Phase Structure of Causal Dynamical Triangulations.Pdf

Phase Structure of Causal Dynamical Triangulations

Andrzej Görlich

Niels Bohr Institute, University of Copenhagen

Jena, November 5th, 2015

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 1 / 27 1 Introduction to CDT

2 Phase diagram

3 De Sitter phase

4 New bifurcation phase

5 Phase transitions

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 2 / 27 What is Causal Dynamical Triangulation? Causal Dynamical Triangulation (CDT) is a background independent approach to quantum gravity.

Z EH X R D[g]eiS [g] e−S [T ] → T CDT provides a lattice regularization of the formal gravitational path integral via a sum over causal triangulations.

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 3 / 27 Quantum mechanics can be described by Path Integrals: All possible trajectories contribute to the transition amplitude. To deﬁne the functional integral, we discretize the time coordinate and approximate any path by linear pieces.

Path integral formulation of quantum mechanics A classical particle follows a unique trajectory.

space

classical trajectory

t1 time t2

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 4 / 27 To deﬁne the functional integral, we discretize the time coordinate and approximate any path by linear pieces.

Path integral formulation of quantum mechanics A classical particle follows a unique trajectory. Quantum mechanics can be described by Path Integrals: All possible trajectories contribute to the transition amplitude.

quantum trajectory

space

classical trajectory

t1 time t2

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 4 / 27 Path integral formulation of quantum mechanics A classical particle follows a unique trajectory. Quantum mechanics can be described by Path Integrals: All possible trajectories contribute to the transition amplitude. To deﬁne the functional integral, we discretize the time coordinate and approximate any path by linear pieces.

quantum trajectory

space

classical trajectory

t1 time t2

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 4 / 27 The role of a trajectory plays now the geometry of four-dimensional space-time. All space-time histories contribute to the transition amplitude.

Path integral formulation of quantum gravity

General Relativity: gravity is encoded in space-time geometry.

1+1D Example: State of system: one-dimensional spatial geometry

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 5 / 27 All space-time histories contribute to the transition amplitude.

Path integral formulation of quantum gravity

General Relativity: gravity is encoded in space-time geometry. The role of a trajectory plays now the geometry of four-dimensional space-time.

1+1D Example: Evolution of one-dimensional closed universe

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 5 / 27 Path integral formulation of quantum gravity

General Relativity: gravity is encoded in space-time geometry. The role of a trajectory plays now the geometry of four-dimensional space-time. All space-time histories contribute to the transition amplitude.

Sum over all two-dimensional surfaces joining the in- and out-state

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 5 / 27 Transition amplitude

Our aim is to calculate the amplitude of a transition between two geometric states: Z iSEH [g] G(gi , gf , t) D[g]e ≡ gi →gf

To deﬁne this path integral we have to specify the measure D[g] and the domain of integration - a class of admissible space-time geometries joining the in- and out- geometries.

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 6 / 27 2 2D space-time surface is built from equilateral triangles. 3 Curvature (angle defcit) is localized at vertices.

Regularization by triangulation. Example in 2D Dynamical Triangulations uses one of the standard regularizations in QFT: discretization. 1 One-dimensional state with a topology S1 is built from links with length a.

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 7 / 27 3 Curvature (angle defcit) is localized at vertices.

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 7 / 27 Regularization by triangulation. Example in 2D Dynamical Triangulations uses one of the standard regularizations in QFT: discretization. 1 One-dimensional state with a topology S1 is built from links with length a. 2 2D space-time surface is built from equilateral triangles. 3 Curvature (angle defcit) is localized at vertices.

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 7 / 27 Causality Causal Dynamical Triangulations assume global proper-time foliation. Spatial slices (leaves) have ﬁxed topology and are not allowed to split in time. Foliation distinguishes between time-like and spatial-like links. Such setup does not introduce causal singularities, which lead to creation of baby universes. CDT deﬁnes the class of admissible space-time geometries which contribute to the transition amplitude.

time

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 8 / 27 Fundamental building blocks of CDT d-dimensional simplicial manifold can be obtained by gluing pairs of d-simplices along their (d 1)-faces. − Lengths of the time and space links are constant. Simplices have a fixed geometry. The metric is flat inside each d-simplex. The angle deficit (curvature) is localized at (d 2)-dimensional sub-simplices. −

