Causal Dynamical Triangulations and Transfer Matrix

Andrzej G¨orlich

Niels Bohr Institute, University of Copenhagen

Jena, November 29th, 2012

Andrzej G¨orlich Causal Dynamical Triangulations Outline

1 Introduction to Causal Dynamical Triangulations 2 Phase diagram 3 Emergence of background geometry 4 Quantum ﬂuctuations 5 Eﬀective action 6 Transfer matrix 7 Conclusions

Andrzej G¨orlich Causal Dynamical Triangulations 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 G¨orlich Causal Dynamical Triangulations Path integral formulation of quantum gravity

Quantum mechanics can be formulated in terms of path integrals: All possible trajectories contribute to the transition amplitude. General Relativity: gravity is encoded in space-time geometry. The role of a trajectory is now played by the geometry of four-dimensional space-time. All space-time histories contribute to the transition amplitude. 1+1D Example: State of system: one-dimensional spatial geometry

Andrzej G¨orlich Causal Dynamical Triangulations Path integral formulation of quantum gravity

Quantum mechanics can be formulated in terms of path integrals: All possible trajectories contribute to the transition amplitude. General Relativity: gravity is encoded in space-time geometry. The role of a trajectory is now played by 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 G¨orlich Causal Dynamical Triangulations 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 G¨orlich Causal Dynamical Triangulations Causality - diﬀerence between DT and CDT

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. In Euclidean DT one cannot avoid introducing causal singularities, which lead to creation of baby universes. EDT and CDT diﬀer in a class of admissible space-time geometries.

Andrzej G¨orlich Causal Dynamical Triangulations Regularization by triangulation

Discretization is a the standard regularization used in QFT. 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 made of 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 Euclidean DT 2D 3D 4D

Andrzej G¨orlich Causal Dynamical Triangulations Regularization by triangulation

4D space-time with topology S3 × S1

Andrzej G¨orlich Causal Dynamical Triangulations Regularization by triangulation

3D spatial slices with topology S3

Andrzej G¨orlich Causal Dynamical Triangulations Regularization by triangulation

Andrzej G¨orlich Causal Dynamical Triangulations 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

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 G¨orlich Causal Dynamical Triangulations Regge action

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

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

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 G¨orlich Causal Dynamical Triangulations 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]

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 G¨orlich Causal Dynamical Triangulations 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 G¨orlich Causal Dynamical Triangulations 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 G¨orlich Causal Dynamical Triangulations Numerical setup

Monte Carlo algorithm probes the space of configurations 1 −S[T ] with the probability P[T ] = Z e . 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 hO[g]i = D[g]O[g]e−S[g] Z ↓ 1 X hO[T ]i = O[T ]e−S[T ] Z T ↓ K 1 X hO[T ]i ≈ O[T (i)] K i=1

Andrzej G¨orlich Causal Dynamical Triangulations Monte Carlo simulations

Random walk over conﬁguration space, consisting of a series of Monte Carlo moves. Ergodicity - all conﬁgurations can be generated by the set of Pachner moves. Detailed balance condition- P(A)W (A → B) = P(B)W (B → A) Fixed topology - moves don’t change the topology. Causality - moves preserve the foliation. 4D CDT - set of 7 moves. Moves in 3D

Andrzej G¨orlich Causal Dynamical Triangulations Spatial slices

The simplest observable giving information about the geometry, is the spatial volume ni deﬁned as a number of tetrahedra building a three-dimensional slice i = 1 ... T .

Restricting our considerations to the spatial volume ni we reduce the problem to one-dimensional quantum mechanics.

3D spatial slices with topology S3

Andrzej G¨orlich Causal Dynamical Triangulations Phase diagram

S[T ] = −K0N0 + K4N4 +∆( N14 − 6N0)

0.8

0.6

0.4 C

∆ A 0.2

0 Triple point B -0.2 0 1 2 3 4 5

Andrzej G¨orlich Causal Dynamical Triangulations Phase diagram

S[T ] = −K0N0 + K4N4 +∆( N14 − 6N0)

0.8

0.6

0.4 C

∆ A 0.2

0 Triple point B -0.2 0 1 2 3 4 5

Andrzej G¨orlich Causal Dynamical Triangulations Phase diagram

S[T ] = −K0N0 + K4N4 +∆( N14 − 6N0)

