Entanglement & area thermodynamics of Rindler space Entanglement & area

1. Ram Brustein אוניברסיטת בן-גוריון Entanglement, thermodynamics & area Series of papers with Amos Yarom, BGU (also David Oaknin, UBC) hep-th/0302186 + to appear • Entanglement & area thermodynamics of Rindler space • Entanglement & area • Entanglement & dimensional reduction (holography) sorry, not today!

2. Bekenstein, Hawking Bekenstein Fichler & Susskind, Bousso Brustein & Veneziano Thermodynamics, Area, Holography • Black Holes • Entropy Bounds • BEB • Holographic • Causal • Holographic principle: Boundary theory with a limited #DOF/planck area ‘thooft, Susskind

3. Rindler space

4. horizon Lines of constant x -constant acceleration Addition of velocities in SR proper acceleration

5. out = z < 0 in = z > 0 Compare two expressions for rin (by writing them as a PI) 1. 2. TFD Minkowski vacuum is a Rindler thermal state(Unruh effect)

6. 1. In general:

8. out in Result

9. 2. Heff – generator of time translations Time slicing the interval [0,b0]:

10. Guess: result

11. Results If • The boundary conditions are the same • The actions are equal • The measures are equal Then

12. out in out in For half space Heff=HRindler , HRindler= boost

13. Rindler area thermodynamics Susskind Uglum Callan Wilczek Kabat Strassler De Alwis Ohta Emparan …

14. In 4D: Go to “optical” space Compute using heat kernel method High temperature approximation Volume of optical space

15. Optical metric Compute: Euclidean Rindler In 4D

16. הפוך

17. S M S,T unitary S

18. M M 1 o M M M M S S

21. Ram Brustein אוניברסיטת בן-גוריון Entanglement, thermodynamics & area Series of papers with Amos Yarom, BGU (also David Oaknin, UBC) hep-th/0302186 + to appear • Entanglement & area thermodynamics of Rindler space • Entanglement & area • Entanglement & dimensional reduction (holography) sorry, not today!

22. out in out in For half space Heff=HRindler , HRindler= boost

23. הפוך

24. (DEV)2 • System in an energy eigenstate  energy does not fluctuate • Energy of a sub-system fluctuates “Entanglement energy” fluctuations Connect to Rindler thermodynamics

25. EV= For free fields

26. X For a massless field Vanishes for the whole space! Geometry F(x) Operator

27. F(x) = F(x) UV cutoff!! In this example Exp(-p/L)

28. For half space

30. Rindler specific heat @ h=0

31. E+ = …  contributions from the near horizon region

32. Other shapes y t z Heffcomplicated, time dependent, no simple thermodynamics, area dependence o.k. For area thermodynamics need – Thermofield double

33. |0> is not necessarily an eigenstate of |0> is an entnangled state w.r.t. V Show: Entanglement and area Non-extensive!, depends on boundary (similar to entanglement entropy)

34. Proof:

35. is linear in boundary area Show that R is the radius of the smallest sphere containing V

36. Need to evaluate • Ia  ka • General cutoff Numerical factors depend on regularization

37. V F(x) DV(x)= (DEV)2for a d-dimensional sphere

38. Kd K27 =

39. Fluctuations live on the boundary V2 V1 V1 V1 V2 V3 Covariance

40. DE The “flower” Circles 5 < R < 75 R=40, dR=4, J R=20, dR=2, J R=10, dR=1, J Increasing m

41. Express as a double derivative and convert to a boundary expression Boundary theory ? This is possible iff which is generally true for operators of interest

42. di+dj = 2  logarithmic di+dj = d  d-function

43. Boundary* correlation functions Show (massless free field, V half space, large # of fields N)

44. First, n-point functions of single fields

45. Then, show that in the large N limit equality holds for all correlation functions Only contribution in leading order in N comes from

46. Summary • Entanglement & area thermodynamics of Rindler space • Entanglement & area • Entanglement & dimensional reduction