Complete Einstein Equa?on from Generalized 1 st law of Entanglement Sang-Jin Sin (Hanyang) 2017.11@KEK arXiv:1709.05752 Es.Oh, Iy. Park + SJS 1 2017.11. KEK
disClaim We will not derive holography. We assume holography: CFT dual exist and the dual of vacuum is AdS. Question : when a physical configuration satisfy certain entanglement entropy relation, what is the equation that is satisfied by its gravity dual? 2 2017.11. KEK
Quantum Entanglement = connectivity by quantum superposition Non-local connec?on If No superposi?on, à No wave func?on collapse à No affec?on by measurement à No entanglement 3 2017.11. KEK
Classical connec?vity from quantum superposi?on Raamsdonk : S pacetime is also consequence of quantum entanglement. [arXiv:1005.3035] |Ψ ⟩ = |Ψ1 ⟩ � |Ψ2 ⟩ ßà disconnected pair of space?mes. |ψ(β) ⟩ = Σ i exp{−βE i /2 }|E i ⟩ � |E i ⟩ ßà connected Eternal BH = E −β Σ e i E E i i Space-Time connec?vity by entangling the d.o.f living in the two components. e − β E i | E i ⟩⟨ E i | = ρ T . � Tr 2 ( | ψ ⟩⟨ ψ | ) = 4 i 2017.11. KEK
Ryu-Takayanagi proposal Quantification of entanglement by Ent. Entropy ρ A = Tr B ( | Ψ ⟩⟨ Ψ | ), S ( A ) = − Tr ( ρ A log ρ A ) S(A)= Minimal surface/4G N 5 2017.11. KEK
Holography on quantum Entanglement (Raamdonk) Decreasing Ent. à Decreasing Area. è Pinch off if Ent à 0 A B A B ~ A A B 6 2017.11. KEK
First Law of Entanglement H = H A � H B , H B is the Hilbert space of local fields over ball B. S (B) = − Tr B ( ρ B log ρ B ) , delete B if no confusion. Take a reference state ρ 0 =vacuum Modular Hamiltonian : H 0 = − L og ρ 0 ⟨ H 0 ⟩ = − Tr ρ l og ρ 0 call it `energy’ ∆ E � ∆ S = S ( ρ | ρ 0 ) , (1) where ∆ E = � Tr( ρ � ρ 0 ) ln ρ 0 , (2) ∆ S = � Tr ρ ln ρ + Tr ρ 0 ln ρ 0 , (3) S ( ρ | ρ 0 ) = Tr ρ ln( ρ / ρ 0 ) . (4) 7 2017.11. KEK
First Law of Entanglement Positivity of relative entropy à S (ρ|ρ 0 ) is minimal at ρ . ∆ E − ∆ S = S ( ρ | ρ 0 ) , ρ 0 E xtremality condi?on : δE − δS = 0 à 1 st law 8 2017.11. KEK
Entanglement First Law in gravity version δE − δS = 0 Express δE , δS in terms of gravity (geometry). δE : Use Casini et.al, δS : Ryu-Takayanagi. d d xR 2 − r 2 d � δ E hyp = δ H 00 (0 , x ) A 16 G N R A d d x ( δ ij − 1 R � δ S = R 2 x i x j ) H ij 8 G N ds 2 = 1 d z 2 ( dz 2 + dx µ dx µ + z d H µ ν ( x, z ) dx µ dx ν ) t µ ν ( x ) = H µ ν ( z = 0 , x ) 16 π G N 9 2017.11. KEK
Ent. First law & Einstein Eq. Lashkari+Macdermott+Raamsdonk [ arXiv : 1308 . 3716 ] δE − δS = 0 is ßà Linearized Einstein equa?on. 3 � ˆ ∂ x 0 ∂ y 0 δ E = A 2 ∂ D R � = − 3 ( xH xx dx + yH yy dy ) ∂ D R = ∂ x 0 ∂ y 0 δ S � dx + � − 2 yH yy + ( R 2 − x 2 − y 2 ) ∂ y H xx � dy . ˆ A = � − 2 xH xx + ( R 2 − x 2 − y 2 ) ∂ x H yy 1 α = 0 ∂ µ H µ ν = 0 � z 4 ∂ z H µ ν � + ∂ 2 H µ ν = 0 H α z 4 ∂ z W µ ν = R µ ν − 1 2 g µ ν R − 3 g µ ν = 0 , which is equiv. to linearization of 10 2017.11. KEK
