Gerard ’t Hooft

1. Remembering Utrecht University Sidney Coleman Gerard ’t Hooft Erice Opening Lecture August 2008

3. June 30 - July 6, 1971: Amsterdam International EPS Conference. This was the first occasion for me to present my theories about the renormalizability of non-Abelian gauge theories with spontaneous symmetry breakdown. My then advisor, Martinus Veltman, had given me the opportunity to explain this in 10 minutes to the audience. “Let me introduce you to two American gangsters”, he said to me. I was puzzled by the remark at that time, but later realized that, in Veltman’s view, anybody who ignored his work, and did physics differently, quickly turned into a gangster. Anyway, talking to these two people, Shelly Glashow and Sidney Coleman, made me realize that these must be the smartest gangsters on the planet. Sidney immediately saw the most essential ingredients of the theory. He later explained that Quantum Field Theory had been an ugly monster, until it received a kiss by our work, at which it turned into a beautiful and wealthy prince. July 11 - August 1, 1975: Erice “The Whys of Subnuclear Physics” I met Sidney many times since, and I vividly remember the lectures he gave in the 1975 Erice School. At that school, not only the ‘best student’ and ‘best secretary’ were chosen, but also ‘best lecturer’. This was Sidney Coleman, who, at the farewell party, also came as the ‘most fancy dressed’ participant, in his fluorescent pink jacket. His lectures had been about topology in QFT. Previously, when renormalizing the theory, we had always limited ourselves to perturbation expansion, where the topology is trivial.

4. Sidney would illustrate vividly what topology is, using the cord of his microphone, by which he was connected to the loudspeaker system. Winding the cord around his neck he explained what winding number was. The students actually worried that he might strangle himself, so they never forgot the definition of winding number. October 1988, Harvard During one of my later encounters at Harvard, I visited him at his home, with my two little children. There, we would play the game of “Charades”; he was very good at it. My children loved it. As emphasized by others here, Sidney was a great story teller: One of his stories was about how he rescued mankind from a pending planetary disaster, while sitting at a bar, with his friend Carl Sagan. With Sagan, Sidney shared his love for science-fiction. Behind a good glass of whiskey, or two, Sagan showed a problem to Sidney: “Suppose that there exist alien, totally unknown, life forms on Mars, or, for that matter, on the Moon as well. The Moon would soon be visited by astronauts. What should space agencies do to avoid the danger that these life forms would infect the Earth, and destroy mankind? How do we avoid contamination of one planet by another in general, when space flight becomes routine? This became a vivid discussion.

5. But a month later, Sidney was highly surprised when a manuscript arrived in his mail: Spacecraft Sterilization Standards and Contamination of Mars by S. Coleman and C. Sagan, Journal of Astronautics and Aeronautics 3, 22 (1965). The norm for the possibility of cross contamination from one celestial body to another should be less than 0.1%. Further, astronauts should be put in quarantine, and the chances of survival for every individual organism of a life form from one planet to the next should be less than one in 10,000. Now, this was the only existing publication in this field, so, NASA had to follow the advice. Committee of Space Research COSPAR decided that astronauts returning from the Moon would have to go into quarantine for a certain amount of time. → the Lunar Receiving Laboratory (LRL) in Houston, Texas Dangerous life forms were fortunately never found.

6. Some effects of in QCD Utrecht University INSTANTONS Gerard ’t Hooft Erice Interlude August 2008

7. x x x x x x x x x x x x x x boundary of Universe x We can map the gauge transformations on the boundary of space-time, but this mapping cannot continuously be extended to the interior Applying this gauge transformation to the vacuum configuration, gives a vacuum at infinity, but continuity in the fields demands a region in space-time that differs from the vacuum: the instanton.

8. + Energy 0 instanton _ Instantons can exist in Minkowski space-time as well as in Euclidean space-time, since they are topological. Instantons are time-dependent, therefore fields such as fermion (quark) fields, can have modes where the energy before and after an instanton is different. There is one mode (for each chiral flavor), that hops from the antiparticle sea to the particle states: This fermion came out of the vacuum ! time

9. time Now, consider Euclidean time Therefore, this effect happens if and only if there is a bounded solution of the Dirac equation in Euclidean space:

10. L R L instanton R L R Under parity, instanton goes to anti-instanton, and left goes to right. Therefore left-helicity fermions do the opposite of right helicity fermions: If a left-helicity appears, then a right-helicity disappears, and vice versa. One instanton causes one flip left → right for every flavor: up down strange

11. Effective instanton action for fermions: Note: also singlet in color space In QCD, this has important effects for ― pseudoscalar mesons ― scalar mesons

12. Pseudoscalars: s Exactly the quantum numbers of the η mass term; it violates chiral symmetry because left ↔ right

13. Scalars: why does the scalar meson appear to be lighter than the non-strange mesons? Is it a “tetra-quark”? No! But it mixes with the tetraquark:

14. GtH, Isidori, Maiani, Polosa and Riquer, arXiv:0801.2288, To appear in Phys Lett B.

15. Gerard ‘t Hooft Erice Lectures August 2008 Utrecht University CRYSTALLINE GRAVITY

16. What will space-time look like at scales much smaller than the Planck length? No physical degrees of freedom anymore Gravity may become topological It may well make sense to describe space-time there as if made of locally flat pieces glued together (as in “dynamical triangulation” or “Regge calculus”) The dynamical degrees of freedom are then pointlike defects The dynamical degrees of freedom are then line-like defects [??]

