EPR = ER and Scattering Shigenori Seki Research Institute for - PowerPoint PPT Presentation

EPR = ER and Scattering Shigenori Seki � Research Institute for Natural Science, Hanyang University � � This talk is based on the work: “EPR = ER, Scattering Amplitude and Entanglement Entropy Change,” Shigenori Seki and Sang-Jin Sin, Phys. Lett. B735 (2014) 272. at YITP , Kyoto on 9 May 2014 1

EPR = ER conjecture 2

EPR pair Einstein-Podolsky-Rosen pair [Einstein-Podolsky-Rosen, Phys.Rev. 47 (1935) 777] entangled two particles e.g. a spin-0 particle decays to A B two spin-1/2 particles. 1 ⌃ | Ψ ⇧ = 2 ( | ⇤⇧ A ⇥ | ⌅⇧ B � | ⌅⇧ A ⇥ | ⇤⇧ B ) Separate them from each other at long distance A B A and B are still entangled. (If the state of A is observed, then the one of B is determined.) S A = − tr( ρ A log ρ A ) = log 2 3

EPR = ER conjecture [Maldacena-Susskind, Fortsch.Phys. 61 (2013) 781] entangled two particles A B Separate them from each other at long distance A B ER bridge (or wormhole) Quantum mechanics Gravity geometric two entangled systems are entanglement interpretation connected by ER bridge 4


 
 From the viewpoint of AdS/CFT correspondence, let us see two examples supporting the EPR = ER conjecture. • Accelerating quark and anti-quark 
 [Jensen-Karch, Phys.Rev.Lett. 111 (2013) 211602] • Scattering gluons 
 [SS-Sin, Phys.Lett. B735 (2014) 272] Entanglement Wormhole on world-sheet 5

Accelerating quark and anti-quark 6

The holographic surface of accelerating quark and anti-quark anti2quark! quark! [Xiao, Phys.Lett. B 665 (2008) 173] AdS bulk metric � � � ! � � � ! World!volume!horizons! ds 2 = 1 z 2 ( − dt 2 + dx 2 + dz 2 ) Minimal surface x 2 = t 2 + b 2 − z 2 t x z 7

[Jensen-Karch, Phys.Rev.Lett. 111 (2013) 211602] Static gauge t = τ , z = σ World-sheet induced metric 1 − ( b 2 − σ 2 ) d τ 2 + ( τ 2 + b 2 ) d σ 2 − 2 τσ d τ d σ ds 2 ⇥ ⇤ ws = σ 2 ( τ 2 + b 2 − σ 2 ) t x x 2 = t 2 + b 2 ( z = 0) z The trajectories of quark and anti-quark are causally disconnected on the world-sheet. wormhole The quark and anti-quark are entangled by the x 2 = t 2 ( z = b ) wormhole that the open string goes through. 8

The entanglement of The interaction between quark and anti-quark (EPR pair) quark and anti-quark ER bridge (wormhole) on world-sheet We can naturally guess that the entanglement of final states is different from that of initial states due to interaction. Therefore the scattering process induces the entanglement entropy change. Are other interacting particles also related to a wormhole on world-sheet? Fortunately, we know the minimal surface in AdS that describes a gluon-gluon scattering. 9

Scattering gluons 10

Minimal surface solution for gluon scattering AdS 5 (momentum space) [Alday-Maldacena, JHEP 0706 (2007) 064] ds 2 = R 2 r 2 ( η µ ν dy µ dy ν + dr 2 ) IR boundary condition r = 0 ∆ y µ = 2 π k µ k µ 3 k µ The solution of Nambu-Goto action 4 k µ 1 + β 2 sinh u 1 sinh u 2 � 2 α k µ 1 y 0 = , cosh u 1 cosh u 2 + β sinh u 1 sinh u 2 α sinh u 1 cosh u 2 y 1 = , cosh u 1 cosh u 2 + β sinh u 1 sinh u 2 Mandelstam variables: α cosh u 1 sinh u 2 y 2 = , 8 α 2 cosh u 1 cosh u 2 + β sinh u 1 sinh u 2 − s (2 π ) 2 = (1 − β ) 2 , y 3 = 0 , 8 α 2 α − t (2 π ) 2 = r = (1 + β ) 2 . , cosh u 1 cosh u 2 + β sinh u 1 sinh u 2 11

AdS 5 (momentum space) [Kallosh-Tseytlin, JHEP 9810 (1098) 016] “T-dual” transformation: ⇥ m y µ = R 2 z = R 2 z 2 � mn ⇥ n x µ , r AdS 5 (position space) ds 2 = R 2 z 2 ( η µ ν dx µ dx ν + dz 2 ) The Alday-Maldacena solution is mapped to x 0 = − R 2 1 + β 2 sinh u + sinh u − , p 2 α = − R 2 x + := x 1 + x 2 2 α [(1 + β ) u − + (1 − β ) cosh u + sinh u − ] , √ √ 2 2 R 2 x − := x 1 − x 2 = 2 α [(1 − β ) u + + (1 + β ) sinh u + cosh u − ] , √ √ 2 2 x 3 = 0 , z = R 2 2 α [(1 + β ) cosh u + + (1 − β ) cosh u − ] where . For later convenience, we introduce u ± := u 1 ± u 2 X µ := α Z := α ( µ = 0 , + , − , 3) , R 2 z ( ≥ 1) R 2 x µ 12

