1. A causal alternative to the c=0 string Jan AmbjornNiels Bohr and Univ. Utrecht Willem Westra Univ. Of Iceland Stefan Zohren Imperial College London Renate LollUniv. Utrecht Yoshiyuki WatabikiTokyo Inst. Tech. Zakopane 16 06 2008

2. Publications and preprints Putting a cap on causality violations in CDT arXiv:0709.2784 JHEP 0712:017,2007 A String Field Theory based on Causal Dynamical Triangulations arXiv:0802.0719 Topology change in causal quantum gravity arXiv:0802.0896 Conference proceedings of JGRG17 Nagoya, Japan A Matrix Model for 2D Quantum Gravity defined by Causal Dynamical Triangulations arXiv:0804.0252 provisional titles for papers to come that are covered in this talk: Loop equations for CDT The causal continuum limit for matrix model quantum gravity

3. What and Why? Two dimensional quantum gravity Non critical (bosonic) string theory = Strings living in target space with D≠26 Toy model for 4d quantum gravity

4. How? Dynamical triangulations (DT): Path integral over geometries  Discrete statistical sum over triangulations  Manifold is discretized with equilateral triangles  Geometry is encoded in the way triangles are glued together

5. Geometry of DT Flat space: Positively curved space: a

6. Two different theories? Euclidean 2D quantum gravity Causal 2D quantum gravity

7. Causal 2D quantum gravity Euclidean 2D quantum gravity Not on the discrete level

8. Causal 2D quantum gravity Euclidean 2D quantum gravity On the discrete level: Euclidean DT  Causal DT

9. Discrete: Euclidean DT  Causal DT On the discrete level: EDT = CDT + spatial topology change

10. In the continuum: EDT ≠CDT + spatial topology change Continuum: Euclidean DT ≠ Causal DT

11. EDT Hausdorff dimension = 4 Time scales non canonically Spatial topology changes are everywhere and dominate the dynamics No single string states Continuum: CDT is better behaved CDT Hausdorff dimension = 2 Time is measured in seconds as should be Spatial topology changes controlled by a coupling constant Fock space of multistring states can be explicitly defined

12. The Transfer matrix The old construction of CDT

13. Causal quantum gravity What do we compute? The disc function W(L,T) L T Probability amplitude

14. Causal Dynamical Triangulations Discrete path integral Transfer matrix The disc function T=1

15. Causal Dynamical Triangulations Discrete path integral Transfer matrix The disc function T=2

16. Causal Dynamical Triangulations Discrete path integral Transfer matrix The disc function T=3

17. Causal Dynamical Triangulations Discrete path integral Transfer matrix The disc function T=4

18. Causal Dynamical Triangulations Discrete path integral Transfer matrix The disc function T=5

19. Causal Dynamical Triangulations Discrete path integral Transfer matrix The disc function T=6

20. NEW for CDT: Loop equations

21. The new CDT loop equations N NN

22. N N+1

23. An example

77. CDT with spatial topology change N N+1 N

78. The coupling constant N N+1 g Coupling constant important to obtain CDT N

79. Scaling coupling constant N+1 N g Non scaling coupling constant  EDT limit If g = a 3 g continuum  CDT with topology change

80. Let’s compare to EDT

81. The “old” EDT matrix model N N+1 N

82. Our new CDT matrix model N N+1 N

83. Scaling coupling constant The continuum limit of our new loop equation can be described by a matrix model with a continuum interpretation Completely unlike the continuum limit of the EDT loop equations, they instead give the KPZ equations

84. Scaling coupling constant The new continuum limit can also be obtained from a matrix model with a conventional discrete interpretation In fact any potential with a linear term in the potential and positive powers can give the new limit This proves universality

85. Conclusions We have generalized CDT to include spatial topology changes The essential ingredient is a coupling constant to control the topology fluctuations We have introduced more powerful techniques to derive CDT amplitudes: loop equations matrix models Our CDT loop equations completely clarify the relation between EDT and CDT

86. Outlook The more powerful methods allow us to study matter coupling to CDT analytically Ising model Minimal models Scalar fields.... Coupling scalar field = adding a target space what are the implications to noncritical string theory?

87. To be continued...