17. The vacuum has This theory will have a clear vacuum state: flat Minkowski space-time “Matter fields” are identified with the defects. There are no gravitons: all curvature comes from the defects, therefore: Gravity = matter. Furthermore: What are the dynamical rules? What is the “matter Lagrangian”?

18. First do this in 2 + 1 DIMENSIONS

19. x x x x A gravitating particle in 2 + 1 dimensions: A A moving particle in 2 + 1 dimensions: A′ A A′

20. A many-particle universe in 2+1 dimensions can now be constructed by tesselation (Staruszkiewicz, G.’t H, Deser, Jackiw, Kadar) There is no local curvature; the only physical variables are N particle coordinates, with N finite.

21. 2 + 1 dimensional cosmology is finite and interesting Quantization is difficult. CRUNCH BANG

22. How to generalize this to 3 +1 (or more) dimensions ? straight strings are linelike grav. defects

23. In units such that : Deficit angle → Calculation deficit angle:

24. A static universe would contain a large number of such strings But what happens when we make them move ?

25. One might have thought: New string ?? But this cannot be right !

26. Static string: pure rotation, Lorentzboost: Movingstring: Holonomies on curves around strings: Q : member of Poincaré group: Lorentz trf. plus translation C All strings must have holonomies that are constrained by

27. C 1 2 1 2 1 2 In general, Tr (C) = a + ib can be anything. Only if the angle is exactly 90° can the newly formed object be a string. What if the angle is not 90°?

28. a d b c Can this happen ? This is a delicate exercise in mathematical physics Answer: sometimes yes, but sometimes no !

29. Notation and conventions:

30. Two quadratic conditions remain: Re (det( )) = 1 Im (det( )) = 0 The Lorentz group elements have 6 degrees of freedom. Im(Q) = 0 is one constraint for each of the 4 internal lines → We have a 2 dimensional space of solutions. Strategy: write in SL(2,C) Im(Tr(Q)) = 0 are four linear constraints on the coefficients a, b . |Re(Tr(Q))| < 2 gives a 4 - dimensional hypercube.

31. In some cases, the overlap is found to be empty ! • relativistic velocities • sharp angles, • and others ...

32. N 2 1 where we expect free parameters. We were unable to verify wether such solutions always exist In those cases, one might expect

33. Assuming that some solutions always exist, we arrive • at a dynamical, Lorentz-invariant model. • When two strings collide, new string segments form. • 2. When a string segment shrinks to zero length, • a similar rule will give newly formed string segments. • 3. The choice of a solution out of a 2- or more • dimensional manyfold, constitutes the • matter equations of motion. • However, there are no independent gravity • degrees of freedom. “Gravitons” are composed of • matter. • 5. it has not been checked whether the string constants • can always be chosen positive. This is probably • not the case. hence there may be negative energy.

34. The model cannot be quantized in the usual • fashion: • a) because the matter - strings tend to generate • a continuous spectrum of string constants; • b) because there is no time reversal (or PCT) • symmetry; • c) because it appears that strings break up, • but do not often rejoin. • However, there may be interesting ways to • arrive at more interesting schemes. For instance:

35. crystalline gravity ❖ Replace 4d space-time by a discrete, rectangular lattice: This means that the Poincaré group is replaced by one of its discrete subgroups Actually, there are infinitely many discrete subgroups of the Poincaré group

36. One discrete subgroup: Compose SO(3) rotations over 90° and powers of

37. But, there are (probably infinitely) many more scarce discrete subgroups. Take in SL(2,C) with and are rotated by 90° along z and y axis. Then, generically, diverge as stronger and stronger boosts.

38. ❖add defect lines : A′ identify A

39. N 2 1 ❖postulate how the defect lines evolve: Besides defect angles, surplus angles will probably be inevitable.

40. ❖pre - quantize... ❖quantize... Define the basis of a Hilbert space as spanned by ontological states / equivalence classes .. Diagonalize the Hamiltonian (evolution operator) to find the energy eigenstates ... and do the renormalization group transformations to obtain and effective field theory with gravity At this point, this theory is still in its infancy.

41. THE END

43. Utrecht University DISCRETE STRINGS IN GRAVITY Gerard ’t Hooft Meeting on Integrability in Gauge and String Theory Utrecht, August 11-15, 2008

44. Utrecht University A DISCRETE MODEL of GRAVITY with CAUSALITY Gerard ’t Hooft Perimeter Institute, 1/5/08

45. Utrecht University A DISCRETE MODEL of GRAVITY with CAUSALITY Gerard ’t Hooft Stueckelberg Lectures Pescara, July 7-19, 2008