REVIEW:

Bounds on Energy, Entropy,and Transport

Tom Hartman
Cornell University
Strings 2018 ✦ Okinawa, Japan1

Various local energy conditions are often discussed in classicalgeneral relativity:
Strong
WeakDominant
Null
These play a key role in our basic
understanding of the universe:
Singularity theorems, no traversable
wormholes, no time machines, etc.

Is it true?

Violated byStrongnot even close

free scalar

Weaknonegative CC

Dominant

nonegative CC

not quite

quantum eﬀects

Null

They are all violated in quantum field theory.
[Epstein, Glaser, Jaﬀe '65]

QFTs do satisfy• Non-local energy conditions• Energy-entropy bounds
This talk: Progress on understanding these bounds in QFT(in flat spacetime, without gravity)

Motivation• Fundamental properties of unitary QFT• Impose constraints on couplings constants, anomaly coeﬃcients• Predictions for real-world systems, e.g. 3d Ising• Connect information-theoretic approaches to QFT with more
traditional observables, i.e. correlation functions• Tractable limit of AdS/CFT: calculate everything on both sides

Focus on: null energy

TuuOutlineI.

The ANECII. Holographic CFTs
III. Energy vs. entropy

The ANEC: Averaged null energy condition

ZduTuuNull coordinates

0u, v = t ± x= null ray
Some background:
80's and 90's
• ANEC is suﬃcient for many of the GR theorems
e.g. [Borde ’87]

• Derived in various free theories by explicit calculation of Tµ⌫[Klinkhammer ’91; Wald, Yurtsever’91; Folacci ’92; Ford, Roman ’95]

[Hofman and Maldacena '08]

2000's: Conformal Collider Bounds

Evaluate the ANEC in a state created by the stress tensor itself,hT (p)|ZduTuu |T (p)i

Constrains the OPE coeﬃcients

4d :0CT T T

1a31
 3c18

Holographic dual:

S=Zpg R22⇤ + ↵R + · · ·Einstein gravity has a=c; so this constrains higher curvature couplings
Same constraints follow from bulk causality[Brigante, Liu, Myers,Shenker, Yaida '07]
[Hofman '09]

But is the ANEC true in interacting QFTs?
Or an extra constraint we should impose?
Answer: The ANEC is a consequence of unitarity.

Recently, three derivations in interacting QFT in d > 2 :•Holographic•

Relative entropy•

Causality of correlation functions[postpone…]

ANEC from Causality
Consider a 4-point correlation function

G = h (x3 )O(x1 )O(x2 ) (x4 )i
Unitarity + Causality + Boostrap methods
Conformal collider bounds[TH, Jain, Kundu ’15, ’16]
[Komargodski, Kulaxizi, Parnachev, Zhiboedov ’16]
[Hofman, Li, Meltzer, Poland, Rejon-Barrera '16]

and the full ANEC
[TH, Kundu, Tajdini '16]

Ingredients:(1) OPE in the lightcone limit gives

OO ⇠ 1 +Z 1duTuu

1This relates the ANEC to properties of 4-point functions,G⇠1+h |

Z 1

1

duTuu | i(2) Causality:[O(x1 ), O(x2 )] = 0(spacelike)I
G=0tells us the domain where G is analytic.

[TH, Kundu, Tajdini '16]

This leads to a sum rule for the integrated null energy:

h |

Z 1

1

duTuu | i =Zh[ , O][ , O]i0

(3) The "double discontinuity" on the right is positive by unitarity.
This implies the ANEC.

compare: Lorentzian OPE inversion formula [Caron-Huot '17]

[Kelly, Wall '14]

ANEC from Holography

The particle stays at fixed z
close to the AdS boundary.

ticleTime delay:

parv⇠⇠bouyrandZ 1

1Z 101duhuu
duhTuu iThis proves the ANEC in holographic CFTs (large N, etc.)
But clearly the logic is very similar to the general CFT argument, which
did not require large N.
Both use causality in the lightcone limit

Applications and Extensions of the ANEC

Applications and Extensions of the ANEC
[TH, Kundu, Tajdini ’16]
see also:[Komargodski, Kulaxizi, Parnachev, Zhiboedov ’16]

1) The Higher Spin ANEC

Positive sum rule for integrated higher spin operators (J even)h |Z 1

1Xuuuu··· | i =Zdz z J 2 h[ , O][ , O]i0This is an experimental prediction for systems in the lab— e.g. 3dIsing or the O(2) model

Confirmed by numerical bootstrap in 3d Ising for J  40
[Simmons-Duﬃn '16]

Applications and Extensions of the ANEC
2) The Continuous Spin ANEC

[Kravchuk, Simmons-Duﬃn '18]

What if J is not an integer?

h |?| i=Z

dz|z|J 2 h[ , O][ , O]i

The right-hand side is still positive.
But does this correspond to some positive operator on the left?
Yes: a nonlocal "light ray operator" with non-integer spin

Z00dudu X(u, u )These operators naturally appear in the OPE data upon analytic
continuation in spin, and control high-energy (Regge) scattering.

