Energy Condition, Modular Flow, and AdS/CFT Black Holes and - - PowerPoint PPT Presentation
Energy Condition, Modular Flow, and AdS/CFT Black Holes and Holography Workshop,TSIMF, Jan 7-11, 2019 Huajia Wang Kavli Institute for Theoretical Physics u v v ( z ) u X + B ( z ) v O z = 0 z arXiv:1806.10560; arXiv:1706.09432; JHEP
arXiv:1806.10560; arXiv:1706.09432; JHEP 1609 038(2016)
X+
B(z)
∆v(z)
z = 0
u
v
O
z
Black Holes and Holography Workshop,TSIMF, Jan 7-11, 2019
Rn dxn E(x) ≥ 0
Hawking, Ellis, 1973
✓ Tab − 1 2Tgab ◆ ζaζb ≥ 0
Tab kakb ≥ 0
x
t
k
x
t
ζ
t
affine parameter
λSA(λ)(ψ)
A(0)
λ
A(λ)
1995; A. Folacci, 1992
causality constraint in the bulk. entanglement wedge nesting (EWN)
monotonicity of relative entropy causality of correlation functions in light-cone limit causality of correlation function under modular flow
−∞
−∞
−∞
−∞
−∞
−∞
leading order constraint in F. G. gauge expansion
uSEE
uSEE
uSEE
uSEE
uSEE
uSEE
uSEE
u
uSEE
u
A
spacelike/null
uSEE
spacelike/null
near boundary expansion: (F-G gauge)
Xi
Σ ˜
A(z) = Xi
∂ ˜ A +
4G dRd−1 zd∂iSEE( ˜ A) + O(zd+1)
Xi
ΣA(z) = Xi ∂A +
4G dRd−1 zd∂iSEE(A) + O(zd+1)
u
A
uSEE
spacelike/null
u
A
uSEE 0
−∞
−∞
−∞
A = − ln ρΨ A ⊗ 1Ac + 1A ⊗ ln ρΨ Ac = HΨ A − HΨ Ac
A : Hfull → Hfull
A|Ψi = 0
−∞
A = − ln ρΨ A ⊗ 1Ac + 1A ⊗ ln ρΨ Ac = HΨ A − HΨ Ac
A : Hfull → Hfull
A|Ψi = 0
A = 2π
x+
x−
~ y
−∞
˜ A iψ hKvac A iψ
x+
x−
~ y
−∞
A(ψ|φ) ≤ SA(ψ|φ)
˜ A iψ hKvac A iψ
˜ A iψ hKvac A iψ
x+
x−
~ y
−∞
˜ A iψ hKvac A iψ
A
˜ A iψ hKvac A iψ
uSEE
uSEE
uSEE
uSEE
uSEE
AOe−is Kψ A
uSEE
˜ A 1 (s)OA 2 (s)|ψi
˜ A 1 (s) = eis Kψ
˜ AO1e−is Kψ ˜ A
2 (s) = eis Kψ
AO2e−is Kψ A
uSEE
: von Neumann algebra associated with A, i.e. operators supported in D(A)
˜ A 1 (s)OA 2 (s)|ψi
˜ A 1 (s) = eis Kψ
˜ AO1e−is Kψ ˜ A
2 (s) = eis Kψ
AO2e−is Kψ A
uSEE
˜ A 1 (s)
˜ A 1 (s)
˜ A 1 (s)OA 2 (s)|ψi
˜ A 1 (s) = eis Kψ
˜ AO1e−is Kψ ˜ A
2 (s) = eis Kψ
AO2e−is Kψ A
2 (s)
2 (s)
uSEE
˜ A 1 (s)
˜ A 1 (s) ,
2 (s)
˜ A 1 (s)OA 2 (s)|ψi
˜ A 1 (s) = eis Kψ
˜ AO1e−is Kψ ˜ A
2 (s) = eis Kψ
AO2e−is Kψ A
uSEE
˜ A A Ac ˜ Ac
uSEE
˜ A A Ac ˜ Ac
uSEE
˜ A A Ac ˜ Ac
T esu(−uv)
d−2 2 IQ + ...
