The causal matrix model

Willem Westra and Stefan Zohren

Leipzig 01 04 2008

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

Two different theories??

Euclidean 2D quantum gravity

Causal2D quantum gravity

Worldsheet definition of (non critical) string theory

Toy model for 4d causal quantum gravity

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Euclidean 2D quantum gravity

Causal2D quantum gravity

Worldsheet definition of (non critical) string theory

Toy model for 4d causal quantum gravity

Two different theories?? Euclidean quantum gravity

Conformal field theory (CFT): Conformal gauge Liouville theory

Euclidean dynamical triangulations (EDT)

Matrix models Loop equations Saddle point methods Orthogonal polynomials

Causal quantum gravity

?? Propertime gauge Quantum mechanics

Causal dynamical triangulations (CDT)

Transfer matrix

Continuum formulation

Discrete formulation

Technology

The big stuff: Gravity

Einstein’s theory of general relativity, 1918

Gravity: curvature of space-time Matter curves space-time Curved space-time tells matter how to

move

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are needed to see this picture.

Gravity = Geometry

The small: Quantum mechanics

Schrödinger’s equation, 1927

Quantum Mechanics Everything that can happen...

does happen

What we see is the stuff that is

most probable to happen

big + small

gravity + quantum

curved space-time + probability

Curved space + quantum Task: count all possible curved space-times Compute the most probable ones

Solution: build space time from little “Lego blocks”

Count the number of ways the Lego blocks can be put together

1+1 dimensional space times One dimension of space One dimension of time Simplest Lego blocks: triangles

Task: count the number of ways one can glue triangles together

Counting the triangles

N N N

Counting the triangles

N N+1 N+1

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Splitting the universe

N N+1 N+1 N

The standard equation

N N+1 N

Missing a move?

N N+1 N