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.

Publications

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

Coming soon:

A New Continuum Limit of Matrix Models

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?

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

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

Common features Quantum theory is defined by a Path integral over

Euclidean metrics modulo diffeomorphisms

Crucial difference Euclidean 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

Example

The disc does not a globally Lorentzian metric

time

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 CDTAmbjorn, Loll, Westra, Zohren arXiv:0709.2784 JHEP 0712:017,2007

A simple amplitude

Feynman diagram

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

A String Field Theory based on Causal Dynamical Triangulations arXiv:0802.0719

String Field Theory (SFT)

Loop creation and annihilation operators

Schwinger Dyson equations:

taking derivatives of the partition function w.r.t. J

The SFT Hamiltonian

propagate split merge end

Schwinger Dyson equation

SFT = Matrix Model?!

Schwinger Dyson eqs. of the string field theory = loop equations of a matrix model:

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

arXiv:0804.0252

Stochastic quantization

Gives time dependent versions of the loop equations

White noise

Gaussian correlations

“Stochastic quantization of the causal matrix model”

in preparation

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

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

FP Hamiltonian The matrix Fokker Planck Hamiltonian:

Loop variables: SFT

Change to “loop variables”:

the Schrödinger functional

The matrix Fokker Planck Hamiltonian:

Loop space FP Hamiltonian

CDT: a new continuum limit for the one matrix model

N N+1 N+1

N

gs

A New Continuum Limit of Matrix Models

coming soon

Discrete sft or loop equation

N N+1 N+1

N

gs

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:

The disc function

N N+1 N+1

N

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

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

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

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...