Quantum Null Energy Condition In two dimensions

Quantum Null Energy Condition

In two dimensions

Daniel Grumiller

Institute for Theoretical Physics TU Wien

ULB/Leuven/Mons/VUB Seminar, November 2018

Outline

Inequalities

QNEC

Holographic QNEC in 4d

Holographic QNEC in 2d

Inequalities in mathematics

I Inequalities are a core part of mathematics

Boring inequalities (type 1): true, but could be sharpened p² ≥ −p² ∀p∈R

Boring inequalities (type 2): true, but actually an equality p²≥p² ∀p∈R

Fine inequalities: true, cannot be sharpened, not always an equality p² ≥0 ∀p∈R

I Many inequalities stem from simple observation that squares of real numbers cannot be negative

p² ≥0 ∀p∈R I Many inequalities are of Cauchy–Schwarz type

|u||v| ≥ |u·v|

hereu, v are some vector,|| is their length and·the inner product I Many inequalities from convexity (Jensen’s inequality)

Inequalities in mathematics

I Inequalities are a core part of mathematics

I Many inequalities stem from simple observation that squares of real numbers cannot be negative

p² ≥0 ∀p∈R

I Many inequalities are of Cauchy–Schwarz type

|u||v| ≥ |u·v|

hereu, v are some vector,|| is their length and·the inner product I Many inequalities from convexity (Jensen’s inequality)

Inequalities in mathematics

I Inequalities are a core part of mathematics

I Many inequalities stem from simple observation that squares of real numbers cannot be negative

p² ≥0 ∀p∈R

Example: given two positive real numbersa, b algebraic mean≥geometric mean

Proof: takep=a−band get from inequality above (a−b)² =a²−2ab+b² ≥0 add on both sides4ab

a²+ 2ab+b²= (a+b)² ≥4ab take square root and then divide by 2

a+b 2 ≥√

ab

I Many inequalities are of Cauchy–Schwarz type

|u||v| ≥ |u·v|

hereu, v are some vector,|| is their length and·the inner product I Many inequalities from convexity (Jensen’s inequality)

Inequalities in mathematics

I Inequalities are a core part of mathematics

I Many inequalities stem from simple observation that squares of real numbers cannot be negative

p² ≥0 ∀p∈R

Example: given two positive real numbersa, b algebraic mean≥geometric mean Proof: takep=a−band get from inequality above

(a−b)² =a²−2ab+b² ≥0

add on both sides4ab

a²+ 2ab+b²= (a+b)² ≥4ab take square root and then divide by 2

a+b 2 ≥√

ab

I Many inequalities are of Cauchy–Schwarz type

|u||v| ≥ |u·v|

hereu, v are some vector,|| is their length and·the inner product I Many inequalities from convexity (Jensen’s inequality)

Inequalities in mathematics

I Inequalities are a core part of mathematics

I Many inequalities stem from simple observation that squares of real numbers cannot be negative

p² ≥0 ∀p∈R

Example: given two positive real numbersa, b algebraic mean≥geometric mean Proof: takep=a−band get from inequality above

(a−b)² =a²−2ab+b² ≥0 add on both sides4ab

a²+ 2ab+b²= (a+b)² ≥4ab

take square root and then divide by 2 a+b

2 ≥√ ab

I Many inequalities are of Cauchy–Schwarz type

|u||v| ≥ |u·v|

hereu, v are some vector,|| is their length and·the inner product I Many inequalities from convexity (Jensen’s inequality)

Inequalities in mathematics

I Inequalities are a core part of mathematics

I Many inequalities stem from simple observation that squares of real numbers cannot be negative

p² ≥0 ∀p∈R

Example: given two positive real numbersa, b algebraic mean≥geometric mean Proof: takep=a−band get from inequality above

(a−b)² =a²−2ab+b² ≥0 add on both sides4ab

a²+ 2ab+b²= (a+b)² ≥4ab take square root and then divide by 2

a+b 2 ≥√

ab

I Many inequalities are of Cauchy–Schwarz type

|u||v| ≥ |u·v|

hereu, v are some vector,|| is their length and·the inner product I Many inequalities from convexity (Jensen’s inequality)

Inequalities in mathematics

I Inequalities are a core part of mathematics

I Many inequalities stem from simple observation that squares of real numbers cannot be negative

p² ≥0 ∀p∈R I Many inequalities are of Cauchy–Schwarz type

|u||v| ≥ |u·v|

hereu, v are some vector,|| is their length and·the inner product

I Many inequalities from convexity (Jensen’s inequality)

Inequalities in mathematics

I Inequalities are a core part of mathematics

I Many inequalities stem from simple observation that squares of real numbers cannot be negative

p² ≥0 ∀p∈R I Many inequalities are of Cauchy–Schwarz type

|u||v| ≥ |u·v|

hereu, v are some vector,|| is their length and·the inner product

.

numerous consequences e.g. triangle inequality

|u|+|v| ≥ |u+v| graphic proof evident

I Many inequalities from convexity (Jensen’s inequality)

Inequalities in mathematics

I Inequalities are a core part of mathematics

I Many inequalities stem from simple observation that squares of real numbers cannot be negative

p² ≥0 ∀p∈R I Many inequalities are of Cauchy–Schwarz type

|u||v| ≥ |u·v|

hereu, v are some vector,|| is their length and·the inner product I Many inequalities from convexity (Jensen’s inequality)

Inequalities in mathematics

I Inequalities are a core part of mathematics

I Many inequalities stem from simple observation that squares of real numbers cannot be negative

p² ≥0 ∀p∈R I Many inequalities are of Cauchy–Schwarz type

|u||v| ≥ |u·v|

hereu, v are some vector,|| is their length and·the inner product I Many inequalities from convexity (Jensen’s inequality)

.

special case of Jensen’s inequality:

secant always above convex curve between intersection points x1,x2

Inequalities in physics

Interesting physical consequences from mathematical inequalities I Positivity inequalities: probabilities non-negative, P ≥0

I Cauchy–Schwarz inequalities: Heisenberg uncertainty, ∆x∆p≥ ¹₂ I Convexity inequalities: second law of thermodynamics, δS≥0 I In gravitational context: energy inequalities

I Definition: (local) inequalities on the stress tensorTµν

e.g. Null Energy Condition (NEC)

Tkk=Tµνk^µk^ν ≥0 ∀k^µkµ= 0 I Physically plausible (positivity of energy fluxes)

I Mathematically useful (singularity theorem, area theorem [2^nd law])

However: all classical energy inequalities violated by quantum effects! Are there quantum energy conditions?

