1. A new continuum limit for the one matrix model
String Field Theory and Matrix Models for Causal Dynamical Triangulations
A new continuum limit for the one matrix model
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.

2. Publications Putting a cap on causality violations in CDT
arXiv: JHEP 0712:017,2007
A String Field Theory based on Causal Dynamical Triangulations
arXiv:
Topology change in causal quantum gravity
arXiv: Conference proceedings of JGRG17 Nagoya, Japan
A Matrix Model for 2D Quantum Gravity defined by Causal Dynamical Triangulations arXiv:
Coming soon:
A New Continuum Limit of Matrix Models

3. Random Surfaces, why? Toy model for quantum gravity in 3+1
How to deal with theories with coordinate transformations as a gauge symmetry?
What are the diffeomorphism invariant observables?
Toy model for non critical string theory
Strings are random surfaces coupled to scalar fields
Is it possible to construct a consistent string theory in D<26?
Can one break the c=1 barrier?

4. Type 1 Euclidean random surfaces Continuum:
Polyakov’s induced action `81
Conformal gauge: Liouville conformal field theory
Recent advances eg. Fateev Zamolodchikov2 Teschner
Discrete:
Dynamical triangulations (DT) or random planar maps
Generating functions: Tutte `62
Matrix integrals: Brezin,Itzykson, Parisi, Zuber `78
using `t Hooft’s large N limit `74

5. Type 2 Causal random surfaces Continuum: Polyakov’s induced action `81
Propertime gauge analysis Nakayama `93
Discrete:
Ambjorn & Loll: Causal Dynamical Triangulations (CDT) `98
Originally solved by transfer matrix techniques
Matrix integrals:
A new scaling of matrix models,
Ambjorn, Loll, Watabiki, Westra, Zohren

6. Common features
Quantum theory is defined by a Path integral over Euclidean metrics modulo diffeomorphisms

7. Crucial difference Euclidean random surfaces Causal random surfaces
All two dimensional metrics contribute to the path integral
Causal random surfaces
Only geometries that can be obtained from a Lorentzian geometry by a Wick rotation are included in the path integral
By comparing the results of the causal and Euclidean theories one concludes that the class of strictly causal geometries is much smaller than the class of all random surfaces
The two theories belong to a different universality class

8. Example
The disc does not a globally Lorentzian metric
time

9. Our new idea
We reintroduce the geometries that do not admit a globally Lorentzian metric in the causal path integral
But..
A coupling constant is associated with the signature violations
Putting a cap on causality violations in CDT
Ambjorn, Loll, Westra, Zohren
arXiv: JHEP 0712:017,2007

10. A simple amplitude

11. Feynman diagram

12. Hamiltonian formulation
A String Field Theory based on Causal Dynamical Triangulations
arXiv:
Hamiltonian formulation
String field theory (SFT)
Describes time evolution of spatial loops
Time coordinate is defined as the geodesic distance to the initial loop
Four processes contribute to the time evolution:
propagate
split
merge
end

13. String Field Theory (SFT)
Loop creation and annihilation operators
Schwinger Dyson equations:
taking derivatives of the partition function w.r.t. J

14. The SFT Hamiltonian
propagate
split
merge
end

15. Schwinger Dyson equation

16. A Matrix Model for 2D Quantum Gravity defined by Causal Dynamical Triangulations
arXiv:
SFT = Matrix Model?!
Schwinger Dyson eqs. of the string field theory = loop equations of a matrix model:

17. Stochastic quantization
“Stochastic quantization of the causal matrix model”
in preparation
Stochastic quantization
Gives time dependent versions of the loop equations
White noise
Gaussian correlations

18. Stochastic quantization II
One “quantizes the equations of motion”
Planck’s constant determines
“how easy the system can go off shell”
In our case the new coupling constant takes the role of Planck’s constant

19. Fokker Planck = Schrödinger
From Langevin to Fokker Planck:
Change to variables in stochastic calculus: Itô’s Lemma
Itô’s Lemma + averaging:
from the Langevin equation to the
Schrödinger = time dependent Schwinger Dyson equation

20. FP Hamiltonian
The matrix Fokker Planck Hamiltonian:

21. Loop variables: SFT The matrix Fokker Planck Hamiltonian:
Change to “loop variables”:
the Schrödinger functional

22. Loop space FP Hamiltonian

23. CDT: a new continuum limit for the one matrix modelgs
A New Continuum Limit of Matrix Models
coming soon

24. Discrete sft or loop equationgs

25. Continuum limit
After the new double scaling limit one recovers the SFT that we found by sewing CDT amplitudes at the beginning of the talk
Essentially new aspect of the scaling:

26. The disc function N
N+1

27. Critical behaviour: very different from the standard continuum limit
Discrete disc function
1) Both factors under the square root become critical
2) v`(z) also becomes critical at the same point

28. Back to pure CDT With spatial topology change: the matrix model result
If we fix the spatial topology (S1):
take Gs to zero: cut shrinks to a pole: CDT

29. Conclusion 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

30. 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 space
what are the implications to noncritical string theory?

31. To be continued...