0D simplex - point

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 9 / 27 Fundamental building blocks of CDT d-dimensional simplicial manifold can be obtained by gluing pairs of d-simplices along their (d 1)-faces. − Lengths of the time and space links are constant. Simplices have a fixed geometry. The metric is flat inside each d-simplex. The angle deficit (curvature) is localized at (d 2)-dimensional sub-simplices. −

1D simplex - link

2D simplex - triangle

3D simplex - tetrahedron

4D simplex - 4-simplex

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 9 / 27 Spatial states are 3D geometries with a topology S3. Discretized states are build from equilateral tetrahedra. The metric is ﬂat inside each 4-simplex.

Length of time links at and space links as is constant. Curvature is localized at triangles.

Regularization by triangulation 4D simplicial manifold is obtained by gluing pairs of 4-simplices along their 3-faces.

4D space-time with topology S 3 S 1 ×

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 10 / 27 The metric is ﬂat inside each 4-simplex.

Length of time links at and space links as is constant. Curvature is localized at triangles.

Regularization by triangulation 4D simplicial manifold is obtained by gluing pairs of 4-simplices along their 3-faces. Spatial states are 3D geometries with a topology S3. Discretized states are build from equilateral tetrahedra.

3D spatial slices with topology S 3

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 10 / 27 Regularization by triangulation 4D simplicial manifold is obtained by gluing pairs of 4-simplices along their 3-faces. Spatial states are 3D geometries with a topology S3. Discretized states are build from equilateral tetrahedra. The metric is ﬂat inside each 4-simplex.

Length of time links at and space links as is constant. Curvature is localized at triangles.

Fundamental building blocks of 4D CDT - two types

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 10 / 27 N0 number of vertices

N4 number of simplices

N14 number of simplices of type 1, 4 { } K0 K4 ∆ bare coupling constants( G, Λ, at /as )

Regge action The Einstein-Hilbert action has a natural realization on piecewise linear geometries called Regge action

1 Z Z SE [g] = dt dD x√g(R 2Λ) −G −

The partition function Z EH X R D[g]eiS [g] e−S [T ] → T

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 11 / 27 Regge action The Einstein-Hilbert action has a natural realization on piecewise linear geometries called Regge action

R S [ ] = K0N0 + K4N4 +∆( N14 6N0) T − −

N0 number of vertices

N4 number of simplices

N14 number of simplices of type 1, 4 { } K0 K4 ∆ bare coupling constants( G, Λ, at /as )

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 11 / 27 To make sense of the gravitational path integral one uses the standard method of regularization - discretization. The path integral is written as a nonperturbative sum over all causal triangulations . T Wick rotation is well deﬁned due to global proper-time foliation. (at iat ) → Using Monte Carlo techniques we can approximate expectation values of observables.

Causal Dynamical Triangulations

The partition function of quantum gravity is deﬁned as a formal integral over all geometries weighted by the Einstein-Hilbert action.

Z EH Z = D[g]eiS [g]

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 12 / 27 Wick rotation is well deﬁned due to global proper-time foliation. (at iat ) → Using Monte Carlo techniques we can approximate expectation values of observables.

Causal Dynamical Triangulations

The partition function of quantum gravity is deﬁned as a formal integral over all geometries weighted by the Einstein-Hilbert action.

X R Z = eiS [g[T ]] T

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 12 / 27 Using Monte Carlo techniques we can approximate expectation values of observables.

Causal Dynamical Triangulations

The partition function of quantum gravity is deﬁned as a formal integral over all geometries weighted by the Einstein-Hilbert action.

X R Z = e−S [T ] T

To make sense of the gravitational path integral one uses the standard method of regularization - discretization. The path integral is written as a nonperturbative sum over all causal triangulations . T Wick rotation is well deﬁned due to global proper-time foliation. (at iat ) →

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 12 / 27 Causal Dynamical Triangulations

The partition function of quantum gravity is deﬁned as a formal integral over all geometries weighted by the Einstein-Hilbert action.