[How is 2^ndlaw saved?]

Inequalities in physics

Interesting physical consequences from mathematical inequalities I Positivity inequalities: probabilities non-negative, P ≥0

Example: unitarity constraints on physical parameters in quark mixing matrix if Standard Model correct then measurements must reproduce unitarity triangle

I Cauchy–Schwarz inequalities: Heisenberg uncertainty, ∆x∆p≥ ¹₂ I Convexity inequalities: second law of thermodynamics, δS≥0 I In gravitational context: energy inequalities

I Definition: (local) inequalities on the stress tensorTµν

e.g. Null Energy Condition (NEC)

Tkk=Tµνk^µk^ν ≥0 ∀k^µkµ= 0 I Physically plausible (positivity of energy fluxes)

I Mathematically useful (singularity theorem, area theorem [2^nd law])

However: all classical energy inequalities violated by quantum effects! Are there quantum energy conditions?

[How is 2^ndlaw saved?]

Inequalities in physics

Interesting physical consequences from mathematical inequalities I Positivity inequalities: probabilities non-negative, P ≥0

I Cauchy–Schwarz inequalities: Heisenberg uncertainty, ∆x∆p≥ ¹₂

I Convexity inequalities: second law of thermodynamics, δS≥0 I In gravitational context: energy inequalities

I Definition: (local) inequalities on the stress tensorTµν

e.g. Null Energy Condition (NEC)

Tkk=Tµνk^µk^ν ≥0 ∀k^µkµ= 0 I Physically plausible (positivity of energy fluxes)

I Mathematically useful (singularity theorem, area theorem [2^nd law])

However: all classical energy inequalities violated by quantum effects! Are there quantum energy conditions?

[How is 2^ndlaw saved?]

Inequalities in physics

Interesting physical consequences from mathematical inequalities I Positivity inequalities: probabilities non-negative, P ≥0

I Cauchy–Schwarz inequalities: Heisenberg uncertainty, ∆x∆p≥ ¹₂

-3 -2 -1 1 2 3

0.2 0.4 0.6 0.8 Gaussian

-3 -2 -1 1 2 3

0.2 0.4 0.6 0.8 Fourier transformed Gaussian

green: localized in coordinate space (x), delocalized in momentum space (p) blue: mildly (de-)localized in coordinate and momentum space

orange: delocalized in coordinate space (x), localized in momentum space (p)

I Convexity inequalities: second law of thermodynamics, δS≥0 I In gravitational context: energy inequalities

I Definition: (local) inequalities on the stress tensorTµν

e.g. Null Energy Condition (NEC)

Tkk=Tµνk^µk^ν ≥0 ∀k^µkµ= 0 I Physically plausible (positivity of energy fluxes)

I Mathematically useful (singularity theorem, area theorem [2^nd law])

However: all classical energy inequalities violated by quantum effects! Are there quantum energy conditions?

[How is 2^ndlaw saved?]

Inequalities in physics

Interesting physical consequences from mathematical inequalities I Positivity inequalities: probabilities non-negative, P ≥0

I Cauchy–Schwarz inequalities: Heisenberg uncertainty, ∆x∆p≥ ¹₂ I Convexity inequalities: second law of thermodynamics, δS≥0

I In gravitational context: energy inequalities

I Definition: (local) inequalities on the stress tensorTµν

e.g. Null Energy Condition (NEC)

Tkk=Tµνk^µk^ν ≥0 ∀k^µkµ= 0 I Physically plausible (positivity of energy fluxes)

I Mathematically useful (singularity theorem, area theorem [2^nd law])

However: all classical energy inequalities violated by quantum effects! Are there quantum energy conditions?

[How is 2^ndlaw saved?]

Inequalities in physics

Interesting physical consequences from mathematical inequalities I Positivity inequalities: probabilities non-negative, P ≥0

I Cauchy–Schwarz inequalities: Heisenberg uncertainty, ∆x∆p≥ ¹₂ I Convexity inequalities: second law of thermodynamics, δS≥0 I In gravitational context: energy inequalities

I Definition: (local) inequalities on the stress tensorTµν

e.g. Null Energy Condition (NEC)

Tkk=Tµνk^µk^ν ≥0 ∀k^µkµ= 0 I Physically plausible (positivity of energy fluxes)

I Mathematically useful (singularity theorem, area theorem [2^nd law])

However: all classical energy inequalities violated by quantum effects! Are there quantum energy conditions?

[How is 2^ndlaw saved?]

Inequalities in physics

Interesting physical consequences from mathematical inequalities I Positivity inequalities: probabilities non-negative, P ≥0

I Cauchy–Schwarz inequalities: Heisenberg uncertainty, ∆x∆p≥ ¹₂ I Convexity inequalities: second law of thermodynamics, δS≥0 I In gravitational context: energy inequalities

I Definition: (local) inequalities on the stress tensorTµν

e.g. Null Energy Condition (NEC)

Tkk=Tµνk^µk^ν ≥0 ∀k^µkµ= 0

I Physically plausible (positivity of energy fluxes)

I Mathematically useful (singularity theorem, area theorem [2^nd law])

However: all classical energy inequalities violated by quantum effects! Are there quantum energy conditions?

