THE CONFORMAL GROUP

1. Utrecht University PROBING THE SHORT DISTANCE STRUCTURE OF CANONICAL QUANTUM GRAVITY using THE CONFORMAL GROUP Gerard ’t Hooft Spinoza Institute, Utrecht University

2. The Schwarzschild Solution to Einstein’s equations Karl Schwarzschild 1916 “Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie”

3. “Time”stands still at the horizon So, one cannot travel from one universe to the other Black Hole or wormhole? UniverseI Universe II

5. Stephen Hawking’s great discovery: the radiating black hole

6. Small region near black hole horizon: Rindler space A quantum field in space splits into two parts, and . The vacuum in flat space corresponds to an entangled Rindler state: time II I space

7. Entropy = ln ( # states ) = ¼ (area of horizon) Are black holes just “elementary particles”? Are elementary particles just “black holes”? Imploding matter Hawking particles Black hole “particle”

8. Black hole complementarity

9. As seen by distant observer As experienced by astro- naut himself Time stands still at the horizon Continues his way through They experience time differently. Mathematics tells us that, consequently, they experience particles differently as well

10. Black hole complementarity An observer going in experiences the original vacuum, Hence sees no Hawking particles, but does observe objects behind horizon An observer staying outside sees no objects behind horizon, but does observe the Hawking particles. They both look at the same “reality”, so there should exist a mapping from one picture to the other and back.

11. Hawking radiation Penrose diagram (according to Hawking) singularity imploding matter

12. Cauchy surface out horizon out out out singul -arity in in imploding matter imploding matter

13. out decay out in imploding matter implosion

14. Hawking radiation imploding matter

15. Hawking radiation imploding matter Penrose diagram ?

16. Extreme version of complementarity Ingoing particles visible; Horizon to future, Hawking particles invisible time space

17. Extreme version of complementarity Outgoing particles visible; Horizon to past, Ingoing particles invisible time space

18. Starting principle: causality is the same for all observers This means that the light cones must be the same Light cone: The two descriptions may therefore differ in their conformal factor. The only unique quantity is

19. Invariance under scale transformations May serve as an essential new ingredient to quantize gravity describes alllightlike geodesics describes scales

20. The transformations that keep the equation unchanged are the conformaltransformations.

21. out The transform-ation from the ingoing matter description “in ” to the outgoing matter description “out ” is a conformal transform- ation in

22. In one conformal frame, an observer sees Hawking radiation, and in an other (s)he does not. For the black hole, the transformation “in” ⇔ “out” is no longer a conformal one when we include in- and out going matter. But by using local conformal transformations, which generate curvature, one can describe all of space-time in one coordinate frame.

23. out out flat Schwarzschild Schwarzschild time in in space

24. THE LOCAL CONFORMAL GROUP IN CANONICAL QUANTUM GRAVITY

25. Exact local conformal invariance emerges formally in canonical quantum gravity: does not depend on ω , therefore is conformallyinfariant ! But this would make renormalizable !!? Unfortunately no !? The conformal anomaly

27. The integral over ω = 1+iα can be done “exactly” (determinant) but it diverges ! Divergent part (pole part): let n = 4 - ε̶

28. The one loop divergence in QFT

29. is conformally invariant in 4 dimensions ! But At n ≠ 4 this “local term” is notconformally invariant.

30. Lessons in conformal invariance Riemann curvature Ricci curvature Ricci scalar Weyl curvature Under the transformation the Riemann and Ricci curvature transform non-trivially, but the Weyl curvature is conformal:

31. Furthermore Is a pure derivative, hence the integral is a topological invariant . Use this to rewrite

32. Under a conformal transformation is invariant and is invariant In our effective action (after integrating out ω ), the overall coefficient is infinite. Hence, it has to be renormalized. A non-local, finite effective action will then result. The (massive) scalar matter field will require a quartic renormalized interaction – fine ! But how to renormalize the Weyl action ?????????

33. Renormalizing the Weyl action would violate unitarity !!!! It introduces unacceptable fourth derivative terms ! Not allowed !?!?!?!? A proposal: don’t renormalize the effective Weyl action. Can one cancel that infinity ? No! Coefficients: grav. ω field: Scalars: Dirac Spinors: Vector fields:

34. The coefficient in front of the entire action is the inverse quantum coupling constant for the fields in question. If this is infinite, the quantum constant is formally zero. But that simply means that these field, here the field, will behave classically ! This may be interesting, but is it physically possible? Failure of action = reaction The classical field affects the quantized matter field, but not vice versa … This would then be a problem. Unless …

35. The classical field equations from the Weyl action: conformal transformation “massless” matter This can be interpreted as a classical background metric; indeed, this was the way we handled In the black hole case …

36. To handle the conformal divergence Counter term 2 point amplitude small distances large distances Landau ghost Large “classical” action

37. THE END