Causal structure on world-sheet The induced metric on world-sheet [SS-Sin, Phys.Lett. B735 (2014) 272] ds 2 ws = R 2 � g ++ du 2 + + 2 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 2(1 − β 2 ) sinh u + sinh u − g + − = [(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” q (1 + β ) 2 sinh 2 u + + 1 + β 2 g ++ = 0 : (1 − β ) cosh u − = (1 + β ) cosh u + + 2 q (1 − β ) 2 sinh 2 u − + 1 + β 2 g −− = 0 : (1 + β ) cosh u + = (1 − β ) cosh u − + 2 13

X ± := 2 ˆ X ± ∈ ( −∞ , + ∞ ) 0 ≤ β < 1 π arctan X ± ∈ [ − 1 , 1] β = 0 1.0 10 Z = 1 0.5 5 Z = ∞ � 1.0 � 0.5 0.5 1.0 � 10 � 5 5 10 � 0.5 � 5 thick red: , g −− = 0 � 1.0 dashed red: , � 10 g ++ = 0 dotted blue: β = 1 / 2 � g ++ = g −− � 1.0 10 blue & red: constant X 0 0.5 green: constant 5 Z � 1.0 � 0.5 0.5 1.0 � 10 � 5 5 10 � 0.5 � 5 � 1.0 � 10 14

Regge limit: with fixed. β = 1 − t − s → ∞ X 0 = − 1 X + = − 1 1 2 sinh u + sinh u − , X − = 2 sinh u + cosh u − , 2 u − , √ √ √ X 3 = 0 , Z = cosh u + . 1.0 10 0.5 5 � 1.0 � 0.5 0.5 1.0 � 10 � 5 5 10 � 0.5 � 5 � 1.0 � 10 While is positive definite, vanishes on . √ cosh u + = 2 g ++ g −− 15

EPR = ER in gluon scattering? { g 1 , g 2 } → { g 3 , g 4 } Incoming gluons: 1.0 c (1) X | g 1 ( t 1 ) i i = ij | A Li ( t 1 ) i ⌦ | A Rj ( t 1 ) i i,j 0.5 c (2) X | g 2 ( t 1 ) i i = ij | B Li ( t 1 ) i ⌦ | B Rj ( t 1 ) i i,j � 1.0 � 0.5 0.5 1.0 Outgoing gluons: � 0.5 c (3) X | g 3 ( t 2 ) i i = ij | A Li ( t 2 ) i ⌦ | B Rj ( t 2 ) i i,j � 1.0 c (4) X | g 4 ( t 2 ) i i = ij | B Li ( t 2 ) i ⌦ | A Rj ( t 2 ) i i,j We can see two types of entanglement which are interpreted to wormholes. 16

1. Internal entanglement c (1) X | g 1 ( t 1 ) i i = ij | A Li ( t 1 ) i ⌦ | A Rj ( t 1 ) i i,j 1.0 The open string endpoints in each gluon are entangled by the open string going through the wormhole. 0.5 This is in the same way as the entanglement of quark and anti-quark. 17

2. Entanglement of gluons Any paths connecting the gluons must go through the wormhole region. There are two channels. 1.0 1.0 0.5 0.5 � 1.0 � 0.5 0.5 1.0 � 1.0 � 0.5 0.5 1.0 � 0.5 � 0.5 � 1.0 � 1.0 1.0 0.5 � 1.0 � 0.5 0.5 1.0 � 0.5 � 1.0 18

How can we measure the change of entanglement in gluon scattering process? i ) (naively) log of scattering amplitude [Lewkowycz-Maldacena, JHEP 1405 (2014) 025] The scattering amplitude corresponds to the Wilson loop which is given by the area of minimal surface. And naively . S = (1 � n ∂ n ) log h W i | n → 1 √ ◆ 2 ◆ 2 ✓ λ log 1 + β ✓ 1 + β s A ∼ e − Area , ∆ S ∼ log A = t = 2 π 1 − β 1 − β ii) the length between boundaries at the contacting points Z + u + ∞ Z + u −∞ � � ⇥ + ( � ) = R ⇥ − ( � ) = R du + √ g ++ u − =0 , du − √ g −− � � u + =0 − u + ∞ − u −∞ where we introduced the cutoff, . z ∞ ( → ∞ ) 2 α z ∞ = (1 + β ) cosh u + ∞ + 1 − β = (1 − β ) cosh u −∞ + 1 + β R 2 ✓ 1  √ 6 log 2 � z ∞ 1 ◆� √ ⇤ ± ( ⇥ ) = R + 6 log 1 ± ⇥ + O R 2 z ∞ ∆ S ∼ ( ` + ( � ) − ` − ( � )) 2 diverges at the Regge limit, , and vanishes at . β = 0 β = 1 ∆ S 19

Scattering vs Entanglement Scattering process S-matrix | ini i = | p 1 , p 2 i { p 1 , p 2 } → { k 1 , k 2 } X | fin i = | k 1 , k 2 ih k 1 , k 2 | S | ini i k i The entanglement of particles is changed from the initial state to the final one. e i ( H 0 + H int ) t Problem: we need to understand the relation between S-matrix theory and entanglement entropy both in the quantum field theory directly and in holography. 20