2. Gerard ’t Hooft Utrecht April 15, 2008 Utrecht University

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

4. A many-particle universe in 2+1 dimensions can now be constructed by tesselation ( Staruszkiewicz, G.’tH, Deser, Jackiw, Kadar ) There is no local curvature; the only physical variables are N particle coordinates, with N finite.

5. 2 + 1 dimensional cosmology is finite and interesting BANG CRUNCH Quantization is difficult.

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

7. Calculation deficit angle: Deficit angle → I n units such that :

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

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

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

11. C 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°? 1 21 2 12

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

13. Notation and conventions:

14. 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 #1: write in SL(2,C) Im(Tr(Q)) = 0 are four linear constraints on the coefficients a, b. |Re(Q)| < 2 gives a 4 - dimensional hypercube. Two quadratic conditions remain: Re (det( )) = 1 Im (det( )) = 0

16. It is not clear whether or when these two quadratic curves have points within our hypercube. but we can do better. 1.Choose the Lorentz frame where string #1 is static and in the z direction: In the frame string a is static as well. We must impose: because is the boost giving the velocity of the joint.

17. 2.next, go to the frame where string #3 is static and in the z direction: There, one must have: If one would choose and, then this gives 4 new conditions, of which only 3 are new, since we knew already that the determinant is real. One would expect this to fix the coefficients a, b, p and q. However, something else happens... !

18. We found that for all initial string configurations there are four coefficients, A, B, C, D, depending only on the external string holonomies, such that So one cannot choose freely ! Instead, the coefficients a, b,... are still underdetermined. 3. This leaves us the option to choose instead. One finds (from the symmetry of the system):

19. If and only if all μ ’s are positive, the strings and the joints between the strings ar all moving with subluminal velocities. This is necessary for consistency of the scheme. In we must demand that this is consistent. Consequently, the case must be excluded. But we found that this will happen if strings move fast !

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

21. Assuming that solutions always exist, we have a model. 1.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. 4.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.

22. 6.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. 7.However, there may be interesting ways to arrive at more interesting schemes. For instance, by limiting ourselves to a discrete subgroup of the Poincaré group: a (relativistic) crystal group.

23. Then, space - time is viewed as a crystalline lattice inbedded in a curved Riemann space. Such models may well be quantizable, if we allow ourselves a deterministic quantization scheme. In crystalline gravity, this lattice has defect lines (and perhaps also defect planes) that propagate according to locally causal laws of physics. The details of such rules are difficult to formulate, but this appears to be a technical rather than a fundamental problem.