Unruh effect and Holography

Transcription

1 nd Mini Workshop on String KEK Unruh effect and Holography Shoichi Kawamoto (National Taiwan Normal University) with Feng-Li Lin(NTNU), Takayuki Hirayama(NCTS) and Pei-Wen Kao (Keio, Dept. of Math.) 009 November KEK

2 Uniformly accelerated observer and Unruh Effect The world-line of the observer with a constant acceleration a is given by solving d dt mx 1 x = F = ma and the solution is given by hyperbolas The observer feels the temperature Unruh Davies De Witt Fulling effect Let us discuss this phenomenon first. 009 Nov. 10 nd Mini Workshop on String KEK

3 Uniformly accelerated observer and the indler coordinates It is convenient to work with the following coordinates choice Starting from the Minkowski metric ds = dt + dx1 + dx + Coordinate transform dx 3 t 1 ξ a e a 1 ξ = sinh aτ x = a e a cosh aτ 1 indler coordinates: ds ( ) dτ + d + dx + dx aξ = e ξ 3 Why it is nice? 009 Nov. 10 nd Mini Workshop on String KEK 3

4 The indler coordinates as a comoving frame t EDK indler coordinates: L ds ( ) dτ + d + dx + dx aξ = e ξ x 1 It covers the region (ight indler wedge) 3 The time translation is generated by the Killing vector CDK t x ~ x1 1 t τ t = a x 1 1 e a ξ sinh aτ 1 ξ = a e a cosh aτ The world line with a constant ξ has a constant acceleration. Accelerated observer in Minkowski space = est observer in indler space (Comoving frame) 009 Nov. 10 nd Mini Workshop on String KEK 4

5 Vacuum, Particles and Observers Let us briefly discuss how the accelerated observer feels a finite temperature. Vacuum is observer dependent. Klein-Gordon equation: (1) () two complete sets of solutions: { f i ( x) } { ( x) } complete sets f = [ f f ] (1) * () ()* α β i I Ii I Ii I f I * ( fi, f j ) = ( f * i, f j ) = δij ( f, * i f j ) = 0 space-like hypersurface 009 Nov. 10 nd Mini Workshop on String KEK 5

6 Vacuum, Particles and Observers II Quantum field can be expanded as positive frequency modes (1) () * () Bogolubov transformation: aˆ = ( α aˆ + β aˆ ) i I Ii I Ii I Bogolubov coefficients Vacuum: 01 ˆ 1 ˆ (1) () 0 defined by ai 0 = 0 a I 0 = 0 VEV of the number operator N (1) i ( 1) ( 1) (1) = aˆ aˆ is 0 0 i i 01 N i 1 = But, (1) 0 N i 0 = β Ii I 0 is an excited states with respect to the particles of (1). 009 Nov. 10 nd Mini Workshop on String KEK 6

7 Quantum Field Theory on Minkowski space D massless scalar field theory: An example ( ) KG equaton: φ( x) = 0 t x M 1 ikx iω t f k = e 0 < ω = k < 4 πω right mover: φ( x) ( M M M M aˆ f aˆ f ) * = k k 0 dk k k + Minkowski vacuum: M 0 aˆ k 0M = 0 M 009 Nov. 10 nd Mini Workshop on String KEK 7

8 t = Quantum Field Theory on indler space Move to indler coordinates: KG eq. a 1 x = a e 1 aξ e sinh aτ aξ cosh aτ ds = e aξ ( dτ + dξ ) ( ) φ( τ, ξ ) = 0 τ ξ t = a x = a 1 e e 1 a ξ aξ sinh aτ cosh aτ L 1 f k = e 4 πω ik ξ iω τ L x 1 1 f k = e 4 πω ik ξ iω τ L L L L* ( aˆ f aˆ f ) φ( x) = dk + k k k k * ( aˆ f aˆ f ) φ( x) = dk + k k k k The indler vacuum L 0 is defined by aˆ 0 = ˆ k ak 0 = Nov. 10 nd Mini Workshop on String KEK 8

9 Minkowski vacuum as a thermal state Each of them cannot be written as Minkowski operators. L The set { k fk } f, can be related to Minkovski ones. Bogolubov transformation:, L = (, L M, L* M aˆ dk α aˆ + β aˆ ) k k k k k k k So the expectation value of the number operators (assume now the energy levels are discrete ω = i k i ) L 1 0 M Ni 0M 0M Ni 0M = πωi / e = a 1 ik / a πk / a ie a ik = Γ α k k 1 π k k k a α L k k ie = π k / a a k k k π ik / a 1 β β L k k k k = e = e ik Γ + a α πk / a * k k α πk / a L* k k It represents the heat bath with the temperature T = a π 009 Nov. 10 nd Mini Workshop on String KEK 9