Applications and Extensions of the ANEC
3) Interference in the Conformal Collider
Evaluate the ANEC in superpositions| i = ↵T (p)|0i + O(p)|0i✓◆cT T T cT T O
0cT T O cOT O
Constrains the oﬀ-diagonal (TTO) couplings by the diagonal (TTT, OTO)
couplings:

Xifi |cT T Oi |2  cT T T

Eigenvalue repulsion makes the ANEC stronger[Cordova, Maldacena, Turiaci '17]

Applications and Extensions of the ANEC4) Bounds on Transport
Apply ANEC to all states in the microcanoncal ensemble at energy E0

B
hi| duT |ji ⇠B@Zuuoﬀ-diagonal:hydrodynamic
fluctuations

diagonal:
equilibrium thermodynamics

1C 0CA[Delacrétaz, TH, Hartnoll, Lewkowycz ’18]

Applications and Extensions of the ANEC
Constraints on transport coeﬃcients vs. thermalization length⇤hydro . f✓

⌘ ⇣, , csound , · · ·s s◆[Delacrétaz, TH,Hartnoll, Lewkowycz’18]

Other (non-ANEC) bounds
Upper bounds on transport coeﬃcients also imposed by causality ofhydrodynamic 2-point functions.
These also have been derived rigorously
in non-relativistic QM.
[Han, Hartnoll ’18]

Quark-gluon plasma?

[Baier, Romatschke, Son,Starinets, Stephanov ’07]
[Romatschke ’09]
[TH, Hartnoll, Mahajan ’17]

Applications and Extensions of the ANEC
5) New Unitarity Bounds in 4d[Cordova, Diab '17]

For (k, k̄) Lorentz reps, the ANEC givesmax(k, k̄)(partly conjecture)~2x stronger for chiral reps!
6) Parity-violating OPE coeﬃcients

3d :hT T T i,[Chowdhury, David, Prakash '17]

hJJT i

Saturated by large-N Chern-Simons-Matter theories

OutlineI.

The ANECII. Holographic CFTs
III. Energy vs. entropy

The goal is to show that energy conditions become muchstronger in large-N, holographic theories, and that this can help
us understand the emergence of gravity in these theories.

[Maldacena, Shenker, Stanford '15]First some comments on the chaos bound:
The chaos bound constrains thermal 4-point functions (OTOCs),

G ⇠ h (0)O(t) (0)O(t)i1 t⇠1e + ···Nmaximal Lyapunov
sign constraint

2⇡This also constrains vacuum correlators, because they can be viewed
thermal correlators in Rindler

ReggeLightcone

Chaos

ANEC

Dominated by high-spin
exchanges

Dominated by low-twist
exchanges(Large N)

OPE in any CFT[Brower, Polchinski, Strassler, Tan ’06]
[Cornalba ’07]
[Cornalba, Costa, Penedones ’08]
[Costa, Goncalves, Penedones ’12]

Two limits in the bulk of AdS:
Lightcone limit

Regge limit

We will now venture into the Regge regime. This requires large-N.
The goal is to derive aspects of local physics in AdS, directly from CFT.

Can we derive gravity in AdS directly from CFT?
In 2d, often “yes” using Virasoro. But in higher dimensions?
(Set d=4).
At low energies, we expect roughly

Large #d.o.f.

+ Large gap —>emergent gravity

specific conjecture in:[Heemskerk, Polchinski, Penedones, Sully ]But what exactly do you need to show?

Any theory of quantum gravity looks like Einstein gravity at lowenergies,

1
S=GNZ ⇣c2 2
2⇤ + R + 2 R + · · ·M

With M = string mass and⌘e.g. 5d Gauss-Bonnet

c2 ⇠ O(1)[Camanho, Edelstein, Maldacena, Zhiboedov ’14]

To "derive AdS gravity from CFT" means to show that
all consistent CFTs (in some class) correspond to bulk
theories with higher derivative terms suppressed by
the string scale.
This suppression is the hallmark of a local bulk.

Ultimately non-perturbative; but for now, correlators in vacuum
A big piece of this was proved by bootstrap in:[Heemskerk, Polchinski, Penedones, Sully ’09]
[Penedones ’10]
[Fitpatrick, Kaplan, Penedones, Raju, van Rees ’11]
etc.

Lagrangians in AdS

But this leaves unanswered:
Why Einstein gravity in the bulk?

