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

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

3. Rindler space

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

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

6. 1. In general:

8. Result inout

9. H eff – generator of time translations Time slicing the interval [0,  0 ]: 2.

10. Guess: result

11. 1. The boundary conditions are the same 2.The actions are equal 3.The measures are equal Results If Then

12. inout inout For half space H eff =H Rindler, H Rindler = boost

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

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

15. Optical metric In 4D Euclidean Rindler Compute:

16. הפוך

17. S MS S,T unitary

18. MM MM M SS M 1 o

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

22. inout inout For half space H eff =H Rindler, H Rindler = boost

23. הפוך

24. (EV)2(EV)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=EV= For free fields

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

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

28. For half space

30. Rindler specific heat @ 

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

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

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

34. Proof:

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

36. Need to evaluate I    k  General cutoff Numerical factors depend on regularization

37. F(x) (  E V ) 2 for a d-dimensional sphere V D V (x)=

38. K 27 = KdKd

39. Fluctuations live on the boundary Covariance V1V1 V2V2 V3V3 V1V1 V1V1 V2V2

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

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

42.  i +  j = 2  logarithmic  i +  j = d   -function

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

44. First, n-point functions of single fields

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

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