[How is 2^ndlaw saved?]

Inequalities in physics

Interesting physical consequences from mathematical inequalities I Positivity inequalities: probabilities non-negative, P ≥0

I Cauchy–Schwarz inequalities: Heisenberg uncertainty, ∆x∆p≥ ¹₂ I Convexity inequalities: second law of thermodynamics, δS≥0 I In gravitational context: energy inequalities

I Definition: (local) inequalities on the stress tensorTµν

e.g. Null Energy Condition (NEC)

Tkk=Tµνk^µk^ν ≥0 ∀k^µkµ= 0

For instance: Penrose singularity theorem from Raychaudhuri eq.

d²area

dk² =−d area dk

2

−shear²−8πG Tkk≤ −8πG Tkk NEC≤ 0

IfTkk≥0(NEC)⇒focussing!

(negative acceleration of area)

For experts:

d area

dk =θis null expansion

I Physically plausible (positivity of energy fluxes)

I Mathematically useful (singularity theorem, area theorem [2^nd law])

However: all classical energy inequalities violated by quantum effects! Are there quantum energy conditions?

[How is 2^ndlaw saved?]

Inequalities in physics

Interesting physical consequences from mathematical inequalities I Positivity inequalities: probabilities non-negative, P ≥0

I Cauchy–Schwarz inequalities: Heisenberg uncertainty, ∆x∆p≥ ¹₂ I Convexity inequalities: second law of thermodynamics, δS≥0 I In gravitational context: energy inequalities

I Definition: (local) inequalities on the stress tensorTµν

e.g. Null Energy Condition (NEC)

Tkk=Tµνk^µk^ν ≥0 ∀k^µkµ= 0 I Physically plausible (positivity of energy fluxes)

I Mathematically useful (singularity theorem, area theorem [2^nd law])

However: all classical energy inequalities violated by quantum effects! Are there quantum energy conditions?

[How is 2^ndlaw saved?]

Inequalities in physics

Interesting physical consequences from mathematical inequalities I Positivity inequalities: probabilities non-negative, P ≥0

I Cauchy–Schwarz inequalities: Heisenberg uncertainty, ∆x∆p≥ ¹₂ I Convexity inequalities: second law of thermodynamics, δS≥0 I In gravitational context: energy inequalities

I Definition: (local) inequalities on the stress tensorTµν

e.g. Null Energy Condition (NEC)

Tkk=Tµνk^µk^ν ≥0 ∀k^µkµ= 0 I Physically plausible (positivity of energy fluxes)

I Mathematically useful (singularity theorem, area theorem [2^nd law])

However: all classical energy inequalities violated by quantum effects! Are there quantum energy conditions?

[How is 2^ndlaw saved?]

Inequalities in physics

Interesting physical consequences from mathematical inequalities I Positivity inequalities: probabilities non-negative, P ≥0

I Cauchy–Schwarz inequalities: Heisenberg uncertainty, ∆x∆p≥ ¹₂ I Convexity inequalities: second law of thermodynamics, δS≥0 I In gravitational context: energy inequalities

I Definition: (local) inequalities on the stress tensorTµν

e.g. Null Energy Condition (NEC)

Tkk=Tµνk^µk^ν ≥0 ∀k^µkµ= 0 I Physically plausible (positivity of energy fluxes)

I Mathematically useful (singularity theorem, area theorem [2^nd law])

However: all classical energy inequalities violated by quantum effects!

NEC violated by Casimir energy, accelerated mirrors, Hawking radiation, ...

Are there quantum energy conditions?

[How is 2^ndlaw saved?]

Inequalities in physics

Interesting physical consequences from mathematical inequalities I Positivity inequalities: probabilities non-negative, P ≥0

I Cauchy–Schwarz inequalities: Heisenberg uncertainty, ∆x∆p≥ ¹₂ I Convexity inequalities: second law of thermodynamics, δS≥0 I In gravitational context: energy inequalities

I Definition: (local) inequalities on the stress tensorTµν

e.g. Null Energy Condition (NEC)

Tkk=Tµνk^µk^ν ≥0 ∀k^µkµ= 0 I Physically plausible (positivity of energy fluxes)

I Mathematically useful (singularity theorem, area theorem [2^nd law])

However: all classical energy inequalities violated by quantum effects!

Are there quantum energy conditions?

[How is 2^ndlaw saved?]

Quantum energy conditions

I Definition: quantum energy condition = convexity condition forhTµνi valid for any state and any (reasonable) quantum field theory (QFT)

I Example: Averaged Null Energy Condition (ANEC) Z

dx^λk_λhT_µνk^µk^νi ≥0

valid∀k^µ(withk^µk_µ= 0) and∀states |i in any (reasonable) QFT I ANEC proved under rather generic assumptions

I ANEC sufficient for focussing properties used in singularity theorems I ANEC compatible with quantum interest conjecture

I However: ANEC is non-local (R dx⁺)

Is there a local quantum energy condition?

Quantum energy conditions

I Definition: quantum energy condition = convexity condition forhTµνi valid for any state and any (reasonable) quantum field theory (QFT) I Example: Averaged Null Energy Condition (ANEC)

Z

dx^λk_λhT_µνk^µk^νi ≥0

valid∀k^µ(withk^µk_µ= 0) and∀states |i in any (reasonable) QFT

I ANEC proved under rather generic assumptions

I ANEC sufficient for focussing properties used in singularity theorems I ANEC compatible with quantum interest conjecture

I However: ANEC is non-local (R dx⁺)

Is there a local quantum energy condition?

