a causal alternative to the c=0 string
DESCRIPTION
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 PresentationTRANSCRIPT
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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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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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An example
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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...