X R Z = e−S [T ] T

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 12 / 27 Numerical setup Monte Carlo algorithm performs a random walk in the space of triangulations, it generates configurations with the probability P[ ] = 1 e−S[T ]. T Z The walk consists of a series of 7 Pachner moves, which preserve topology and causality, are ergodic and fulfill the detailed balance condition. It is enough to know the probability functional P( ) up to the normalization. T To calculate the expectation value of an observable, we approximate the path integral by a sum over a finite set of Monte Carlo configurations 1 Z [g] = [g] [g]e−S[g] hO i Z D O

↓ K 1 X 1 X [ ] = [ ]e−S[T ] [ (i)] hO T i Z O T ≈ K O T T i=1

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 13 / 27 Spatial slices

The simplest observable giving information about the geometry, is the spatial volume N(i) deﬁned as a number of tetrahedra building a three-dimensional slice i = 1 ... T . Restricting our considerations to the spatial volume N(i) we reduce the problem to one-dimensional quantum mechanics.

3D spatial slices with topology S 3

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 14 / 27 Phase diagram

S[ ] = K0N0 + K4N4 +∆( N14 6N0) T − − 0.8 Two coupling constants 0.6 Tuned cosmological constant

0.4 C (de Sitter)

∆ A 0.2 D (bifurcation)

0 Quadruple point B 0.2 − 0 1 2 3 4 5

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 15 / 27 Phase diagram

S[ ] = K0N0 + K4N4 +∆( N14 6N0) T − − 0.8

0.6

0.4 C (de Sitter)

∆ A 0.2 D (bifurcation)

0 Quadruple point B 0.2 − 0 1 2 3 4 5

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 15 / 27 Phase diagram

S[ ] = K0N0 + K4N4 +∆( N14 6N0) T − − 0.8

0.6

0.4 C (de Sitter)

∆ A 0.2 D (bifurcation)

0 Quadruple point B 0.2 − 0 1 2 3 4 5

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 15 / 27 Phase diagram

S[ ] = K0N0 + K4N4 +∆( N14 6N0) T − − 0.8

0.6

0.4 C (de Sitter)

∆ A 0.2 D (bifurcation)

0 Quadruple point B 0.2 − 0 1 2 3 4 5

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 15 / 27 Phase diagram

S[ ] = K0N0 + K4N4 +∆( N14 6N0) T − − 0.8

0.6

0.4 C (de Sitter)

∆ A 0.2 D (bifurcation)

0 Quadruple point B 0.2 − 0 1 2 3 4 5

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 15 / 27 Phase A

Phase A 12000

10000 )

t 8000 ( 41 N 6000

4000 Volume,

2000

0 0 10 20 30 40 50 60 70 80 Time, t

Triangulations disintegrate into uncorrelated and irregular sequences of small "universes". This phase is related to the branched polymers phase which is present in Euclidean DT. The "geometry" oscillates in the time direction - analogy to the helicoidal phase of Lifshitz scalar model ( ∂t [g] > 0). | |

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 16 / 27 Phase B

Phase B 160000

140000

120000 ) t ( 100000 41 N 80000

60000 Volume, 40000

20000

0 0 10 20 30 40 50 60 70 80 Time, t

Time dependence (of conﬁgurations) is reduced to a single time slice. The universe has neither time extension nor spatial extension. Hausdorﬀ dimension dh = . ∞ Related to the crumpled phase in Euclidean DT. No geometry in classical sense - analogy to the paramagnetic phase of Lifshitz scalar model ([g] = 0).

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 17 / 27 The average volume N(i) is with high accuracy given by formula h i i N(i) = H cos3 h i W a classical vacuum solution.

De Sitter phase (C) In phase C the time translation symmetry is spontaneously broken and the three-volume proﬁle N(i) is bell-shaped.

