hep-lat/9508006 7 Aug 1995 commissie in Simplicial ten overstaan van Bastiaan op het op enbaar ter woensdag aan op ACADEMISCH verkrijging gezag de prof een geb oren Universiteit van Quantum Valentijn do or van dr te verdedigen in september PWM de van de het te Amsterdam do or PROEFSCHRIFT Rector Magnicus de college graad Amsterdam Meijer de van van de te dekanen ingestelde Bakker Aula do ctor Gravity der uur Universiteit

Promotor prof dr J Smit

Promotiecommissie prof dr PJ van Baal

prof dr ir FA Bais

dr JW van Holten

prof dr G t Ho oft

prof dr B Nienhuis

prof dr HL Verlinde

Faculteit der Wiskunde Informatica

Natuur en Sterrenkunde

Instituut voor Theoretische Fysica

CIPGEGEVENS KONINKLIJKE BIBLIOTHEEK DEN HAAG

Bakker Bastiaan Valentijn de

Simplicial quantum gravity Bastiaan Valentijn de Bakker

Sl sn Ill

Pro efschrift Universiteit van Amsterdam Met index lit opg

Met samenvatting in het Nederlands

ISBN

NUGI

Trefw quantumgravitatie

ú—â® ¡ë ï å®â¥« ¯®¦¥« âì ¬®«®¤ñ¦¨

¬®¥© ®¤¨­ë ¯®á¢ï⨢襩 ᥠ¡ï ­ 㪥

‚â®à®¥ íâ® áªà®¬­®áâì ¨ª®£¤

­¥ ¤ã¬ ©â¥ çâ® ¡ë 㦥 ¢áñ §­ ¥â¥

ˆ ª ª ¡ë ¢ë᮪® ­¨ ®æ¥­¨¢ «¨ ¢ á

¢á¥£¤ ¨¬¥©â¥ ¬ã¦¥á⢮ ᪠§ âì ᥠ¡ ¥

Ÿ ­¥¢¥¦¤ û

ˆ¢ ­ ¥â஢¨ç  ¢«®¢

¨á쬮 ª ¬®«®¤ñ¦¨

Table of contents

Table of contents

Introduction

Quantum gravity

Problems

Lattice eld theory

Simplicial gravity

Discretization of gravity

Is this welldened

Two dimensions

Euclidean approach

Monte Carlo simulations

Outline

Phases

Partition function

Canonical partition function

Phase transition

Clusters

Curvature and scaling

Introduction

Curvature

Dimension

Scaling

Summary

Discussion

Table of contents

Correlations

Introduction

Curvature and volume

Twopoint functions

Connected part

Discussion

Binding

Description of the mo del

Implementation

Results

Noncomputability

Noncomputability

Barriers

Results

Discussion

Fluctuating top ology

Introduction

Denition of the mo del

Tensor mo del

Monte Carlo simulation

Results

Discussion

Simulations

Moves

Detailed balance

Program

Initialization

tests

A move

Double subsimplices

Outlo ok

Simplicial spaces

Eective action

Table of contents

A Summary

B Samenvatting

Summary in Dutch

C Uitleg voor nietfysici

Explanation for nonphysicists in Dutch

C Zwaartekracht

C Ruimtetijd

C Simplices

C Quantumfysica

C Simplicial Quantum Gravity

C Waarom

Dankwoord

Acknowledgements in Dutch

Publications

References

Index

Pro duction notes

Chapter One

Introduction

uantum gravity is one of the greatest unsolved problems in the

physics of fundamental interactions In this introduction I discuss

some of the well known problems of quantum gravity and dene

simplicial quantum gravity which is a relatively new approach to

quantum gravity that tries to deal with some of these problems Q

I will also explain some general concepts of lattice eld theory and Monte Carlo

metho ds

Quantum gravity

Quantum gravity is the name given to various theories that attempt to dene a

quantum theory of gravity that is consistent and gives the right answers in the

classical limit Although many p eople have b een working on this problem no

satisfactory solution has yet b een found A few of the many attempts that have

b een made will b e mentioned in section

As is widely known general relativity is a classical theory of gravity that is

b oth very elegant and gives accurate predictions of all observed phenomena of

gravity Well known examples of such phenomena are Newtonian gravity in the

weak eld limit of general relativity the p erihelion movement of Mercury and a

more recent discovery the change in frequency of a binary pulsar due to the loss

of energy to gravitational waves

The action of general relativity without matter is called the EinsteinHilb ert

action

Z

p

d x g R 11 S

G

Introduction

where G is the gravitational constant is the cosmological constant g is the

determinant of the metric g and R is the scalar curvature which is a nasty

function of the metric This function is usually written in terms of the connection

g g g g 12

as

R 13

R g R 14

Because this function is so complicated nding classical solutions to the action

11 is very dicult General relativity allows wavelike solutions which in a

quantum theory would corresp ond to particles called gravitons For more details

on general relativity see one of the many b o oks on this sub ject like Misner et

al

The question arises whether quantum gravity is necessary at all Considering

that we have no exp eriments that detect quantum corrections to general relativity

one might try to describ e gravity using only a classical theory It is very hard

however to conceive of a world where gravity is classical while all the matter is

quantized What would the metric couple to If it would couple to the exp ectation

value of the matter strange situations would o ccur if after a measurement the

matter conguration that we have measured is far away from its exp ectation value

Therefore it seems likely that the fundamental theory of gravitation is a quantum

theory Thus the desire to nd the fundamental laws of nature leads to a study

of quantum gravity

Another somewhat more practical reason to study quantum gravity is that

general relativity predicts singularities but breaks down at those same singulari

ties In particular the Big Bang could have pro duced anything as far as classical

general relativity is concerned Quantum gravity might b e able to pro duce some

b oundary conditions that can tell us why the universe lo oks like it do es Very

roughly sp eaking this is done by calculating the transition probabilities from

nothing to the p ossible early universes For the theory to have predictive p ower

the probabilities should b e strongly p eaked around one or more states of the uni

verse

As a related issue quantum uctuations in the very early universe may have

pro duced uctuations in the cosmic background radiation such as were measured

by the COBE satellite Quantum gravity may predict the b ehaviour of these

Problems

uctuations Because ination the rapid expansion of the early universe would

change the observed uctuations relating the preination quantum gravity pre

dictions to the observations could provide information on the feasibility of ination

scenarios

We will from the b eginning restrict ourselves to Euclidean gravity that is

gravity where the metric is p ositive denite In this approach the action 11

b ecomes

Z

p

S g R 15 d x

G

Strictly sp eaking Riemannian gravity would b e a b etter name b ecause space cer

tainly do es not need to b e at but Euclidean gravity is the commonly used name

for this approach We will discuss the consequences of this change in section

Problems

The usual approach to quantum eld theories is p erturbation theory One of the

main problems of quantizing Einstein gravity that is gravity with the normal Ein

stein action 15 is that p erturbation theory do es not work b ecause the resulting

p erturbation expansion is not renormalizable This can b e quite easily seen from

the fact that the coupling constant G has a negative mass dimension This means

that higher order lo op corrections will generate an innity of counterterms of ever

higher dimensions

One hop e for standard p erturbation theory might b e that the theory is nite

in every order due to cancellations of divergences In p erturbation theory around

at space this do es indeed happ en for the one lo op diagrams t Ho oft Veltman

but not for the two lo op diagrams Goro Sagnotti van de Ven

Also the divergences no longer cancel at one lo op if the background space

is not at or a scalar particle is added to the theory

Not b eing renormalizable do es not make the theory useless Einstein gravity

could b e a low energy eective theory for some unknown underlying theory This

can b e compared to the situation for pions One can write down an eective theory

for pions but this theory is not renormalizable In this case the underlying theory

is QCD which is renormalizable At the mass of the particle the eective pion

theory is not valid anymore and new physics arises

In path integral quantization one tries to calculate the Euclidean path integral

Introduction

over metrics which formally lo oks like

Z

Z Dg expSg 16

The main problems in this case are that it is not clear how to dene the measure

Dg and that the action Sg is not b ounded from b elow which dep endent on

the denition of the measure might lead to a diverging integral

Many alternatives have b een created All of them have their attractions and

problems I will briey mention some of them This is not meant to imply anything

ab out those I have left out

In what is called R gravity Stelle one adds a term prop ortional to R to

the action This makes the action b ounded from b elow making the path integral

converge assuming the measure can b e dened In p erturbation theory one can

use the R term to mo dify the propagator in such a way that the Feynman integrals

b ecome convergent This theory however app ears to b e nonunitary Stelle

Johnston

The Ashtekar variables Ashtekar Ashtekar result from a way to

cast the gravity action in Hamiltonian form which is then canonically quantized

This results in a large number of constraints a solution of which is given by the

lo op representation where states are represented as functionals on sets of closed

lo ops This is a nonp erturbative approach

Sup ergravity see Nieuwenhuizen for a review adds to all elds a partner

eld with opp osite statistics To the graviton would corresp ond a spin particle

called the gravitino The hop e which has now diminished somewhat was that all

the divergences would exactly cancel those of the partner elds

A more radical approach are the famous strings and their sup ersymmetric

counterparts the sup erstrings see Green et al for one of the many b o oks

on this sub ject In string theory the fundamental ob jects are not particles but

onedimensional extended ob jects called strings This allows the theory to b e

nite The particles as we know them are excitations of the string One of the

imp ortant attractions of string theory is that gravity comes out of it in a natural

way as the interaction of a spin excitation on the string which is identied with

the graviton

Last but not least there is simplicial gravity which is what this thesis is all

ab out It is a nonp erturbative lattice formulation of quantum gravity Several

theories have b een developed where spacetime itself is discrete in some way eg

t Ho oft Garay but we will take a more traditional p oint of view and

consider the lattice only as a regularization which eventually has to b e removed

Lattice eld theory

Before explaining what simplicial gravity is in section I will rst review some

basic information ab out lattice theories

Lattice eld theory

As an example of lattice eld theory we will consider a system with a single scalar

eld x and the action

Z

m

S d x x x x x 17

where m is the bare mass and the bare selfcoupling This system is commonly

known as theory We can put this mo del on a fourdimensional hypercubic

lattice with lattice spacing a We can then discretize the action by replacing

derivatives with dierences and integrals with sums over p oints times the volume

of a lattice hypercub e This gives us the discretized action

X X

m

x+a x

18 S a a

x x

a

x x

The bare parameters in the action have a priori little to do with the energy of

the excited states of the system The energy gap of the rst excited state of the

system is interpreted as the mass of a single particle This mass is the renormalized

mass m The connected twopoint function of the scalar eld b ehaves for large

R

distances like

jx yj

hxyi exp 19

c

a

The correlation length which is a dimensionless number of lattice spacings is

related to the renormalized mass m as

R

110

m a

R

Similarly we can dene a renormalized coupling from the connected fourp oint

R

function We see from 110 that in the continuum limit where the lattice spacing

a go es to zero the correlation length at xed renormalized mass m needs to go

R

to innity This means that the system must b ecome critical If the system

shows no critical b ehaviour we cannot dene a physical continuum limit As the

Introduction

Figure Qualitative phase diagram of lattice theory

m

hi

hi

system b ecomes critical the short distance lattice details b ecome irrelevant and

the b ehaviour of the system only dep ends on a few parameters The large scale

b ehaviour do es not change if eg we start with a dierent lattice or discretize

the action in another way This phenomenon is known as universality

The mo del we are considering do es indeed have a critical line which is the

curved line sketched in gure Ab ove this line the system is in a symmetric

phase with hi and b elow the line in a sp ontaneously broken phase with

hi

In this particular mo del if we take the continuum limit the renormalized cou

pling vanishes no matter where or how we approach the critical line The theory

b ecomes a free theory This phenomenon is known as triviality We can how

ever still get a go o d interacting theory by not taking the continuum limit but

-

only approaching it closely The coupling only go es to zero as ln a In

R

renormalization group language it is a marginal op erator Other lattice eects

corresp onding to irrelevant op erators go to zero as a This means that by taking

the lattice distance a suciently small we get prop er continuum b ehaviour while

still having the freedom to choose within some b ounds

R

Simplicial gravity

Simplicial gravity is an attempt to dene quantum gravity as a path integral over

metrics by discretizing spacetime henceforth often called space Usually this is

Simplicial gravity

done in terms of simplices but this is not necessary A in d dimensions

is a solid ob ject a p olytop e with d vertices which are pairwise connected by

edges It is in some sense the simplest d dimensional ob ject In two dimensions

it is a triangle and in three dimensions a tetrahedron Each dsimplex has d

faces See for an illustration gure C on page

We can glue a number of simplices together at the faces to get a simplicial

a space which is piecewise at This idea originated with Regge Regge

for classical general relativity The idea was to approximate a smo oth man

ifold although in the quantum gravity version we do not think of a particular

simplicial space as approximating any particular smo oth space

Simplicial quantum gravity attempts to dene the path integral over metrics

by a suitable integral or sum over simplicial There are two main trends

within this scheme which are commonly known as Regge calculus and dynamical

In Regge calculus one takes a and considers the integral over

all edge lengths keeping the connections b etween the simplices xed Similar to

the continuum case one of the main problems is what measure to use in this

integral A review of work in this direction can b e found in Williams Tuckey

Recently there have b een cast some doubts on this metho d Holm Janke

Bo ck Vink at least for the measures normally used

A similar formulation Ponzano Regge sp ecic for three dimensions

lab els the edges of a xed simplicial complex with representations of the rotation

group The partition sum then consists of a sum over lab elings of edges of a

pro duct of the jsymbols formed by the six edges of each simplex This metho d

has b een extended to four dimensions using jsymbols in Carfora et al

In dynamical triangulation Weingarten David Kazakov et al

Agishtein Migdal a Amb jrn Jurkiewicz the metho d used through

out this thesis one takes equilateral simplices of the same size and considers the

sum over all p ossible ways to connect the simplices One could conceive of com

bining the approaches and taking b oth the sum over connections and the integral

over edge lengths An approach that do es this was put forward in Shamir

In almost all cases one only considers sums over simplicial complexes with

the same top ology This is not b ecause this is b elieved to b e the right thing to

do physically but b ecause such sums are easier to dene and to simulate on the

computer In chapter seven I will briey discuss the generalization to sums over

top ologies

Dynamical triangulation has several nice features It is welldened at least

Introduction

Figure A conguration with two edges b etween the same pair of vertices

A

C

B

at the regularized level see section It is nonp erturbative It allows changes

in the top ology of spatial sections ie spacetime do es not have to b e of the form

M R and might b e extendible to uctuating spacetime top ologies It has the

p otential to give actual predictions through computer simulations

Discretization of gravity

The partition function of dynamical triangulation can b e written as a sum over

triangulations T These should not b e considered as triangulations of smo oth

manifolds but just as ways to glue equilateral simplices together such that the

resulting space has the top ology S

X

Z expST 111

T

We only allow those glueings where no two simplices of the same dimension

have the same set of vertices Eg there cannot b e two distinct edges b etween one

pair of vertices For an example of such a conguration in two dimensions see

gure Two triangles have b een glued along the sides AB and AC creating

a small cone There are now two distinct edges b etween the vertices B and C

Although congurations violating this constraint are not simplicial complexes in

the mathematical sense there do es not seem to b e a physical argument in favour of

or against allowing them In the continuum limit it should not matter whether we

include those congurations or not but this has not b een tested in four dimensions

The reason we do make this restriction is mainly for numerical convenience

Discretization of gravity

Figure Two congurations with dierent orders of symmetry

There is a natural measure on the space of triangulations use each trian

gulation just once There is some ambiguity in the word once ie in what

triangulations are considered the same In practice symmetrical congurations

are counted less often by a factor of the order of their symmetry Let us again

illustrate this in two dimensions To simplify the example we temp orarily drop

the condition explained in the previous paragraph There are two ways to make a

twosphere out of two triangles I have drawn these in gure The symmetry

group of the left hand one has only two elements the identity and turning it up

side down The right hand one can also b e rotated over or degrees and its

symmetry group which is D has six elements In the simulations the rst one

will b e counted three times as often as the second one Considering that for a large

number of simplices only a very small fraction of the triangulations is symmetrical

this is not exp ected to make a dierence Anyway the current simulation practice

is identical to the situation in the matrix mo dels explained in section

The next question is what the action ST should b e To nd the answer we

need to nd the discrete version of the EinsteinHilb ert action for the continuum

15 The term with the cosmological constant is easy as it is prop ortional

to the total volume In the discrete system the volume is simply the number of

simplices times the volume of a single simplex We can therefore substitute

Z

p

gd x Z= N V 112

where N is the number of i dimensional simplices and V is the ivolume of one

i i

Introduction

such simplex If the edge length equals the volumes are

p

i

i

p

113 V

i

i

i

We now need to know what the scalar curvature is in the discrete system We

cannot assign a nite curvature to any p oint of the complex b ecause the simplices

are at inside and the curvature is concentrated in a function like way But we

can determine the integrated curvature Curvature is dened from the rotation

of a vector which is parallel transp orted around a small closed lo op For a small

closed lo op not to b e at it needs to go around a d dimensional ob ject in

our case a triangle The rotation of a vector will b e prop ortional to the decit

angle around this triangle that is the angle that is missing from Also the

curvature integrated over a region around such a triangle will b e prop ortional to

the volume of a triangle V The result is that this integrated curvature around a

single triangle is given by

Z

p

R gd x V n 114

4

4

where n is the number of simplices around a triangle and is the angle b etween

4

two faces of a four simplex which is

arccos 115

I have given only a plausibility argument and not a real derivation as this would b e

quite long See for details Cheeger et al Friedberg Lee Summing

the relation 114 over all triangles in the simplicial complex tells us what to

substitute for the total curvature in the discrete system

Z

p

R gd x Z= V N N 116

where we used the fact that every foursimplex contains triangles and therefore

P

n N Performing the substitutions 112 and 116 in the action 15

4

gives us the discrete action

V N V N V N 117 ST

G

which is usually written as

ST N N 118

Discretization of gravity

where

V

119

G

and

V V

120

G

This action is known as the ReggeEinstein action sp ecialized to equilateral sim

plices

Due to conditions on the glueing of the simplices there are several relations

b etween the N making them dep endent First of all each simplex has ve faces

i

which are glued pairwise giving the relation

N N 121

Second we want the triangulations to have a xed top ology and therefore a xed

Euler number which can b e expressed in the N as

i

N N N N N 122

For the hypersphere S that we usually consider we have

The third condition is known as the manifold condition We cannot just glue

together simplices at the faces and exp ect the result to b e a simplicial mani

fold We have to imp ose the restriction that the neighbourho o d of each p oint is

homeomorphic to a simplicial ball To see how this could b e violated lo ok at

gure a on the following page for an illustration in three dimensions Two

tetrahedrons are glued together Four outer sides come together in the upp er

vertex Now take these four faces and glue together each pair of opp osite ones

in b oth cases such that the upp er vertex is glued to itself A neighbourho o d of

this vertex will not b e a ball but something with a twodimensional torus as its

b oundary This is illustrated in gure b which depicts such a b oundary It

is the intersection of gure a with a horizontal plane just b elow the upp er

vertex

Consider now in four dimensions the set of p oints which have a xed small

compared to the lattice spacing distance from a particular vertex of the triangu

lation Because the intersection of this set with each foursimplex surrounding this

vertex is equivalent to a threesimplex this set of p oints will b e a threedimensional