0.8

0.6

0.4 C

∆ A 0.2

0 Triple point B -0.2 0 1 2 3 4 5

Andrzej G¨orlich Causal Dynamical Triangulations Phase diagram

S[T ] = −K0N0 + K4N4 +∆( N14 − 6N0)

0.8

0.6

0.4 C

∆ A 0.2

0 Triple point B -0.2 0 1 2 3 4 5

Andrzej G¨orlich Causal Dynamical Triangulations De Sitter space-time as background geometry

In phase C the time translation symmetry is spontaneously broken and the distribution ni is bell-shaped. The average volume hni i is with high accuracy given by formula i hn i = H cos3 i W It describes Euclidean de Sitter space (S4), a classical vacuum solution. 9000 ni 8000 n h ii 7000

6000

5000 i n 4000

3000

2000

1000

0 -40 -30 -20 -10 0 10 20 30 40 i Andrzej G¨orlich Causal Dynamical Triangulations De Sitter space-time as background geometry

6000

5000 i n 4000

3000

2000

1000

0 -40 -30 -20 -10 0 10 20 30 40 i Andrzej G¨orlich Causal Dynamical Triangulations Hausdorﬀ dimension

The time coordinate i and spatial volume hni i scale with total volume N4 as a genuine four-dimensional Universe,

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

n v¯(t) h ii 9000 1.2 20k 20k 8000 40k 40k 80k 1 80k 7000 120k 120k 160k 160k 6000 0.8 Fit 5000 0.6 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 G¨orlich Causal Dynamical Triangulations Spectral dimension

Simulations of the diﬀusion process allow to compute spectral dimension ds . 4 Measurement Fit 3.8

3.6

S 3.4 d

3.2

2.8 0 100 200 300 400 500 σ

Extrapolation of the results gives short and long range behavior

ds (σ → 0) = 1.95 ± 0.10, ds (σ → ∞) = 4.02 ± 0.10, where σ is a fictitious diffusion time. Andrzej Görlich Causal Dynamical Triangulations Minisuperspace model

v¯(t) 1.2 20k 40k 1 80k 120k 160k 0.8 Fit v¯(t) = 3 cos3 t 0.6 4ω ω 0.4

0.2

0 -1 -0.5 0 0.5 1 t Classical trajectoryv ¯(t) corresponds to Euclidean de Sitter space (S4), a spatially homogeneous and isotropic vacuum solution. We ,,freeze” all degrees of freedom except spatial volume and assume that metric on S3 × S1 space-time has form

2 2 2 2 3 ds = dt + a (t) dΩ3, v(t) = a (t) In this particular case, the Einstein-Hilbert action takes form

1 Z Z √ S = dt dΩ g(R − 6λ) G

with classical solutionv ¯(t). Andrzej G¨orlich Causal Dynamical Triangulations Minisuperspace model

v¯(t) 1.2 20k 40k 1 80k 120k 160k 0.8 Fit v¯(t) = 3 cos3 t 0.6 4ω ω 0.4

0.2

2 2 2 2 3 ds = dt + a (t) dΩ3, v(t) = a (t) In this particular case, the Einstein-Hilbert action takes form

2 1 Z v˙ 1 S = + v 3 − λvdt G v

with classical solutionv ¯(t). Andrzej G¨orlich Causal Dynamical Triangulations Minisuperspace model

v¯(t) 1.2 20k 40k 1 80k 120k 160k 0.8 Fit v¯(t) = 3 cos3 t 0.6 4ω ω 0.4 Question? 0.2 0 -1 -0.5 0 0.5 1 t Classical trajectoryv ¯(t) corresponds to Euclidean de Sitter space (S4), a spatially homogeneous and isotropic vacuum solution. How well does the minisuperspace model describe the quantum We ,,freeze” all degrees of freedom except spatial volume and assume ﬂuctuations computed from Monte Carlo simulations? that metric on S3 × S1 space-time has form

2 2 2 2 3 ds = dt + a (t) dΩ3, v(t) = a (t) In this particular case, the Einstein-Hilbert action takes form

2 1 Z v˙ 1 S = + v 3 − λvdt G v

with classical solutionv ¯(t). Andrzej G¨orlich Causal Dynamical Triangulations Quantum ﬂuctuations

We can measure the correlation matrix of spatial volume ﬂuctuations around the classical solutionn ¯ = hni,

Cij ≡ h(ni − hni i)(nj − hnj i)i

The propagator C appears in the semiclassical expansion of the eﬀective action describing quantum ﬂuctuations