9000 N(i) h i 8000 N(i)

7000

6000 Average proﬁle

) 5000 i (

N 4000 3000 Snapshot conﬁguration

2000 center of volume

1000

0 -40 -30 -20 -10 0 10 20 30 40 i

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 18 / 27 De Sitter phase (C) In phase C the time translation symmetry is spontaneously broken and the three-volume proﬁle N(i) is bell-shaped. The average volume N(i) is with high accuracy given by formula h i i N(i) = H cos3 h i W a classical vacuum solution. 9000 N(i) h i 8000 δN(i) h i 7000

6000 Average proﬁle i

) 5000 i ( N

h 4000 3000 Quantum ﬂuctuations 2000

1000

0 -40 -30 -20 -10 0 10 20 30 40 i

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 18 / 27 Hausdorﬀ dimension The time coordinate i and spatial volume N(i) scale with total volume h i N4 as a genuine four-dimensional Universe,

1 − /4 t = N4 i, −3/ 3 t v¯(t) = N 4 N(i) = cos3 . 4 h i 4ω ω

9000 1.2 20k 20k 8000 40k 40k 80k 1 80k 7000 120k 120k 160k 160k 6000 0.8 Fit i

) 5000 ) i t ( ( 0.6 ¯ v N

h 4000

3000 0.4

2000 0.2 1000

0 0 -40 -30 -20 -10 0 10 20 30 40 -1 -0.5 0 0.5 1 i t

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 19 / 27 Effective action The average volume profile 3 t v(t) = cos3 4ω ω corresponds to an Euclidean de Sitter space. It is a classical (maximally symmetric) solution of the minisuperspace action 1 Z v˙ 2 S[v] = + v 1/3 λv dt, G v − which is obtained from the Einstein-Hilbert action by „freezing” all degrees of freedom except the scale factor. Simulations inside phase C show that its discrete form 1 X (N(i + 1) N(i))2 S[N]= − + µN(i)1/3 λN(i) Γ N(i + 1) + N(i) − i also properly describes quantum fluctuations of the three-volume N(i).

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 20 / 27 Effective action The average volume profile Background space-time geometry 3 t v(t) = cos3 4ω ω corresponds to an Euclidean de Sitter space. It is a classical (maximally symmetric) solution of the minisuperspace action 1 Z v˙ 2 S[v] = + v 1/3 λv dt, G v − Couples only adjacent slices which is obtained from the Einstein-Hilbert action by „freezing” all degrees of freedom except the scale factor. Simulations inside phase C show that its discrete form 1 X (N(i + 1) N(i))2 S[N]= − + µN(i)1/3 λN(i) Γ N(i + 1) + N(i) − i also properly describes quantum fluctuations of the three-volume N(i).

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 20 / 27 Effective transfer matrix The effective action suggests the existence of an effective transfer matrix M labeled by the scale factor Directly measured 1 P( N(i) ) = N(1) M N(2) N(2) M N(3) N(T ) M N(1) { } Z |h | | ih | {z| i · · · h | | }i e−S[N] 2 1 (n−m) n+m 1/3 n+m Product of − Γ n+m +µ( 2 ) −λ 2 n M m = e matrix elements h | | i N Potential term

200 400 600 1 1000 1400 0.5 1800 2200 i n | M | n h 0.1 log 0.05

0.01

0 500 1000 1500 2000 2500 n

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 21 / 27 Effective transfer matrix The effective action suggests the existence of an effective transfer matrix M labeled by the scale factor Directly measured 1 P( N(i) ) = N(1) M N(2) N(2) M N(3) N(T ) M N(1) { } Z |h | | ih | {z| i · · · h | | }i e−S[N] 2 1 (n−m) n+m 1/3 n+m Product of − Γ n+m +µ( 2 ) −λ 2 n M m = e matrix elements h | | i N Kinetic term, gaussian

0.008 c = 2000 c = 2800 c = 3600

0.006 i n − c | 0.004 M | n h

0.002

0 800 1000 1200 1400 1600 1800 2000 n

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 21 / 27 However, the transfer matrix bifurcates into two branches. At some volume the kinetic term splits into a sum of two shifted Gausses. In every second slice, there emerges a vertices of very high order. Periodic clusters of volume around singular vertices form a tube structure. Not captured by global properties of the triangulation.

New bifurcation phase (D)

14000 ∆ = 0.3 12000 ∆ = 0.4

10000 i ) i (

N 8000 h

6000 Volume, 4000

2000

0 0 10 20 30 40 50 60 70 80 Time, i

The spatial volume proﬁle N(i) is similar as in phase C. h i

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 22 / 27 In every second slice, there emerges a vertices of very high order. Periodic clusters of volume around singular vertices form a tube structure. Not captured by global properties of the triangulation.