Quantum energy conditions

I Definition: quantum energy condition = convexity condition forhT_µνi valid for any state and any (reasonable) quantum field theory (QFT) I Example: Averaged Null Energy Condition (ANEC)

Z

dx^λkλhTµνk^µk^νi ≥0

valid∀k^µ(withk^µkµ= 0) and∀states |i in any (reasonable) QFT I ANEC proved under rather generic assumptions

Faulkner, Leigh, Parrikar and Wang 1605.08072 Hartman, Kundu and Tajdini 1610.05308

I ANEC sufficient for focussing properties used in singularity theorems I ANEC compatible with quantum interest conjecture

I However: ANEC is non-local (R dx⁺)

Is there a local quantum energy condition?

Quantum energy conditions

I Definition: quantum energy condition = convexity condition forhTµνi valid for any state and any (reasonable) quantum field theory (QFT) I Example: Averaged Null Energy Condition (ANEC)

Z

dx^λk_λhT_µνk^µk^νi ≥0

valid∀k^µ(withk^µk_µ= 0) and∀states |i in any (reasonable) QFT I ANEC proved under rather generic assumptions

I ANEC sufficient for focussing properties used in singularity theorems

I ANEC compatible with quantum interest conjecture I However: ANEC is non-local (R

dx⁺)

Is there a local quantum energy condition?

Quantum energy conditions

I Definition: quantum energy condition = convexity condition forhTµνi valid for any state and any (reasonable) quantum field theory (QFT) I Example: Averaged Null Energy Condition (ANEC)

Z

dx^λk_λhT_µνk^µk^νi ≥0

valid∀k^µ(withk^µk_µ= 0) and∀states |i in any (reasonable) QFT I ANEC proved under rather generic assumptions

I ANEC sufficient for focussing properties used in singularity theorems I ANEC compatible with quantum interest conjecture

I However: ANEC is non-local (R dx⁺)

Is there a local quantum energy condition?

Quantum energy conditions

I Definition: quantum energy condition = convexity condition forhTµνi valid for any state and any (reasonable) quantum field theory (QFT) I Example: Averaged Null Energy Condition (ANEC)

Z

dx^λk_λhT_µνk^µk^νi ≥0

valid∀k^µ(withk^µk_µ= 0) and∀states |i in any (reasonable) QFT I ANEC proved under rather generic assumptions

I ANEC sufficient for focussing properties used in singularity theorems I ANEC compatible with quantum interest conjecture

I However: ANEC is non-local (R dx⁺)

Is there a local quantum energy condition?

Quantum energy conditions

I Definition: quantum energy condition = convexity condition forhTµνi valid for any state and any (reasonable) quantum field theory (QFT) I Example: Averaged Null Energy Condition (ANEC)

Z

dx^λk_λhT_µνk^µk^νi ≥0

valid∀k^µ(withk^µk_µ= 0) and∀states |i in any (reasonable) QFT I ANEC proved under rather generic assumptions

I ANEC sufficient for focussing properties used in singularity theorems I ANEC compatible with quantum interest conjecture

I However: ANEC is non-local (R dx⁺)

Is there a local quantum energy condition?

Outline

Inequalities

QNEC

Holographic QNEC in 4d

Holographic QNEC in 2d

Quantum null energy condition (QNEC)

Proposed byBousso, Fisher, Leichenauer and Wallin1506.02669

QNEC (inD >2) is the following inequality hT_kki ≥ ~

2π√γ S⁰⁰

Physical motivation from focussing properties and second law:

Replace area by area + 4G(entanglement entropy) Modified Raychaudhuri eq., schematically:

d²area

dk² + 4G S⁰⁰=−8πG T_kk+ 4G S^00QNEC≤ 0 requires for focussing property (= 2^nd law) QNEC

fineprint: above we set expansion to zero, ^{d area}= 0, and shear to zero; we also set the area to unity,√ γ= 1

I T_kk=T_µνk^µk^ν with k_µk^µ= 0and hidenotes expectation value I S⁰⁰: 2^nd variation of EE for entangling surface deformations alongkµ

I √γ: induced volume form of entangling region (black boundary curve)

Quantum null energy condition (QNEC)

Proposed byBousso, Fisher, Leichenauer and Wallin1506.02669

QNEC (inD >2) is the following inequality hT_kki ≥ ~

2π√γ S⁰⁰

Obvious observations:

I if r.h.s. vanishes: semi-classical version of NEC I if r.h.s. negative: weaker condition than NEC

(NEC can be violated while QNEC holds) I if r.h.s. positive: stronger condition than NEC

(if QNEC holds also NEC holds)

I T_kk=T_µνk^µk^ν with k_µk^µ= 0and hidenotes expectation value I S⁰⁰: 2^nd variation of EE for entangling surface deformations alongkµ

I √γ: induced volume form of entangling region (black boundary curve)

Quantum null energy condition (QNEC)

Proposed byBousso, Fisher, Leichenauer and Wallin1506.02669

QNEC (inD >2) is the following inequality hT_kki ≥ ~

2π√γ S⁰⁰

I Tkk=Tµνk^µk^ν with kµk^µ= 0and hidenotes expectation value

I S⁰⁰: 2^nd variation of EE for entangling surface deformations alongkµ

I √γ: induced volume form of entangling region (black boundary curve)

Quantum null energy condition (QNEC)

Proposed byBousso, Fisher, Leichenauer and Wallin1506.02669

QNEC (inD >2) is the following inequality hT_kki ≥ ~

2π√γ S⁰⁰

I Tkk=Tµνk^µk^ν with kµk^µ= 0and hidenotes expectation value I S⁰⁰: 2^nd variation of EE for entangling surface deformations alongkµ

I √γ: induced volume form of entangling region (black boundary curve)

Quantum null energy condition (QNEC)

Proposed byBousso, Fisher, Leichenauer and Wallin1506.02669

QNEC (inD >2) is the following inequality hT_kki ≥ ~

2π√γ S⁰⁰

I Tkk=Tµνk^µk^ν with kµk^µ= 0and hidenotes expectation value

Proofs (D >2)