Reformulation T. Faulkner, M. Guica, T. Hartman, R. C. Myers and M. Van Raamsdonk, “Gravita?on from Entangle- ment in Holographic CFTs,” JHEP 1403, 051 (2014) [arXiv :1312.7856 ]. There exists a d − 1 form χ t Z Z � = � S grav � = � E grav , and X B B z ˜ B B ξ A Z Z � E grav � � S grav = � = d � , B B B B − ˜ B Σ v A Σ t d � = � 2 ⇠ a B � E ab ✏ b , ˜ B Entanglement first law ßà δE ab = 0. 11 2017.11. KEK
Brief review Z S = dtL [ q, ˙ q ] t1 t2 Symmetry ⇒ δ S = 0 Z dt δ q · ˆ p f − p i = E B B~ where, p = ∂ L E = d ∂ L q − ∂ L ˆ q , ∂ ˙ ∂ ˙ ∂ q dt • This is off-shell Noether theorem • Eq. of mo?on ßà conserva?on law 12 2017.11. KEK
Full version of the story? • δE − δS = 0 ßà Linearized Einstein eq. δE ab = 0. • Full Einstein eq? it has been Ques?oned during 2013-2017. • Most recencent work: proof for second order in δg. By Faulkner et.al 1705.03026 T. Ugajin’s talk Yesterday 1705.01486 • Claim : Full eq. can be inferred from ∆ E � ∆ S = S ( ρ | ρ 0 ) , • And above eq. is the finite difference version of first law 13 2017.11. KEK
Noether identity of Holland-Wald and Full Einstein Eq. Z Z ˆ � E grav � � S grav ! ( g 0 ; � g, � ξ B g ) = � E B B Σ Σ bc [ g 0 ] � g bc + � C a where ˆ ✏ a E g E = � ⇠ a � � . B On-shell à E^ =0. Then Z � E grav � � S grav = ! ( g 0 ; � g, � ξ B g ) B B Σ Conversely, for metric satisfying above eq à E^=0 or bc [ g 0 ] � g bc + � C a = 0 ✏ a E g for all order in " . = AdS metric. g C [ g 0 + � g ] � C [ g 0 ] = 0 . Since C a = 2 E g ab ✏ b . E [ g ( " )] = E [ g 0 ] = 0 . ε g 0 for all order in " . 14 2017.11. KEK = AdS metric. g
Deriva?on of Holland-Wald L ⌘ L [ � ] ✏ � L [ � ] = E φ �� + d Θ [ �� ] , ariation is a di ff eomorphism generat d by a ector field ⇠ , ⇠ , � ξ L = d ( ⇠ · L ) since � ξ = i ξ d + di ξ he top form. d L is ion 1 form J [ ⇠ ] = Θ [ � ξ � ] � ⇠ · L , d J [ ⇠ ] = � E [ � ] � ξ � , � at J is the closed form for the fields at on-shell. J [ ⇠ ] = d Q [ ⇠ ] + ⇠ a C a , � J [ ⇠ ] = ! ( �� , � ξ � ) + d ( ⇠ · Θ ) � ⇠ · E φ �� ! ( � ; � 1 � , � 2 � ) = � 1 Θ ( � 2 � ) � � 2 Θ ( � 1 � ) d � = ! ( �� , � ξ � ) � ⇠ · ( � C + E φ �� ) , 1 C a = 2 E g r a ⇠ b ✏ ab , ab ✏ b . h � = � Q [ ⇠ ] � ⇠ · Θ . Q = 16 ⇡ G N 1 ( R ab � 1 he explicit form of No E g 2 g ab R ) � T m ab = ab . 8 ⇡ G N 15 2017.11. KEK