2 1 X −1 3 −1 ∂ S[n] S[n =n ¯+η] = S[¯n]+ ηi [C ]ij ηj +O(η ), [C ]ij = 2 ∂ni ∂nj i,j n=¯n 9000 ∆ni 8000 ni hFiti 7000

6000

5000 i n 4000

3000

2000

1000

0 -40 -30 -20 -10 0 10 20 30 40 i Andrzej G¨orlich Causal Dynamical Triangulations Quantum ﬂuctuations

We can measure the correlation matrix of spatial volume ﬂuctuations around the classical solutionn ¯ = hni,

Cij ≡ h(ni − hni i)(nj − hnj i)i

The propagator C appears in the semiclassical expansion of the eﬀective action describing quantum ﬂuctuations

2 1 X −1 3 −1 ∂ S[n] S[n =n ¯+η] = S[¯n]+ ηi [C ]ij ηj +O(η ), [C ]ij = 2 ∂ni ∂nj i,j n=¯n 9000 ∆ni 8000 ni hFiti 7000

6000

5000 i n 4000

3000

2000

1000

0 -40 -30 -20 -10 0 10 20 30 40 i Andrzej G¨orlich Causal Dynamical Triangulations Quantum ﬂuctuations

We can measure the correlation matrix of spatial volume ﬂuctuations around the classical solutionn ¯ = hni,

Cij ≡ h(ni − hni i)(nj − hnj i)i

The propagator C appears in the semiclassical expansion of the eﬀective action describing quantum ﬂuctuations

2 1 X −1 3 −1 ∂ S[n] S[n =n ¯+η] = S[¯n]+ ηi [C ]ij ηj +O(η ), [C ]ij = 2 ∂ni ∂nj i,j n=¯n 9000 ∆ni 8000 ni hFiti 7000

6000

5000 i n 4000

3000

2000

1000

0 -40 -30 -20 -10 0 10 20 30 40 i Andrzej G¨orlich Causal Dynamical Triangulations Eﬀective action

2 1 Z v˙ 1 Minisuperspace action S[v]= + v 3 − λvdt G v ⇓ 2 1 X (nt+1 − nt ) 1/3 Discretization S[n]= + µn −λn Γ n + n t t t t+1 t Inverse of propagator C

2 5 −1 (ηi+1 − ηi ) 2 − /3 ηi [C ]ij ηj = − ui ηi , ki ∝ n¯i , ui ∝ n¯i ki

Andrzej G¨orlich Causal Dynamical Triangulations Eﬀective action

2 1 Z v˙ 1 Minisuperspace action S[v]= + v 3 − λvdt G v ⇓ 2 1 X (nt+1 − nt ) 1/3 Discretization S[n]= + µn −λn Γ n + n t t t t+1 t Inverse of propagator C

2 5 −1 (ηi+1 − ηi ) 2 − /3 ηi [C ]ij ηj = − ui ηi , ki ∝ n¯i , ui ∝ n¯i ki

160k 8000 120k 80k 40k 6000 20k i k 1 g , ) i ( 4000 ¯ N

2000

0 0 10 20 30 40 50 60 70 80 i Andrzej G¨orlich Causal Dynamical Triangulations Eﬀective action

2 1 Z v˙ 1 Minisuperspace action S[v]= + v 3 − λvdt G v ⇓ 2 1 X (nt+1 − nt ) 1/3 Discretization S[n]= + µn −λn Γ n + n t t t t+1 t Inverse of propagator C

2 5 −1 (ηi+1 − ηi ) 2 − /3 ηi [C ]ij ηj = − ui ηi , ki ∝ n¯i , ui ∝ n¯i ki

6e-06 5/3 Fit av− Monte Carlo, 160k

4e-06 i u

2e-06

0 1000 2000 3000 4000 5000 N¯ (i) Andrzej G¨orlich Causal Dynamical Triangulations Transfer matrix

The model of Causal Dynamical Triangulations is completely determined by transfer matrix M labeled by 3D triangulations τ of spatial slices, X R ! Z = e−S [T ] = TrMT , T 1 P(T )(τ , . . . , τ ) = hτ |M|τ ihτ |M|τ i ... hτ |M|τ i. 1 T Z 1 2 2 3 T 1 Matrix element hτ1|M|τ2i denotes the transition amplitude from state |τ1i to |τ2i.

