Unruh effect and Holography

1. HEP-QIS Joint Seminar at CYCU Unruh effectand Holography Shoichi Kawamoto (National Taiwan Normal University) with Feng-Li Lin(NTNU), Takayuki Hirayama(Kyoto Sangyo U.) and Pei-Wen Kao (Keio, Dept. of Math.) (Based on arXiv:1001.1289) 2010 October 19 @ CYCU

2. String theory is a candidate of quantum gravity with strings being fundamental d.o.f. Recently, strong coupling regime of a class of (conformal) field theory can be probed by using string theory (supergravity) on a curved b.g. AdS/CFT correspondence This is a best understood example of holography. D-dimensional QFT vs. D+1 dimensional gravity We start with a brief review of them. String Theory and AdS/CFT correspondence Shoichi Kawamoto (NTNU)

3. String Theory D-dim. A tiny string is propagating in D-dimensional space-time (target space) Consistent quantization D=10 (with supersymmetry) Strings are very small (almost Planck length): Looking like “particles” But it has more internal degrees of freedom: diversity of particles Closed strings: Gravity multiplets (graviton, dilaton, ….) supergravity (SUGRA) Open strings: gauge multiplets (gauge fields, gaugino,…) Shoichi Kawamoto (NTNU)

4. Dp-branes as a boundary of open strings Dirichlet p-brane (Dp-brane) is (1+p) dimensional object on which open strings can end. 9 dim p dim Open strings: In (1+p) dim. with gauge fields closed strings (SUGRA) (1+p) dim. U(1) (SUSY) gauge theory 9-p dim Gauge theory (open string) There are also freely moving closed strings (SUGRA on flat 10D) Note: it is a source of RR (p+2)-form field strength If N number of Dp-brane are on top of each other, (Witten) N Dp Gauge symmetry is enhanced: We have U(N) Super Yang-Mills (SYM) in (1+p) dim. on N Dp-branes. Shoichi Kawamoto (NTNU)

5. Dp-brane as a classical solution of gravity The same charged object can be constructed as a classical solution of SUGRA. Curved space (Black p-brane) Black p-brane solution It... • has the same RR-flux • preserves the same SUSY in extremal case. • does not have gauge symmetry (no open strings) • and has only gravity d.o.f. (closed strings) • is extended version of black “hole”. RR-flux Open-Closed duality: It is believed that these two descriptions are the different viewpoints for the same object. It leads to the following “duality” Shoichi Kawamoto (NTNU)

6. AdS/CFT correspondence (p=3 case) Best known case: D3-branes and N=4 super Yang-Mills theory N D3-branes 10D SUGRA Curved space (Black 3-brane) Flat space decouple Maldacena limit (N!1 , a’ ! 0) decouple Gauge theory (open string) z=0 : AdS boundary z=1 String (SUGRA) in AdS5 * S5 N=4 Supersymmetric Yang-Mills (weak curvature: L4=la’2=1) (Strong coupling regime: l=gYM2N=1) Correspondence: Symmetry, States, correlation functions,.... Shoichi Kawamoto (NTNU)

7. AdS/CFT correspondence 2 Conformal Symmetry and R-symmetry Symmetry: Isometry of AdS5 and S5. (Actually, full superconformal symmetry matches.) (GKPW) Correlation functions: source of an operator boundary value of gravity fields A special example: A fundamental charge An open string boundary bulk Shoichi Kawamoto (NTNU)

8. Finite temperature We can put a blackhole in the AdS background (AdS-BH solution). The blackhole is at a finite temperature (Hawking temperature). The corresponding gauge theory becomes finite temperature as well (with the same temperature). Finite temperature quantum field theory. In quantum field theory, a temperature is measured for an accelerated observer. Unruh Effect!! Q: How is it looking like in the gravity side? Before then, we will recall the Unruh effect... Shoichi Kawamoto (NTNU)

9. Unruh(-Davies-De Witt-Fulling) effect The world-line of the observer with a constant acceleration a is given by solving the solution is given by hyperbolas The observer feels the temperature There is a convenient choice of coordinates. Coordinate transform Rindler coordinates: Shoichi Kawamoto (NTNU)

10. The Rindler coordinates as a comoving frame t EDK Rindler coordinates: LR RR x1 It covers the region (Right Rindler wedge) The “time” translation is generated by the Killing vector CDK The world line with a constant x has a constant acceleration. Accelerated observer in Minkowski space = Static observer in Rindler space (Comoving frame) Shoichi Kawamoto (NTNU)

11. Vacuum, Particles and Observers Let us briefly discuss how the accelerated observer feels a finite temperature. Vacuum is observer dependent. Klein-Gordon equation: Assume two complete sets of solutions: complete sets space-like hypersurface Shoichi Kawamoto (NTNU)

12. Vacuum, Particles and Observers II Quantum field can be expanded as positive frequency modes Bogolubov coefficients Bogolubov transformation: Vacuum: defined by VEV of the number operator is But, is an excited states with respect to the particles of (1). Shoichi Kawamoto (NTNU)

13. Quantum Field Theory on Minkowski space 2D massless scalar field theory: An example KG equaton: right mover: Minkowski vacuum: Shoichi Kawamoto (NTNU)

14. Quantum Field Theory on Rindler space Move to Rindler coordinates: KG eq. LR RR x1 The Rindler vacuum is defined by Shoichi Kawamoto (NTNU)