Introduction

Figure Simplices glued together but not creating a simplicial manifold

a b

simplicial complex For this threedimensional complex with numbers of simplices

M we demand

i

M M M M 123

b ecause the Euler number of an o dddimensional manifold is zero We can now

sum this relation over all vertices of the fourdimensional complex taking into

account that a d simplex of the threemanifold is part of a d simplex of the

fourmanifold and that each such a d simplex joins d vertices This results

in

N N N N 124

Note that this equation is a necessary but not a sucient condition for the simpli

cial complex to b e a simplicial manifold As I will explain on page in chapter

seven a manifold which satises 124 do es satisfy 123 at each vertex But

b ecause the Euler number of any three dimensional manifold is zero the neigh

b ourho o d of a vertex that satises 123 do es not have to b e a simplicial ball

The equations 121 122 and 124 are three indep endent relations b etween

ve numbers N i Therefore only two of the N are indep endent which

i i

we can choose to b e N and N This shows that the action 118 is the most

Is this welldened

general action linear in the N We can also imp ose a manifold condition on the

i

twospheres that we can draw around an edge of the complex but the relation we

get from this is dep endent on the other three

These manifold conditions can easily b e extended to other dimensions In d

dimensions they b ecome

d

X

j

d-j

N N 125

i j

i

j=i

which are known as the DehnSommerville relations For xed top ology there is

also the relation that xes the Euler number

d

X

i

N 126

i

i=

For o dd number of dimensions d the Euler number is always zero In that case

this relation is not indep endent from 125 but follows from them by taking the

i

sum over i with an additional factor after which the right hand side of the

sum can b e shown to vanish

Is this welldened

The question is now whether the theory is welldened This question has two

parts The rst part is whether the partition function is welldened in the discrete

system The second part is whether the theory can give any continuum predictions

The rst part is the easiest The question is whether the partition sum

X

Z exp N N 127

T

converges for some values of the coupling constants It do es if the number of trian

gulations is b ounded by an exp onential function of N This has b een investigated

b oth analytically and in numerical simulations The result is that it do es indeed

converge Amb jrn Jurkiewicz Br ugmann Marinari Barto cci et

al I will discuss this extensively in chapter two

The hard part of the question is whether the theory can give continuum pre

dictions in other words whether the theory shows scaling This means that for

small lattice spacings the b ehaviour of the system b ecomes indep endent of the

lattice spacing Some encouraging results are presented in chapter three

Introduction

One might think that a theory which is p erturbatively nonrenormalizable will

not have a sensible continuum limit This do es not need to b e true The canonical

counterexample is the fourdimensional O nonlinear sigma mo del It is a theory

a a a

with a four comp onent scalar eld that satises the condition In

the continuum the action is

Z

a a

S f d x 128

This theory also has a coupling constant with a dimension just like gravity For

the same reasons as in gravity it is not renormalizable in p erturbation theory As

a nonp erturbative lattice mo del however it has a welldened continuum limit

L usher Weisz L usher Weisz Heller

One might also think that b ecause the continuum action is not b ounded from

b elow the mo del will not make sense b ecause the congurations with the largest

curvature will dominate the partition function Those congurations however

may have very low entropy In continuum language the measure may b ehave

in such a way as to make the path integral converge despite the action b eing

unbounded from b elow This is indeed what happ ens in the simulations Except

for large values of the typical congurations are not those with very high

curvature Similar results are rep orted in Regge calculus Berg The eects

of the measure on the semiclassical uctuations have also b een investigated in the

continuum Mottola

Although the simulations seem to indicate that the mo del is alright this do es

not imply that it actually describ es gravity In lattice eld theory there is no

guarantee that the mo del that comes out in the continuum limit is the same

mo del as the one that we started discretizing We will discuss this further in

chapter nine

Two dimensions

An imp ortant indication that this simplex stu might have something to do with

gravity comes from work in two dimensions In two dimensions many results have

b een found analytically b oth in the continuum theory Polyakov Knizhnik

et al and in dynamical triangulation The latter through what are called

matrix mo dels

In its simplest form a matrix mo del is a mo del of a matrix eld in zero di

mensions In other words a single matrix M of dimension N This matrix is

m

ab

Two dimensions

Figure Feynman diagram of a matrix mo del

usually taken to b e hermitian The partition function of this mo del is Brezin et

al

Z

Z dM expSM 129

ab ab

with the action

g



p

SM M M M M M 130

ca

ab ab ab bc

ab

N

m

where g is a coupling constant Due to the cubic term in the action this partition

function is not well dened for real g We can however still dene it by its

p erturbation expansion If we expand this partition function in p owers of g we

generate Feynman diagrams like the one shown in gure with solid lines The

propagators carry two matrix indices and are therefore drawn as double lines The

two p oint function is

hM M i 131

ab cd ad bc

Introduction

which translates into the connection of the double lines from two vertices in such

a way that an outgoing arrow from one vertex matches up with an ingoing one

from the other vertex and vice versa

The dual of such a Feynman diagram shown in the gure using dashed lines is

precisely a triangulation of a twodimensional orientable surface The contribution

of a diagram is

N

g

N N

p

N g N 132

m m

N

m

where N and N are the numbers of vertices and triangles on the dual lattice

corresp onding to lo ops and vertices of the Feynman digram The parameter is

the Euler number of the two dimensional surface Taking the limit N keeps

m

only the congurations with highest which are those that have the top ology of

the twosphere S The partition function 129 b ecomes equal to the sum over all

triangulated surfaces weighted by the Boltzmann weight with the ReggeEinstein

action This sum includes disconnected diagrams corresp onding to disconnected

surfaces Taking the logarithm of 129 results in the usual way in a sum over only

connected diagrams This sum is exactly the partition function of twodimensional

dynamical triangulation with cosmological constant lng

All the results which exist in b oth the continuum and the matrix mo del ap

proach coincide This is not only true for pure gravity but also for gravity coupled

to conformal matter In particular we have what is called KPZscaling Knizhnik

et al In dynamical triangulation this means that the partition function at

xed number of triangles N b ehaves like

-

c

N 133 exp Z N

where the string susceptibility dep ends on the central charge c of the matter

and the Euler number of the surface as

p

c c c

134

c

in 133 is not universal and dep ends for instance on whether we The constant

use triangles or squares in the mo del

It has b een suggested that the mo del works b etter in two than in four dimen

sions b ecause in two dimensions one can build at space and lo cal deviations from

it out of triangles In four dimensions one cannot ll at space with equilateral

simplices This argument do es not hold b ecause the twodimensional mo del works

Euclidean approach

equally well using p entagons heptagons or other regular p olygons Kazakov

Also one could use hypercub es instead of simplices in four dimensions Weingarten

although it has not b een veried that this indeed gives the same results

Obviously there can b e other reasons why the mo del in two dimensions is dierent

from the one in four dimensions and the fact that the mo del works in the former

case is no guarantee for the latter one It is often suggested that the top ological

nature of the twodimensional action is such a dierence I do not think this is a

very comp elling argument First the twodimensional mo del also works if we add

matter making the action no longer top ological Second the dicult part is not

the action but the measure and that comes out correctly in two dimensions

Euclidean approach

As was mentioned in section what we are discussing is Euclidean quantum

gravity as opp osed to Lorentzian quantum gravity This means that we consider

metrics with signature instead of The main reason to do so

is that we do not know how to simulate Lorentzian gravity Although it might b e

p ossible to dene a discretization by glueing pieces of Minkowski space together

P

we cannot simulate the partition sum expiS b ecause we cannot do Monte

Carlo simulations see section with complex probabilities

In lattice gauge theory the Minkowskian and Euclidean formulations are related

by a Wick rotation where time is replaced by imaginary time In quantum gravity

the situation is far from clear We cannot simply analytically continue Lorentzian

metrics to Riemannian ones but p erhaps it is p ossible to analytically continue the

Green functions in some way

It has even b een suggested Hawking b that reality is describ ed by a

Euclidean path integral and that that is the formulation that others should b e

proven equivalent to instead of vice versa What seems less implausible is that

b oth signatures of the metric can exist and that one is chosen by some dynamical

pro cess A mechanism by which the Lorentzian signature as well as the dimension

of spacetime comes out was describ ed in Greensite If this is the case this

mechanism is probably not describ ed by the dynamical triangulation metho d used

here

We mentioned in section that the unboundedness from b elow of the Eu

clidean action could b e comp ensated by the measure There is however still a

problem with the eective action describing semiclassical uctuations around the

Introduction

average spacetime we measure in the simulations We will discuss this problem in

chapter nine

Monte Carlo simulations

Almost no analytic results are known for four dimensional dynamical triangula

tion Most of our knowledge stems from computer simulations using Monte Carlo

metho ds In these simulations we measure the exp ectation value of observables A

X

A expST

T

X

hAi 135

expST

T

by generating a suitably weighted ie with their relative probabilities prop ortional

to expST set of triangulations In Monte Carlo jargon these are usually

referred to as congurations We then calculate or measure the value of the

observable A on each triangulation The average is an approximation for hAi that

b ecomes b etter as we take more congurations

The sets of congurations are generated by starting with an arbitrary congu

ration and generating new ones by p erforming lo cal changes in the conguration

These changes are called moves If the probabilities which are used to choose a

particular move to p erform satisfy a sucient but not necessary condition called

detailed balance the probability distribution of the congurations will converge

to the desired distribution where the probabilities are prop ortional to expST

This pro cess of convergence is called thermalization

When the conguration has thermalized we can start to p erform the measure

ments Two congurations that are only one move apart however will usually

show almost the same value of the observable In other words the measure

ments are highly correlated To get a go o d estimation of the exp ectation value of

the observable and the statistical error we make we need to b e able to pro duce

uncorrelated congurations The number of moves that is needed b etween two

congurations to make them uncorrelated is called the auto correlation time To

b e more precise the correlation b etween two observables b ehaves like

0 0

hOtOt i hOi expjt t j 136

where is the auto correlation time This time can dep end very much on the

observable that is used

Outline

One of the ma jor practical problems in these simulations is a phenomenon

called critical slowing down This means that the auto correlation time dened

ab ove b ecomes very large when the system b ecomes critical As was explained

in section this is just where we want to study the system Critical slowing

down can b e explained intuitively as follows If the Monte Carlo moves are lo cal

any change in the system will propagate through the system like a random walk

Before the system can b ecome uncorrelated from its previous state this change will

need to propagate for a distance of the order of the correlation length Because

the distance travelled by a random walk go es like the square ro ot of its length

the auto correlation time of the system will b ehave like which by denition

b ecomes very large for a critical system

Ways to get around this problem dep end very much on the system under con

sideration Most of these ways consist of dening Monte Carlo moves which make

nonlo cal changes in the system For the particular case of dynamical triangula

tion I will briey discuss this on page in chapter eight

Outline

The reader who is interested in the generation of the triangulations on the com

puter can nd the details in chapter eight Most of the other chapters deal with

the results of the various measurements done on these triangulations and do not

require knowledge of these metho ds

Chapter two describ es the region of parameter space where the mo del is dened

and the two phases that o ccur in the mo del It discusses the distinguishing features

of those phases and the nature of the transition b etween them

Chapter three describ es the spaces generated in the mo del by dening a cur

vature and an eective global dimension at scales large compared to the lattice

spacing Both features indicate that at the transition the space resembles the hy

p ersphere S This chapter also presents evidence for scaling ie the idea that as

we go to small lattice spacings the features of the mo del b ecome indep endent of

the lattice spacing

In chapter four we measure twopoint correlation functions of the scalar curva

ture and something we call the lo cal volume Peculiar eects o ccur b ecause b oth

the lo cal observables and the distance b etween them dep end on the geometry It

turns out that at the transition and in the elongated phase the correlations have

a long range ie they fall o like a p ower of the distance

Chapter ve go es into the measurement of gravitational attraction and binding

Introduction

of particles It presents a p ossible way of dening the renormalized gravitational

p

G in lattice units constant G and thereby measuring the Planck length

R R P

An interesting feature of the dynamical triangulation mo del is that the results

are noncomputable in the recursion theoretical sense of the word This could

create problems for the numerical simulations Chapter six tries to quantify the

eects of this feature which seem to b e negligible

A mo del that do es not x the top ology of the four dimensional space is briey

considered in chapter seven It is equivalent to a tensor mo del generalization of

the matrix mo del describ ed in section

Finally chapter nine gives a general discussion of the current status of dynam

ical triangulation as a mo del for fourdimensional quantum gravity

Chapter Two

Phases

ritical b ehaviour of a lattice system is necessary to dene a go o d

continuum limit Such b ehaviour is generally found at a second

order phase transition This chapter describ es the phase structure

of the dynamical triangulation mo del of four dimensional quantum

gravity Numerical simulations show that the mo del has two phases C

with very dierent prop erties We will usually just call these phases but in chapter

three we will discuss the p ossibility that the system has an entire critical line and

that therefore these phases are not phases in the usual sense

Partition function

As has b een explained in chapter one the partition function of four dimensional

dynamical triangulation without matter is

X

Z exp N N 21

T

The sum is over all the ways T to glue together foursimplices such that the

resulting simplicial complex is a simplicial manifold with the top ology of the four

sphere S and N is the number of isimplices in the resulting complex To facilitate

i

the discussion b elow we dene a canonical partition function ZN at xed

volume N

X

ZN exp N 22

T (N )

Phases

which we can use to rewrite 21 as

X

Z ZN exp N 23

N

The question is now whether the sum in 21 converges for some values of

the coupling constants We see in 23 that this will b e the case if the canonical

partition function has an exp onential b ound

c

N 24 ZN exp

c

for some critical This will make the grand canonical partition function

c c

Because the N simplices and p erhaps if converge for

have only N faces to connect in pairs and each connection can b e made in at

 N

most ways ZN certainly has the factorial b ound N

The existence of an exp onential b ound for the canonical partition function

22 do es not dep end on This can b e shown as follows Because each simplex

has triangles and each triangle is shared by at least simplices the number of

triangles is b ounded as N N Therefore if

X

ZN exp N

T (N )

X

exp N ZN exp N 25

T (N )

and if

X X

ZN exp N ZN 26

T (N ) T (N )

Or in words if the exp onential b ound exists for or any other sp ecic

value it exists for all To investigate the question further we dene an N

dep endent critical as

ln ZN

c

N 27

N

c

and the exp onential If this ob ject converges as N its limit will b e

b ound exists If on the other hand it diverges there is no exp onential b ound



n!! stands for n(n - )(n - )

Partition function

c

N we simulate on the computer the To measure the value of this

mo died system

X

0

Z V exp N N N V 28

T

X

ZN exp N N V 29

N

The extra term N V in the action certainly makes the sum converge for

p ositive b ecause the number of triangulations can never rise faster than fac

A constant term in the action is torially in N which is slower than expN

irrelevant which allows us to get rid of by a suitable shift in V or vice versa

but we will not do so

We can now make a saddle p oint expansion of 29 Let us dene N as the

value of N that gives the most imp ortant contribution to the sum 29 Using

27 we see that

c

N

N V 210

N shows that in this system hN i The saddle p oint expansion of 29 around

- c

N using N ON Consequently we can measure

ln ZN

c

N

N

hN i V O 211

hN i

In gure on the following page we have plotted the measured value of

c

N as a function of the volume hN i at where all congurations

of a particular volume contribute equally to the partition function Within the

errors the data at show just a horizontal line This means that at large

c

N on N we measure no dep endence of

I tted the data at to a logarithm and a converging p ower The

c

N and no exp onential b ound while logarithm corresp onds to a diverging

the p ower corresp onds to the existence of the exp onential b ound The results of

the logarithmic t were

c

hN i a b lnhN i 212

a 213

Phases

c

as a function of the volume at Figure The value of

N c

a b ln N

-c

a bN

hN i

b 214

at dof 215

and those of the p ower t were

-c c

hN i a bhN i 216

a 217

b 218

c 219

at dof 220

These results are not conclusive The of the p ower law is somewhat lower

but considering that it has more parameters this is not surprising Data that has

b een found by other groups Amb jrn Jurkiewicz Br ugmann Marinari

c

converging according with N up to now quite strongly favors a

to 216 which means that there is an exp onential b ound on the number of

Partition function

c

as a function of for various volumes Figure The value of

c

c

congurations and that the grand canonical partition function is well dened A

handwaving argument for an exp onent of c was presented in Amb jrn

Jurkiewicz From the latest data one now nds an exp onent of

Br ugmann Marinari

Recently there has also app eared an analytical pro of that the number of con

gurations has an exp onential b ound for all dimensions and xed top ologies

Barto cci et al This pro of however applies to metric ball coverings

While some arguments can b e made that this also provides a b ound for our trian

gulations this is not yet completely clear

c

I have plotted its We will now take a closer lo ok at the dep endence of

value at various xed volumes in gure To increase the vertical resolution a

linear function has b een subtracted from the measured values The black

dots are the critical values of discussed b elow in section As we have already

c

mentioned ab ove at large values of there is virtually no dep endence of on

the volume

There is a close relation b etween the slop es in the gure and the average

Phases

Figure The average curvature hN N i as a function of for various volumes

i

N N

h

curvature We see from equation 116 that up to an additive and a multiplicative

constant the curvature p er unit volume is simply N N We have plotted this

quantity in gure Its relation with the slop es of gure on the page b efore

is

c

N

ln ZN hN i

221

N N

Using this relation we can see that those slop es are b ounded by deriving b ounds

on the number of triangles It follows from the relations b etween the N which

i

were derived in chapter one that

N

N N 222

Therefore as N and we have N N On the other hand

as we have shown ab ove N N In the simulations it turns out that in

fact N N O but I do not know how to prove this from geometrical

c

arguments These b ounds t with the slop es of In Br ugmann Marinari

Canonical partition function

a larger range of values was used than in our simulations and the authors

concluded that

c

constant 223

c

constant 224

Although the grand canonical partition function would not b e welldened if

this exp onential b ound did not exist this need not invalidate the mo del as dis

cussed in de Bakker Smit One can also dene the mo del using the

canonical partition function and then take the number of simplices to innity Al

ternatively one could try to make a suitable expansion in the inverse gravitational

constant In the case of two dimensions and varying top ology the latter approach

is known as the double scaling limit Brezin Kazakov Douglas Shenker

Gross Migdal

Canonical partition function

Because the grand canonical ensemble 21 is very dicult to simulate simula

tions are always done using the canonical ensemble 22 We cannot however

directly simulate the canonical ensemble on the computer The reason is that no

set of usable moves is known that keeps the volume N constant and is ergo dic in

the canonical ensemble It seems probable that no such set of moves exists see

chapter six for more discussion on this topic

To get around the problem explained ab ove we can rewrite the canonical par

tition function 22 as

X

ZN expSN exp N SN T 225

N [T ]N

T

where N is xed and N T means the number of simplices in the triangulation

T This allows us to calculate observables in the canonical ensemble using grand

canonical moves The precise form of SN is not imp ortant as long as it makes

the sum without the function factor convergent In practice we use

SN N N V 226

where and V are free parameters We should now ideally only use the

congurations with N T N in the measurements In practice we dont do

Phases

this but use all the congurations We can control the volume uctuations by

changing the parameter according to

-

c

227 i hN i N hN

N

This follows from the saddle p oint approximation of 28 Due to the dep endence

of N on the right hand side always stays p ositive In our simulations we used

- c

N Thus the volume uctuations were This makes N

approximately

It is often argued that using the canonical ensemble for large volumes ap

c c

proximates taking the limit b ecause the volume diverges for

Unfortunately this is not at all clear Let us write the canonical partition function

