temperature correlators, intriguingly leads to the same
constraints [13] but is also incomplete as it requires the use of the Euclidean OPE beyond its usual
regime of validity. A recent discussion suggesting that the average null energy condition might be
violated in acceptable theories can be found [14]. Significant progress on this question was also made
very recently in [15], where it was shown that the Hofman-Maldacena constraints follow from the
opticaltheoreminadeepinelasticscatteringexperiment,assumingthattheamplitudeobeysastrong
enough growth condition at large |s|.
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dimensions of certain high-spin operators, and, barring some unexpected cancellations,
we expect this statement to actually be stronger than the inequalities above. The
anomalous dimensions can be computed by the lightcone bootstrap, but the calcula-
tion has not yet been done (in the required regime). Further remarks on the connection
to the lightcone bootstrap are in section 5. It would also be very interesting to better
understand the relationship between the present work and other recent work on related
constraints, such as [15,27,28].
One motivation for this work was the recent demonstration, using causality on the
gravity side, that holographic CFTs have a ≈ c [10] (see also [29,30]). As discussed at
length in the introduction to [1], the weaker (but universal) constraint (1.9) may be a
step toward deriving this surprising feature of large-N sparse CFTs directly from the
conformal bootstrap.
1.4 Overview
In the next section, we calculate the leading corrections to correlators across a shock-
wave. The result is proportional to (1.2), and specializing to O = J and O = T with
various polarizations leads to the combinations in (1.7) and (1.8). Requiring these
corrections to be negative gives the constraints.
In the rest of the paper (which is almost independent from section 2) we show why
these corrections must be negative. First we derive positivity conditions for spinning
correlators, from reflection positivity, in section 3. This essentially reduces the problem
of spinning causality constraints to the scalar case. The derivation of the causality
constraints is in section 4. We conclude with a discussion of the relationship to the
lightcone bootstrap and anomalous dimensions in section 5.
2 Correlators across a shockwave
In this section, we summarize how the constraints in the introduction follow naturally
from computing correlators in a ‘shockwave state’,
|Ψ(cid:105) = ψ(t = −iδ)|0(cid:105) . (2.1)
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Viewed on scales much larger than δ, the energy in this state (cid:104)Ψ|T |Ψ(cid:105) is supported
µν
on an expanding null shell centered at the origin.3 Consider the 4-point function
(cid:104)Ψ|ε·O(x )ε∗·O(x )|Ψ(cid:105) (2.2)
2 3
where
ε·O = ε Oµν... . (2.3)
µν...
In the limit where the probe operators are almost null separated (x −x )2 → 0, the
2 3
leading correction comes from stress tensor exchange,
OO → T → ψψ . (2.4)
This interaction is controlled by the 3-point couplings in (cid:104)OOT(cid:105), and by the scaling
dimension of ψ, which fixes the coupling (cid:104)ψψT(cid:105) via the Ward identity,
Γ(d/2)d
c = − ∆ . (2.5)
ψψT 2(d−1)πd/2 ψ
The magnitude of this correction grows with the energy of the shock E ∼ 1. For
shock δ
scalar probes, it was shown in [1] that such corrections must be negative, or causality
is violated. Below, we will show that the same is true for spinning probes: unitary
theories must obey
(cid:104)ψε·Oε∗·Oψ(cid:105) |x −x |d/2−1
2 3
≈ 1−(positive) (2.6)
(cid:104)ε·Oε∗·O(cid:105)(cid:104)ψψ(cid:105) δ
in the limit 0 < (x − x )2 (cid:28) δ (cid:28) 1. The basic reason is that if the correction has
2 3
the wrong sign, then analyticity forces the correlator to have a singularity before the
Minkowski lightcone, and this singularity produces a non-zero commutator at spacelike
separation. For now, we just assume (2.6) and work out the resulting constraints. In
a theory with a low-dimension scalar appearing in the OO OPE, d −1 < ∆ < d−2,
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the leading term is not of the form (2.6) and is not constrained, so our analysis does
not apply.
3ForotherworkonshockwavesinCFT,especiallyinrelationtotheAdS/CFTcorrespondence,see
[31–33]. The shockwave state considered there is apparently different, though the analytic properties
of the correlator are similar.
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Figure1: Kinematicsfortheshockwavecorrelator. Oisaspinningprobeinthestateproduce
by the ψ insertion. The two O’s are inserted symmetrically across the shock, and the two
ψ’s are offset into the imaginary time direction (not drawn), t = ±iδ.