−
uSEE
˜ A A Ac ˜ Ac
Im s=±π/2
uSEE
˜ A A Ac ˜ Ac
Im s=±π/2
uSEE
uSEE
e.g. T. Faulkner, A.Lewkowycz, 2017
e.g. T. Faulkner, A.Lewkowycz, 2017
e.g. T. Faulkner, A.Lewkowycz, 2017
˜ A A Ac ˜ Ac
δu
˜ A 1 (s)OA 2 (s)|ψi
˜ A A Ac ˜ Ac
δu
˜ A 1 (s)OA 2 (s)|ψi
˜ AO1e−isKψ ˜ A eisKψ AO2e−isKψ A|ψi
˜ A A Ac ˜ Ac
δu
Kψ
A = Hψ A ⊗ 1Ac − 1A ⊗ Hψ Ac
recall
Hψ
A,Ac = half-sided modular Hamiltonian
˜ AO1e−isKψ ˜ A eisKψ AO2e−isKψ A|ψi
˜ A−isHψ ˜ Ac O1e−isHψ ˜ A+isHψ ˜ Ac eisHψ A−isHψ Ac O2e−isHψ A+isHψ Ac |ψi
˜ A 1 (s)OA 2 (s)|ψi
˜ A A Ac ˜ Ac
δu
Kψ
A = Hψ A ⊗ 1Ac − 1A ⊗ Hψ Ac
recall
Hψ
A,Ac = half-sided modular Hamiltonian
˜ AO1e−isKψ ˜ A eisKψ AO2e−isKψ A|ψi
˜ A−isHψ ˜ Ac O1e−isHψ ˜ A+isHψ ˜ Ac eisHψ A−isHψ Ac O2e−isHψ A+isHψ Ac |ψi
Ac+isHψ ˜ A O1 O2 e−isHψ ˜ A+isHψ Ac |ψi
˜ A 1 (s)OA 2 (s)|ψi
[Hψ
Ac, ˜ Ac, O1] = 0, [Hψ A, ˜ A, O2] = 0
using
˜ A A Ac ˜ Ac
δu
Kψ
A = Hψ A ⊗ 1Ac − 1A ⊗ Hψ Ac
recall
Hψ
A,Ac = half-sided modular Hamiltonian
˜ AO1e−isKψ ˜ A eisKψ AO2e−isKψ A|ψi
˜ A−isHψ ˜ Ac O1e−isHψ ˜ A+isHψ ˜ Ac eisHψ A−isHψ Ac O2e−isHψ A+isHψ Ac |ψi
Ac+isHψ ˜ A O1 O2 e−isHψ ˜ A+isHψ Ac |ψi
˜ A 1 (s)OA 2 (s)|ψi
[Hψ
Ac, ˜ Ac, O1] = 0, [Hψ A, ˜ A, O2] = 0
using
AeisHψ ˜ AO1 O2e−isHψ ˜ AeisHψ A|ψi
Kψ
A|ψi = 0 ! Hψ A|ψi = Hψ Ac|ψi
recall etc
˜ A A Ac ˜ Ac
δu
Kψ
A = Hψ A ⊗ 1Ac − 1A ⊗ Hψ Ac
recall
Hψ
A,Ac = half-sided modular Hamiltonian
˜ A−isHψ ˜ Ac O1e−isHψ ˜ A+isHψ ˜ Ac eisHψ A−isHψ Ac O2e−isHψ A+isHψ Ac |ψi
Ac+isHψ ˜ A O1 O2 e−isHψ ˜ A+isHψ Ac |ψi
˜ A 1 (s)OA 2 (s)|ψi
[Hψ
Ac, ˜ Ac, O1] = 0, [Hψ A, ˜ A, O2] = 0
using
AeisHψ ˜ AO1 O2e−isHψ ˜ AeisHψ A|ψi
Kψ
A|ψi = 0 ! Hψ A|ψi = Hψ Ac|ψi
recall etc
A(s)|O1 O2|ψA, ˜ A(s)i
A(s)i = e−isHψ
˜ AeisHψ A|ψi
where
˜ AO1e−isKψ ˜ A eisKψ AO2e−isKψ A|ψi
˜ A A Ac ˜ Ac
δu
A(s)i = e−isHψ
˜ AeisHψ A|ψi
where
A(s)|O1O2|ψA, ˜ A(s)i
˜ A A Ac ˜ Ac
δu
A(s)i = e−isHψ
˜ AeisHψ A|ψi