Andrzej G¨orlich Causal Dynamical Triangulations Measurements

Assuming that 1 P(T )(n ,..., n ) = hn |M|n ihn |M|n i ... hn |M|n i 1 T Z 1 2 2 3 T 1 we can measure elements of M q P(3)(n = n, n = m) hn|M|mi = P(2)(n, m) or hn|M|mi = 1 2 p (4) P (n1 = n, n3 = m) and check that it is consistent minisuperspace model

» 2 1/3 – − 1 (n−m) +µ( n+m ) −λ n+m hn|M|mi = N e Γ n+m 2 2

Andrzej G¨orlich Causal Dynamical Triangulations Kinetic part

The kinetic term

» 2 1/3 – − 1 (n−m) +µ( n+m ) −λ n+m hn|M|mi = N e Γ n+m 2 2 causes a Gaussian behavior for n + m = c

2 − (2n−c) hn|M|c − ni = N (c)e k(c) , k(c) = Γ · c Γ ≈ 26.1 is constant for all ranges of n.

0.008 c = 2000 200 c = 2800 120000 400 c = 3600 600 100000 1000 1400 0.006 1800 2200 i 80000 n − ) c c | (

0.004 k

M 60000 | n h 40000 0.002 20000

0 0 800 1000 1200 1400 1600 1800 2000 0 1000 2000 3000 4000 n c

Andrzej G¨orlich Causal Dynamical Triangulations Potential part

The potential term

» 2 1/3 – − 1 (n−m) +µ( n+m ) −λ n+m hn|M|mi = N e Γ n+m 2 2 can be extracted from gathered data for n = m 1 loghn|M|ni = − µn1/3 − λn + const Γ

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 G¨orlich Causal Dynamical Triangulations The stalk

For small volumes n we observe strong discretization eﬀects, 1 P(3)(n) = hn|M3|ni. TrM3 Split into three families. We smooth out M by summing over the families.

n = 3k + 1 0.08 n = 3k + 2 n = 3k + 3 n M3 n h | | i 0.06 ) n ( (3)

P 0.04

0.02

0 20 40 60 80 100 120 n

Andrzej G¨orlich Causal Dynamical Triangulations The stalk

For small volumes n we observe strong discretization eﬀects, 1 P(3)(n) = hn|M3|ni. TrM3 Split into three families. We smooth out M by summing over the families.

Grouped 0.08

0.06 ) n ( (3)

P 0.04

0.02

0 20 40 60 80 100 120 n

Andrzej G¨orlich Causal Dynamical Triangulations The stalk

The eﬀective action for the stalk has the same form as for the blob

» 2 – 1 (n−m) +v( n+m ) hn|M|mi = N e Γ n+m 2 , v(x) = µx1/3 − λx + δx−ρ Gaussian for n + m = c

The kinetic term is in complete agreement, k(c) = Γ · c

` k c 0.025 14 000

0.020 12 000 È È H L 10 000 0.015 8000

0.010 6000

4000 0.005 2000

0.000 0 50 100 150 200 250 300 350n 100 200 300 400 500 c

Andrzej G¨orlich Causal Dynamical Triangulations The stalk

The eﬀective action for the stalk has the same form as for the blob

» 2 – 1 (n−m) +v( n+m ) hn|M|mi = N e Γ n+m 2 , v(x) = µx1/3 − λx + δx−ρ

minisuperspace possible curvature potential corrections The potential term is slightly modiﬁed for small volumes

` log

-2 È È -3

-4

-5

-6 100 200 300 400 n

Andrzej G¨orlich Causal Dynamical Triangulations Summary

1 The model of Causal Dynamical Triangulations is manifestly diffeomorphism-invariant. No coordinates are introduced and only geometric invariants, like length and angle, are involved. 2 Phase diagram consists of three phases. In phase C emerges a four-dimensional universe with well defined time and space extent. 3 The background geometry corresponds to the Euclidean de Sitter space, i.e. classical solution of the minisuperspace model. 4 The effective action obtained from both the correlation matrix and transfer matrix corresponds to the minisuperspace action and properly describes the emergent background geometry and quantum fluctuations. In CDT no degrees of freedom are frozen. 5 For small volumes we observe strong discretization effects. Despite different nature, after the smoothing procedure, the effective action for small volumes is basically the same as for large volumes, with a small modification in the potential.

Andrzej G¨orlich Causal Dynamical Triangulations Thank You!

Andrzej G¨orlich Causal Dynamical Triangulations