Solutions of CFT crossing equation
order by order in 1/N

The simplest version of this question is for 3-point functions:

Why does some class of CFTs have universal <TTT> correlators?
In a general CFT, there are 3 free coeﬃcients:

hT T T icf t = ahT T T i1 + chT T T i2 + t4 hT T T i3

But Einstein gravity is very special:hT T T i =a=candt4 = 0

Similar questions apply to matter couplings:
Why is hT T Oi universal?
Bulk:

S ⇠ ··· +Zp

g

2
Cµ⌫↵

So eﬀective field theory in the bulk predicts

hT T Oi ⇠ 0
Or more accurately,hT T Oi ⇠1#gap

These universal 3-point functions have now been derived from CFT.
The basic idea:
Unitarity + Causality + Bootstrap methods
ANEC in the lightcone limit[first part of talk]universality of spinning 3-point functions in the Regge limit(in large-N theories with a gap)
[Afkhami-Jeddi, TH, Kundu, Tajdini ’16, ’17]
[Li, Meltzer, Poland ’17]
[Kulaxizi, Parnachev, Zhiboedov ’17]
[Costa, Hansen, Penedones ’17]
[Meltzer, Perlmutter ’17]
[Afkhami-Jeddi, Kundu, Tajdini ’18]

These papers use several diﬀerent methods, looking at diﬀerent parts of
the amplitude.

Conclusion: The ANEC gets replaced by a stronger condition inholographic CFTs.
This stronger condition interpolates from the Hofman-Maldacenaconstraints

1a31
 3c18
to the Einstein gravity result,a1 1ci.e. a = c(Several constraints from diﬀerent polarizations.)
Universality of <TTT> and <TTO>

OutlineI.

The ANECII. Holographic CFTs
III. Energy vs. entropy

Bekenstein Bound[Bekenstein ’81]

S  2⇡RE
Reformulated as relative entropy:Srel (⇢|⇢vacuum ) =hHi

S0

[Marolf, Minic, Ross ’03]
[Marolf ’04]
[Casini ’08]

Measures distinguishability from vacuum.
Monotonicity
Relative entropy decreases under deformations that “shrink” the region

AdSrel  0d“‘harder to distinguish”

Consider a null deformationof Rindler half-space:

AFor this deformation, monotonicity gives the “half-ANEC”:

dSrel =dZ 1
01 0duTuu +SA2⇡

0

Adding the same formula for the complement region gives the regularANEC. (Entanglement terms cancel.)[Faulkner, Leigh, Parrikar, Wang ’16]
see also: [Wall ’10], [Blanco, Casini ’13]

Relative entropy is positive and monotonic in any quantum system.
A surprise: In QFT, there is a stronger condition,

d2Srel2d0dSrel ! 0 at infinity, this implies monotonicity
Since
dThis parallels the derivation of the second law of black holethermodynamics:focusing)00

Area  0)

Area

0

0

[Bousso, Fisher, Leichenauer, Wall ’15]
[Bousso, Fisher, Leichenauer, Koeller, Wall ’15]
[Balakrishnan, Faulkner, Khandker, Wang ’17]

The 2nd derivative has local and non-local terms along the horizon.
The non-local terms are positive by strong subadditivity.
The local term is a new local energy condition:
The QNEC: Quantum Null Energy Condition

hTuu i

1 00SA2⇡

Recall the theorem: “no local positive energy in QFT”
This evades the theorem because the r.h.s. is not an operator.

[Bousso, Fisher, Leichenauer, Wall ’15]

Status of the QNEC
• Motivated by coupling to gravity and asking for a quantum
analogue of the focusing equation; the QNEC survives as
Newton’s constant —> 0

[Bousso, Fisher, Leichenauer, Wall ’15]

• Derived in free QFT[Bousso, Fisher, Leichenauer, Koeller, Wall ’15]

• Derived in holographic theories from a local causality condition(cf. ANEC and a=c from boundary causality)[Koeller, Leichenauer ’15]

• Derived in interacting QFT[Balakrishnan, Faulkner, Khandker, Wang ’17]

• Found to be saturated in interacting theories!

1 00
hTuu i =SA
2⇡[Ecker, Grumiller, van der Schee,Stanzer ’17]
[Leichenauer, Levine, ShahbaziMoghaddam ’18]
[Khandker, Kundu, Li ’18]
[Balakrishnan et al, work in progress]

QNEC vs ANEC
Integrating gives the ANEC,

Z 1

duTuu =1=Z 1

d2 Sreld2d1

Z 1d (SSA + QN EC)

1

If the QNEC is saturated, the integrand comes entirely from the nonlocalterms. Therefore
Strong subadditivityof a “skinny” region

ANEC

And the proof of the QNEC by [Balakrishnan, Faulkner, Khandker, Wang]
becomes a field-theoretic proof of SSA.
[Leichenauer, Levine, Shahbazi-Moghaddam ’18]

Conclusions• ANEC<— Causality in lightcone limit• Universal 3-point functions @ large N• QNEC<— Causality in Regge limit<— Local causality in the bulk

Holography
Bootstrap
Quantum
information

These are fundamental properties of QFT.
But each was first discovered by coupling the QFT to gravity, or from
holography.