15. Minkowski vacuum as a thermal state Each of them cannot be written as Minkowski operators. The set can be related to Minkovski ones. Bogolubov transformation: So the expectation value of the number operators (assume now the energy levels are discrete ) It represents the heat bath with the temperature For operator with Right Rindler modes: details Shoichi Kawamoto (NTNU)

16. Notion of “vacuum” and “particles” are observer dependent. Observer in an accelerated frame (Rindler observer) sees the vacuum of the inertia observer (Minkowski vacuum) as a thermal b.g. This is due to having a “horizon” (Rindler horizon) and loosing the access to the other part of spacetime. How is this effect looking like in the holographic dual theory? Summary of the introduction Shoichi Kawamoto (NTNU)

17. Plan 1. Introduction: Review of AdS/CFT & Unruh Effect 2. Uniformly accelerated string and comoving frame 3. Investigating various quantities 4. Conclusion Shoichi Kawamoto (NTNU)

18. 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!!! Shoichi Kawamoto (NTNU)

19. Uniformly accelerated string in AdS space (2/3) We wand to take a “comoving frame” for the open string. First determine the configuration. Consider AdS part of the metric a boundary with boundary condition: and solve the e.o.m. Exact solution to NG action has been found (Xiao) Shoichi Kawamoto (NTNU)

20. Comoving coordinates for uniformly accelerated string (Xiao) Uniformly accelerated string in AdS space (3/3) Now the open string configuration: with a r Shoichi Kawamoto (NTNU)

21. Generalized Rindler space (Xiao’s metric) Illustrate how the new coordinates covers a part of the original AdS5 (horizon = Rindler horizon + AdS horizon) right Rindler wedge with 0 < r < a-1 constant r surface Shoichi Kawamoto (NTNU)

22. 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 temperature = Bulk Blackhole temperature Note: The horizon appears in the radial direction. Different from the effect of the heavy object on the accelerated direction. We will examine the thermal properties, and see what are similar to the case with BH. Shoichi Kawamoto (NTNU)

23. Boundary stress tensor Quark-anti Quark potential Some phase transition involving mesons III. Calculation of various quantities Shoichi Kawamoto (NTNU)

24. Boundary stress tensor We first look at the boundary stress tensor. (Balasubramanian-Kraus, Myers) r where trace of the extrinsic curvature of the boundary counter term After eliminating the divergences, we get (HKKL) Xiao’s metric (generalized Rindler): Conformal thermal gas with the temperature Shoichi Kawamoto (NTNU)

25. Quark – anti Quark potential L comoving frame e r0 1/a We may calculate quark – anti quark potential in the accelerated frame. Energy is given by that of the stretched string String profile Solution is given by configuration satisfying the boundary conditions Shoichi Kawamoto (NTNU)

26. Limiting acceleration and screening Compare the energy to the straight line configuration (green ones). L Energy(=total length) r0 So at some critical distance (=critical acceleration difference), the force between quark-anti quark may be screened (and no limiting acceleration difference). Maybe, energy cannot reach the other end due to causality. Shoichi Kawamoto (NTNU)

27. A look at mesons: Introducing D7-brane Now we come to investigate the meson physics. Introducing meson in AdS5 is achieved by putting a probe D7-brane. 0,1,2,3 D7 “meson” excitations 8,9 First we argue, what is the appropriate setup for accelerated mesons? fundamental matters 4,5,6,7 1. Moved to Xiao’s metric (generalized Rindler coord.) D3 2. Then embedding D7-brane to be static on this coordinate system. (will be replaced with curved b.g.) (static for accelerated observers) This will define : An operator corresponding to an accelerated meson. Holographic calculation will be thermal one. Shoichi Kawamoto (NTNU)

28. D7-brane embedding (1/2) We work with the following coordinates. D7 brane extends these 8 directions. Ansatz: Then we solve the equation of motion with boundary conditions. Shoichi Kawamoto (NTNU)

29. D7-brane embedding (2/2) D7-brane profile z(r) Asymptotic solution near the boundary is Minkowskiembedding In general, starting with arbitrary m and n, the solution will diverge. BH embedding horizon m Regular solution: D7 reaches to the center: Minkowski embedding D7 terminates at horizon: Blackhole embedding Shoichi Kawamoto (NTNU)

30. One point function and Chiral condensate (Hirayama-Kao-SK-Lin) T/M Solution near the boundary: Parameters may be identified with It shows the phase transition behavior corresponding to “meson” melting. Minkowski embedding AdS-BH result Mateos et al. JHEP 0705:067, Fig. 4 BH embedding Shoichi Kawamoto (NTNU)

31. Fluctuations on D7-brane w5(r) Now we consider transverse fluctuations of D7. 1/T rh r We can calculate the retarded Green’s function for this fluctuation modeand then derive the spectral function: Shoichi Kawamoto (NTNU)

32. Only results: Spectral functions for mesons temperature low In low temperature,there are sharp peaks (stable mesons) high For higher temperature, spectral function becomes featureless (meson dessociation) Shoichi Kawamoto (NTNU)

33. (Review) To describe string with accelerated end point, the generalized Rindler coordinates is useful. Checked that it has the boundary stress tensor corresponding to thermal conformal matter. Wilson loop shows screening behavior. We have calculated various quantities of holographic QCD-like model in the generalized Rindler space. The results quite resemble AdS-BH results. Conclusion Shoichi Kawamoto (NTNU)

34. Thank you Shoichi Kawamoto (NTNU)