Energy Positivity from Holographic Volume Susceptibility Zachary - PowerPoint PPT Presentation

Energy Positivity from Holographic Volume Susceptibility Zachary Fisher with R. Myers, A. Speranza, W. Wieland June 18, 2019

Energy conditions from geometry and causality From bulk causality to boundary energy conditions: • Average Null Energy Condition ◮ Holographically, from Gao-Wald • Strong Subadditivity of Entanglement Entropy ◮ From extremality of HRT ◮ Also implies boundary energy conditions • Quantum Null Energy Condition ◮ Holographically, from nesting of entanglement wedges near boundary Bulk geometry, causality → energy positivity relations.

Diagonal and o ff -diagonal variations Quantum Null Energy Condition 󰄂 S ′′ 〈 T kk ( p ) 〉 ≥ √ out 2 π h S out = S out [ X a ] is a functional of the coordinates X a of the entangling surface. What does S ′′ out mean?

Diagonal and o ff -diagonal variations Quantum Null Energy Condition 󰄂 S ′′ 〈 T kk ( p ) 〉 ≥ √ out 2 π h S out = S out [ X a ] is a functional of the coordinates X a of the entangling surface. What does S ′′ out mean? The second (variational) derivative of S out is a matrix; use a local basis for variations. • O ff -diagonal variations: sup ( δ 1 X a ) ∩ sup ( δ 2 X a ) = ∅ . • Diagonal variations: δ 1 X a , δ 2 X a ∝ δ ( X a − y a ) (at same pt). • S ′′ out means a diagonal variation (in a null direction).

O ff -diagonal variations of entanglement entropy O ff -diagonal variations ( entanglement density or entanglement susceptibility ) are also interesting: Entanglement Susceptibility 󰀐 δ 2 S 󰀐 S ′′ 󰀐 o ff -diagonal ( y 1 , y 2 ) := s a s b 󰀐 δ X a ( y 1 ) δ X b ( y 2 ) y 1 ∕ = y 2 Why? Every o ff -diagonal matrix element is non-positive, because of strong-subadditivity.

O ff -diagonal variations of entanglement entropy O ff -diagonal variations ( entanglement density or entanglement susceptibility ) are also interesting: Entanglement Susceptibility 󰀐 δ 2 S 󰀐 S ′′ 󰀐 o ff -diagonal ( y 1 , y 2 ) := s a s b 󰀐 δ X a ( y 1 ) δ X b ( y 2 ) y 1 ∕ = y 2 Why? Every o ff -diagonal matrix element is non-positive, because of strong-subadditivity. This can be enough to prove energy conditions in some perturbative cases (involving shockwaves) [Khandker, Kundu, Li].

The famous holographic argument for SSA [Headrick, Takayanagi]

The famous holographic argument for SSA [Headrick, Takayanagi] = +

The famous holographic argument for SSA [Headrick, Takayanagi] = + ≥ +

The famous holographic argument for SSA [Headrick, Takayanagi] = + ≥ + S ( AB ) + S ( BC ) ≥ S ( B ) + S ( ABC )

The famous holographic argument for SSA [Headrick, Takayanagi] = + ≥ + S ( AB ) + S ( BC ) ≥ S ( B ) + S ( ABC ) 0 ≥ S ( ABC ) − S ( AB ) − S ( BC ) + S ( B ) 󰀐 δ 2 S 󰀐 󰀐 ⇒ s a s b A , C tiny perturbations of B = ≤ 0 󰀐 δ X a ( y 1 ) δ X b ( y 2 ) y 1 ∕ = y 2

Maximal volume slices Maximal volume slices are also believed to play an important role in holography, e.g. CV conjecture: C ∼ V max G N ℓ Some similar properties to entanglement entropy: • Leading divergences are local, geometrical • Obeys nesting • Strong super additivity relation [Carmi, Myers, Rath; Carmi; Couch, Eccles, Jacobson, Nguyen]

Holographic Volume Superadditivity

Holographic Volume Superadditivity = +

Holographic Volume Superadditivity = + = +

Holographic Volume Superadditivity = + = + ≤ +

Holographic Volume Superadditivity = + = + ≤ + V 1 + V 2 ≤ V 1+2 + V 0 󰀐 δ 2 V max 󰀐 󰀐 ⇒ t a t b tiny perturbations = ≥ 0 󰀐 δ X a ( y 1 ) δ X b ( y 2 ) y 1 ∕ = y 2

O ff -diagonal variations of volume Perform a similar variation of boundary conditions for maximal volume surfaces. Holographic Volume Susceptibility The non-local contribution to the volume at one point, due to variations at another point, as we did for entropy. 󰀐 δ 2 V max 󰀐 󰀐 V ′′ o ff -diagonal ( y 1 , y 2 ) := t a t b 󰀐 δ X a ∂ ( y 1 ) δ X b ∂ ( y 2 ) y 1 ∕ = y 2 Now the maximal volume susceptibility is positive , by strong super additivity.

Example: Perturbations around vacuum Start in AdS vacuum, and inject some energy, perturbatively.

