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

Publications and preprints Putting a cap on causality violations in CDT arXiv: JHEP 0712:017,2007 A String Field Theory based on Causal Dynamical Triangulations arXiv: JHEP 0805:032,2008 Topology change in causal quantum gravity arXiv: Conf. Proc. JGRG17 Nagoya, Japan A Matrix Model for 2D Quantum Gravity defined by Causal Dynamical Triangulations arXiv: t.a. Phys. Lett. B 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

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 Talks by J. Ambjorn and A. Goerlich

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

Geometry of DT Flat space: Positively curved space: a

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

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

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

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

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

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

The Transfer matrix The old construction of CDT

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

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

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

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

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

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

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

NEW for CDT: Loop equations

The new CDT loop equations N NN

N N+1

An example

CDT with spatial topology change N N+1 N

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

The “old” EDT matrix model N N+1 N

Our new CDT matrix model N N+1 N g

Solution of the disc-function

Continuum limit

Very different from EDT! Here both factors under square root contribute to the continuum limit: No non scaling contributions in the continuum!

Looks familiar?!

Continuum Matrix Model Close to ’t Hooft’s original idea: N controls the topological expansion only

Conclusions We have generalized CDT to include spatial topology changes String coupling constant controls spatial topology fluctuations too We have introduced more powerful techniques to derive CDT amplitudes: loop equations matrix models Our matrix model makes the relation between EDT and CDT very clear

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? Gauge-string duality at N=2?

To be continued...