Holographic Entanglement and Interaction


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1 Holographic Entanglement and Interaction Shigenori Seki RINS, Hanyang University and Institut des Hautes Études Scientifiques Intrication holographique et interaction à l IHES le 30 janvier
2 Contents 1. Quantum entanglement and holography [S. Ryu and T. Takayanagi, Phys.Rev.Lett. 96 (2006) ] 2. EPR = ER conjecture [J. Maldacena and L. Susskind, arxiv: ] 3. Accelerating quark and antiquark [K. Jensen and A. Karch, Phys.Rev.Lett. 111 (2013) ] 4. Gluon scattering [SS and S.J. Sin, to appear] 2
3 Quantum entanglement and holography 3
4 Quantum entanglement Let s consider two systems: A and B Hilbert space: H A H B { i A } { i B } basis A general state in total system is denoted by ψ tot = i,j c ij i A j B If If c ij = c A i c B j c ij = c A i c B j, it is a nonentangled state., it is an entangled state. What is an order parameter of entanglement? Entanglement Entropy 4
5 Density matrix: for Ψ H A H B Reduced density matrix ρ tot = ΨΨ ρ A := tr B ρ tot = j Bj ( ΨΨ ) j B The entanglement entropy is defined by Von Neumann entropy as S A := tr(ρ A log ρ A ) 5
6 EPR pair EinsteinPodolskyRosen pair [EinsteinPodolskyRosen, Phys.Rev. 47 (1935) 777] entangled two particles A B e.g. a spin0 particle decays to two spin1/2 particles. Separate them from each other at long distance A B Observe A The state of B is determined. A and B are still entangled. 6
7 e.g. Consider the state Ψ = 1 2 ( A B A B ) Since the reduced density matrix is ρ A = 1 2 ( A A + A A ) we obtain the entanglement entropy S A = tr(ρ A log ρ A ) = log 2 Note that the pure state ρ P = 1 2 ( A + A )( A + A ) has zero entanglement entropy S P = tr(ρ P log ρ P )=0 So the entanglement entropy works as an order parameter. 7
8 Holographic entanglement entropy AdS/CFT correspondence (Holography) Field theory on boundary Gravity theory in bulk Partition function Wilson loop Z CFT = e S grav W = e Area(γ) Entanglement entropy S A = 1 4G N Area(γ) A z AdS A z A γ A γ boundary minimal surface A [RyuTakayanagi, PRL 96 (2006) ] 8
9 EPR = ER conjecture 9
10 ER bridge [MaldacenaSusskind, arxiv: ] Consider the eternal AdSSchwarzschild black hole. There are two boundaries and two CFTs. H L,R n L,R = E n n L,R n L Future interior This eternal BH is described by the entangle state, Ψ = n e βe n/2 n L n R We can interpret this state in two ways: 1. a single black hole in thermal equilibrium [Israel, Phys.Lett.A57 (1976) 107] Left boundary Left exterior Right exterior Right boundary n Past interior R Penrose diagram 2. two black holes in disconnected spaces with a common time [Maldacena, hepth/ ] 10
11 1. a single black hole in thermal equilibrium Define a fictitious thermofield Hamiltonian, H tf = H R H L, u(i which generates boosts. u() Ψ = n e βe n/2 n L n R is an eigenvector of H tf with eigenvalue zero. U t = const 2. two black holes in disconnected spaces with a common time The time evolution is generated by H = H R + H L Ψ(t) = n e βe n/2 e 2iE nt n L n R The state Ψ describes the two black holes at a specific instant, t =0. Even though the two black holes exist in separate noninteracting worlds, their geometry is connected by an EinsteinRosen bridge (a wormhole). [EinsteinRosen, Phys.Rev. 48 (1935) 73] 11
12 EPR = ER conjecture From the example of eternal black hole, we studied Quantum mechanics entanglement geometric interpretation Gravity two systems connected by ER bridge By extending this concept, Maldacena and Susskind conjectured pairs of entangled particles separate at a long distance ER bridge BH BH 12
13 Further extension Even for an entangled pair of particles, in a quantum theory of gravity, there must be a Planckian bridge between them. ER bridge Black Black hole hole Hawking radiation. 13
14 Accelerating quark and antiquark 14
15 The holographic surface of accelerating quark and antiquark 0.1,23(0$4! 3(0$4! x 2 = t 2 + b 2 z 2 [Xiao, PLB 665 (2008) 173]! "#$%&!'#%()*!+#$,#./!! The trajectories of quark and antiquark are causally disconnected on the worldsheet. t x z The quark and antiquark are entangled by the wormhole that the open string goes through. [JensenKarch, PRL 111 (2013) ] 15
16 The entanglement of quark and antiquark (EPR pair) The interaction between quark and antiquark ER bridge (wormhole) on worldsheet Do other interacting particles also have ER bridge on worldsheet? Fortunately, we know the minimal surface in AdS that describes a gluongluon scattering.