Generalized first law i) gravity version Z When and Why � E grav � � S grav = ! ( g 0 ; � g, � ξ B g ) B B Σ Claim: This is the gravity version of the identity ∆ E � ∆ S = S ( ρ | ρ 0 ) , interpreted as the finite difference version of the first law . R Z We need and ! ( g 0 ; � g, � ξ B g ) = S ( ⇢ | ⇢ 0 ) , Σ S ( ⇢ | ⇢ 0 ) = W ξ [ M ] � W ξ [ AdS ] := W, Z � W ξ = ! ( g 0 ; � g, � ξ B g ) , 15–17]. If W ξ is defined by Σ N. Lashkari, J. Lin, H. Ooguri, B. Stoica and M. Van Raamsdonk [arXiv:1605.01075] . Smarr Rela?on: E grav − S grav = W ξ 16 2017.11. KEK
Generalized 1 st Law of Entanglement: ii) CFT version Suppose the density opereator depends on parameters R 1 , R 2 , · · · , R M , which we simbolically denote by a vec- tor R . Then ρ 0 = ρ ( R 0 ) and ρ = ρ ( R 1 ) for some R 0 , R 1 . ∂ − 1 0 0 1 Z ∂ g F α = r α ln ρ = ρ − 1 ∂ R α ρ , S ( ρ | ρ 0 ) = h d R · F i , space, we can express the rel C Work done on the system ∆ E � ∆ S = W. all ‘generalized entanglement first law’. gravity version, as we will see later. O 17 2017.11. KEK
Remarks Z 1. I assumed ! ( g 0 ; � g, � ξ B g ) = S ( ⇢ | ⇢ 0 ) , Σ following [arXiv:1605.01075; 1508.00897] However, it is worth to prove it by using bulk-boundary correspondence and actually this is done to second order in metric perturba?on by Sarosi+Ugajin:1611.02959, 1705.01486 and also by Faulkner Haehl, Hijano,Parrikar,Rabideau, Raamsdonk 1705.03026 2. I used Wald-Holland gauge: form invariance of the B~ and killing field. 18 2017.11. KEK
Entanglement Vector Field Goal: construct a vector field V E such that Z Z B 0 V a V a E d Σ a = E d Σ a = S B . ˜ B 16 ⇡ G N Q = r a ⇠ b ✏ ab = � 2 r a ⇠ t p� g tt ✏ a := V a ✏ a . ( R 2 � z 2 � ~ x 2 h i V = 4 ⇡ h + z ) dz + x i dx i i . 2 z Rz B V a ✏ a = 4 ⇡ Area[ ˜ R It is easy to check that B ]. R ˜ it is tempting to call V as entanglement ve r a V a = 2 ⇡ d Rz ( z 2 + ~ x 2 � R 2 ) = ( � 2 d ) n · ⇠ , 19 2017.11. KEK
Entanglement Vector Field Goal: construct a vector field V 0 such that Therefore we look for a at r a ( V a � V a 0 ) = 0 ( r � R ) 2 V 0 = 2 ⇡ d Answer : r 2 cos 3 ✓ dr, R h r 2 + R 2 r 2 cos ✓ dr � ( R 2 � r 2 ) V E = 2 ⇡ tan ✓ i Finally cos ✓ d ✓ � V 0 R r 20 2017.11. KEK
Flux lines of V and V E 2.0 t X z 1.5 ξ A 1.0 z B v A Σ t 0.5 ˜ B 0.0 - 2 - 1 0 1 2 x 2.0 B 1.5 B 1.0 z 0.5 0.0 - 2 - 1 0 1 2 21 2017.11. KEK x
Sewing the space with flux line of V E flux lines of VE look like B sewing the two regions B and B ̄ B along their interface. Wanted feature in ref. M. Freedman and M. Headrick, “Bit threads and holo- graphic entanglement,” arXiv:1604.00354 [hep-th]. 22 2017.11. KEK
Conclusion 1. Generalized Entanglement fist law, çè Full Einstein equation. Z Z ˆ � E grav � � S grav � ! ( g 0 ; � g, � ξ B g ) = E B B Σ Σ bc [ g 0 ] � g bc + � C a where ˆ ✏ a E g E = � ⇠ a � � . B B 2. entanglement mostly through the edge B 23 2017.11. KEK
감사합니다 . 24 2017.11. KEK
Recommend
More recommend
