A causal alternative to the c=0 string

Jan Ambjorn Niels Bohr and Univ. Utrecht

Willem Westra Univ. Of Iceland

Stefan Zohren Imperial College London

Renate Loll Univ. Utrecht

Yoshiyuki Watabiki Tokyo Inst. Tech.

Zakopane 16 06 2008

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 TriangulationsarXiv:0802.0719

Topology change in causal quantum gravityarXiv: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

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

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

Causal2D quantum gravity

Causal2D quantum gravity

Euclidean

2D quantum gravity

Not on the discrete level

Causal2D 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(L,T)

L

T

Probability amplitude

Causal Dynamical Triangulations Discrete path integral Transfer matrix

The disc functionT=1

Causal Dynamical Triangulations Discrete path integral Transfer matrix

The disc functionT=2

Causal Dynamical Triangulations Discrete path integral Transfer matrix

The disc functionT=3

Causal Dynamical Triangulations Discrete path integral Transfer matrix

The disc functionT=4

Causal Dynamical Triangulations Discrete path integral Transfer matrix

The disc functionT=5

Causal Dynamical Triangulations Discrete path integral Transfer matrix

The disc functionT=6

NEW for CDT:

Loop equations

The new CDT loop equations

N N N

The new CDT loop equations

N N+1 N+1

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CDT with spatial topology change

N N+1 N+1

N

N+1

The coupling constant

N N+1 g

Coupling constant important to obtain CDT

N

Scaling coupling constant

N+1N N+1N+1g

Non scaling coupling constant EDT limit

If g = a3 gcontinuum CDT with topology change

Let’s compare to EDT

The “old” EDT matrix model

N N+1

N

Our new CDT matrix model

N N+1

N

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

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

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

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 spacewhat are the implications to noncritical string

theory?

To be continued...