I For free QFTs: Bousso, Fisher, Koeller, Leichenauer and Wall,1509.02542

I For holographic CFTs: Koeller and Leichenauer,1512.06109

I For general CFTs: Balakrishnan, Faulkner, Khandker and Wang,1706.09432

QNEC (in CFT2) is the following inequality hT_kki ≥S⁰⁰+ 6

cS⁰² c >0is the central charge of the CFT2

I S like anomalous operator with conformal weights(0,0)

⇒ construct vertex operator V = exp [⁶_cS]

I QNEC saturation equivalent to vertex operator solving Hill’s equation V⁰⁰− LV = 0 L ∼ hT_kki

I QNEC saturated for vacuum, thermal states and their descendants I QNEC not saturated in hol. CFT₂ with positive bulk energy fluxes I QNEC can be violated in hol. CFT2 with negative bulk energy fluxes

Proofs and non-saturation (D= 2)

Ongoing work withEcker, Sheikh-Jabbari, Stanzer and van der Schee

QNEC (in CFT₂) is the following inequality hTkki ≥S⁰⁰+ 6

cS⁰² c >0is the central charge of the CFT₂

I S like anomalous operator with conformal weights(0,0)

⇒ construct vertex operator V = exp [⁶_cS]

I QNEC saturation equivalent to vertex operator solving Hill’s equation V⁰⁰− LV = 0 L ∼ hTkki

I QNEC saturated for vacuum, thermal states and their descendants I QNEC not saturated in hol. CFT₂ with positive bulk energy fluxes I QNEC can be violated in hol. CFT2 with negative bulk energy fluxes

Proofs and non-saturation (D= 2)

Ongoing work withEcker, Sheikh-Jabbari, Stanzer and van der Schee

QNEC (in CFT₂) is the following inequality hTkki ≥S⁰⁰+ 6

cS⁰² c >0is the central charge of the CFT₂ I S like anomalous operator with conformal weights(0,0)

⇒ construct vertex operator V = exp [⁶_cS]

I QNEC saturation equivalent to vertex operator solving Hill’s equation V⁰⁰− LV = 0 L ∼ hTkki

I QNEC saturated for vacuum, thermal states and their descendants I QNEC not saturated in hol. CFT₂ with positive bulk energy fluxes I QNEC can be violated in hol. CFT2 with negative bulk energy fluxes

Proofs and non-saturation (D= 2)

Ongoing work withEcker, Sheikh-Jabbari, Stanzer and van der Schee

QNEC (in CFT₂) is the following inequality hTkki ≥S⁰⁰+ 6

cS⁰² c >0is the central charge of the CFT₂ I S like anomalous operator with conformal weights(0,0)

⇒ construct vertex operator V = exp [⁶_cS]

I QNEC saturation equivalent to vertex operator solving Hill’s equation V⁰⁰− LV = 0 L ∼ hTkki

I QNEC saturated for vacuum, thermal states and their descendants I QNEC not saturated in hol. CFT₂ with positive bulk energy fluxes I QNEC can be violated in hol. CFT2 with negative bulk energy fluxes

Proofs and non-saturation (D= 2)

Ongoing work withEcker, Sheikh-Jabbari, Stanzer and van der Schee

QNEC (in CFT₂) is the following inequality hTkki ≥S⁰⁰+ 6

cS⁰² c >0is the central charge of the CFT₂ I S like anomalous operator with conformal weights(0,0)

⇒ construct vertex operator V = exp [⁶_cS]

I QNEC saturation equivalent to vertex operator solving Hill’s equation V⁰⁰− LV = 0 L ∼ hTkki

I QNEC saturated for vacuum, thermal states and their descendants

I QNEC not saturated in hol. CFT₂ with positive bulk energy fluxes I QNEC can be violated in hol. CFT2 with negative bulk energy fluxes

Proofs and non-saturation (D= 2)

Ongoing work withEcker, Sheikh-Jabbari, Stanzer and van der Schee

QNEC (in CFT₂) is the following inequality hTkki ≥S⁰⁰+ 6

cS⁰² c >0is the central charge of the CFT₂ I S like anomalous operator with conformal weights(0,0)

⇒ construct vertex operator V = exp [⁶_cS]

I QNEC saturation equivalent to vertex operator solving Hill’s equation V⁰⁰− LV = 0 L ∼ hTkki

I QNEC saturated for vacuum, thermal states and their descendants

I QNEC can be violated in hol. CFT2 with negative bulk energy fluxes

Proofs and non-saturation (D= 2)

Ongoing work withEcker, Sheikh-Jabbari, Stanzer and van der Schee

QNEC (in CFT₂) is the following inequality hTkki ≥S⁰⁰+ 6

cS⁰² c >0is the central charge of the CFT₂ I S like anomalous operator with conformal weights(0,0)

⇒ construct vertex operator V = exp [⁶_cS]

I QNEC saturation equivalent to vertex operator solving Hill’s equation V⁰⁰− LV = 0 L ∼ hTkki

I QNEC saturated for vacuum, thermal states and their descendants I QNEC not saturated in hol. CFT₂ with positive bulk energy fluxes I QNEC can be violated in hol. CFT2 with negative bulk energy fluxes

Calculating QNEC holographically

calculating CFT observable holographically = some gravity calculation AdS/CFT:

Maldacena hep-th/9711200

Gubser, Klebanov and Polyakov hep-th/9802109 Witten hep-th/9802150

holographic stress tensor:

Henningson and Skenderis hep-th/9806087 Balasubramanian and Kraus hep-th/9902121 Emparan, Johnson and Myers hep-th/9903238 de Haro, Solodukhin and Skenderis hep-th/0002230 holographic entanglement entropy (HEE):