22 as

c

N 228 ZN fN exp

c

the grand canonical partition function 23 b ehaves like At

X

c

Z fN 229

N

and therefore

X

N fN

N

X

hN i 230

fN

N

Dep ending on fN there are two scenarios First hN i might diverge and the

argument is alright Second hN i might converge to a nite value In this case

there is no way to tune such that hN i b ecomes large Recent data Amb jrn

Jurkiewicz b show that fN b ehaves approximately like

( )-

c

if N

fN 231

c

if expcN

with values of This would imply the second scenario This might

change if we add matter andor change the top ology of the system In two dimen

sions this happ ens according to 133 and 134

Phase transition

Figure The curvature susceptibility as a function of for and

simplices R is dened as N N

i R

h i R

h i

N

h

Phase transition

We have seen that we need to tune or equivalently V to keep the system

around the volume we would like This leaves us with only one parameter to play

around with which is Remember that this constant is inversely prop ortional

to the bare Newton constant G by equation 119

A strong indication that there is a phase transition comes from the curva

ture susceptibility which turns out to have a diverging p eak This curvature

susceptibility is dened as

h i

hN i ln ZN

i hN i hN 232

R

N N N

But b ecause we let the volume uctuate the actual uctuations in N are dom

inated by the volume uctuations We therefore study the uctuations in the

Phases

curvature p er unit volume

N N

hN i 233

R

N N

This quantity is plotted in gure on the preceding page Although the pic

ture is somewhat messy the p eak in this susceptibility clearly increases with the

volume and also moves somewhat to the right A more detailed analysis using

multihistogramming metho ds Catterall et al a has shown that the p eak

increases as

c

hN i 234

R

with a critical exp onent This indicates a second or higher order

phase transition A rst order transition would have

c

changes with the volume These values The value of the critical coupling

c

N one have b een drawn as black dots in gure on page As with

R

c

N Using a dierent metho d to determine can investigate the b ehaviour of

c

N it was found in Amb jrn Jurkiewicz b that it moves like

- c c

j N 235 N j

with an exp onent Again a rst order transition would have

There are several features that qualitatively distinguish the system on b oth

sides of the phase transition The most notable is that the phase at low is

highly connected the average distance b etween two simplices is small Therefore

this phase is called the crumpled phase The other phase has long thin branches

and is called the elongated phase In the elongated phase the system resembles a

branched p olymer

Figure on the facing page shows the average distance b etween two simplices

as a function of The distance b etween two simplices is dened to b e the

minimum number of steps we need to take from simplex to directly connected

simplex to get from the rst simplex to the second To distinguish this denition

from others this distance is often called the geo desic distance The gure shows

a sharp crossover b etween the regions of small and large distance On the left

c

the average distance increases as the logarithm side of the transition at

of the volume while in the elongated phase it increases as the square ro ot of the

volume The latter b ehaviour supp orts the statement that the system b ehaves

like a branched p olymer in that phase b ecause the internal fractal dimension of a

branched p olymer is two David It is remarkable that due to the change of

Phase transition

Figure The average distance b etween two simplices as a function of for

various volumes

i r

h

c

with the volume for some values of and volume ranges the average distance

can actually decrease by adding more simplices

Not only the distance itself but also its uctuations show a very marked dif

ference b etween the phases By uctuations in the average distance we mean that

we take the average of the distances of one conguration and then measure its

uctuations across congurations As we can see in gure on the next page

in the elongated phase the average distance uctuates wildly The squared uc

are approximately equal to hri On the other hand in the tuations hr i hri

crumpled phase there is hardly any change Note the logarithmic vertical scale

in the picture Unlike the uctuations in the curvature there is no p eak and the

c

uctuations remain high for

A nice illustration of the nature of the elongated phase can b e made by cal

culating the massless scalar propagator on such a conguration by solving the

Laplace equation

236

x

x

Phases

Figure Fluctuations in the average distance as a function of

-

-

i

N h

i - r h i r h

-

-

for some origin We can then plot for each simplex the value of the propagator

against the geo desic distance This has b een done in gure on the facing page

for simplices and If there are long tub es sticking out of the space

the geo desic distance along these tub es will increase but the propagator will b e

constant Such eects can b e clearly seen in the gure

Clusters

This section is ab out an idea that is still rather sp eculative and its arguments are

mostly intuitive Considering that there is a phase transition in the mo del one

might wonder whether there is a symmetry that is sp ontaneously broken in one

of the phases creating long range order

It will b e suggested in this section that this symmetry is the reversal of the

orientation Popularly sp eaking the symmetry of turning the conguration inside

out Imagine a part of a two dimensional triangulated surface consisting of a

vertex with less than six triangles around it ie less than in at space Keeping

Clusters

Figure The massless propagator against the geo desic distance in the elongated

phase

x

r

x

the triangles rigid but allowing them to turn around the edges it is not p ossible

to ip the vertex through the circle of edges of the conguration to the other

side With more than six triangles around the vertex this is p ossible If two

vertices with less then six triangles each are connected by an edge b oth sets of

triangles must p oint in the same direction b ecause the triangles next to this edge

are contained in b oth sets This is illustrated in gure on the next page

This means that if most of the vertices with less than six triangles are connected

by edges in a large cluster all these vertices will p oint in the same general direction

outside or inside The conguration will not b e exible enough to continuously

deform it to turn it inside out If however those vertices exist only in isolated

patches such a deformation seems p ossible As the space is not embedded imagine

it to b e able to freely pass through itself Let us compare this with the familiar

Ising mo del In the symmetric phase we can ip the whole conguration ie

turn all the spins upside down by lo cally changing spins without going through

a large energy barrier On the other hand in the broken phase the phase with

Phases

Figure Two neighbouring vertices with less than six triangles must p oint in

the same direction

sp ontaneous magnetization this is not p ossible Similarly not b eing able to turn

the dynamical triangulation conguration inside out is an indication of symmetry

breaking

In four dimensions we can now similarly lo ok for clusters of vertices with less

simplices around them than in at space To do this we rst have to calculate

how many simplices there would b e around a vertex to completely ll the spatial

angle around it

We see from equation 116 that for a space with zero average curvature we

have

N

237

N

and therefore using 222

N

238

N

where the term in 222 vanishes b ecause we are considering at space Con

sidering that each simplex has ve vertices the number of simplices around each

vertex at zero curvature will b e times 238 or approximately We get

this same number but not as easily by calculating the four dimensional spatial

angle of a simplex at one vertex and dividing the total spatial angle by

In gure on the facing page I have plotted the average cluster size as a

function of for volumes of and simplices The average cluster

Clusters

Figure Average cluster size as a function of for and

simplices

i cluster S

h

size is dened as

X

0

S x

cluster

x

hS i 239

X

cluster

0

x

where the sum is taken only over those vertices x that can b e in clusters all vertices

of order and S x is the size ie the number of vertices of the cluster

cluster

containing the vertex x We see that in the crumpled phase the average size is

much smaller and decreases with the volume while in the elongated phase it is

much larger and increases with the volume Extrap olating this to large volumes we

see that the change in hS i b etween the phases will b ecome very sharp This is

cluster

even more remarkable if we consider that in the crumpled phase the connectedness

is much higher making any kind of p ercolation pro cess much easier

One should keep in mind that there are various mo dels where the p ercolation

transition do es not coincide with the phase transition For instance although the

Phases

twodimensional Ising mo del on a square lattice b ecomes p ercolating at the phase

transition this is no longer true on a triangular lattice It can therefore b e dan

gerous to draw conclusions ab out phase transitions from p ercolation arguments

Nevertheless considering the geometrical nature of dynamical triangulation it

seems likely that in this case the two are related

A completely dierent symmetry breaking mechanism was describ ed in Shamir

in the mo del mentioned in section where b oth the connectivity and the

edge lengths of the simplicial complex were varied In this mo del there is a SL

symmetry which could sp ontaneously break to O The Goldstone b osons of

this symmetry breaking would then b e the gravitons

Chapter Three

Curvature and scaling

0

e study the average number of simplices N r at geo desic dis

tance r in the dynamical triangulation mo del of Euclidean quan

0

tum gravity in four dimensions We use N r to explore deni

tions of curvature and of eective global dimension An eective

curvature R go es from negative values for low the inverse W

V

c

Far bare Newton constant to slightly p ositive values around the transition

ab ove the transition R is hard to compute This R dep ends on the distance

V V

scale involved and we therefore investigate a similar explicitly r dep endent run

ning curvature R r This increases from values of order R at intermediate

e V

distances to very high values at short distances

A global dimension d go es from high values in the region with low to d at

high At the transition d is consistent with We present evidence for scaling of

0

N r and introduce a scaling dimension d which turns out to b e approximately

s

in b oth weak and strong coupling regions We discuss p ossible implications of the

results the emergence of classical euclidean spacetime and a p ossible triviality

of the theory

Introduction

The dynamical triangulation mo del is a very interesting candidate for a nonp er

turbative formulation of fourdimensional euclidean quantum gravity Agishtein

Migdal a Agishtein Migdal b Amb jrn Jurkiewicz The

congurations in the mo del are obtained by glueing together equilateral four

dimensional simplices in all p ossible ways such that a simplicial manifold is ob

tained A formulation using hypercub es was pioneered in Weingarten The

Curvature and scaling

simplicial mo del with spherical top ology is dened as a sum over triangulations T

with the top ology of the hypersphere S where all the edges have the same length

The partition function of this mo del is

X

ZN exp N 31

T N =N

Here N is the number of simplices of dimension i in the triangulation T The

i

R

p

weight exp N is part of the Regge form of the EinsteinHilb ert action gR

G

X

V N N 32 S

4

G

4

V

33

G

arccos

34

Here V is the volume of simplices triangles and is the decit angle

4

around The volume of an isimplex is

q

i

i

i V i 35

i

Because the N i satisfy three constraints only two of them are indep en

i

dent We have chosen N and N as the indep endent variables For comparison

with other work we remark that if N is chosen instead of N then the corre

sp onding coupling constant is related to by This follows from the

relations b etween the N which imply that

i

N N 36 N

where is the Euler number of the manifold which is for S

Average values corresp onding to 31 can b e estimated by Monte Carlo meth

o ds which require varying N Agishtein Migdal a Agishtein Migdal

b Amb jrn Jurkiewicz One way to implement the condition N N

is to base the simulation on the mo died partition function Agishtein Migdal

a

X

0

Z N exp N N N N 37

T

Introduction

where is chosen such that hN i N and the parameter controls the volume

uctuations The precise values of these parameters are irrelevant if the desired

N are picked from the ensemble describ ed by 37 This is not done in practice

but the results are insensitive to reasonable variations in We have chosen

- -

the parameter to b e giving hN i hN i ie the

uctuations in N are approximately

Numerical simulations Agishtein Migdal a Agishtein Migdal b

Amb jrn Jurkiewicz Amb jrn et al c Br ugmann Catterall et



al a Varsted have shown that the system 37 can b e in two phases

c

For N weak bare coupling G the system is in an elongated phase with

high h Ri where hRi is the average Regge curvature

R

p

gR

V N

R

38 R

p

V N

g

In this phase the system has relatively large baby universes Amb jrn et al b

c

N strong coupling the system and resembles a branched p olymer For

is in a crumpled phase with low h Ri This phase is highly connected ie the

average number of simplices around a p oint is very large The transition b etween

the phases app ears to b e continuous

As an example and for later reference we show in gure on page the

susceptibility

i h

N N V

R i hRi V 39 N h

N N V

where V is the total volume N V The three curves are for N and

simplices

The data for N are consistent with those published in reference Cat

terall et al a where results are given for

N 310 i hN i hN

N

For xed N this is of our curvature susceptibility 39 as can b e seen from

36

The b ehavior of Z N as a function of N for large N has b een the sub ject

of recent investigations Amb jrn Jurkiewicz de Bakker Smit



In section on page we discuss the p ossibility that these are not phases in the sense of

conventional statistical mechanics

Curvature and scaling

Br ugmann Marinari Catterall et al b In de Bakker Smit

c

might move to innity as N and argued we discussed the p ossibility that

that this need not invalidate the mo del So far a nite limit is favoured by the

data Amb jrn Jurkiewicz Br ugmann Marinari however

It is of course desirable to get a go o d understanding of the prop erties of the

euclidean spacetimes describ ed by the probability distribution expS A very

interesting asp ect is the proliferation of baby universes Amb jrn et al b

Here we study more classical asp ects like curvature and dimension extending pre

vious work in this direction Agishtein Migdal a Agishtein Migdal b

Amb jrn Jurkiewicz Amb jrn et al c Br ugmann Catterall et

al a Varsted The basic observable for this purp ose is the average

0

number of simplices at a given geo desic distance from the arbitrary origin N r

We want to see if this quantity can b e characterized approximately by classical

prop erties like curvature and dimension and if for suitable bare couplings there

0

is a regime of distances where the volumedistance relation N r can b e given a

classical interpretation It is of course crucial for such a continuum interpretation

0

of N r that it scales in an appropriate way

In section we investigate the prop erties of an eective curvature for a dis

tance scale that is large compared to the basic unit but small compared to global

distances Eective dimensions for global distances are the sub ject of section

and scaling is investigated in section We summarize our results in section

and discuss the p ossible implications in section

Curvature

A straightforward measure of the average lo cal curvature is the bare curvature

p

-

at the triangles ie h Ri hN N i this follows from 38 and

35 It has b een well established that at the phase transition hN N i

practically indep endent of the volume Agishtein Migdal

a Agishtein Migdal b Amb jrn Jurkiewicz Amb jrn et al

-

c This means that hRi has to b e divergent in the continuum limit

The curvature at scales large compared to the lattice distance however

is not necessarily related to the average curvature at the triangles One could

imagine eg a spacetime with highly curved baby universes which is at at large

scales

The scalar curvature at a p oint is related to the volume of a small hypersphere

around that p oint Expanding the volume in terms of the radius of the hypersphere

Curvature

Table Number of congurations used in the various calculations

results in the relation

R r

V

n

Or 311 V r C r

n

n

n

312 C

n

n

for an ndimensional manifold We have written R here to distinguish it from

V

the small scale curvature at the triangles Dierentiating with resp ect to r results

in the volume of a shell at distance r with width dr

R r

V

0 n-

V r C nr Or 313

n

n

We explore this denition of curvature as follows We take the dimension

n assuming that there is no need for a fractional dimension diering from

at small scales For r we take the geo desic distance b etween the simplices that is

the lowest number of hops from foursimplex to neighbour needed to get from one

foursimplex to the other Setting the distance b etween the centers of neighbouring

simplices to corresp onds to taking a xed edge length in the simplicial complex

p p

0

of ie we will use lattice units with For V r and V r we take

V r V Nr 314

e

0 0

V r V N r 315

e

Curvature and scaling

0

N r Nr Nr 316

where Nr is the average number of foursimplices within distance r from the

0

arbitrary origin N r is the number of simplices at distance r and we have

allowed for an eective volume V p er simplex which is dierent from V Since

e

R is to b e a long distance in lattice units observable we shall call it the eective

V

curvature

0

The eective curvature was determined by tting the function N r to

0

N r ar br 317

It then follows from 313 and 315 that R is determined by

V

b

318 R

V

a

and V by

e

C

V 319

e

a

The constant C in equation 311 is In at space we would have V V

e

giving

p

C

a 320

V

p p

for Such a space cannot b e formed from where we used V

equilateral simplices b ecause we cannot t an integer number of simplices in an

angle of around a triangle

0

Figure on the facing page shows N r for simplices Three dierent

values of are shown in the crumpled phase close to the transi

tion and in the elongated phase These curve can also b e interpreted as

the probability distribution of the geo desic length b etween two simplices Such

distributions were previously presented in Amb jrn Jurkiewicz Amb jrn

et al c

0

We determined N r by rst averaging this value p er conguration successively

using each simplex as the origin We then used a jackknife metho d leaving out

one conguration each time to determine the error in R

V

Our results in this pap er are based on N and simplices Congu

rations were recorded every sweeps where a sweep is dened as N accepted

Curvature

0

Figure The number of simplices N r at distance r from the origin at

and for N

r

0

N

r

moves The time b efore the rst conguration was recorded was also sweeps

We estimated the auto correlation time in the average distance b etween two sim

plices to b e roughly sweeps for N and For other values of

the auto correlation time was lower The number of congurations at the values

of used are shown in table on page

Figures show eective curvature ts continuous lines in the crumpled

phase near the transition and in the elongated phase

together with global dimension ts see the next section at longer

distances for N The curves are extended b eyond the tted data range

r to indicate their region of validity A least squares t was used which

0

suppresses the lattice artefact region where N r is small b ecause it is sensitive

to absolute errors rather than relative errors

c

the ts were go o d even b eyond the range of r used to determine For

the t except obviously when this range already included all the p oints up to

0

the maximum of N r This can b e seen in gure on page where the t

Curvature and scaling

Figure Eective curvature t and global dimension t in the crumpled phase

at

r

0

N

r

is go o d up to r The t with ar br do es not app ear sensible anymore

0 c

b ecause then N r go es for values somewhat larger than the critical value

roughly linear with r down to small distances This can b e seen quite clearly in

gure on page so the R t in this gure should b e ignored

V

Figure on page shows the resulting eective curvature R as a function

V

of for and simplices For N the tting range was r

We see that R starts negative and then rises with going through zero In

V

contrast the Regge curvature at the triangles h Ri is p ositive for all values in

the gure

The value of a in 318 varied from ab out at to near the

transition These numbers are much smaller than the at value of in equation

320 indicating an eective volume p er simplex V much larger than

e

V This is at least partly due to the way we measure distances The

distances are measured using paths which can only go along the dual lattice and

will therefore b e larger than the shortest paths through the simplicial complex

Curvature

Figure Eective curvature t and global dimension t near the transition

r 0

N

r

There is a strong systematic dep endence of R on the range of r used in the

V

t For example for the N data in gure on page we used a least

squares t in the range r If the range is changed to r the data for

R have to b e multiplied with a factor of ab out We can enhance this eect

V

by reducing the tting range to only two r values and thereby obtain a running

curvature R r at distance r We write

e

0

N r arr brr 321

0

N r arr brr 322

and dene

brar 323 R r

e

which gives

0 0

r r N r N r

R r 324

e

0 0

r r N r N r

Curvature and scaling

Figure Eective curvature t and global dimension t in the elongated phase

at the eective curvature t is not appropriate here

r 0

N

r

Figure on page shows the b ehavior of R r for various It drops rapidly

e

from large values which is near h Ri at r to small values at r

c

the curves have a minimum and around this minimum the values of For

R average approximately to the R s displayed earlier

e V

The idea of R r and R is to measure curvature from the correlation function

e V

0

N r by comparing it with the classical volumedistance relation for distances r

going to zero as long as there is reasonable indication for classical b ehavior at

these distances Clearly we cannot let r go to zero all the way b ecause of the

huge increase of R This seems to indicate a Planckian regime where classical

e

b ehavior breaks down

Dimension

One of the interesting observables in the mo del is the dimension at large scales

