a causal alternative to the c=0 string jan ambjornniels bohr and univ. utrecht willem westra univ....
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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...