Ryu and Takayanagi hep-th/0603001

Hubeny, Rangamani and Takayanagi0705.0016

I need holographic computation of hT_kki

I need holographic computation of (deformations of) EE

Calculating QNEC holographically

calculating CFT observable holographically = some gravity calculation I need holographic computation of hTkki

well-known AdS/CFT prescription: extract boundary stress tensor from bulk metric expanded near AdS boundary

Example: AdS₃/CFT₂ ds²= `²

z² dz²+2 dx⁺dx⁻

+hT₊₊i dx⁺2

+hT−−i dx⁻2

+O z²

AdS3 boundary: z→0

O(1)terms in metric: flux components of stress tensorhT±±i (trace vanishes, hT+−i= 0)

`: so-called AdS-radius (cosmological constantΛ =−1/`²)

I need holographic computation of (deformations of) EE

Calculating QNEC holographically

calculating CFT observable holographically = some gravity calculation I need holographic computation of hTkki

I need holographic computation of (deformations of) EE HEE = area of extremal surface

simple to calculate!

also: simple proof of strong subadditivity inequalities

Outline

Inequalities

QNEC

Holographic QNEC in 4d

Holographic QNEC in 2d

Thermal case

see work withEcker, Stanzer and van der Schee1710.09837

thermal states in CFT4 = black holes in AdS5

I paper-and-pencil calculation starts with Schwarzschild black brane ds²= 1

z² −f(z) dt²+ dz²

f(z)+ dy²+ dx²₁+ dx²₂ with f(z) = 1−π⁴T⁴z⁴

I determine area of minimal surfaces for small temperature,T `1, and extract HEE (`= width of strip)

1

2πS⁰⁰≈ −0.065

`<sup>4</sup> + 0.019π<sup>4</sup>T<sup>4</sup>−0.083`⁴π⁸T⁸+O `<sup>8</sup>T<sup>12</sup> I do same for large temperatures, T `1

1

2πS⁰⁰≈ −0.364π⁴T⁴e⁻

√

6`πT +O e⁻²

√ 6`πT

I use numerics for intermediate values of temperature

Thermal case

see work withEcker, Stanzer and van der Schee1710.09837

thermal states in CFT4 = black holes in AdS5

I paper-and-pencil calculation starts with Schwarzschild black brane ds²= 1

z² −f(z) dt²+ dz²

f(z)+ dy²+ dx²₁+ dx²₂ with f(z) = 1−π⁴T⁴z⁴

I determine area of minimal surfaces for small temperature,T `1, and extract HEE (`= width of strip)

1

2πS⁰⁰≈ −0.065

`<sup>4</sup> + 0.019π<sup>4</sup>T<sup>4</sup>−0.083`⁴π⁸T⁸+O `⁸T¹²

I do same for large temperatures, T `1 1

2πS⁰⁰≈ −0.364π⁴T⁴e⁻

√

6`πT +O e⁻²

√ 6`πT

I use numerics for intermediate values of temperature

Thermal case

see work withEcker, Stanzer and van der Schee1710.09837

thermal states in CFT4 = black holes in AdS5

I paper-and-pencil calculation starts with Schwarzschild black brane ds²= 1

z² −f(z) dt²+ dz²

f(z)+ dy²+ dx²₁+ dx²₂ with f(z) = 1−π⁴T⁴z⁴

I determine area of minimal surfaces for small temperature,T `1, and extract HEE (`= width of strip)

1

2πS⁰⁰≈ −0.065

`<sup>4</sup> + 0.019π<sup>4</sup>T<sup>4</sup>−0.083`⁴π⁸T⁸+O `<sup>8</sup>T<sup>12</sup> I do same for large temperatures, T `1

1 S⁰⁰≈ −0.364π⁴T⁴e⁻

√

6`πT +O e⁻²

√ 6`πT

I use numerics for intermediate values of temperature

Thermal case

see work withEcker, Stanzer and van der Schee1710.09837

thermal states in CFT4 = black holes in AdS5

I paper-and-pencil calculation starts with Schwarzschild black brane ds²= 1

z² −f(z) dt²+ dz²

f(z)+ dy²+ dx²₁+ dx²₂ with f(z) = 1−π⁴T⁴z⁴

I determine area of minimal surfaces for small temperature,T `1, and extract HEE (`= width of strip)

1

2πS⁰⁰≈ −0.065

`<sup>4</sup> + 0.019π<sup>4</sup>T<sup>4</sup>−0.083`⁴π⁸T⁸+O `<sup>8</sup>T<sup>12</sup> I do same for large temperatures, T `1

1

2πS⁰⁰≈ −0.364π⁴T⁴e⁻

√

6`πT +O e⁻²

√ 6`πT

I use numerics for intermediate values of temperature

Thermal case

see work withEcker, Stanzer and van der Schee1710.09837

thermal states in CFT4 = black holes in AdS5

0 1 2 3 4

-10 -0.1 - 0.001 -10

1 2π



'' / π

T

0 1 2

0 0.01 0.02

Thermal-vac.

