1. Quantum Gravity and Quantum Entanglement (lecture 2) Dmitri V. Fursaev Joint Institute for Nuclear Research Dubna, RUSSIA Talk is based on hep-th/0602134 hep-th/0606184 Dubna, July 26, 2007 Helmholtz International Summer School on Modern Mathematical Physics Dubna July 22 – 30, 2007

2. definition of entanglement entropy

3. some results of 1 st lecture entanglement entropy in relativistic QFT’s path-integral method of calculation of entanglement entropy entropy of entanglement in a fundamental gravity theory -the value of the entropy is given by the “Bekenstein- Hawking formula” (area of the surface playing the role of the area of the horizon)

4. effective action approach to EE in a QFT -effective action is defined on manifolds with cone-like singularities - “inverse temperature” - “partition function”

6. effective action on a manifold with conical singularities is the gravity action (even if the manifold is locally flat) curvature at the singularity is non-trivial: derivation of entanglement entropy in a flat space has to do with gravity effects!

7. entanglement entropy in a fundamental theory

8. CONJECTURE (Fursaev, hep-th/0602134) - entanglement entropy per unit area for degrees of freedom of the fundamental theory in a flat space

9. Open questions: ● Does the definition of a “separating surface” make sense in a quantum gravity theory (in the presence of “quantum geometry”)? ● Entanglement of gravitational degrees of freedom? ● Can the problem of UV divergences in EE be solved by the standard renormalization prescription? What are the physical constants which should be renormalized? the geometry was “frozen” till now:

10. assumption the Ising model: “fundamental” dof are the spin variables on the lattice low-energies = near-critical regime low-energy theory = QFT (CFT) of fermions

11. at low energies integration over fundamental degrees of freedom is equivalent to the integration over all low energy fields, including fluctuations of the space-time metric

12. This means that: (if the boundary of the separating surface is fixed) the geometry of the separating surface is determined by a quantum problem fluctuations of are induced by fluctuations of the space-time geometry

13. entanglement entropy in the semiclassical approximation a standard procedure

14. fix n and “average” over all possible positions of the separating surface on - entanglement entropy of quantum matter - pure gravitational part of entanglement entropy - some average area

15. “Bekenstein-Hawking” formula for the “gravitational part” of the entropy Note: - the formula says nothing about the nature of the degrees of freedom - “gravitational” entanglement entropy and entanglement entropy of quantum matter fields (EE of QFT) come together; - EE of QFT is a quantum correction to the gravitational part; -the UV divergence of EE of QFT is eliminated by renormalization of the Newton coupling;

16. renormalization the UV divergences in the entropy are removed by the standard renormalization of the gravitational couplings; the result is finite and is expressed entirely in terms of low-energy variables and effective constants like G

17. what are the conditions on the separating surface?

18. conditions for the separating surface the separating surface is a minimal (least area) co-dimension 2 hypersurface

19. - induced metric on the surface - normal vectors to the surface - traces of extrinsic curvatures Equations

20. NB: we worked with Euclidean version of the theory (finite temperature), stationary space-times was implied; In the Lorentzian version of the theory space-times: the surface is extremal; Hint: In non-stationary space-times the fundamental entanglement may be associated to extremal surfaces A similar conclusion in AdS/CFT context is in (Hubeny, Rangami, Takayanagi, hep-th/0705.0016)

21. a Killing vector field - a constant time hypersurface (a Riemannian manifold) is a co-dimension 1 minimal surface on a constant-time hypersurface Stationary spacetimes: a simplification the statement is true for the Lorentzian theory as well !