Example: Perturbations around vacuum Start in AdS vacuum, and inject some energy, perturbatively. In Fe ff erman-Graham coordinates, ds 2 = dz 2 − dt 2 + d 󰂔 y 2 + h αβ dx α dx β z 2 〈 T tt 〉 z d − 2 + . . . . where we will work to fi rst order in h αβ = 16 π G N d

Example: Perturbations around vacuum Start in AdS vacuum, and inject some energy, perturbatively. In Fe ff erman-Graham coordinates, ds 2 = dz 2 − dt 2 + d 󰂔 y 2 + h αβ dx α dx β z 2 〈 T tt 〉 z d − 2 + . . . . where we will work to fi rst order in h αβ = 16 π G N d • Let the maximal volume surface σ be given by t = T ( 󰂔 y , z ) . • Assume time re fl ection symmetry: T = 0 is our initial maximal volume surface.

Example: Perturbations around vacuum Start in AdS vacuum, and inject some energy, perturbatively. In Fe ff erman-Graham coordinates, ds 2 = dz 2 − dt 2 + d 󰂔 y 2 + h αβ dx α dx β z 2 〈 T tt 〉 z d − 2 + . . . . where we will work to fi rst order in h αβ = 16 π G N d • Let the maximal volume surface σ be given by t = T ( 󰂔 y , z ) . • Assume time re fl ection symmetry: T = 0 is our initial maximal volume surface. • Perturb the boundary conditions δ T ( 󰂔 y , z = 0) ≡ δ t ( 󰂔 y ) , and expand volume functional around 0 : 󰁞 󰁞 󰁞 √ δ V [ δ t ] = δ H = ( · · · ) δ T + ( · · · ) δ T δ T + . . . σ [ δ t ] σ [ δ t ] σ [ δ t ] • Impose e.o.m.: δ T drops out, δ T δ T term localizes to boundary (Hamilton-Jacobi).

Perturbations around vacuum Result: Susceptibility in perturbed space 󰀖 󰀗 δ 2 V δ t ( y 1 ) δ t ( y 2 ) = 1 1 − 8 π G N ρ ( y 1 ) z d g ( y 1 , z | y 2 ) 0 2 d 󰁞 + 1 d d − 1 y dz z 2 − d ( δ ab h tt + 2 δ ac δ be h ce ) × 4 × ∂ a G ( y , z | y 1 , z 0 ) ∂ b G ( y , z | y 2 , z 0 ) ( g = bulk-bd'y prop; G = bulk-bulk; ρ = 〈 T tt 〉 , z 0 = cuto ff .) First term: diagonal ( δ in the limit z 0 → 0 ). ⇒ ≥ 0 . Second term: o ff -diagonal, = Integrate second term over y 1 to obtain 󰁞 󰀖 󰀗 δ 2 V 8 π G N d d − 1 y 1 = z d 0 ≤ ρ ( y 2 ) + ( subleading ) 0 δ t ( y 1 ) δ t ( y 2 ) d o ff -diag

How did we get 〈 T tt 〉 ≥ 0 ? • Famously, in all QFTs, ∃ states that violate 〈 T tt 〉 ≥ 0 . ◮ Re fl ections o ff moving mirrors ◮ Hawking radiation ◮ Casimir e ff ect

How did we get 〈 T tt 〉 ≥ 0 ? • Famously, in all QFTs, ∃ states that violate 〈 T tt 〉 ≥ 0 . ◮ Re fl ections o ff moving mirrors ◮ Hawking radiation ◮ Casimir e ff ect • In many cases, violations involve dynamics, so nontrivial to check if WEC is violated in sense of this calculation.

How did we get 〈 T tt 〉 ≥ 0 ? • Famously, in all QFTs, ∃ states that violate 〈 T tt 〉 ≥ 0 . ◮ Re fl ections o ff moving mirrors ◮ Hawking radiation ◮ Casimir e ff ect • In many cases, violations involve dynamics, so nontrivial to check if WEC is violated in sense of this calculation. • Casimir e ff ect? ◮ Solution: conformal anomaly is subtracted from boundary stress tensor in the Fe ff erman-Graham expansion ◮ This bound concerns additional excitations above the anomalous (Casimir) energy.

Summary Holographic volume susceptibility • A new quantity to study in fi eld theory and holography. • Strong superadditivity = ⇒ Positivity of susceptibility • Leads to a geometrical proof of weak energy condition in symmetric cases.

Summary Holographic volume susceptibility • A new quantity to study in fi eld theory and holography. • Strong superadditivity = ⇒ Positivity of susceptibility • Leads to a geometrical proof of weak energy condition in symmetric cases. To do • Relation to [Lashkari, Lin, Ooguri, Stoica, Raamsdonk]? • Study diagonal terms, à la QNEC. (Complexity density?) De fi nite sign, related to nesting near boundary, like QNEC? • Relation to boundary symplectic form? • Braneworld: bounds on energy density in weak gravity?

Summary Holographic volume susceptibility • A new quantity to study in fi eld theory and holography. • Strong superadditivity = ⇒ Positivity of susceptibility • Leads to a geometrical proof of weak energy condition in symmetric cases. To do • Relation to [Lashkari, Lin, Ooguri, Stoica, Raamsdonk]? • Study diagonal terms, à la QNEC. (Complexity density?) De fi nite sign, related to nesting near boundary, like QNEC? • Relation to boundary symplectic form? • Braneworld: bounds on energy density in weak gravity? Thank you!