A common way to dene a fractal dimension is by studying the b ehaviour of the

Dimension

Figure The eective curvature R as a function of for and

V

simplices

V

R

volume within a distance r and identifying the dimension d if the volume b ehaves

like

d

V r const r 325

Such a measurement has b een done in Agishtein Migdal a Agishtein

Migdal b using the geo desic distance and a distance dened in terms of the

massive scalar propagator David Arguments against the necessity of such

use of the massive propagator were raised in Filk Although we feel that the

issue is not yet settled we shall use here the geo desic distance as in the previous

section

If the volume do es go like a p ower of r the quantity

d ln V ln Nr ln Nr

d 326

d ln r lnr lnr

would b e a constant Figure on page shows this quantity for some values

of for a system with simplices We see a sharp rise at distances r

Curvature and scaling

Figure The running curvature R r for various and simplices

e

r e

R

r

indep endent of which is presumably a lattice artefact The curves then continue

to rise until a maximum where we may read o an eective dimension which is

clearly dierent in the crumpled phase and in the elongated phase

Instead of a lo cal maximum one would of course like a plateau

of values where d ln Vd ln r is constant and may b e identied with the dimension

d Only for b eyond the transition a range of r exists where d ln Vd ln r lo oks

like a plateau In this range d

Similar studies have b een carried out in twodimensional dynamical triangu

lation where it was found that plateaus only app ear to develop for very large

numbers of triangles Agishtein Migdal Amb jrn et al Kawamoto

et al Our fourdimensional systems are presumably much to o small for

d ln Vd ln r to shed light on a fractal dimension at large scales if it exists We

feel however that the approximate plateau in the elongated phase with d

should b e taken seriously

As we are studying a system with the top ology of the sphere S it seems

d

reasonable to lo ok whether it b ehaves like a ddimensional sphere S For such a

Dimension

Figure lnNrNr lnrr d ln Vd ln r as a function of r

for some values of

r

ln Vd

ln

d

r

hypersphere with radius r the volume b ehaves like

r

d- 0 d-

327 V r dC r sin

d

r

This prompts us to explore 327 as a denition of the dimension d we shall call

it the global dimension For small rr this reduces to 325 On the other hand

for d n the form 327 is compatible with the eective curvature form 313

with

nn

R 328

V

r

0

To determine the dimension d we t the data for N r to a function of the

form 327

r

d- 0

329 N r c sin

r

Curvature and scaling

The free parameters are r d and the multiplicative constant c It is a priori not

clear which distances we need to use to make the t At distances well b elow the

0

maximum of N r cf gure on page the eective curvature t app ears to

give a reasonable description of the data but it will b e interesting to try 327 also

for these distances Small distances are of course aected by the discretization

This is most pronounced at low where the range of rvalues is relatively small

On the other hand for small the ts turn out to b e go o d up to the largest

distances indicating a close resemblance to a hypersphere while at larger the

0

values of N r are asymmetric around the p eak cf gure on page and

ts turn out to b e go o d only up to values of r not much larger than this p eak

0

The system b ehaves like a hypersphere up to the distance where N r has its

maximum which would corresp ond to halfway the maximum distance for a real

hypersphere Ab ove that distance it starts to deviate except for small where

0

N r is more symmetric around the p eak and the likeness remains

For gure on page the global dimension t 329 was p er

formed to the data at r for gure on page to r

and for gure on page to r

The two descriptions eective curvature at lower distances and eective di

mension at intermediate and larger distances app ear compatible Notice that at

in the crumpled phase the lo cal eective curvature R is negative while

V

the global structure resembles closely a p ositive curvature sphere with radius

0

r r r with r the value where N r is maximal At

m m

near the transition the eective curvature and eective dimension descriptions ap

p ear to coincide At deep in the elongated phase the eective curvature

t do es not make sense anymore its rregion of validity has apparently shrunk

to order or less The eective dimension t on the other hand is still go o d in

this phase and the p ower b ehavior 325 with d is extended by 329 to

0

intermediate distances including the maximum of N r

Figure on the next page shows the global dimension as a function of

for the total volumes of and simplices For small values of it is

high and increases with larger volumes For values of b eyond the transition

it quickly go es to two conrming the statement made earlier that in this region

0

N r is approximately linear with r down to small r

A most interesting value of the dimension is the one at the phase transition To

c

where the transition takes place we lo ok at the curvature determine the value of

susceptibility of the system gure on page For simplices the p eak in

the susceptibility is b etween and where the dimensions we measured

Scaling

Figure The global dimension as a function of for and simplices

dimension

are and For simplices the p eak is b etween and where

the dimensions are and Therefore the dimension at the transition

is consistent with As can b e seen from these numbers the largest uncertainty

c

The eective dimensions have in the dimension is due to the uncertainty in

some uncertainty due to the ambiguity of the range of r used for the t Near the

transition this generates an extra error of approximately

Scaling

To get a glimpse of continuum b ehavior it is essential to nd scaling b ehavior in

the system We found a b ehavior like a d dimensional hypersphere for r values up

0

to the value r where N r is maximal This suggests scaling in the form

m

r

0 d-

r r N f d 330

m

r

m

0

ie N r dep ends on and N through d d N and r r N The

m m

o ccurrence of d in this formula is unattractive however since it is obtained by

Curvature and scaling

Figure The scaling function for at N and at

N

rr

m

d-

0

comparing N r to sin rr at intermediate scales which is a somewhat im

precise concept We would like a mo del indep endent test of scaling

Consider the probability for two simplices to have a geo desic distance r

0

N r

pr 331

N

Z

1

X

pr dr pr 332

r=

which dep ends parametrically on and N It seems natural to assume scaling

for this function in the form

r

pr 333

r r

m m

Z

dx x 334

where x rr and is a parameter playing the role of d in 330 which lab els

m

the dierent functions obtained this way For instance could b e the value

Scaling

Figure The scaling function for at N at

N and at N

rr

m

at the maximum of This may give give problems if do es

m m

not change appreciably with similar to d in the elongated phase Other

P

k

k

k k

p ossibilities are rpr or r r for some k with r prr In

m

r

c

N at some standard choice of N practice we may also simply take

and compare the probability functions with the pr at this N

Matched pairs of x for N and and one curve at N

are shown in gures on pages resp ectively far in the crumpled

phase near the transition and in the elongated phase For clarity we have left out

part of the errors Scaling app ears to hold even for values we considered far

away from the transition

The values of N of the matched pairs in gures are increasing

with N in the crumpled phase and decreasing with N in the elongated phase

c

N as N increases For current This suggests convergence from b oth sides to

c

N is still very much dep endent on N a p ower extrap olation system sizes

c

estimate in Catterall et al a gives while one in Amb jrn

Curvature and scaling

Figure The scaling function for at N and at

N

rr

m

Jurkiewicz b gives although the latter one uses a dierent criterion

to determine the place of the transition

We can use the matched pairs of to dene a scaling dimension d by

s

d

s

335 N r

m

where and d dep end only on Using nonlattice units replacing the integer

s

p

r by r where r is now momentarily dimensionful we can interpret

m m m

335 as

d

s

r

m

N 336

which shows that the scaling dimension characterizes the dimensionality of the

system at small scales with r xed Returning to lattice units taking

m

0

for r the integer value of r where N r is maximal and using an assumed error

m

of these scaling dimensions would b e and resp ectively

Scaling

for gure on page gure on page and gure on the preceding

page The scaling dimension b etween and simplices in gure was

The largest errors in these gures probably arise due to the uncertainty in

the values of we need to take to get matching curves ie to get the same value

of As we do not have data for a continuous range of values we have to make

do with what seems to match b est from the values we do have Nevertheless the

values of d far away from the transition are strikingly close to when compared

s

to the values of the global dimension d which are and for gures on

page and on the preceding page

The scaling form 330 is in general incompatible with 333 except for d

s

d The evidence for d p oints instead to a scaling b ehavior of the form

s

r

0

N r r f 337

m

r

m

0

with fx x This further suggests scaling of the volume V r at

distance r with an eective volume V p er simplex cf 315 dep ending only on

e

A precise denition of the scaling form x may b e given by

x r pr x N such that r pr N 338

m m m m

where we used for illustration Intuitively one would exp ect the con

m

vergence to the scaling limit 338 to b e nonuniform with the large x region

converging rst and there may b e physical asp ects to such nonuniformity

The scaling analogue of running curvature 324 is given by

d ln d ln x

e

339 R xr Rx r

m

e

m

x d ln d ln x

Figure on the following page shows this function for the matched pair of

gure on page in the transition region We have also included the curve for

at N The curves do not match in the region around x and

R r is apparently a sensitive quantity for scaling tests Still gure on the

e

following page suggests reasonable matching for a value of somewhere

b etween and and even the steep rise as far as shown app ears to b e scaling

e e

approximately with R somewhat b elow R We nd similar scaling

b ehavior for the matched pair in the elongated phase The number of our values

in the crumpled phase is to o limited to b e able to draw a conclusion there

The steep rise app ears to move to the left for increasing N a scaling violation

A most interesting question is whether the onset of the rise eg the x value where

Curvature and scaling

Figure Scaling form r R xr versus x rr near the the transition for

m m

e

m

N middle and N lower

and upp er The hypothetical limiting form corresp onding to S is

also shown

S

m

r e

R

rr

m

e

R continues to move towards x as N Such b ehavior is needed for

a classical region to op en up from x around towards the origin x In other

words the Planckian regime would have to shrink in units of the size r of the

m

e

R could b e dened as a limiting value of R x universe Then r

V

m

Since we are lo oking for classical b ehavior in the transition region it is instruc

e

tive to compare with the classical form of R corresp onding to the sphere S for

which sin x

m

tan

e

R x 340

tan

This function is also shown in gure Our current data are evidently still far

from the hypothetical classical limiting form 340

Summary

Summary

0

The average number of simplices N r at geo desic distance r gives us some basic

information on the ensemble of Euclidean spacetimes describ ed by the partition

0 0

function 31 The function N r is maximal at r r and r N rN shows

m m

scaling when plotted as a function of rr We explored a classical denition

m

of curvature in the small to intermediate distance regime based on spacetime

dimension four the eective curvature R We also explored a description at

V

0

global distances by comparing N r with a sphere of eective dimension d Judged

by eye the eective curvature ts and eective dimension ts give a reasonable

0

description of N r in an appropriate distance regime gures on pages

The resulting R dep ends strongly on the tting range which led us to an ex

V

plicitly distance dep endent quantity the running curvature R r This dropp ed

e

rapidly from lattice values of order of the Regge curvature at r to scaling values

of order of the R found in the eective curvature ts at intermediate distances A

V

preliminary analysis of scaling b ehavior then suggested the p ossibility of a classi

cal regime with a precise denition of r R in the limit of large N We shall now

V

m

summarize the results further keeping in mind the ambiguity in R as derived

V

0

from the ts to N r

For small the eective curvature is negative Furthermore the system re

sembles a dsphere with very large dimension d which increases with the volume

This suggests that no matter how large the volumes we use there will never b e a

d

region of r where the p ower law V r r gives go o d ts over large ranges of r

In other words the curves for d ln Vd ln r in gure on page will never have

a plateau This b ehavior is consistent with that of a space with constant negative

curvature where the volume rises exp onentially with the geo desic distance for

distances larger than the curvature radius and if we lo ok at large enough scales

the intrinsic fractal dimension equals innity The resulting euclidean spacetime

cannot b e completely describ ed as a space with constant negative curvature as

such a space with top ology S do es not exist and nite size eects take over at

still larger distances

At the transition the spacetime resembles a fourdimensional sphere with small

p ositive eective curvature up to intermediate distances

For large the system has dimension In this region it app ears to b ehave like

a branched p olymer which has an intrinsic fractal dimension of David

Moving away from the transition the curvature changes much more rapidly than in

p

the small phase and the eective curvature radius r R so on b ecomes

V V

Curvature and scaling

of the order of the lattice distance A priori two outcomes seem plausible In the

rst the system collapses and r b ecomes of order in lattice units reecting the

V

unboundedness of the continuum action from b elow In the second the spacetime

remains fourdimensional and r can still b e tuned to large values in lattice units

V

but very small compared to the global size r for a suciently large system So

m

far the second outcome seems to b e favored for two reasons Firstly the function

0

N r lo oks convex for r slightly ab ove the transition eg for

at simplices In other words the linear b ehavior as seen in gure on

page do es not set in immediately ab ove the transition Secondly the system

shows scaling and the scaling dimension d as dened in 335 is approximately

s

even far into the elongated phase indicating fourdimensional b ehavior at small

scales

High and low dimensions in the crumpled and elongated phases with the value

four at the transition were rep orted earlier in Agishtein Migdal b This

dimension was apparently interpreted as a small scale dimension whereas instead

we nd a small scale dimension of four in all phases Similar results are also found

in the Regge calculus approach to quantum gravity Hamber Beirl et al

a where one also nds two phases a strong bare Newton coupling phase

with negative curvature and fractal dimension four and using an R term in the

action for stabilization a weak coupling phase with fractal dimension around two

Discussion

The evidence for scaling indicates continuum b ehavior This brings up a number

of issues which need to b e addressed in a physical interpretation of the mo del

One guideline in this work is the question whether there is a regime of distances

0 0

where V r V N r b ehaves classically for suitable bare Newton constant G

e

The connection with classical spacetime can b e strengthened by identifying the

0

geo desic distance r with a cosmic time t and V r with the scale factor at in

a Euclidean Rob ertsonWalker metric The classical action for ar is given by

Z Z

da

S a dr a dr a V 341

G dr

where is a Lagrange multiplier enforcing a total spacetime volume V It plays

the role of a cosmological constant which is just right for getting volume V For

p ositive G the solution of the equations of motion following from 341 is a

r sin rr which represents S with R r G For negative G the

Discussion

solution is a r sinh rr which represents a space of constant negative curvature

G cut o at a maximal radius to get total volume V R r

The form 341 serves as a crude eective action for the system for intermediate

-

0

distances around the maximum in V r at r and couplings G in the

m

transition region At larger distances the uctuations of the spacetimes grow

causing large baby universes and branching and averaging over these may b e the

0

reason for the asymmetric shap e of V r Because of this the Rob ertsonWalker

metric cannot give a go o d description at these distances

Intuitively one exp ects also strong deviations of classical b ehavior at distances

p

of order of the Planck length G assuming that a Planck length exists in the

mo del A prop osal for measuring it will b e put forward in chapter ve The

steep rise in the running curvature R r at smaller r indeed suggests such a

e

Planckian regime It extends to rather large r but it app ears to shrink compared

to r as the lattice distance decreases ie N increases for a given scaling curve

m

lab elled by This suggests that the Planck length go es to zero with the lattice

spacing Gr as N at xed This do es not necessarily mean that

m

the Planck length is of order of the lattice spacing although this is of course quite

p ossible The theory however may also scale at Planckian distances and b elong

to a universality class It might then b e trivial

At this p oint it is instructive to recall other notorious mo dels with a dimension

ful coupling as in Einstein gravity the fourdimensional nonlinear sigma mo dels

The lattice mo dels have b een well studied in particular the O mo del for low

energy pion physics see for example L usher Weisz L usher Weisz

which corresp onds to the renor Heller It has one free parameter f

malized dimensionful coupling f f is the pion decay constant or the electroweak

scale in the application to the Standard Mo del With this one bare parameter it

is p ossible to tune two quantities fm and f where m is the mass of the sigma

particle or the Higgs particle This trick is p ossible b ecause the precise value of



is unimp ortant as long as it is suciently small In the continuum limit f

however triviality takes its toll mf and the mo del b ecomes noninteracting

-

The analogy f suggests that we may b e lucky and there G m r

m

is a scaling region in N space for a given scaling curve given where the

theory has universal prop erties and where we can tune Gr to a whole range of

m

desired values Taking the scaling limit however might lead to a trivial theory



It is go o d to keep the numbers in p ersp ective for example in the Standard Mo del f =

GeV and for a Higgs mass m = GeV or less is orders of magnitude smaller than the

Planck length or even much smaller

Curvature and scaling

with Gr In case this scenario fails it is of course p ossible to introduce more

m

parameters eg as in R gravity to get more freedom in the value of Gr This

m

then raises the question of universality at the Planck scale

We really would like to replace r by R in the reasoning in the previous

V

m

paragraph since we view R as the lo cal classical curvature provided that a

V

classical regime indeed develops as N

Next we discuss the nature of the elongated phase Even deep in this phase

we found evidence for scaling Furthermore for given scaling curve increasing N

means decreasing Hence increasing N at xed brings the system deep er in

the elongated phase This leads to the conclusion that there is nothing wrong with

the elongated phase It describ es very large spacetimes which are two dimensional

on the scale of r but not necessarily at much smaller scales It could b e eectively

m

classical at scales much larger than the Planck length but much smaller than r

m

This reasoning further suggests that N and primarily serve to sp ecify the

c

shap e of the spacetime The tuning is apparently not needed for obtain

ing criticality but for obtaining a type of spacetime The p eak in the susceptibility

of the Regge curvature could b e very much a reection of shap e dep endence Most

imp ortantly this suggests that the physical prop erties asso ciated with general co

ordinate invariance will b e recovered automatically in fourdimensional dynamical



triangulation as in two dimensions with xed top ology



A eld theory analogue is Z lattice gauge theory which for n > has b een found to p ossess

n

a Coulomb phase a whole region in bare parameter space with massless photons see for example