0.0191-0.083L⁴ 0.376ⅇ^{- 6 L}

Vacuum Thermal

Colliding gravitational shockwaves and QNEC saturation

see work withEcker, Stanzer and van der Schee1710.09837

plasma formation in CFT4 = colliding gravitational shock waves in AdS5

toy model for quark-gluon plasma formation in heavy ion collisions

Colliding gravitational shockwaves and QNEC saturation

see work withEcker, Stanzer and van der Schee1710.09837

plasma formation in CFT4 = colliding gravitational shock waves in AdS5

toy model for quark-gluon plasma formation in heavy ion collisions I paper-and-pencil calculations with Romatschke0803.3226

I δ-like shocks

I particle production in forward lightcone of shocks

I shortly after collision anisotropic pressure: PL/E=−3,PT/E= +2 confirmed numerically for thin shocks byCasalderrey-Solana, Heller, Mateos and van der Schee1305.4919

I close to shockwaves negative energy fluxes⇒NEC violation!

confirmed numerically and interpreted as absence of local rest frame by Arnold, Romatschke and van der Schee1408.2518

I consider finite width gravitational shockwaves

(pioneered numerically byChesler and Yaffe 1011.3562) I extract metric, holographic stress tensor and HEE numerically

I check QNEC and its saturation, particularly in region of NEC violation

Colliding gravitational shockwaves and QNEC saturation

see work withEcker, Stanzer and van der Schee1710.09837

plasma formation in CFT4 = colliding gravitational shock waves in AdS5

toy model for quark-gluon plasma formation in heavy ion collisions I paper-and-pencil calculations with Romatschke0803.3226

I δ-like shocks

I particle production in forward lightcone of shocks

I shortly after collision anisotropic pressure: PL/E=−3,PT/E= +2 confirmed numerically for thin shocks byCasalderrey-Solana, Heller, Mateos and van der Schee1305.4919

I close to shockwaves negative energy fluxes⇒NEC violation!

confirmed numerically and interpreted as absence of local rest frame by Arnold, Romatschke and van der Schee1408.2518

I consider finite width gravitational shockwaves

(pioneered numerically byChesler and Yaffe 1011.3562)

I extract metric, holographic stress tensor and HEE numerically

I check QNEC and its saturation, particularly in region of NEC violation

Colliding gravitational shockwaves and QNEC saturation

see work withEcker, Stanzer and van der Schee1710.09837

plasma formation in CFT4 = colliding gravitational shock waves in AdS5

toy model for quark-gluon plasma formation in heavy ion collisions I paper-and-pencil calculations with Romatschke0803.3226

I δ-like shocks

I particle production in forward lightcone of shocks

I shortly after collision anisotropic pressure: PL/E=−3,PT/E= +2 confirmed numerically for thin shocks byCasalderrey-Solana, Heller, Mateos and van der Schee1305.4919

I close to shockwaves negative energy fluxes⇒NEC violation!

confirmed numerically and interpreted as absence of local rest frame by Arnold, Romatschke and van der Schee1408.2518

I consider finite width gravitational shockwaves

(pioneered numerically byChesler and Yaffe 1011.3562) I extract metric, holographic stress tensor and HEE numerically

I check QNEC and its saturation, particularly in region of NEC violation

Colliding gravitational shockwaves and QNEC saturation

see work withEcker, Stanzer and van der Schee1710.09837

plasma formation in CFT4 = colliding gravitational shock waves in AdS5

toy model for quark-gluon plasma formation in heavy ion collisions I paper-and-pencil calculations with Romatschke0803.3226

I δ-like shocks

I particle production in forward lightcone of shocks

I shortly after collision anisotropic pressure: PL/E=−3,PT/E= +2 confirmed numerically for thin shocks byCasalderrey-Solana, Heller, Mateos and van der Schee1305.4919

I close to shockwaves negative energy fluxes⇒NEC violation!

confirmed numerically and interpreted as absence of local rest frame by Arnold, Romatschke and van der Schee1408.2518

I consider finite width gravitational shockwaves

(pioneered numerically byChesler and Yaffe 1011.3562) I extract metric, holographic stress tensor and HEE numerically

I check QNEC and its saturation, particularly in region of NEC violation

Colliding gravitational shockwaves and QNEC saturation

see work withEcker, Stanzer and van der Schee1710.09837

plasma formation in CFT4 = colliding gravitational shock waves in AdS5

toy model for quark-gluon plasma formation in heavy ion collisions

Colliding gravitational shockwaves and QNEC saturation

see work withEcker, Stanzer and van der Schee1710.09837

plasma formation in CFT4 = colliding gravitational shock waves in AdS5

toy model for quark-gluon plasma formation in heavy ion collisions

+''/2π

-''/2π

++

--

-1.0 -0.5 0.0 0.5 1.0 1.5

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8

QNEC for L → ∞ (μ y = -0.5)

Outline

Inequalities

QNEC

Holographic QNEC in 4d

Holographic QNEC in 2d

Non-equilibrium and quantum equilibrium

A state is in quantum equilibrium when QNEC saturates for all times and all entangling regions

Definition:

Consequences:

I Far-from-(thermal)-equilibrium state can be in quantum equilibrium I All states dual to Ba˜nados geometries are in quantum equilibrium I Natural to introduce “quantum equilibration time”:

For a given separation of the entangling interval quantum equilibration time = smallest time after which normalized QNEC non-saturation lower than prescribed bound (e.g. 1%)

Quantum equilibrium hopefully a useful notion

Non-equilibrium and quantum equilibrium

A state is in quantum equilibrium when QNEC saturates for all times and all entangling regions

Definition:

Consequences:

I Far-from-(thermal)-equilibrium state can be in quantum equilibrium

Figure 2 from1311.3655(Nature Phys.)