2.1 J and T in d = 4
We start with a direct, full calculation of the stress tensor contribution to the shock-
wave correlator for conserved current probes in d = 4. This approach is conceptually
straightforward but requires a computer to do the algebra. It also hides a dramatic
simplification that occurs in the lightcone limit which we return to below.
The full conformal blocks for stress tensor exchange are calculated following [34].
Details, andallofourconventions, areintheappendices. Theoutputofthiscalculation
is the spinning conformal block expressed as derivatives of the scalar conformal block.
The scalar block cannot be written in closed form in odd dimensions, but we will only
require the leading terms in the lightcone limit, so this could be done in any d.
We work in R1,3, with coordinates (t,y,x ,x ) and place all insertions in the ty-
3 4
plane. The operators creating the shockwave are inserted at
x = (t ,y ) = (iδ,0), x = (t ,y ) = (−iδ,0) , (2.7)
1 1 1 4 4 4
and the probes are inserted symmetrically across the shock:
(cid:16) η η (cid:17)
x = 1 1−D+ ,1+D+ (2.8)
2 2 D D
(cid:16) η η (cid:17)
x = 1 1+D− ,1−D− .
3 2 D D
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We will also use the lightcone coordinates
y± = y ±t . (2.9)
D is the distance to the shockwave, and the lightcone limit is y+ ∼ η → 0, as
23 D
illustrated in figure 1. With these kinematics, we calculate the contribution from the
identity and the stress tensor to the correlator
(cid:104)ψ(x )ε·O(x )ε∗·O(x )ψ(x )(cid:105)
1 2 3 4
G(ε,η,D) = , (2.10)
(cid:104)ε·O(x )ε∗·O(x )(cid:105)(cid:104)ψ(x )ψ(x )(cid:105)
2 3 1 4
in the limit η (cid:28) δ (cid:28) 1, and compare to (2.6). The conformal cross ratios in the
configuration (2.7-2.8) are
(D+iδ)2 (D+iδD−η)(D−iδD+η)
z = , z¯= . (2.11)
(D−iδ)2 (D−iδD−η)(D+iδD+η)
In the limit η (cid:28) δ (cid:28) 1, the cross-ratios are both near 1:
4iδ 4iδ
z = 1+ , z¯= 1+ η . (2.12)
D D
An important subtlety is that the conformal blocks are multivalued, and must be
evaluated on the correct sheet. The prescription for the shockwave correlator, with
operators ordered as in (2.6), was worked out in [1]. The prescription to deal with
branch cuts amounts to sending z → ze−2πi (with z¯ held fixed) before evaluating the
conformal blocks (see figure 2).
2.1.1 Results for (cid:104)ψJJψ(cid:105)
For a spin-1 probe O = J in d = 4, the two-point function in the denominator is
µ
D2 (cid:20) 2η η2 (cid:21)
(cid:104)ε·J(x )ε∗·J(x )(cid:105) = −c |ε+|2 − |ε⊥|2 + |ε−|2 , (2.13)
2 3 J128η4 D2 D4
where ε⊥ is in the 3,4-directions. It is clear from this expression that different po-
larizations will contribute at different orders in the lightcone limit η → 0. For any
given polarization, the stress tensor corrections to the four-point function must be sup-
pressed compared to the identity, so it will have a similar polarization dependence in
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Figure 2: The analytic continuation appropriate to the ordering ψOOψ is shown here in
real spacetime (left) and on the z plane (right). As we move the O operator into the future
lightcone of the shockwave operator insertions, we cross a branch cut in the correlator. In
terms of the cross-ratios, crossing this branch cut (with the correct i(cid:15) prescription for this
ordering) leads to z circling the origin.
the η expansion.
Addingthestresstensorconformalblocktotheidentitycontributionandevaluating
this in the lightcone limit, for a generic polarization the leading term in (2.10) comes
from the J J polarization and is
+ +
5D∆ η
ψ
G(ε,η,D) = 1− · n . (2.14)
s
πc c δ
J T
Therefore n ≥ 0. If we instead choose a polarization with ε+ = 0, this eliminates the
s
leading terms in both the numerator and denominator of (2.10), giving
5D∆ η
ψ
G(ε,η,D) = 1− · (n +24n ) . (2.15)
s f
2πc c δ
J T
So n +24n ≥ 0. There is no new constraint from the sub-sub-leading terms.
s f
2.1.2 Results for (cid:104)ψTTψ(cid:105)
Forthestresstensorind = 4,choosewithoutlossofgeneralityasymmetricpolarization
tensor with ε = εµ = 0. For the configuration (2.7-2.8), the two-point function of
34 µ
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