22. the black hole entropy is a particular case for stationary black holes the cross-section of the black hole horizon with a constant-time hypersurface is a minimal surface: all constant time hypersurfaces intersect the horizon at a bifurcation surface which has vanishing extrinsic curvatures due to its symmetry

23. remarks ● the equation for the separating surface ㅡ may have a different form in generalizations of the Einstein GR (the dilaton gravity, the Gauss-Bonnet gravity and etc) ● one gets a possibility to relate variations of entanglement entropy to variations of physical observables ● one can test whether EE in quantum gravity satisfy inequalities for the von Neumann entropy

24. some examples of variation formulae for EE - change of the entropy per unit length (for a cosmic string) - string tension -change of the entropy under the shift of a point particle -mass of the particle - shift distance

25. subadditivity strong subadditivity equalities are applied to the von Neumann entropy and are based on the concavity property check of inequalities for the von Neumann entropy

26. entire system is in a mixed state due to the presence of a black hole B 2 1 black hole Araki-Lieb inequality: - entropy of the entire system

27. strong subadditivity: a b cd f ab cd f 12

28. rest of the talk ● the Plateau problem ● entanglement entropy in AdS/CFT: “holographic formula” ● some examples: EE in SYM and in 2D CFT’s

29. the Plateau Problem (Joseph Plateau, 1801-1883) It is a problem of finding a least area surface (minimal surface) for a given boundary soap films: - the mean curvature - surface tension -pressure difference across the film - equilibrium equation

30. the Plateau Problem there are no unique solutions in general

31. the Plateau Problem simple surfaces The structure of part of a DNA double helix catenoid is a three-dimensional shape made by rotatingdimensionalshape a catenary curve (discovered by L.Euler in 1744)catenarycurve helicoid is a ruled surface, meaning that it is a trace of a lineruled surface

32. the Plateau Problem Costa’s surface (1982) other embedded surfaces (without self intersections)

33. the Plateau Problem A minimal Klein bottle with one end Non-orientable surfaces A projective plane with three planar ends. From far away the surface looks like the three coordinate plane

34. the Plateau Problem Non-trivial topology: surfaces with hadles a surface was found by Chen and Gackstatter a singly periodic Scherk surface approaches two orthogonal planes

35. the Plateau Problem a minimal surface may be unstable against small perturbations

36. more evidences: entanglement entropy in QFT’s with gravity duals

37. Consider the entanglement entropy in conformal theories (CFT’s) which admit a description in terms of anti-de Sitter (AdS) gravity one dimension higher N=4 super Yang-Mills 

38. Holographic Formula for the Entropy 4d space-time manifold (asymptotic boundary of AdS) (bulk space) separating surface extension of the separating surface in the bulk (now: there is no gravity in the boundary theory, can be arbitrary)

39. Holographic Formula for the Entropy Ryu and Takayanagi, hep-th/0603001, 0605073 CFT which admit a dual description in terms of the Anti-de Sitter (AdS) gravity one dimension higher Let be the extension of the separating surface in d-dim. CFT 1) is a minimal surface in (d+1) dimensional AdS space 2) “holographic formula” holds: is the area of is the gravity coupling in AdS

40. a simple example 2 2 1 – is IR cutoff

41. the holographic formula enables one to compute entanglement entropy in strongly coupled theories by using geometrical methods

42. entanglement in 2D CFT ground state entanglement for a system on a circle is the length of c – is a central charge

43. example in d=2: CFT on a circle - AdS radius A is the length of the geodesic in AdS - UV cutoff -holographic formula reproduces the entropy for a ground state entanglement - central charge in d=2 CFT

44. Some other developments ● D.Fursaev, hep-th/0606184 (proof of the holographic formula) R. Emparan, hep-th/0603081 (application of the holographic formula to interpretation of the entropy of a braneworld black hole as an entaglement entropy) M. Iwashita, T. Kobayashi, T. Shiromizu, hep-th/0606027 (Holographic entanglement entropy of de Sitter braneworld) T.Hirata, T.Takayanagi, hep-th/0608213 (AdS/CFT and the strong subadditivity formula) M. Headrick and T.Takayanagi, hep-th/0704.3719 (Holographic proof of the strong subadditivity of entanglement entropy) V.Hubeny, M. Rangami, T.Takayanagi, hep-th/0705.0016 (A covariant holographic entanglement entropy proposal )

45. conclusions and future questions there is a deep connection between quantum entanglement and gravity which goes beyond the black hole physics; entanglement entropy of fundamental degrees of freedom in quantum gravity is associated to the area of minimal surfaces; more checks of entropy inequalities are needed to see whether the conjecture really works; variation formulae for entanglement entropy, relation to changes of physical observables (analogs of black hole variation formulae)