T Steady'state'

AdS'boosted'black'hole'

JE6= 0

Bhaseen, Doyon, Lucas, Schalm I Far from equilibrium transport in

strongly coupled CFT I Long-time energy transport

universally via steady-state I In AdS3/CFT2: specific Ba˜nados

geometry with step function

I All states dual to Ba˜nados geometries are in quantum equilibrium I Natural to introduce “quantum equilibration time”:

For a given separation of the entangling interval quantum equilibration time = smallest time after which normalized QNEC non-saturation lower than prescribed bound (e.g. 1%)

Quantum equilibrium hopefully a useful notion

Non-equilibrium and quantum equilibrium

A state is in quantum equilibrium when QNEC saturates for all times and all entangling regions

Definition:

Consequences:

I Far-from-(thermal)-equilibrium state can be in quantum equilibrium

TL=0.04 T_L=0.1 T_L=0.4

TL=1.0

0 1 2 3 4

0.0 0.5 1.0 1.5 2.0

t SEE-SEE,div

TR=0.2,(x1,x₂) = (1, 3) (solid),(x1,x₂) = (-3,-1) (dashed)

TL=0.04 TL=0.1 TL=0.4 TL=1.0 T₊₊2π

0.0 0.5 1.0 1.5 2.0

0.05 0.10 0.50 1 5 10 50

λ

S''+6/cS'2

T_R=0.2,(t₁,x₁) = (0.4, 0.2),(t₂,x₂) = (0.8+λ,-2+λ)

Left: HEE Right: QNEC saturation

I All states dual to Ba˜nados geometries are in quantum equilibrium I Natural to introduce “quantum equilibration time”:

For a given separation of the entangling interval quantum equilibration time = smallest time after which normalized QNEC non-saturation lower than prescribed bound (e.g. 1%)

Quantum equilibrium hopefully a useful notion

Non-equilibrium and quantum equilibrium

A state is in quantum equilibrium when QNEC saturates for all times and all entangling regions

Definition:

Consequences:

I Far-from-(thermal)-equilibrium state can be in quantum equilibrium I All states dual to Ba˜nados geometries are in quantum equilibrium

I Natural to introduce “quantum equilibration time”: For a given separation of the entangling interval quantum equilibration time = smallest time after which normalized QNEC non-saturation lower than prescribed bound (e.g. 1%)

Quantum equilibrium hopefully a useful notion

Non-equilibrium and quantum equilibrium

A state is in quantum equilibrium when QNEC saturates for all times and all entangling regions

Definition:

Consequences:

I Far-from-(thermal)-equilibrium state can be in quantum equilibrium I All states dual to Ba˜nados geometries are in quantum equilibrium I Natural to introduce “quantum equilibration time”:

For a given separation of the entangling interval quantum equilibration time = smallest time after which normalized QNEC non-saturation lower than prescribed bound (e.g. 1%)

Quantum equilibrium hopefully a useful notion

Non-equilibrium and quantum equilibrium

A state is in quantum equilibrium when QNEC saturates for all times and all entangling regions

Definition:

Consequences:

I Far-from-(thermal)-equilibrium state can be in quantum equilibrium I All states dual to Ba˜nados geometries are in quantum equilibrium I Natural to introduce “quantum equilibration time”:

For a given separation of the entangling interval quantum equilibration time = smallest time after which normalized QNEC non-saturation lower than prescribed bound (e.g. 1%)

Half-saturation of QNEC in Vaidya

I Vaidya = simple model for bulk matter; mass function M(t) ds² = 1

z² (−(1−M(t)z²) dt²−2 dtdz+ dx²)

I Numerical studies show curious “half-saturation” for large entangling regionsl

liml1

S⁰⁰+⁶_c(S⁰)² hT_kki ≈ 1

2

I Can be derived perturbatively forM(t) =θ(t) with 1 I If size of entangling region much larger than time,lt0 we find

QNEC half-saturation

ltlim₀

S⁰⁰+ ⁶_c(S⁰)² hT_kki = 1

2± t0

l +O(t²₀/l²) +O()

I If time is much larger than entangling region we find QNEC saturation

tlim0l

S⁰⁰+⁶_c(S⁰)²

hT_kki = 1 +O()

Half-saturation of QNEC in Vaidya

I Vaidya = simple model for bulk matter; mass function M(t) ds² = 1

z² (−(1−M(t)z²) dt²−2 dtdz+ dx²)

I Numerical studies show curious “half-saturation” for large entangling regionsl

liml1

S⁰⁰+⁶_c(S⁰)² hT_kki ≈ 1

2

I Can be derived perturbatively forM(t) =θ(t) with 1 I If size of entangling region much larger than time,lt0 we find

QNEC half-saturation

ltlim₀

S⁰⁰+ ⁶_c(S⁰)² hT_kki = 1

2± t0

l +O(t²₀/l²) +O()

I If time is much larger than entangling region we find QNEC saturation

tlim0l

S⁰⁰+⁶_c(S⁰)²

hT_kki = 1 +O()

Half-saturation of QNEC in Vaidya

I Vaidya = simple model for bulk matter; mass function M(t) ds² = 1

z² (−(1−M(t)z²) dt²−2 dtdz+ dx²)

I Numerical studies show curious “half-saturation” for large entangling regionsl

liml1

S⁰⁰+⁶_c(S⁰)² hT_kki ≈ 1

2

I Can be derived perturbatively forM(t) =θ(t) with 1

I If size of entangling region much larger than time,lt0 we find QNEC half-saturation

ltlim₀

S⁰⁰+ ⁶_c(S⁰)² hT_kki = 1

2± t0

l +O(t²₀/l²) +O()

I If time is much larger than entangling region we find QNEC saturation

tlim0l

S⁰⁰+⁶_c(S⁰)²

hT_kki = 1 +O()

Half-saturation of QNEC in Vaidya

I Vaidya = simple model for bulk matter; mass function M(t) ds² = 1

z² (−(1−M(t)z²) dt²−2 dtdz+ dx²)

I Numerical studies show curious “half-saturation” for large entangling regionsl

liml1

S⁰⁰+⁶_c(S⁰)² hT_kki ≈ 1

2

I Can be derived perturbatively forM(t) =θ(t) with 1 I If size of entangling region much larger than time,lt0 we find

QNEC half-saturation

ltlim₀

S⁰⁰+⁶_c(S⁰)² hT_kki = 1

2± t0

l +O(t²₀/l²) +O()

I If time is much larger than entangling region we find QNEC saturation

tlim0l

S⁰⁰+⁶_c(S⁰)²

hT_kki = 1 +O()