New bifurcation phase (D)

0.0020 0.10 0.15 0.0015 0.20 0.0010 0.25 0.30 0.0005

-4000 -2000 0 2000 4000 m-n=s-2n

The spatial volume proﬁle N(i) is similar as in phase C. h i However, the transfer matrix bifurcates into two branches. At some volume the kinetic term splits into a sum of two shifted Gausses.

New bifurcation phase (D)

0.0020 0.10 0.15 0.0015 0.20 0.0010 0.25 0.30 0.0005

-4000 -2000 0 2000 4000 m-n=s-2n

The spatial volume proﬁle N(i) is similar as in phase C. h i However, the transfer matrix bifurcates into two branches. At some volume the kinetic term splits into a sum of two shifted Gausses.

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 22 / 27 Periodic clusters of volume around singular vertices form a tube structure. Not captured by global properties of the triangulation.

New bifurcation phase (D)

16000 N(t) Maxh o(t)i 14000 h i

12000

i 10000 ) t i ( ) o t

8000 ( N h Max h 6000

4000

2000

0 20 25 30 35 40 45 50 55 60 t

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 22 / 27 New bifurcation phase (D)

The spatial volume proﬁle N(i) is similar as in phase C. h i However, the transfer matrix bifurcates into two branches. At some volume the kinetic term splits into a sum of two shifted Gausses. In every second slice, there emerges a vertices of very high order. Periodic clusters of volume around singular vertices form a tube structure. Not captured by global properties of the triangulation.

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 22 / 27 Phase transistion lines 0.8 First order transition 0.6

0.4 C (de Sitter)

∆ A 0.2 D (bifurcation)

0 Quadruple point B 0.2 − 0 1 2 3 4 5

Order parameter: N0

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 23 / 27 Phase transistion lines 0.8

0.6

0.4 C (de Sitter)

∆ A 0.2 D (bifurcation)

0 Quadruple point B 0.2 − 0 1 2 3 4 5

Second order transition K0

Order parameter: N32

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 23 / 27 Phase transistion lines 0.8

0.6 Second or higher-order transition

0.4 C (de Sitter)

∆ A 0.2 D (bifurcation)

0 Quadruple point B 0.2 − 0 1 2 3 4 5

Order parameter: Max o(p)

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 23 / 27 D/C transition

crit N4,1 = 80k Δ χ(OP2 ) / 0.35

0.30 600 0.25

0.20 400 0.15

0.10 200 0.05

Δ0.00 N41 0.25 0.30 0.35 0.40 50 100 150

OP2 = Max ot Max ot+1 | − |

Peak of the susceptibility χ(OP2) gives a clear signal of the phase transition. The details of the microscopic geometry play important role in the bifurcation transition. Preliminary measurements of the position of critical point ∆crit for diﬀerent total volumes N4, suggest a second or higher-order transition. (estimate of the cricial exponent ν 2.6 0.6) ≈ ± crit crit −1/ν ∆ (N4) = ∆ ( ) α N ∞ − · 4 Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 24 / 27 Summary

The model of Causal Dynamical Triangulations is a lattice approach to quantum gravity. In phase C a four-dimensional background geometry emerges dynamically. It corresponds to the Euclidean de Sitter space, i.e. classical solution of the minisuperspace model. The superimposed quantum ﬂuctuations of the scale factor are described by the minisuperspace model. We have presented an up-to-date phase diagram. It includes a recently discovered bifurcation phase. The new phase is characterized by a bifurcation in the transfer matrix, vertices of high order and a propagating structure of clusters. Importance of the microscopic details of the geometry explaining why the transition went unnoticed.

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 25 / 27 Thank You!

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 26 / 27 Comparison of Lifshitz scalar and CDT phases Landau free-energy-density:

2 4 2 2 2 2 F [φ(x)] = a2φ + a4φ + c2(∂αφ) + d2(∂βφ) + e2(∂βφ)

A m = 0 a > 0 φ = 0 Helicoidal 2

B Para m > 0 d2 < 0 ∂t φ > 0 | |

C Ferro m = 0 a2 < 0 φ > 0 | |

Andrzej Görlich (NBI) Causal Dynamical Triangulations Jena, November 5th, 2015 27 / 27