10 Unruh effect and QCD In QCD, there is a critical temperature T c at which the chiral symmetry is restored. Chiral symmetry can be restored by acceleration? An interesting work by Ohsaku (PLB599, 004 ). Consider chiral restoration in 4D Nambu-Jona-Laisnio model For Λ 1GeV, a c ~ Λ * 10-1 a c ~ cm/sec Too big to test experimentally, but theoretical implication is intriguing. For QCD, we may study the effect of acceleration through holographic correspondence. What are the same (similar)? What s the difference? 009 Nov. 10 nd Mini Workshop on String KEK 10

11 Plan 1. Introduction: Unruh effect in field theory. Uniformly accelerated string and comoving frame 3. Introducing mesons 4. Conclusion and Discussion 009 Nov. 10 nd Mini Workshop on String KEK 11

12 Uniformly accelerated string in AdS space (1/3) Let us consider a uniformly accelerated particle (quark) on the boundary field theory. a The particle is the end point of an open string. We are going to make a coordinate transformation which gives the comoving frame on the boundary. Infinitely many choices!!! 009 Nov. 10 nd Mini Workshop on String KEK 1

13 Uniformly accelerated string in AdS space (/3) We wand to take a comoving frame for the open string. First determine the configuration. Consider AdS part of the metric boundary a with boundary condition: and solve the e.o.m. Exact solution to NG action has been found (Xiao) 009 Nov. 10 nd Mini Workshop on String KEK 13

14 Uniformly accelerated string in AdS space (3/3) Comoving coordinates for uniformly accelerated string (Xiao) ds AdS 5 ( dt + dx + dx + dx ) + du + Ω 5 = u 1 3 d S t = x u a r e aα sinh ( aτ ) aα 1 = a r e cosh( aτ ) aα = r 1 e u 5 Now the open string configuration: with α 009 Nov. 10 nd Mini Workshop on String KEK 14 r

15 Generalized indler space (Xiao s metric) (horizon = indler horizon + AdS horizon) Illustrate how the new coordinates covers a part of the original AdS 5 right indler wedge with 0 < r < a -1 constant r surface 009 Nov. 10 nd Mini Workshop on String KEK 15

16 Temperature in the comoving frame On the boundary, the observer feels the Unruh temperature Xiao s metric has the horizon. And the Hawking temperature is They coincides Boundary acceleration = Bulk Blackhole The effect of the acceleration is completely equivalent to the gravitational force from BH? Let us calculate some physical quantities and see whether we can see difference. (Later we also discuss another accelerated frame) 009 Nov. 10 nd Mini Workshop on String KEK 16

17 Boundary stress tensor We first look at the boundary stress tensor. (Balasubramanian-Kraus, Myers) T μ ν = lim r δ S γ δγ tot ν μ trace of the extrinsic curvature of the boundary 1 16πG 1 8πG 1 8πG 5 4 where S = tot d x[ Λ] d x γ Θ + γ [ c1 + c( γ )] M M r= After eliminating the divergences, we get (HKKL) counter term Xiao s metric (generalized indler): μ 4 T ν ( 3, 1, 1, 1), p = ε / 3 N T Conformal thermal gas with the temperature a T = π 009 Nov. 10 nd Mini Workshop on String KEK 17

18 Quark anti Quark potential a 1 a a x 1 ε L α a < a < 1 a 1/a We may calculate quark anti quark potential in the accelerated space. Energy given by the Wilson loop r 0 Wilson loop profile X μ = ( τ, r, α( r),0,0) Solution is given by h( r) 0 = h( r0 ) r 1+ h( r) α r0 α h( r) = 1 a r 009 Nov. 10 nd Mini Workshop on String KEK 18

19 Profile of the Wilson loop (1/3) We first look at the profile α r h( r) α 1+ h( r) α = r 0 0 h( r 0 ) r 0 large r The left is the profile function α(r) for various r 0 (=a=1). First they keep similar shape, but for large r 0, the profile becomes more steep. right half of the string profile 009 Nov. 10 nd Mini Workshop on String KEK 19

20 Profile of the Wilson loop (/3) r 0 against L plot: the maximal length does not occur for r 0 =1/a. L = r0 ( α(0) α( r )) = α 0 ( r) 0 dr The energy against quark-anti quark distance. First grows linearly, but finally shows a strange behavior. 009 Nov. 10 nd Mini Workshop on String KEK 0

21 Profile of the Wilson loop (3/3) Compare the energy to the straight line configuration (green ones). L α r 0 So at some critical distance (=critical acceleration difference), the force between quark-anti quark is screened? Finite temperature case, it does happen. However, our horizon is not real one!! Energy cannot reach the other end? Loose causal relation?? Still unclear Nov. 10 nd Mini Workshop on String KEK 1

22 Another choice of the metric? Why do we need to stick to Xiao s choice? We may choose a simpler coordinates. the same transformation as on the boundary 1 ξ t a e a 1 aξ = sinh aτ x = a e cosh aτ 1 ds indler aξ ( e ( dτ + dξ ) + dx + dx ) + du + Ω = u 3 d A naive extension of indler space. u 5 We also check the boundary stress tensor. μ Tν = 0 zero temp. vacuum?? But the boundary theory should be the finite temperature system with T=a/π No clear answer yet Nov. 10 nd Mini Workshop on String KEK

