a causal alternative to the c=0 string

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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 16 06 2008. Publications and preprints. - PowerPoint PPT Presentation

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Page 1: A causal alternative  to the c=0 string
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A causal alternative to the c=0 string

Jan Ambjorn Niels Bohr and Univ. Utrecht

W.W. Univ. Of Iceland

Stefan Zohren Imperial College London

Renate Loll Univ. Utrecht

Yoshiyuki Watabiki Tokyo Inst. Tech.

Zakopane 16 06 2008

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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 JHEP 0805:032,2008

Topology change in causal quantum gravityarXiv:0802.0896 Conf. Proc. JGRG17 Nagoya, Japan

A Matrix Model for 2D Quantum Gravity defined by Causal Dynamical Triangulations arXiv:0804.0252 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

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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

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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

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Geometry of DT

Flat space:

Positively curved space:

a

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Two different theories?

Euclidean

2D quantum gravity

Causal2D quantum gravity

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Causal2D quantum gravity

Euclidean

2D quantum gravity

Not on the discrete level

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Causal2D quantum gravity

Euclidean

2D quantum gravity

On the discrete level:Euclidean DT Causal DT

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Discrete:Euclidean DT Causal DT

On the discrete level:

EDT = CDT + spatial topology change

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In the continuum:

EDT ≠CDT + spatial topology change

Continuum:Euclidean DT ≠ Causal DT

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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

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The Transfer matrix

The old construction of CDT

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Causal quantum gravity What do we compute?

The disc function W(Z,T)

Z

T

Probability amplitude

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Causal Dynamical Triangulations Discrete path integral Transfer matrix

The disc functionT=1

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Causal Dynamical Triangulations Discrete path integral Transfer matrix

The disc functionT=2

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Causal Dynamical Triangulations Discrete path integral Transfer matrix

The disc functionT=3

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Causal Dynamical Triangulations Discrete path integral Transfer matrix

The disc functionT=4

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Causal Dynamical Triangulations Discrete path integral Transfer matrix

The disc functionT=5

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Causal Dynamical Triangulations Discrete path integral Transfer matrix

The disc functionT=6

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NEW for CDT:

Loop equations

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The new CDT loop equations

N N N

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The new CDT loop equations

N N+1 N+1

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An example

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CDT with spatial topology change

N N+1 N+1

N

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N+1

The coupling constant

N N+1 g

Coupling constant important to obtain CDT

N

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The “old” EDT matrix model

N N+1

N

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Our new CDT matrix model

N N+1

N

N+1 g

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Solution of the disc-function

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Continuum limit

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Continuum limit

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Very different from EDT! Here both factors under square root contribute to the continuum limit:

No non scaling contributions in the continuum!

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Looks familiar?!

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Continuum Matrix Model

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

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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

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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?

Gauge-string duality at N=2?

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To be continued...