where
A(s)|O1O2|ψA, ˜ A(s)i
˜ A A Ac ˜ Ac
δu
A(s)i = e−isHψ
˜ AeisHψ A|ψi
where
A(s)|O1O2|ψA, ˜ A(s)i
˜ A A Ac ˜ Ac
δu
A(s)i = e−isHψ
˜ AeisHψ A|ψi
where
A(s)|O1O2|ψA, ˜ A(s)i
A, ˜ A(x1, x2)
A, ˜ A(x1, x2)
t x
t
x
A|ψi
t x
t
x
A|ψi
AOAeisHψ A|ψi = hψ|e−isHψ Ac OAeisHψ Ac |ψi
Ac eisHψ Ac OA|ψi = hψ|OA|ψi
t x
t
x
A|ψi
00ψA(s) ≡ ψ”
00ψA(s) ≡ ψ”
each segment is a geodesic in the
each segment is a geodesic in the
modular flow affects the matching condition at .
each segment is a geodesic in the
modular flow affects the matching condition at . JLMS (2015): .
Hψ
A(bdry) =
ˆ A 4G + Hψ
a (bulk)
each segment is a geodesic in the
modular flow affects the matching condition at . JLMS (2015): . is a constant in EW, .
Hψ
A(bdry) =
ˆ A 4G + Hψ
a (bulk)
A(bdry) ∝ eisHψ a (bulk)
each segment is a geodesic in the
modular flow affects the matching condition at . JLMS (2015): . is a constant in EW, . bulk theory free (leading ordering 1/N): close to , acts like bulk Rindler Hamiltonian and generates boosts.
Hψ
A(bdry) =
ˆ A 4G + Hψ
a (bulk)
A(bdry) ∝ eisHψ a (bulk)
Hψ
a (bulk)
s(x1, x2) = 1 4π ln ✓pu [L(ξ, x1)] pv [L(ξ, x1)] ◆ ✓pv [L(ξ, x2)] pu [L(ξ, x2)] ◆
2 (s∗)iψ = hO1O2iψA(s∗)
A(s)i = e−isHψ
˜ AeisHψ A|ψi
ξA ∈ ΣA
ξ ˜
A ∈ Σ ˜ A
A(s)i
A ∈ Σ ˜ A
1 (s∗)O ˜ A 2 (s∗)iψ = hO1O2iψA, ˜
A(s∗)
A, ξA) + L(ξ ˜ A, x2))]
A(s)i = e−isHψ
˜ AeisHψ A|ψi
ξA ∈ ΣA
ξ ˜
A ∈ Σ ˜ A
A(s)i
A ∈ Σ ˜ A
1 O1iψ ⇡ exp [2mL(ξA, x1)]
1 (iπ)O1iψ
1 O1iψ ⇡ exp [2mL(ξA, x1)]
1 ”
1 (iπ)O1iψ
˜ A 1 (s + iπ)OA 1 (s)iψ
1 O1iψ ⇡ exp [2mL(ξA, x1)] for all s
A ˜ A
ΣA
Σ ˜
A
ξA
ξ ˜
A
A 1 (s + iπ)OA 1 (s)iψ
1 (iπ)O1iψ
A, ξA) + O(δA2)
˜ A 1 (s + iπ)OA 1 (s)iψ
1 O1iψ ⇡ exp [2mL(ξA, x1)] for all s
A 1 (s + iπ)OA 1 (s)iψ
1 (iπ)O1iψ
A ˜ A
ΣA
Σ ˜
A
ξA
ξ ˜
A
˜ A 1 (s + iπ)OA 1 (s)iψ
1 O1iψ ⇡ exp [2mL(ξA, x1)] for all s
A ˜ A
ΣA
Σ ˜
A
ξA
ξ ˜
A
˜ AteiKψ At