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
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
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
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
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(Z,T)
Z
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
The “old” EDT matrix model
N N+1
N
Our new CDT matrix model
N N+1
N
N+1 g
Solution of the disc-function
Continuum limit
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 spacewhat are the implications to noncritical string
theory?
Gauge-string duality at N=2?
To be continued...