23 Comments on other s work ξ Pareres-Peeters-Zamaklar(009) computed the similar system, but using naive inlder space, and found also a bound for the acceleration difference. Fig 1. JHEP 0904:015,009 ds = z ( ξ κ dη dξ dz ) + They also found the maximum of the ratio of the acceleration ( a / a ) ~. 70 L max Also argue that dissociation happens when a = a L. (String reaches the indler horizon) Again, what happens in the rest frame?? 009 Nov. 10 nd Mini Workshop on String KEK 3

24 Introducing D7-brane Now we come to investigate the meson physics. Introducing meson in AdS 5 is achieved by putting a probe D7-brane. D7 0,1,,3 8,9 4,5,6,7 We would like to argue the chiral condensates. First we argue, what is the appropriate setup for accelerated mesons? D3 fundamental matters meson excitations 009 Nov. 10 nd Mini Workshop on String KEK 4

25 Who is moving? There will be two ways to embed D7-brane in the Generalized indler space. Embedding I (I) First, embed D7-brane in the original AdS 5 coord., and make the coordinate transform. In this embedding, the quarks are rest in the Minkowski metric. On the boundary field theory, Minkowski vacuum Fluctuations are the operators on Minkowski space 0 M O M Will be calculating 0 O M M 0M After coordinate transformation, the operator is accelerated. While the vacuum seems to be thermal (to the observer). 0 O still??? M M 0M 009 Nov. 10 nd Mini Workshop on String KEK 5

26 Comoving probe brane Embedding II We want to have a static operator in the accelerated frame. The other way around! 1. Moved to Xiao s metric (generalized indler coord.). Then embedding D7-brane to be static on this coordinate system. This will define O (Note: D7-brane is time dependent in the original frame. But quarks are static.) Holographic calculation will be thermal one. 0 O M 0M 009 Nov. 10 nd Mini Workshop on String KEK 6

27 009 Nov. 10 nd Mini Workshop on String KEK 7 D7-brane embedding (1/) ( ) ( ) [ ] ( ) dw dw d d w dx dx e d h d h w ds a + + Ω = + ρ ρ α τ α We work with the following coordinates. + + = ± = + = ± 6 5 4, 4 1, 4 4 w w w w a h w a w r ρ Ansatz: 0 ), ( 6 5 = = w z w ρ + = ) ( 1 ρ ρ α w h h x e d T S a D D7 brane extends these 8 directions. Then we solve the equation of motion with boundary conditions.

28 D7-brane embedding (/) D7-brane profile z(ρ) Minkowski embedding BH embedding horizon m Asymptotic solution near the boundary is z( ρ) ~ m ν ( 1+ O(ln( ρ) ρ )) + ( 1+ O( ρ )) ρ In general, starting with arbitrary m and ν, the solution will diverge. egular solution: ( m, ν ( m) ) D7 reaches to the center: Minkowski embedding D7 terminates at horizon: Blackhole embedding 009 Nov. 10 nd Mini Workshop on String KEK 8

29 One point function and Chiral condensate (Hirayama-Kao-SK-Lin) 1 ν + mlnm T/M Solution near the boundary: ν z( ρ ) ~ m + ρ Parameters may be identified with 1 M q / T ~ m, qq ~ ν + mlnm It shows the phase transition behavior similar to AdS-BH case. There will be a chiral restoration! Minkowski embedding 009 Nov. 10 BH embedding AdS-BH result Mateos et al. JHEP 0705:067, Fig. 4 nd Mini Workshop on String KEK 9

30 Conclusion (eview) To describe string with accelerated end point, the generalized indler coordinates is useful. Checked that it has the boundary stress tensor corresponding to thermal conformal matter. Wilson loop shows strange behavior. We have calculated various quantities of holographic QCD-like model in the generalized indler space. The results quite resemble AdS-BH results. 009 Nov. 10 nd Mini Workshop on String KEK 30

31 Open problems...(a lot) What is the difference of the coordinates choice? Xiao s metric gives similar results to BH case. Other choices? They do not have to be the same. Equivalence principle? Gravitational force and acceleration are not the same in this setup... How to interpret the Wilson loop and q-q bar potential? We are still checking Behavior of the fluctuations on D7-brane. Drag force? 009 Nov. 10 nd Mini Workshop on String KEK 31

32 Future directions Hagedorn temperature: There is a limiting temperature in string theory. Is there any limiting acceleration?? Acceleration in the finite temperature field theory Generalized indler space from AdS-BH metric? Temperature vs. Acceleration? Acceleration horizon covers or is covered by BH horizon? What happens?? Should be lots more... Interesting to study! 009 Nov. 10 nd Mini Workshop on String KEK 3