Frolich Sp encer Alessandrini Alessandrini Boucaud

Chapter Four

Correlations

n the dynamical triangulation mo del of dimensional Euclidean

quantum gravity we measure twopoint functions of the scalar cur

vature as a function of the geo desic distance To get the corre

lations it turns out that we need to subtract a squared onep oint

function which although this seems paradoxical dep ends on the I

distance At the transition and in the elongated phase we observe a p ower law b e

haviour while in the crumpled phase we cannot nd a simple function to describ e

it

Introduction

In the dynamical triangulation mo del of four dimensional euclidean quantum grav

ity the path integral over metrics on a xed manifold is dened by a weighted sum

over all ways to glue foursimplices together at the faces Amb jrn Jurkiewicz

Agishtein Migdal a This idea was rst formulated in Weingarten

using hypercub es instead of simplices

The partition function of the mo del at some xed volume of N simplices is

X

ZN exp N 41

T (N =N)

The sum is over all ways to glue N foursimplices together such that the resulting

complex has some xed top ology which is usually as well as in this chapter

taken to b e S The N are the number of isimplices in this complex is a

i

coupling constant which is prop ortional to the inverse of the bare Newton constant

-

G

Correlations

It turns out that the mo del has two phases For low the system is in a

crumpled phase where the average number of simplices around a vertex is large

and the average distance b etween two simplices is small In this phase the volume

within a distance r app ears to increase exp onentially with r a b ehaviour like

that of a space with constant negative curvature At high the system is in an

elongated phase and resembles a branched p olymer As is the case with a branched

p olymer the large scale internal fractal dimension is two

c

The transition b etween the two phases o ccurs at a critical value which

dep ends somewhat on N This transition app ears to b e a continuous one making

a continuum limit p ossible Agishtein Migdal b Amb jrn et al c

Catterall et al a see chapter two At the transition the space b ehaves in

several resp ects like the four dimensional sphere see chapter three

Curvature and volume

In the Regge discretization of general relativity all the simplices are pieces of

at space The curvature is concentrated on the subsimplices of co dimension

two in our case the triangles On these triangles it is prop ortional to a two

dimensional delta function From the denition of curvature as the rotation of

a parallel transp orted vector one can nd the integrated curvature over a small

region V around such a triangle

Z

p

R g dx A 42

4 4

V (4)

where A is the area of the triangle and is the decit angle around the triangle

4 4

see eg Hamber The decit angle around a triangle is the angle which is

missing from

X

43

4 d

d2fS(4)g

where fSg are the simplices around the triangle and is the angle b etween

d

those two faces of the simplex that b order the triangle The decit angle can

4

b e negative

In dynamical triangulation all the simplices have the same size and shap e and

the decit angle is a simple function of the number n of simplices around the

4

triangle Then expression 42 reduces to

Z

p

R g dx V n 44

4

V (4)

Twopoint functions

where is the angle b etween two faces of a simplex which for D dimensions equals

arccosD and V A is the now constant area of a twosimplex that is a

4

triangle

For each triangle we can now dene a lo cal fourvolume that b elongs to the

triangle by assigning that part of each adjoining simplex to it which is closer to

the triangle than to any other For our equal and equilateral simplices this just

results in V p er adjoining simplex with V the volume of a four simplex In

other words this lo cal volume is

Z

p

V

V n 45 g dx

4 4

(4)

where is the region of space asso ciated to that triangle It is not clear what

V would mean in the continuum limit We dene it here mainly to compare our

4

results with other work on simplicial quantum gravity

If we view the delta function curvature as the average of a constant curvature

over the region this constant curvature would b e equal to

n

V

4

R 46

4

V n

4

Because neither a constant term nor a constant factor is imp ortant for the b e

haviour of correlation functions we will in the rest of this chapter for convenience

dene the curvature as

-

R n 47

4

4

and the lo cal volume as

V n 48

4 4

Twopoint functions

One of the interesting asp ects of the dynamical triangulation mo del one can inves

tigate is the b ehaviour of twopoint functions of lo cal observables In continuum

language such a correlation function of a lo cal observable Ox at a distance d

would b e dened by

Z Z

p p

dx gx gy OxOydx y d dy

Z Z

hOOid 49

p p

gx dy gy dx y d dx

Correlations

where the average is

Z

Dg A expSg

Z

hAi 410

Dg expSg

where dx y is the geo desic distance b etween the p oints x and y for a given metric

g In other words for each conguration ie for each metric we average over

all pairs of p oints that have geo desic distance d Obviously it makes little sense to

dene such a correlation function for two xed p oints x and y Because of general

co ordinate invariance such a correlation could only dep end on whether x and y

coincide or not

The discrete version of equations 49 and 410 b ecomes

X

OxOy

d(xy)d

xy

hOOid X 411

d(xy)d

xy

with the average

X

A exp N T

T

X 412 hAi

exp N T

T

In gure on the facing page we have plotted the correlation function of the

curvature with the square of its exp ectation value subtracted Most of the data in

this pap er are for a volume N simplices The values of corresp ond to a

system in the crumpled phase near but slightly b elow the transition

and in the elongated phase

Congurations were recorded every sweeps where a sweep is dened as

a number of accepted moves equal to the number of simplices N For

and we used and congurations resp ectively

We dene the geo desic distance b etween two triangles as the smallest number

of steps b etween neighbouring triangles needed to get from one to the other For

this purp ose we dene two triangles to b e neighbours if they are subsimplices of

the same foursimplex and share an edge Other denitions of neighbour are con

ceivable One such a denition would b e to dene two triangles to b e neighbours

Twopoint functions

Figure The correlation function hRRid hRi for various values of

i R h

d i

RR

h

d

if they share an edge irresp ective of whether they are in the same simplex The

one we use has the advantage that it is quite narrow and therefore results in larger

distances

One thing is immediately striking the correlation functions do not go to zero

at long distances To keep the short distance b ehaviour visible the full range in

the elongated phase has not b een plotted but we already see that also in this

phase it crosses the zero axis and indeed this curve do es eventually go to large

p ositive values

The lo cal volume V is prop ortional to the number of simplices n around a

4 i

triangle i We see that in this mo del this observable V is essentially the same

4

as the scalar curvature At rst sight one would therefore exp ect them to have

the same b ehaviour If one is p ositively correlated the other one would also b e

p ositively correlated Figure on the next page shows the correlation of n We

see that quite the opp osite is true With few exceptions n is p ositively correlated

-

where R n is negatively correlated and vice versa

This b ehaviour is similar to that rep orted for the Regge calculus formulation

Correlations

Figure The correlation function hnnid hni for various values of

i n h

d i nn

h

d

of simplicial quantum gravity in Hamber There it is also found that the

curvature correlations are p ositive and the volume correlations negative at large

distances

This dierence in b ehaviour can b e explained intuitively as follows Because

triangles with large n have more neighbours any random triangle will have a large

chance to b e close to a p oint with large n and a small change to b e close to a p oint

with small n So whatever the value of n at the origin the p oints nearby have

large n and the p oints far away have small n The average hnni will then b e large

at small distances and small at large distances Because large n means small R

the situation is reversed if we substitute R for n in this discussion qualitatively

explaining gure on the preceding page and gure

At rst sight one might conclude from this explanation that a p oint with

large n having many neighbours is just an artefact of the mo del This is not

true however Large n corresp onds to large negative curvature and also in the

continuum a p oint with large negative curvature has a larger neighbourho o d To

b e more precise the volume of ddimensional space within a radius r around a

Connected part

Figure The curvature as a function of the distance hRid for various values

of

d

i R

h

d

p oint with scalar curvature R equals

R

d

V R C r r Or 413

d

d

Connected part

The ab ove reasoning leads us to the somewhat unusual concept of a correlation

function which do es not dep end on some observable at the origin We dene such

a correlation as

X

R

x

d(xy)d

xy

X

414 hRid

d(xy)d

xy

Correlations

Figure Comparison b etween the correlation function hRRid upp er curves of

each pair and the squared onep oint function hRid lower curves

at various values of

d

i R

h and d

i RR

h

d

where x and y denote a triangle In the more usual case of a quantum eld theory

on at space this could never dep end on the distance but here it do es The reason

is that we correlate functions of the geometry with the distance which is itself a

function of the geometry

Figure on the preceding page shows this correlation function No average

has b een subtracted The b ehaviour of this onep oint function turns out to b e

very similar to that of the curvature correlation in gure on page This

correlation function again shows that any particular p oint has a large chance to b e

in the neighbourho o d of a p oint with low curvature which can b e simply explained

with the fact that p oints with low curvature have more neighbourho o d

The same plot for n not shown shows the opp osite b ehaviour At small

distances it is larger than average while at large distances it is smaller than

average This is rather obvious b ecause where n is large its inverse is small and

vice versa

Connected part

Figure Corrected correlation function C d at various values of

R

-

-

-

d R

C

-

-

-

-

d

We can now investigate how much of the curvature correlation shown of g

ure on page is due to this eect Figure on the preceding page compares

the curvature correlation hRRid with the square of this onep oint function We

see that except at small distances the two are indistinguishable on this scale In

other words we have not b een measuring any curvature correlations All we have

measured are correlations b etween the curvature and the geo desic distance

It is now easy to explain the dierence in b ehaviour b etween the curvature and

the volume correlations Because they are almost equal to the square of hRid

and hnid resp ectively they b ehave just like them And as we just mentioned it

is easy to understand that these have opp osite b ehaviours

The way to go now is to subtract the two things and see what real curvature

correlations are left This is similar to subtracting a disconnected diagram and

keeping the connected part We get the corrected correlation functions for the

curvature

C d hRRid hRid 415

R

Correlations

Figure Corrected correlation function C d at various values of

V

d

V

C

d

and the volume

C d hnnid hnid 416

V

The results for the curvature are plotted in gure on the preceding page and

those for the volume in gure The error bars were found by a jackknife

metho d each time leaving out one of the congurations in the calculation of

C d and C d Now b oth correlations b ehave almost exactly the same Note

R V

the large dierence in scale b etween these gures and gures and

In the crumpled phase we were not able to t C d to a simple function This

R

is probably due to the fact that we cannot reach very large distances in this phase

Near the transition however it is p ossible to t the correlation function to a p ower

law decay at not to o small distances This is shown in gure on the facing

b

page In the region d it ts nicely to ad with the result

a 417

b 418

Connected part

Figure Power law t to curvature correlation C d near the phase transition

R

at N and

- d R

C

-

-

d

at dof 419

This data was made at a volume of simplices with We used

congurations which were recorded every sweeps

This result should b e taken with caution however One would really like to

have a go o d t over a larger range To get some idea of the typical ranges involved

we consider the number of triangles at distance d

X

d(xy)d

xy

0

420 N d

N

where N is the number of triangles of the conguration The corresp onding

quantity with triangles replaced by foursimplices was studied more closely in

0

chapter three The value of d which is that d for which N d has its maximum

m

is an indication of the distance at which nite size eects might b ecome imp ortant

Correlations

Figure Power law t to curvature correlation C d in the elongated phase at

R

-

- d R C

-

-

d

At this d is only indicating that nite size eects might play a

m

role in the p ower that was measured

The situation is even b etter far in the elongated phase Here a p ower law ts

well as can b e seen in gure This t was done to the p oints d and

the parameters of this t are

a 421

b 422

at dof 423

The value of d with maximum number of triangles was so in this case we are

in a region of small distances compared to the system size This data was made

from congurations of simplices We have also tted the connected RR

correlation at other p oints in the elongated phase and at simplices The

p ower that emerged was within the errors equal to the one given ab ove

Discussion

Discussion

We have investigated the b ehaviour of the curvature and volume correlation func

tions It turned out that the naive correlation functions could b e almost entirely

describ ed by a disconnected part which we therefore subtracted The dierence

turns out to b ehave according to a p ower law in the elongated phase and near the

transition In the latter case the p ower is close to four which is reminiscent of

two graviton exchange

In chapter three we explored the p ossibility of a semiclassical region near the

transition in which the system b ehaves like a foursphere for not to o small or

large distances To this end we dened a scale dep endent eective curvature See

gure on page For near the transition the following picture emerged

At small distances this eective curvature is large indicating a Planckian regime

At intermediate distances there seems to b e a semiclassical regime where the

space b ehaves like a foursphere The uctuations around this approximate S

might then corresp ond to gravitons We consider it therefore encouraging that

the p ower b in 418 is compatible with four

For the volumes in current use the eective curvature shows that the semi

classical regime sets in at a distance roughly of r Here r is the geo desic

m m

0

distance through the simplices where the number of simplices N r has its max

imum Similarly of d turns out to b e the distance where the curvature

m

-

correlations start to b ehave like d We like to think of this as a conrmation of

the p oint of view sketched ab ove

Twopoint functions of curvature and volume have b een studied in the Regge

calculus formulation of simplicial quantum gravity in references Hamber

Beirl et al b In these studies there are only results in what is called the well

dened phase of the Regge calculus approach which corresp onds to our crumpled

phase This makes it hard to do more than the qualitative comparison which was

done in section on page

The curvature correlations have also b een investigated in the continuum In

Antoniadis Mottola a theory is developed for the conformal factor in four

dimensional quantum gravity and from this the curvature correlation is calculated

The conformally invariant phase discussed in Antoniadis Mottola seems

to corresp ond to the elongated phase in the dynamical triangulation mo del Intu

itively this can b e understo o d by visualizing large uctuations in the conformal

factor as generating many baby universes Many baby universes is also a fea

ture of the branched p olymer like elongated phase of simplicial quantum gravity

Amb jrn et al b Furthermore the conformally invariant phase is supp osed

Correlations

to o ccur at very large distance scales In chapter three we argued that the elon

gated phase also describ es scales which are large compared to a typical physical

curvature scale In this conformally invariant phase a p ower law is predicted for

the curvature correlations see also Antoniadis et al with a p ower of

Unfortunately a direct comparison with Antoniadis Mottola Antoniadis

et al is not p ossible b ecause in the continuum the correlation function is

dened as a function of the distance in a xed ducial metric a quantity that is

not yet dened in our mo del

Chapter Five

Binding

ne of the most obvious features of gravity is that it attracts ob jects

to each other A mo del of quantum gravity should therefore b e able

to show gravitational attraction In this chapter we investigate this

in the dynamical triangulation mo del by putting a scalar eld on

the congurations and lo oking for a nonzero binding energy In O

the end we will see that attraction is indeed present

Description of the mo del

We lo ok at the b ehaviour of a free scalar eld with bare mass m in a quantum

gravity background The Euclidean action of this system in continuum language

is a sum of a gravitational and a matter part

S Sg Sg 51

Z

p

D

Sg d x g R 52

G

Z

p

D

g m 53 g Sg d x

where is the bare cosmological constant R is the scalar curvature and G is

the bare Newton constant

We take as a test particle here ie the back reaction of the eld on the

metric is not taken into account This approach is often called the quenched ap

proximation In QCD this approximation turns out to give go o d results see eg

Sharp e In that case it is also called the valence quark approximation

b ecause it neglects diagrams with internal quark in our case lo ops A con

tinuum calculation of the gravitational attraction of a scalar eld in this same

Binding

quenched approximation was done in Mo danese It is seen in other sim

ulations Amb jrn et al a that including matter has little inuence on the

gravity sector of the theory

We will use the following notation for exp ectation values of an observable A

On a xed background geometry we can average over congurations of the matter

eld

Z

D A expSg

Z

hAi 54

D expSg

and we can average over metrics

Z

Dg A expSg

Z

hAi 55

g

Dg expSg

The quenched exp ectation value is then

hAi hhAi i 56

g

We can now lo ok at propagators in a xed geometry The one particle propa

gator denoted by Gx is dened as

Gx h i 57

x

where is an arbitrary p oint The connected two particle propagator will then b e

the square of the one particle propagator

h i Gx 58

x x

conn

Letting the metric uctuate we take the average of the propagators over the

dierent metrics Because of reparametrization invariance the average hGxi

g

will not dep end on the place x Therefore we lo ok at averages at xed geo desic

distance d

Z

p

D

Z

g Gx dx d d x

Z

Dg expSg 59 hGdi

g

p

Z

D

d x g dx d

Implementation

where dx y is the minimal geo desic distance b etween x and y For a massive

particle we exp ect the propagator 59 to fall o exp onentially as

hGdi d expmd 510

g

hGdi d expmd 511

g

with some p ower and the renormalized mass m which in general will not equal

the bare mass m These expressions neglect nite size eects and should probably

b e mo died when lo oking at distances comparable to the size of the system We

will only use the data at relatively short distances so this should not b e imp ortant