17 Gluon scattering 17
18 Minimal surface solution for gluon scattering AdS 5 (momentum space) [AldayMaldacena, JHEP 0706 (2007) 064] ds 2 = R2 r 2 (η µνdy µ dy ν + dr 2 ) y µ =2πk µ The solution of NambuGoto action r =0 IR boundary condition y 0 = y 1 = y 2 = α 1+β 2 sinh u 1 sinh u 2 cosh u 1 cosh u 2 + β sinh u 1 sinh u 2, α sinh u 1 cosh u 2 cosh u 1 cosh u 2 + β sinh u 1 sinh u 2, α cosh u 1 sinh u 2 cosh u 1 cosh u 2 + β sinh u 1 sinh u 2, y 3 =0, α r =, cosh u 1 cosh u 2 + β sinh u 1 sinh u 2 Mandelstam variables: s(2π) 2 = t(2π) 2 = 8α2 (1 β) 2, 8α2 (1 + β) 2. 18
19 AdS 5 (momentum space) AdS 5 Tdual transformation: m y µ = R2 z 2 mn n x µ, (position space) z = R2 r ds 2 = R2 z 2 (η µνdx µ dx ν + dz 2 ) The AldayMaldacena solution is mapped to x 0 = R2 1+β2 sinh u + sinh u, 2α x + := x 1 + x 2 2 = R2 2 2α [(1 + β)u +(1 β) cosh u + sinh u ], x := x 1 x 2 2 = R2 2 2α [(1 β)u + +(1+β)sinhu + cosh u ], x 3 =0, where u ± := u 1 ± u 2 z = R2 2α [(1 + β) cosh u + +(1 β) cosh u ]. For later convenience, we introduce X µ := α R 2 x µ (µ =0, +,, 3), Z := α z ( 1) R2 19
20 Causal structure on worldsheet The induced metric on worldsheet [SSSin, to appear] ds 2 ws = R 2 g ++ du g + du + du + g du 2 g ++ = 4(1 + β)2 sinh 2 u + + 4(1 + β 2 ) [(1 + β) cosh u + (1 β) cosh u ] 2 2[(1+β) cosh u + +(1 β) cosh u ] 2, g + = 2(1 β 2 )sinhu + sinh u [(1 + β) cosh u + +(1 β) cosh u ] 2, g = 4(1 β)2 sinh 2 u + 4(1 + β 2 ) [(1 + β) cosh u + (1 β) cosh u ] 2 2[(1+β) cosh u + +(1 β) cosh u ] 2. Horizons g ++ =0: (1 β) cosh u =(1+β) cosh u + +2 (1 + β) 2 sinh 2 u + +1+β 2 g =0: (1+β) cosh u + =(1 β) cosh u +2 (1 β) 2 sinh 2 u +1+β 2 20
21 0 β < 1 β =0 X ± (, + ) ˆX ± := 2 π arctan X ± [ 1, 1] Z = Z = β =1/ red: g =0, dashed red: g ++ =0, dotted blue: g ++ = g
22 β =1 Regge limit: s with t fixed. X 0 = 1 2 sinh u + sinh u, X + = 1 2 u, X = 1 2 sinh u + cosh u, X 3 =0, Z = cosh u While g ++ is positive definite, g is negative in cosh u + > 2. 22
23 EPR = ER = Interaction? The entanglement is given by a wormhole Incoming gluons: g 1 (t 1 ) = A L (t 1 ) A R (t 1 ) g 2 (t 1 ) = B L (t 1 ) B R (t 1 ) Outgoing gluons: g 3 (t 2 ) = A L (t 2 ) B R (t 2 ) 1.0 g 4 (t 2 ) = B L (t 2 ) A R (t 2 ) The entanglement is induced by gluonic interaction. 23
24 There are two ways to see entanglement. 1. Internal entanglement g 1 (t 1 ) = A L (t 1 ) A R (t 1 ) 1 0 The open string endpoints in each gluon are entangled by the open string going through the wormhole. This is in the same way as the entanglement of quark and antiquark. 24
25 2. Entanglement of gluons There are two channels
26 How can we measure the entanglement of gluons? i ) (naively) log of scattering amplitude The scattering amplitude corresponds to the Wilson loop which is given by the area of minimal surface. λ A e Area, S log A = log 1+β 2 + (divergent part) 2π 1 β ii) the length between boundaries at the contacting points We consider + (β) =R +u+ u + du + g++ u =0, (β) =R where we introduced the cutoff, z ( ). +u u du g u+ =0 2 αz R 2 =(1+β) cosh u + +1 β =(1 β) cosh u +1+β 6 2αz ± (β) =R log R log 1 ± β + O z 6log 1 S + (β)+ (β) =R + (divergent part) 1 β2 Anyway, diverges at the Regge limit, and vanishes at. β =1 S β =0 26
27 EPR = ER = Interaction 27
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