The two particle propagator will b ehave similarly as

hGd i d expE d 512

g c

where E is the energy of the two particle comp ound If this energy turns out to b e

c

less than two times the mass of a single particle the dierence can b e interpreted as

a binding energy b etween the particles This would show gravitational attraction

b etween them Because the average of the square of a uctuating quantity is always

greater than the square of its average it is obvious that hGd i hGdi This

g

g

do es not yet imply anything ab out the way they fall o In particular it is not

guaranteed that E m

c

Implementation

We have run numerical simulations with b oth two and four dimensional dynamical

triangulations

In two dimensions the volume can b e kept constant and for xed top ology no

parameters will b e left We used systems of and triangles with the

top ology of the twosphere

In four dimensions the analogue of the continuum gravitational action is

Z

p

Sg d x g R 513

G

N N 514

where N and N are the number of triangles and foursimplices resp ectively We

used systems of ab out simplices and the top ology of the foursphere To

keep the number of simplices around the desired value we added a quadratic term

to the action as was describ ed in section

Binding

Figure The two particle propagator and the square of the one particle prop

agator versus the geo desic distance for three dierent bare masses m

in two dimensions The vertical scale is logarithmic

i G

h

ln and

i

G h

ln

d

The parameter is inversely prop ortional to the bare gravitational constant

G As we have shown in chapter two we see indications of a second order phase

transition as varies

On each dynamical triangulation conguration we then calculated the propa

gator

-

515 Gx m

x

of the scalar eld using the algebraic multigrid routine AMGR The discrete

Laplacian is dened as

d if x y

516

if x and y are nearest neighbours

xy

otherwise

Results

Figure The eective binding energy E as a function of the geo desic distance

b

for three dierent bare masses in two dimensions

energy

binding

d

Eventually we hop e to b e able to extract the renormalized Newton constant

G near the critical value of the inverse bare Newton constant for example

R

according to the nonrelativistic formula

E m E m 517 G

c

b

R

This formula is just the familiar energy m of the hydrogen atom in the

e

ground state but with the gravitational parameters substituted as G m

R

and taking into account that we now have two particles of equal mass As it is

nonrelativistic it may not suce to t the data

Results

In gure on the preceding page we see the results in two dimensions for three

dierent bare masses Each pair of lines corresp onds to one bare mass In each

Binding

Figure As in gure but in four dimensions which is very

close to the transition

i G h

ln

and i G

h

ln

d

pair the upp er line is lnhGd i the two particle propagator and the lower line

is lnhGdi the pro duct of two single particle propagators

There is clearly a dierence in slop e b etween the lines in each pair This shows

that the energy of the two particle comp ound is less than two times the mass of

a single particle and consequently that there is a p ositive binding energy b etween

the particles

In the two dimensional case one might exp ect not to see any attraction b ecause

of the absence of dynamics in the classical system In the quantum case however

a nontrivial theory results due to the conformal anomaly Polyakov

Figure shows similar data in four dimensions with a coupling constant

This is very close to the phase transition We used congurations

recorded every sweeps For the three masses from lowest to highest we used

and origins resp ectively We calculated hGdi by averaging it over all

p oints at distance d from the origin This corresp onds to taking the propagator

from a source that is not a single p oint but a complete shell around the origin

Results

Figure As in gure but in four dimensions Again

energy

binding

d

The use of what are called smeared sources can improve the data by increasing

the contribution of the ground state and decreasing the contribution of the excited

states

As in the twodimensional case there is a clear dierence in slop e Using this

data we can measure the renormalized mass m The results are

m m mm

518

It was argued in Agishtein Migdal b that the physical mass should vanish

at zero bare mass and that therefore the renormalization would b e only multiplica

tive Our data seem to show that the relation is more complicated Increasing m

p

by a factor of increases m by a factor of ab out

The propagators curve downward towards the origin This is somewhat unex

p ected as it is interpreted as a sum over decaying exp onentials We do not know

how to explain this phenomenon

Binding

Using these data we can now estimate the binding energy of the particles

From 511 and 512 we have

E m E 519

c

b

hGd i

g

-

d 520 d ln

hGdi

g

As we cannot use innite distances we will consider this expression at nite d

and lo ok whether the eective binding energy E d b ecomes constant

b

Figure on page shows this quantity as a function of the geo desic distance

The three curves again corresp ond to the three dierent bare masses Although

the result is not yet very accurate it is clear that the binding energy go es to a

nonzero value

Figure on the preceding page shows the data in the four dimensional sim

ulations These data are still somewhat preliminary Nevertheless this gure

also clearly indicates a nonzero binding energy Unfortunately unlike the two

dimensional case the correlation b etween the mass and the binding energy do es

not app ear to b e strictly p ositive The lowest binding energy b elongs to the high

est mass

Considering that the space lo oks like a foursphere as argued in chapter three

it might b e interesting to compare the propagators to those on a real foursphere

This could p erhaps reduce the nite size eects in the eective binding energies

at large distances

Chapter Six

Noncomputability

ue to the unrecognizability of certain manifolds there must exist

pairs of triangulations of these manifolds that can only b e reached

from each other by going through an intermediate state that is very

large This might reduce the reliability of dynamical triangulation

b ecause there will b e states that will not b e reached in practice D

This problem was investigated numerically for the manifold S in Amb jrn

Jurkiewicz a but it is not known whether S is recognizable We p erform

a similar investigation for the manifold S which is known to b e unrecognizable

but see no sign of any unreachable states

Noncomputability

To generate all the congurations of dynamical triangulation we need an algorithm

that is ergo dic ie a set of moves that can transform any triangulation into any

other triangulation with the same top ology A well known set of moves that

satisfy this condition are the socalled k l moves whose ergo dicity was shown

in Pachner Gross Varsted These moves are extensively discussed

in chapter eight

Unfortunately the number of moves we need to get from one conguration

to another can b e very large To b e more precise the following theorem holds

if the manifold under consideration is unrecognizable then for any nite set of

elementary moves the number of moves needed to get from one conguration of N

simplices to another such conguration is not b ounded by a computable function

of N This was shown in reference Nabutovsky BenAv We will explain

Noncomputability

some of the terms in this theorem in a way that is not mathematically precise

but hop efully intuitively clear See Nabutovsky BenAv for details

A manifold A is unrecognizable if given a triangulation T A of this manifold

there do es not exist an algorithm that given as input an arbitrary triangulation

T B of a manifold B can decide whether A and B are homeomorphic The

denition of unrecognizability is not imp ortant for the rest of this article it is only

imp ortant to know that for some manifolds the ab ove theorem holds Certain four

dimensional manifolds are unrecognizable but for the sphere S which is usually

used in dynamical triangulation this is not known It is known however that the

ve dimensional sphere S is unrecognizable

A computable function is a function from N to N that can b e computed by a

large enough computer Although the computable functions are only an innitesi

mally small fraction of all the functions from N to N most functions one can think

of are computable A fastgrowing example of a computable function would b e

N with N factorial signs

Elementary moves can b e any type of moves that are computable ie any type

that we can actually do in our computer in nite time The authors of Amb jrn

Jurkiewicz a restrict the discussion to lo cal moves Lo cal moves are moves

that involve a number of simplices that is b ounded by a constant in other words

a number that do es not grow with the volume of the conguration But their

reasoning extends to any computable set of moves Because having a computable

b ound on the necessary number of computable moves allows you to compute all

p ossible reachable congurations and in that way recognize unrecognizable mani

folds The conclusion is that implementing nonlo cal moves like the baby universe

surgery of Amb jrn Jurkiewicz b will not allow you to get around this

problem

The pro of of the theorem go es along the following lines Supp ose that such

a computable b ound exists We could then try all p ossible sequences of moves

which are not longer than this b ound starting from the triangulation T A For

each triangulation T A of A that we get this way we can check whether it is

i

equal to T B As this would give us all p ossible triangulations of A this allows

us to check whether A and B are homeomorphic and hence recognize B as b eing

equal to A Given that A is not recognizable this is imp ossible

The ab ove theorem might seem a terrible obstacle for numerical simulation

but the theorem says nothing ab out the number of moves needed to generate all

congurations of any particular size It is therefore far from clear whether it will

have implications for the present simulations

Barriers

Barriers

From the theorem stated ab ove it follows that for an unrecognizable manifold the

maximum size N N of the intermediate congurations needed to interpolate

int

b etween any two congurations of size N is also not b ounded by a computable

function of N If N N did have such a b ound a b ound on the number of p ossible

int

congurations of size less than or equal to N N would b e a b ound on the number

int

of moves needed which would violate the theorem As I explained on page in

chapter two a simple computable b ound on the number of congurations of size

N

N is N where d is the dimension of the simplices

It was p ointed out in Amb jrn Jurkiewicz a that this means that for

such a manifold there must exist barriers of very high sizes b etween certain p oints

in conguration space Although the situation is not clear from the theorem it

seems natural that these barriers o ccur at all volume scales We can then apply

the following metho d which was formulated in Amb jrn Jurkiewicz a

We start from an initial conguration with minimum size For S and S there is

a unique conguration of minimum size with six and seven simplices resp ectively

This is just the b oundary of one ve resp ectively sixdimensional simplex We

increase the volume to some large number and let the system evolve for a while

which might take it over a large barrier Next we rapidly decrease the volume

hoping to trap the conguration on the other side of this barrier

We can check whether this has happ ened by trying to decrease the volume even

more If this brings us back to the initial conguration we have gone full circle

and cannot have b een trapp ed at the other side of a barrier Conversely if we

get stuck we are apparently in a metastable state ie at a p oint in conguration

space where the volume has a lo cal minimum

This was tried in Amb jrn Jurkiewicz a for S but no metastable

states were found To judge the signicance of this it is useful to investigate the

situation for a manifold which is known to b e unrecognizable It is rather dicult

to construct a four dimensional manifold for which this is known but if we go to

ve dimensions this is easy b ecause already the sphere S is not recognizable

Results

Because my program for dynamical triangulation was written for any dimension

see chapter eight it was not dicult to investigate S The ReggeEinstein

Noncomputability

action in the ve dimensional mo del is

S N N 61

where N is the number of simplices of dimension i This is not the most general

i

action linear in N in ve dimensions as this would take three parameters This

i

follows straightforwardly from the DehnSommerville relations 125 which imply

N N 62

N N N 63

N N N N 64

For the purp oses of this chapter this is not relevant however

I generated and congurations at N and simplices

resp ectively These were recorded each sweeps starting already after the

rst sweeps where a sweep is N accepted moves All congurations were

made at curvature coupling making all conguration of the same volume

contribute equally to the partition function in other words making them app ear

equally likely in the simulation The volume was kept around the desired value by

adding a quadratic term to the action as I have explained in section Lo oking

at the number of hinges N the system seemed to b e thermalized after ab out

sweeps irresp ective of the volume

The critical value of the bare cosmological constant b elow which the vol

c

in chapter two See equation 211 ume diverges was measured as explained for

on page The measured values of hN i were not exactly the values I aimed for

In the mo died action 226 I just set to zero and V to the volumes mentioned

The results were

c

V hN i

65

The errors at the largest volume cannot b e trusted b ecause of the low statistics

at this volume

Starting with these congurations the volume was decreased by setting see

equation 226 to zero and to a xed number larger than the critical value We

call this pro cess co oling b ecause it attempts to reach a conguration of minimum

volume and thereby minimum action

Results

Figure A typical co oling run starting at N using The

horizontal units are accepted moves The vertical axis is the

number of vesimplices The inset is a blowup of a small part of the

curve

N

moves

For each conguration we co oled four times with and two times with

For b oth values of used one of the runs is shown in gures and

In the insets we can see the typical volume uctuations that o ccurred These were

of the order at and at The volume would rst decrease very

quickly until it reached roughly a quarter of the starting value and then started to

decrease much slower In all cases the initial conguration of seven simplices was

reached We also tried to use The same b ehaviour of fast and slow co oling

was seen but the latter was so slow that due to CPU constraints these had to b e

stopp ed b efore either a stable situation or the minimal volume was reached

There is an imp ortant dierence b etween four and ve dimensions In four

dimensions there is a move that leaves the volume constant Therefore the system

can evolve at constant volume In ve dimensions this is not p ossible b ecause all

moves change the volume In this case the volume has to uctuate for the con

Noncomputability

Figure As gure but with

N

moves

guration to change This is why much larger values of the cosmological constant

such as were used in Amb jrn Jurkiewicz a would eectively freeze the

system

Initially b efore the system was thermalized there was a strong p ositive corre

lation b etween the time used to evolve the system at large volume and the time

needed to co ol the system back to the initial conguration The rates of fast and

slow co oling did not change but the volume at which the slow co oling set in b e

came larger Some of these times are shown in gure on the next page for the

case of simplices This trend do es not continue and the co oling times seem

to converge

Discussion

So contrary to exp ectation no metastable states were seen in any of the co ol

ing runs Small volume uctuations were necessary but these gave no indication

of the high barriers exp ected

Discussion

Figure Number of co oling moves needed at as a function of the number

of equilibrium sweeps at the large volume of simplices

accepted moves

sweeps

It is not clear why we dont see any metastable states There are several p ossi

bilities First the barriers might b e much larger than and we need to go to

extremely large volumes b efore co oling Second there might b e no barriers much

larger than the volume for the volumes we lo oked at and the size of the intermedi

ate congurations needed still grows very slowly for the volumes considered even

though this size is not b ounded by a computable function Third the metastable

regions in conguration space might b e very small and the chance that we see one

is therefore also very small

It was sp eculated in Amb jrn Jurkiewicz a that the absence of visi

ble metastable states might indicate that S is indeed recognizable The results

shown in this pap er for S which is known to b e unrecognizable indicate that

unfortunately the results for S say nothing ab out its recognizability This of

course in no way invalidates the conclusion of Amb jrn Jurkiewicz a that

the number of unreachable congurations of S seems to b e very small

Recognizability is not the only thing that matters Even if S is recognizable

Noncomputability

this says little ab out the actual number of k l moves needed to interpolate

b etween congurations except that this number might b e but do es not have to

b e b ounded by a computable function

Chapter Seven

Fluctuating top ology

sually one restricts the dynamical triangulation mo del of quan

tum gravity to systems where the spacetime top ology is xed al

though the top ology of space is allowed to change with time In

a path integral formulation of quantum gravity summing over the

top ologies seems a natural thing to do In this chapter I dene U

a mo del where we take this sum over top ologies and describ e some preliminary

Monte Carlo results

Introduction

The previous chapters of this thesis considered a partition function which was a

sum over triangulations with xed fourtop ology This was done mainly for prac

tical reasons Fixing the top ology seems somewhat unnatural to me To see why

take a lo ok at gure on the following page It depicts the splitting and con

verging of space Both pro cesses are allowed with xed spacetime top ology But

the combination a space that splits rst and then reconverges is not Somehow

the space has to remember whether it split or not to see if it can converge To

me this seems strange

A third p ossibility is that the top ology of spacetime is restricted to M R

where M is xed This would not raise the ob jection expressed in the previous

paragraph A mechanism that imp oses such a restriction dynamically is describ ed

in Anderson DeWitt

Because the typical curvature uctuations b ecome larger at smaller scales

allowing an arbitrary top ology will result in a very complicated structure at the

Planck scale Wheeler Such a spacetime which is full of holes is commonly

Fluctuating top ology

Figure Splitting and converging universe is allowed with xed spacetime

top ology Why not b oth

t

called spacetime foam It was estimated in Hawking a that the dominant

contribution to the path integral would come from spacetimes where the Euler

characteristic is of the order of the volume of the spacetime in Planck units

Denition of the mo del

The partition function of the mo del in d dimensions is

X

Z N expN 71

d d-

T (N )

d

This expression is the same as for a xed top ology but here the sum is over

all p ossible ways to glue a xed number N of dsimplices together maintaining

d

orientation ie only identify d dimensional faces with opp osite orientation

Because the faces have to b e glued in pairs the volume N must b e even if the

d

number of spacetime dimensions is even

At this stage we include disconnected congurations Because the action is a

sum of the actions of the connected comp onents the Boltzmann weight factorizes

which means that the lo cal physics will not change Including these congurations

however will tell us something ab out the probability to obtain a particular size

of connected comp onent A universe which is most likely to b e split up into many

small parts seems an unlikely candidate for the real world

Naively gluing together simplices will not result in manifolds but only in

pseudomanifolds Several ways to deal with this problem are conceivable First

as nob o dy knows what spacetime lo oks like at the Planck scale one could argue

Denition of the mo del

that this is not a problem The theory could b e universal such that the inclusion

of pseudomanifolds do es not change the continuum b ehaviour

Second one can at least for d lo cally deform the resulting pseudomanifold

to turn it back into a manifold by removing a small compared to the lattice

spacing region around the singular p oints and pasting in a regular region Taking

out the region around the singular p oints will generate a b oundary which is an

oriented d manifold A well known result from cob ordism theory says that

for d or we can always nd an oriented dmanifold with that d

manifold as its b oundary By scaling this region down by a factor the curvature

-d

will increase like but the volume will decrease like Therefore in three and

four dimensions this can b e done while changing the total curvature only by an

arbitrarily small amount In two dimensions there was no problem to b egin with

b ecause the manifold condition is always satised as the only connected compact

onedimensional manifold is the circle S

Third these congurations might b e unimp ortant in the limit which

will have to b e taken see b elow Eg in three dimensions for each xed N and

volume the number of edges which couples to is maximal if and only if the

conguration is a real ie nonpseudo manifold The explanation is analogous

to the derivation of 124 Around each vertex we have the relation

()

M M M h 72

where h is the number of handles of the twodimensional manifold around the

vertex Summing over the vertices x gives us

X

N N N N h 73

x

x

The sum over h vanishes if and only if all the vertices are surrounded by a

x

simplicial ball Otherwise it is p ositive This proves the claim that N is maximal

if and only if all the h vanish and the complex is a simplicial manifold Using

x

the relation N N the analog of 121 it also follows that

N N N N 74

where the equality holds if and only if the pseudomanifold is a manifold The

situation is less clear in four dimensions though b ecause a similar reasoning only

results in the neighbourho o d of a p oint b eing a nonpseudo manifold but not

necessarily a ball Taking the relation 74 as applying to the neighbourho o d of

Fluctuating top ology

a vertex in the fourdimensional complex and summing it over all vertices gives

us the generalization of 124

N N N N 75

where the equality holds if and only if it holds in 74 for all vertices

At low the connected congurations will contribute most as they have the

highest entropy This is due to the fact that the total number of congurations

increases factorially with the number of simplices while the entropy of the dis

connected ones is extensive and therefore their number only rises exp onentially

The number of disconnected congurations will also pick up a factor P N the

p

number of partitions of N but this b ehaves only like exp N

At high the disconnected congurations will contribute most as they have

the lowest action Because lowest action means highest number of hinges d

dimensional subsimplices where the curvature in concentrated these hinges will

have very few dsimplices around them In fact the conguration with the lowest

action will have every hinge contained in only one simplex making no connections

b etween the simplices and therefore a completely disconnected conguration In

an even dimension d this is not p ossible and the lowest action will o ccur in a cong

uration which completely consists of connected comp onents of only two simplices

It is a priori not clear whether this change b etween domination of connected

and disconnected congurations o ccurs gradually or whether there is a phase tran

c

sition Supp ose for a moment that there is a sharp crossover at some dep ending

on the volume Because as explained ab ove the number of connected congura

tions rises faster with the volume than the number of disconnected ones the value

c

of will increase with the number of simplices This means that in a p ossible

continuum limit the value of will have to b e taken to innity Because the

c

number of connected congurations rises like N expN ln N the value of

will diverge logarithmically with N

Tensor mo del

We can formally write down a tensor mo del which is a generalization of the well

known one matrix mo del of two dimensional quantum gravity see eg Ginsparg

Mo ore and references therein The partition function of this tensor

mo del is written in terms of an ndimensional tensor M of rank d The tensor

M is invariant under an even p ermutation of its indices and go es to its complex

Tensor mo del

conjugate under an o dd p ermutation of its indices The partition function for

three dimensions is

Z

  

M M gM M M M 76 Z dM exp

cef ade bdf abc abc abc abc

and for four dimensions it is

Z

  

M Z dM exp M M gM M M M

cfhj dgij abcd aefg behi abcd abcd abcd

77

a b n 78

Using the prop erties of M the action can easily b e seen to b e real The general

ization to more dimensions should b e obvious from these expressions

These expressions are only formal b ecause the interaction term can b e negative

for any g and is of higher order than the gaussian term Therefore these integrals

will not converge Nevertheless if we expand these expressions in g each term

of the expansion is welldened and the dual of each of its Feynman diagrams is

a dynamical triangulation conguration of size N equal to the order of g used

d

The propagators carry d indices corresp onding to the d dimensional hinges

of the simplices To b e precise

X



i 79 hM M

a a

a (b ) a (b )

b b

d

d d

d

even

where the sum is over the even p ermutations As an example in three dimensions

this b ecomes

if all indices are equal

710 hM M i

if abc is an o dd p ermutation of def

abc def

otherwise

The second clause should only b e considered if the rst one do es not apply

Each vertex of the tensor mo del corresp onds to a dsimplex and each prop

agator b etween them corresp onds to the identication of two d dimensional

faces of simplices For the propagator not to vanish the sets of indices at each

end of a propagator must b e an o dd p ermutation of each other making sure that

the simplicial complex has an orientation As in the matrix mo del each hinge

corresp onds to an index lo op in the diagram giving a factor n while each sim

plex corresp onds to a vertex giving a factor g The contribution of a particular

Fluctuating top ology

diagram will b e

N N

d d-

g n explngN lnnN 711

d d-

which is precisely prop ortional to the Boltzmann weight of the dynamical triangu

lation conguration according to the ReggeEinstein action with the identication

ln n

This mo del has b een discussed for three dimensions in Amb jrn et al

A dierent generalization of the matrix mo del where the dimension of the ma

trix couples to the number of p oints in the dynamical triangulation conguration

which means one do es not get the ReggeEinstein action in more that dimen

sions has b een discussed in Gross

Monte Carlo simulation

A move can most easily b e describ ed in the tensor mo del formulation It consists

of rst cutting two propagators and then randomly reconnecting them Due to the

orientability there are d ways to connect two propagators In the dynamical

triangulation mo del this corresp onds to cutting apart the simplices at each side

of two of the d dimensional faces and pasting these four faces together in two

dierent pairs

Unlike the case of xed top ology with k l moves one can easily see that

these moves are ergo dic We can always get one propagator correct in a single

move The number of moves needed to get from one conguration to another

is ON raising none of the computability problems asso ciated with the xed

d

top ology case Nabutovsky BenAv see chapter six The nonexistence of

a classication of fourtop ologies and their unrecognizability is usually mentioned

as a problem for the summation over top ologies In dynamical triangulation

however it seems to b e more a problem for xing the top ology

We use the standard Metrop olis test to accept or reject the moves This means

that we accept any move that lowers the action and accept a move that raises

the action with a probability expS Because again unlike the xed top ology

case the number of p ossible moves do es not dep end on the conguration detailed

balance see section is easily obtained

One could restrict the simulation to connected congurations by checking con

nectedness for each move accepted by the Metrop olis test This might b e rather

slow b ecause this is not a lo cal test Also although it seems very plausible it is

not clear whether this would b e ergo dic in the space of connected congurations

Results

Figure The number of connected comp onents as a function of in the four

dimensional mo del

comp onents

Results

We have simulated this mo del in four dimensions at a volume N We used

a hot start that is a conguration with completely random connections This

would b e an equilibrium conguration at

The number of connected comp onents is plotted in gure We see that at

low the average number of comp onents is almost one while at high the average

number of comp onents is almost equal to the maximum number of N Already

d

at this low number of simplices the change b etween few and many comp onents

lo oks quite sharp

c

Unfortunately the acceptance rates for these moves are quite low for near

of the order of One of the reasons for this is that the prop osed moves are

not lo cal in the sense that they can connect any two p oints in the complex For

near the transition thermalization to ok much longer than for far away from

it in either direction To illustrate this phenomenon I have plotted the number

Fluctuating top ology

Figure The number of connected comp onents as a function of computer time

for various values of

comp onents

sweeps

of connected comp onents as a function of computer time in gure As usual

a sweep is dened as N succeeded moves The curve at eventually ends

up near but takes much longer than the one at Both eects mean

that simulating much larger systems will probably not b e feasible with only these

moves

Discussion

The main problem to investigate is the existence of a sensible continuum limit

Although due to the factorially increasing number of congurations the grand

canonical partition function in the normal denition do es not exist this do es

not exclude that the lo cal b ehaviour of the system might show scaling for large

volumes

In two dimensions a limit of the grand canonical partition function the so

called double scaling limit is known Brezin Kazakov Douglas Shenker

Discussion

Gross Migdal In this limit we also see that the coupling has to

b e taken to innity We can similarly investigate the problem in two dimensions

with xed volume For simplicity we will now restrict ourselves to connected

congurations The partition sum with uctuating top ology can b e written as a

sum over the number of handles h Using 133 this b ecomes

X X

ZN ZN h fN exp h ln N 712

h h

where we have put the factors that do not dep end on h in the function fN

This function is not relevant for this discussion We see that by increasing with

the volume N as

ln N 713

where is some p ositive constant it is p ossible to keep the relative imp ortance of

each top ology constant in the thermo dynamic limit

I think that the ab ove discussion shows that a factorially increasing number of

congurations do es not by itself preclude a continuum limit A similar mechanism

could apply in four dimensions It would b e nice if we had any idea how the

function fN as dened in 228 b ehaves as a function of the top ology This

would allow a similar analysis in four dimensions Perhaps it is necessary to

introduce another coupling constant like to do this in a meaningful way

Chapter Eight

Simulations

his chapter contains a description of the computer program that

was used to generate the congurations of the dynamical trian

gulation mo del of quantum gravity It consists of three sections

the rst is a description of the moves that are used to generate

the congurations In the second section we calculate the relative T

probabilities with which these moves must b e used The last section describ es in

detail the co de used to implement the moves

Of course generating appropriately weighted congurations is not all that is

needed We need to p erform various measurements on these congurations These

measurements are describ ed although not in the same detail in the chapters that

deal with the results

Unless stated otherwise in this chapter the word simplex will strictly refer to

0

a simplex of the dimension d we are working in A simplex of dimension d d

will b e called a subsimplex

Moves

As explained in chapter one we need to generate a set of simplicial complexes of

some xed dimension and top ology This can b e done by starting with one of

those complexes and randomly p erforming a sequence of moves that take one such

complex into another These moves have to b e ergo dic which means that one

must b e able to reach all p ossible complexes using these moves

Two sets of moves are known that satisfy this condition of ergo dicity These

are called resp ectively the Alexander moves Alexander and the k l moves

These have b een proven to b e ergo dic in resp ectively Alexander and Pach

Simulations

Figure The p ossible k l moves in two dimensions

T T

T T

T T

T T

T T

==

T T

b

b

b

T T

b

b T T

==

ner Gross Varsted in the latter case by proving them to b e equivalent

to the Alexander moves The k l moves are more convenient to implement on a

computer and I will therefore restrict the discussion to these

The k l moves in d dimensions can b e dened by taking the b oundary of a

d dimensional simplex which is itself a d dimensional complex A k l move

consists of replacing a sub complex which is equal to a part of this b oundary by

the other part of this b oundary

Let me illustrate this in two dimensions Figure shows the two p ossible

moves in two dimensions One should imagine these congurations to b e embedded

in much larger simplicial complexes The rst move takes a part of the b oundary

of a tetrahedron in this case a triangle This triangle is replaced by the other

part of the tetrahedron which is a conguration of three triangles This is called

a move b ecause it takes one simplex into three The other move replaces

two triangles with two other triangles and is similarly called a move

The p ossible moves in three dimensions are shown in gure on the facing

page In four dimensions it b ecomes somewhat hard to draw Figure on

page shows the p ossible moves but in the dual picture This means that

the simplices have b een replaced by vertices and if two simplices were adjacent by

sharing a face the vertices of the dual graph have b een connected All information

ab out the relative orientation of the simplices is lost

We see that the moves always consist of replacing all the simplices containing

a given subsimplex In the move this subsimplex is a triangle itself in the

Moves

Figure The p ossible k l moves in three dimensions

==

==

move the two triangles next to an edge are replaced and the move

replaces the three triangles around a vertex We can therefore choose a particular

place in the conguration to apply a move by choosing a subsimplex with the

correct number of simplices containing it For a subsimplex of dimension k this

number is d k The move which might then b e p erformed is a d k k

move

As has b een explained on page in section these lo cal moves can cause

critical slowing down One would like to use moves which are not lo cal ie moves

which change large parts of the system at a time One such set of moves has

b een implemented by others Amb jrn Jurkiewicz b and is known as baby

universe surgery These moves are not ergo dic by themselves so they have to b e

supplemented by the normal k l moves A big move as it is called consists of

cutting the space of simplices in two parts at a neck and reconnecting them A

neck is a place where part of the space is connected to the rest by the smallest

p ossible b oundary In four dimensions this b oundary is a three dimensional ob ject

consisting of ve tetrahedrons which are all connected to each other It is the same

b oundary as we would get when removing a single simplex from the triangulation

Simulations

Figure The p ossible k l moves in four dimensions on the dual graph

A

A

r

b

B

E

b

B

P

b

P

B

P

b E

B

b P r r

P

P

B

P

Z

B

P

B

Z P

B

r P

B Z

B

==

S

Z

B

B

Z

S

B

Z

B

S

B r r Z B

S

S

S C D

S

S

SC D

B

A

B C

L

L

L

A

D

L

a

a

r r

a

L

a

A B

a

a L r

==

r

a

a

L

a

D

C

r r

a

a L

a

D A

L

L

L

LC B

D C

B A

A A

C

b b

b b

b b

r br b

T T

T T

T T

A

B

T T

b b

b b

==

T T

b b

r r T b b r r T

H H

H H

H H

H H

C B B

C C

Detailed balance

Adding these big moves to the normal k l moves is rep orted to drastically

reduce the auto correlation time near the phase transition b oth in numbers of

sweeps and more imp ortantly in CPU time Amb jrn Jurkiewicz b

Detailed balance

To approximate the sum over triangulations correctly we need to generate the

congurations with the correct relative probabilities A sucient but not neces

sary condition for this is detailed balance see eg Creutz This means

that for two congurations A and B which should o ccur with relative probabilities

pA and pB there should hold the equation

pApA B pBpB A 81

where pA B is the probability that if we have conguration A that the next

move will take it to B The implementation of this condition is complicated by

the fact that the number of p ossible moves changes with the conguration

In our case this probability pA B is a pro duct of two probabilities the

probability p that we will try the correct type of move to take A to B times the

kl

probability that we will choose the correct subsimplex whose move will take A to

B

pA B p psubsimplex 82

kl

We will see b elow that the routine that p erforms the moves chooses a random

subsimplex of a particular dimension by rst choosing a random simplex and

then a random subsimplex of this simplex This means that the probability for

a particular subsimplex to b e chosen is prop ortional to the number of simplices

containing it If the geometry is the correct one to p erform a k l move this

d+

number is k The number of idimensional subsimplices in a dsimplex is

i+

Thus the probability to choose the particular subsimplex whose move will take A

to B is

k

psubsimplex 83

d+

N A

d

k-

The desired probabilities pA are prop ortional to expSA The detailed

balance equation 81 then b ecomes

l k

expSBp 84 expSAp

lk kl

d+ d+

N A N B

d d

k- l-

Simulations

Because k l d it follows that

k l k l

85

d+ d+

d

k- l-

and equation 84 can b e reduced to

p N A

kl d

86 expSA SB

p N B

lk d

If as was the case in our simulations the action SA only dep ends on the N the

i

right hand side only dep ends on the current values of N and how they change The

i

latter dep ends only on the type of move not on where it is done The conclusion

is that we just need to try the dierent moves with suitable relative probabilities

which dep end on the current conguration No Metrop olis acceptance step is

needed Note that we must not choose a type of move and then try to p erform

this type of move until it is accepted Instead if the move routine rejects a move

b ecause the geometry do es not t we must again randomly choose a type of

move to try

We also see that only the relative probability of opp osite moves k l and l k

is determined We are free to choose the relative probability of eg a and

move One could try to tune this to minimize auto correlation times I have

not done this but simply used

p p p p p 87

If the b est ratio is unknown this one is certain not to b e more than a factor of

three slower This would b e the case if the b est ratio would only use one type of

move and others would not contribute at all to the decorrelation a very unlikely

event Any other ratio could b e worse as it could try that particular unknown

type less often

Program

In this section I will describ e the routine that p erforms the moves in the simplicial

complex This part of the program can b e used for any dimension by a simple

recompilation It was based on ideas put forward in Br ugmann but devel

op ed indep endent of the very similar program describ ed by Catterall in Catterall

He did however in a private discussion provide me with the idea of writ

ing co de for arbitrary dimensions The version included here has b een shortened

Program

somewhat In particular it lacks all the consistency checks on the conguration

that could b e compiled in conditionally for testing purp oses

There are two basic philosophies in programming dynamical triangulation One

can try to store the minimum amount of information This is called minimal

co ding Minimal co ding makes nding a correct conguration to p erform a move

harder but facilitates up dating and saves memory It is the approach that has

b een taken in Br ugmann Catterall and it is also what we do On

the other hand one can keep track of subsimplices of dierent dimensions for

fast lo okup of places where moves can b e p erformed at the exp ense of larger

up dating complexity and memory usage We call this maximal co ding and it has

b een describ ed in Bilke et al The lattice sizes which are currently used

in practice are b ounded by the CPU time needed for large lattices not by the

available memory size At the moment it is not clear which approach is more

CPU ecient

We start the program by dening the most imp ortant constants the dimension

and the maximum number of subsimplices of some particular dimension that a

simplex can have which is needed for some static arrays The constant DIMEN is

actually the number of dimensions d plus one b ecause this number is used much

more often than d as it equals the number of vertices or faces of a simplex The

number of subsimplices of dimension k is simply the number of ways we can choose

a subset of k p oints of the d p oints in the simplex The constant MSUB is

the maximum over k of this quantity ie

DIMEN

88 MSUB

bDIMEN c

So for four dimensions we have

define DIMEN

define MSUB

Now we dene a type to contain numbers of vertices RAM and disk usage

signicantly increase if we use a four byte type here This will b e necessary when

N b ecomes larger than In four dimensions we have N N the

ratio strongly dep ends on so the volumes we used up to are still far

away from this limit

typedef unsigned short vertex

The most imp ortant denition concerns the representation of a simplex in the

computer We use the following structure to represent a simplex

Simulations

struct ssimplex

struct ssimplex nextDIMEN

vertex pointDIMEN

vertex nxtpntDIMEN

unsigned int flag

typedef struct ssimplex simplex

For each simplex s we keep track of its neighbours in snextDIMEN and its

vertices in spointDIMEN The vertex spointi is always that vertex that

is not contained in the neighbour snexti The array snxtpnti contains

the number of that vertex of neighbour snexti which is not contained in the

simplex s It can easily b e calculated using

DIMEN-

X

nxtpnti simplexnexti poi ntj

j=

DIMEN-

X

simplexpoint j simplexpointi 89

j=

but b ecause it is so often needed we store it separately To save disk space

however it is not saved with the congurations but calculated when reading

them in Finally flag will b e explained later where it is used It is not needed for

the representation of the simplicial complex but only for the routine that p erforms

the moves As such it do es not have to b e stored with the congurations on disk

Because the number of simplices uctuates the simplex structures often b e

come unused This is marked by setting next equal to The free simplices

are kept in a linked list and next is used as a p ointer to the next free simplex

Incidentally the routine that saves the conguration also renumbers the simplices

and vertices such that no free simplices are stored This not only saves a little

disk space but also makes all the measurement routines slightly simpler b ecause

they do not have to take into account the p ossibility of unused simplices in the

array

We see here an imp ortant advantage of the programming language used If this

same program had b een written in FORTRAN instead of C the array of structures

would typically have b een co ded as a set of dierent arrays But we often need

all the information in a particular simplex and hardly ever the information of

contiguously numbered simplices This seems to make no dierence until we

consider the data cache of a typical CPU A simplex structure nicely ts into a

Program

few cache lines while taking elements of several dierent arrays reads entire cache

lines for a few bytes while leaving large parts of those cache lines unused

The next thing we use are several arrays that describ e the simplex These are

initialized in initmove and dont change thereafter A move of type m will

denote a m d m move The number of subsimplices of dimension d i

which is the dimension of the subsimplex that must b e found for a move of type

i that a simplex has is kept in

static int numsubsimDIMEN

We will need lists of all the subsimplices in a simplex so we can choose one to

p erform a move around The rst i members of subsimij contain the neigh

b ours that dene by their intersection with the base simplex the jth subsimplex

of dimension d i where j numsubsimi The other d i members

of subsimij contain the other neighbours in no particular order Of course

one can also dene subsimplices by the vertices they contain but the scheme used

happ ens to b e what we need in the ndmove routine b elow

static int subsimDIMENMS UB DIME N

If we p erform a move of type i then N will change by N This quantity is kept

j j

in deltanij

static int deltanDIMENDI MEN

Now we dene some external variables that contain information ab out the

conguration that other routines will need The rst is simply N

i

int nsimsDIMEN

The next thing is the number of allo cated simplex structures Some of these may

b e unused so this is not N

d

int simalloc

The array that contains these simplices Its size is simalloc

simplex lattice

And the number of vertices allo cated Again some of these may b e unused

vertex pntalloc

Simulations

Initialization

n

The rst routine is a small function that calculates or returns if k or

k

k n It has not b een repro duced here

static int combiint n int k

The next routine initializes the static arrays declared ab ove It should always

b e called b efore ndmove can b e used

void initmovevoid

int i

We start by calculating the number of subsimplices of dimension d i in a simplex

d+

of dimension d It is just

i

numsubsim

for i iDIMEN i

numsubsimi numsubsimi DIMENi i

d+

The next part of this routine calculates those dierent sets and stores them

i

in the subsim array

memsetsubsim sizeofsubsim

for i iDIMEN i

int j

int nowsubsDIMEN

for j ji j

nowsubsj j

for ji jDIMEN j

nowsubsj

for j jnumsubsimi j

int k

int neighbor i

int ssind nsind i

memcpysubsim i j nowsubs sizeofnowsubs

for k kDIMEN k

if subsimijss ind k

ssind

else

Program

subsimij nsin d k

while nowsubsneig hbor DIMEN neighbor i

neighbor

for kneighbor ki k

nowsubsk nowsubsk

The last thing to calculate is N for a move of type i This is of course equal

j

to the number of j dimensional subsimplices that are only in the new d i

simplices minus the number of j dimensional subsimplices that are only in the old

i simplices

for i iDIMEN i

int j

for j jDIMEN j

deltanij combiDIMENi DIMENj

combii DIMENj

The next routine makes the smallest p ossible conguration which is the b ound

ary of a d dimensional simplex

void makesmallvoid

int i

This conguration uses d vertices and d simplices Note that the co de

assumes that the lattice array is at the end of the data segment One should

give the stdio library its buers using setbuf b efore calling this routine or very

strange errors will o ccur later on

simalloc DIMEN

lattice simplex sbrksimallo c sizeofsimplex

pntalloc DIMEN

Just connect all the simplices to all the others We set it up such that sim

plex i do es contain all vertices except vertex i This is not necessary vertex

numbers themselves have no meaning and can b e p ermuted at will Only the

assignment to the elements of point is signicant b ecause the move routine as

sumes that latticeinext k is the neighbour that do es not contain the vertex

latticeipoint k

Simulations

for i iDIMEN i

int j

int k

for j jDIMEN j

if j i

continue

latticeine xtk lattice j

latticeipo int k j

k

for j jDIMEN j

latticeinx tpnt j i

latticeiflag

In this conguration any set of p oints is a subsimplex Therefore N the number

i

of subsimplices of dimension i is just the number of ways to choose i vertices

out of the d

for i iDIMEN i

nsimsi combiDIMEN i

Geometry tests

We nally get to the routine that actually p erforms the moves When called

with the argument movetype which we will denote by m it tries to p erform one

m d m move If it returns the move succeeded otherwise it did not

In the latter case the return value will indicate the test which was failed so we

can keep some statistics

int ndmoveint movetype

int subs

simplex base

simplex movingDIMEN

vertex mpointDIMEN

int i

int heresub

Program

vertex newpoint

static unsigned int recount

static simplex first

static int frstpnt

define MXFRPNT

static vertex freepntsMXFRPN T

First of all we have to choose a random subsimplex of dimension d m to p erform

the move on We do this by choosing a random simplex base a member of the

lattice array that has not b een marked as free and then a random subsimplex

subs of this simplex This means that the probability that a subsimplex will b e

chosen is prop ortional to the number of simplices in which it is contained As was

explained on page in section this has b een accounted for in the calculation

of the probabilities to get the various values of m The routine randbb is

a random number generator that provides an unsigned int Because most of

the allo cated simplices are actually used the vast ma jority of cases will see the

following lo op only iterated once

do

base lattice randbb simalloc

while basenext

subs randbb numsubsimmovet ype

heresub subsimmovetype su bs

The array heresubm will contain the numbers i that mark the m simplices

basenexti that together with the base simplex will b e replaced by new

simplices if the move succeeds The other d m elements those with indices

m i DIMEN of the array heresubDIMEN mark the other neighbours

The d move which has m is always p ossible so none of the tests

are necessary in that case Otherwise we need to check whether the move can b e

p erformed This is done by three geometry tests

if movetype

int frst last

define MXRNDSM

simplex listMXRNDSM

define MAXPNTS

static char repntMAXPNTS

Simulations

Figure Example of trying a move around vertex A If the rst geometry

test is passed the picture on the left turns out to b e the picture on

the right

B newpoint

T

T

B C

F

T

b

a

T

T T

a

b

b

T T

A

F A

b

b

b

T T

Q

a

T

b

base

T T s T

c

e

T T T

c

b

T T T

T T T

D E D C

E

As has b een explained on page in section a move can b e seen as replacing

a part of the b oundary of a d dimensional simplex This means that there are

always d vertices involved in a move One of these is not a vertex of the base

simplex This one is stored in newpoint

newpoint basenxtpnther esub

The rst test we will do is to see if the geometry around the subsimplex is the

correct geometry to p erform the move To understand the co de for this test lo ok

at the example in gure We are trying to p erform a move and have

chosen the neighbours a and b as p otential simplices to replace The move can

only b e p erformed if there are three triangles around the vertex A this vertex is

uniquely determined by the two neighbours This is the case if the vertices B and

C are actually the same vertex making the edges AB and AC the same b ecause

we do not allow two dierent edges to have the same vertices In other words B

and C should b oth b e the newpoint We have now made newpoint equal to B and

need to check if C is the same p oint

for i imovetype i

if basenxtpnt here sub i newpoint

return

The ab ove lo op do es nothing if m This corresp onds to a d move Such

a move must always pass this test b ecause there are always exactly two simplices

that share a face which is a d dimensional subsimplex

Program

Figure Example of a move

newpoint newpoint

A A

T T T T

T T T T

e e

d d

T T T T

a

T T T T

T T T T

y

x

F E F E

T T T T

base

T T T T

T T T T

c c

b b

T T T T

T T T T

C C D D

B B

repnt repnt

A move might create more than one subsimplex consisting of the same vertices

We do not allow these congurations and we have to check whether any of the

subsimplices that will b e created would have the same vertices as any existing

subsimplex

We start by p erforming a quick test to see if any of the other vertices opp osite

the base simplex is the same as the newpoint In the example of gure on

the preceding page we check whether vertex D is equal to vertex B This would

mean that after the move we would get two triangles with the vertices B E and

F One might think that D and B cannot b e equal b ecause this would imply the

existence of two edges BE but this is not true b ecause these can actually b e the

same edge This test is implied by the next test We could therefore just leave it

out but b ecause it is faster and nds a large part of the cases including it sp eeds

up the program

for iDIMEN i

if basenxtpnt here sub i newpoint

return

Because most of the calls of this routine dont even get to this p oint we see that

it is essential to p erform the rst two tests as fast as p ossible This is the reason

we keep track of the nxtpnt array instead of calculating this information when

we need it

Simulations

As has already b een suggested the ab ove test is not complete This cannot b e

seen in the rst example but it can happ en in the second example which we see

in gure on the preceding page Here we are trying to do a move on the

base triangle and triangle a creating a new edge from vertex A to B If however

any of the triangles around vertex B also contains A this would create two edges

b etween the same pair of vertices The test ab ove only checks whether C or D is

equal to A See subsection for a more general explanation

The metho d is now to go through all the simplices which contain the vertex

B and see whether they contain the newpoint To do this we keep a queue of

simplices to visit adding for each visited simplex all its neighbours which also

contain B and have not yet b een visited to the queue I have also tried to

implement this with a recursive routine instead of a queue This turned out to b e

signicantly slower

To mark a simplex as visited we make its flag eld equal to recount a

number that is incremented each time the test is run If the number b ecomes

we have to reset all the ags to make sure none of them is accidentally still equal

to recount b efore the test is run This happ ens only once every times the

test is run

if recount

for i isimalloc i

latticeiflag

recount

baseflag recount

The variables frst and last are used to mark the start and end of the queue

which is kept in listMAXRNDSI M Therefore the constant MAXRNDSIM has to b e

larger than the maximum number of simplices around any particular subsimplex

This is not checked at runtime b ecause this would b e tested so often as to result

in signicant slowdown It do es not matter if it is much to o large though

frst

last

The array repnt is only for those p oints that form the subsimplex around

which we have to lo ok In the example it will only b e for vertex B

for i i movetype i

repntbasep oint her esub i

Program

We start by putting the simplices b and c in the queue Any simplex which is put

in the queue has its flag immediately set equal to recount to mark it such that

it will not b e put in the queue another time We see that the vertices of b and

c will b e lo oked at another time even though they have already b een checked in

the previous test I tried to explicitly add their neighbours to the queue while

not lo oking at their vertices but this did not result in any sp eedup Presumably

the extra co de and resulting instruction cache loading outweigh the advantage of

executing fewer instructions

for i DIMEN i

listlast basenexthe resu bi flag recount

Go through the queue and see if any of the vertices of the simplices lo oked at is

equal to newpoint If that is the case we cannot p erform the move so clear the

repnt array and return to the calling routine

do

simplex thissim listfrst

for i i DIMEN i

if thissimpoint i newpoint

for i i movetype i

repntbasepoi nth eres ubi

return

Remember that vertex pointi is never contained in simplex nexti Therefore

all neighbours that do not corresp ond to vertex B which was marked in repnt

contain B and should b e added to the queue unless they have already b een agged

as b eing in the queue

for i i DIMEN i

if repntthissim poin ti

thissimnexti f lag recount

continue

listlast thissimnexti flag recount

while frst last

for i i movetype i

repntbasep oint her esub i

else

Simulations

Else m and we are dealing with a d move This is a barycentric

sub division of a simplex and creates a new vertex which is now the newpoint

We keep a stack of free vertex numbers in freepntsMXFRPN T if the stack is

empty we use a new one

if frstpnt

newpoint freepntsfrs tpnt

else

newpoint pntalloc

if pntalloc

fprintfstderr

Too many points for vertex data typen

exit

A move

All tests have b een passed We can now actually p erform the move The array

movingDIMEN holds the d simplices in the b oundary of the d simplex

that denes the move The rst m simplices already exist and are replaced by

the others We use free elements of the lattice array for the new simplices These

are kept in a linked list the head of which is p ointed to by the static variable first

It is admittedly somewhat inecient to make a system call for each new allo cated

simplex but this is not imp ortant b ecause the number of times this happ ens is

only of the order N not of the order of the number of moves p erformed Note

d

also that this was run under AIX where brk never returns an error due to

lack of memory

moving base

for i imovetype i

movingi basenexthere sub i

for iDIMEN i

if first

simalloc

brkchar lattice simalloc

first lattice simalloc

firstnext

Program

firstnext

firstflag

movingi first

first firstnext

Similarly the array mpointDIMEN holds the vertices of this d dimensional

simplex b oundary As in a simplex structure mpointi is that vertex which is

not a vertex of movingi

mpoint newpoint

for i iDIMEN i

mpointi basepointher esub i

The new simplices i have to b e given connections at all faces k To do this we go

through all the moving simplices j connecting the simplex i to either another new

simplex if j is new or to one of the simplices which are not themselves moving

if j will b e replaced

In the words of our second example in gure on page we have to

connect the new simplex x to three other simplices To do this we go through all

four simplices involved in this move base a x and y

for imovetype iDIMEN i

int j

int k

for j jDIMEN j

The simplex x must not b e connected to x itself so in this case do nothing

if j i

continue

If j is another new simplex y in the example simply connect to it

if j movetype

movinginext k movingj

movinginxtp ntk mpointi

else

Simulations

If j is an old simplex instead of connecting to it we connect to the simplex that

j is connected to In the example this means that instead of connecting x to a

we connect it to e And instead of connecting x to base we connect it to b The

variable oldsim p oints to simplex a oldtonon is the face of a connecting it to e

nonsim p oints to simplex e and nontoold is the face of e that connects it to a and

which will have to b ecome connected to x

simplex oldsim movingj

int oldtonon

simplex nonsim

int nontoold

We recognize the simplex e in lieu of d as the correct neighbour of a by the fact

that it do es not contain the vertex E the vertex which is also not in simplex x

and therefore has its number contained in mpointi

for oldtonon oldtononDIMEN oldtonon

if oldsimpoint old tono n mpointi

break

nonsim oldsimnexto ldto non

for nontoold nontooldDIMEN nontoold

if nonsimnext nont oold oldsim

break

movinginext k nonsim

movinginxtp ntk nonsimpointno ntoo ld

nonsimnextno ntoo ld movingi

nonsimnxtpnt nont oold mpointj

movingipo int k mpointj

k

The move has b een done and the old simplices that have b een replaced should b e

marked as unused and put at the head of the linked list of free simplices

for i imovetype i

movinginext

movinginext first

first movingi

Program

If this was a d move a vertex has b een deleted and should b e put on top

of the stack of unused vertex numbers

if movetype DIMEN

if frstpnt MXFRPNT

fprintfstder rS ize of freepnt too smalln

exit

freepntsfrstpn t mpointDIMEN

Finally adjust N and return indicating success

i

for i iDIMEN i

nsimsi deltanmovetype i

return

Double subsimplices

~ All that is left is to explain why the third geometry test do es what we

need We will drop the convention used ab ove to call any simplex of fewer

dimensions than d a subsimplex The aim of the test is to avoid creating a new

simplex that consists of the same vertices that already dene an existing simplex

To do this let us divide the d vertices that are involved in the move into

two sets The rst set we shall call a and consists of the vertices that make the

i

simplex around which the move will b e done In the example of gure on

page these are the vertices E and F The second set b are the other vertices

i

In the program these are the vertices newpoint and those that will b e marked

in the repnt array In the example these are the vertices A and B The two

sets are complementary in the sense that their roles are exactly reversed when the

inverse move is done

We will show in the next paragraph that the move can b e p erformed if and

only if there do es not yet exist a simplex formed by the vertices b Because this

i

simplex will b e a subsimplex of a dsimplex all that is necessary is to lo ok if any

dsimplex contains all the vertices b The program do es this by going through

i

all the d simplices that contain the vertices marked in repnt which are all the

b except for newpoint and checking whether one of these contains newpoint

i

Simulations

The implication to the right is trivial The move always creates the simplex

formed by the b so if this simplex do es already exist the move cannot b e done

i

To prove the implication to the left it is sucient to show that any newly created

simplex will contain all the vertices b Because if such a simplex already exists the

i

simplex formed by b must exist as a simplicial complex contains all subsimplices

i

of its simplices Now supp ose there is a newly created simplex S that do es not

contain a particular vertex b Because the a are the intersection of all the old

k i

dsimplices involved in this move there must b e such a dsimplex T that do es not

contain b and therefore contains all the other d vertices involved But this

k

is imp ossible b ecause S already exists as a subsimplex of T and can therefore not

b e newly created by this move This leads to a contradiction which completes

our pro of of the claim that the third test indeed exactly tests what we want it to

test

Chapter Nine

Outlo ok

eaders of this chapter are forewarned that much of it is sp ec

ulation and not proven by the data but only hinted at I will

rst discuss our current view of the simplicial spacetimes that are

generated by the mo del and then try to say something ab out the

eective action of the resulting theory R

Simplicial spaces

It seems that the bare gravitational coupling constant determines an eective

curvature that I will call R A denition of R has b een explored in chapter

V V

three but it would b e nice if a b etter denition could b e made than that one To

this curvature R corresp onds a curvature radius r the magnitude of which is

V V

determined by the equation

s

jr j 91

V

R

V

The phase transition is the place where this radius is just sucient to turn the

space into a foursphere At larger values of the radius r decreases Because

V

the system has to accommo date the whole volume which is larger than that of a

foursphere with radius r it turns into a set of branching tub es with radius r

V V

At large scales compared to r the tub es b ehave like a branched p olymer with

V

fractal dimension two

So it app ears that determines the eective curvature radius r On the other

V

hand is the bare gravitational constant and one would think it determines the

p

G renormalized gravitational constant G and therefore the Planck length

R P R

Outlo ok

If two scales are set by one parameter we exp ect them to b e related and therefore

of the same order of magnitude But in our universe we see that r o by

V P

many orders of magnitude The question is then how to separate the two scales

with only one parameter

A p ossible way out of this problem is the triviality discussed on page in

chapter three This could allow us to set b oth the Planck length and the curvature

radius by tuning not only but also the number of simplices N When the

curvature radius r is xed the shap e of the space is xed Using more simplices

V

then means that the simplices are smaller in physical units that is in units of r

V

So the number of simplices N determines r ie the lattice spacing in physical

V

units In other words by tuning the number of simplices N we are tuning the

lattice spacing

This idea seems to b e in conict with the usual view that the physical b ehaviour

of the system is indep endent of the lattice spacing This b ehaviour would indeed

b ecome indep endent in the continuum limit But at small nite lattice

spacing we could still get continuum b ehaviour where all the parameters that are

irrelevant in the renormalization group sense which go like r have vanished

V

but the marginal parameter r which go es only like lnr can b e varied

P V V

at xed r

V

To check whether something like this happ ens it is necessary to measure the

Planck length Two p ossible ways have b een describ ed in this thesis First the

Planck scale might b e the scale where the running curvature dened in chapter

three b ecomes large Figure shows a very slight hint that this scale b ecomes

smaller compared to the system size but this needs to b e conrmed at other

lattice sizes A problem is that in the ab ove scenario this scale decreases only

logarithmically with the lattice spacing making it very hard to see in the computer

simulations

A second way to measure the Planck length is to measure the renormalized

directly A p ossible way has b een indicated in gravitational constant G

R

P

chapter ve where we measured binding energies An imp ortant problem there is

that the b ehaviour of the binding energy as a function of the particle mass and

the gravitational constant do es not seem to conrm to our guess 517 Perhaps

this is due to the fact that the ratio r is not yet very small

P V

Another measurement that deserves extension to larger lattices is the scaling

analysis in chapter three Compared to the pap er de Bakker Smit where

these ndings where originally published gure has already b een extended

with a larger volume and the matching lo oks even b etter than it did But it would

Eective action

also b e interesting to extend the measurements for values of far away from the

phase transition and esp ecially to conrm that the scaling dimension d which is

s

the dimension at small scales is indeed four

Eective action

As has b een mentioned in chapter one the Euclidean action is not b ounded from

b elow Although the path integral could still converge due to the b ehaviour

of the measure this do es raise a dierent problem Assume for a moment that

we can take a go o d continuum limit as indicated by the evidence for scaling of

chapter three If the eective action describing semiclassical uctuations around

the resulting average space is the Euclidean gravity action there will b e a mo de

with negative eigenvalue making that space unstable Several scenarios seem

p ossible

First the eective action that comes out is not Euclidean gravity as describ ed

by the action 15 There is no guarantee that the eective action that comes

out of a lattice mo del is the one we started out with to discretize For instance

in the continuum limit the nonlinear O sigma mo del mentioned in section

b ecomes the linear sigma mo del The eective action could still b e something

describing gravity p erhaps Euclidean gravity with the conformal mo de rotated

Gibb ons et al Schleich

As a second scenario the instability might b e physical The universe is unstable

at large time scales slowly collapsing to black holes Perhaps the instability is a

manifestation of this phenomenon The main problem with this line of thought

is that naive arguments would predict the instability to b e of the scale of the

parameters of the theory which is the Planck scale The Planck length we would

measure in the S like space we found in chapter three would then b e of the order

of the size of the universe making that universe very small Although the mo del

would in this case not b e a go o d description of quantum cosmology it could still

describ e quantum gravity at small scales

It is interesting to compare this scenario to the minisup erspace mo del devel

op ed in Hartle Birmingham The mo del describ es a universe with a

b oundary of only one simplex with varying complex edge length It turns out that

the mo del has Euclidean and Lorentzian stationary p oints but the Euclidean sta

tionary p oints all have universe sizes less than a critical value while the Lorentzian

ones have sizes greater than this value

Another interesting phenomenon was observed in the jsymbol approach to

Outlo ok

threedimensional simplicial quantum gravity in Barrett Foxon The

authors of this pap er show that the mo del has b oth Lorentzian and Euclidean

stationary p oints in the sense that the geometry of the manifold is Euclidean or

Lorentzian The Lorentzian stationary p oints however are minima in the action

like one normally sees in a Euclidean mo del Conversely the Euclidean stationary

p oints are oscillatory

App endix One

Summary

ravity is the one fundamental force that eludes a quantum de

scription Creating a go o d theory of quantum gravity is one of

the greatest problems in current high energy physics A problem

of quantizing gravity in the usual way is that general relativity is

not renormalizable in p erturbation theory This by itself however G

implies little ab out its nonp erturbative b ehaviour

Simplicial quantum gravity is a candidate for a nonp erturbative denition of

quantum gravity Starting p oint is the creating of a euclidean spacetime by glueing

simplices together at the faces In dynamical triangulation the metho d used in

this thesis the path integral over metrics is dened by a sum over all p ossible

ways to glue the simplices together to form a piecewise linear manifold usually

only taking those conguration with a xed top ology

The mo del has two coupling constants the gravitational constant and the

cosmological constant To take a continuum limit the number of simplices has

to go to innity xing the bare unrenormalized cosmological constant As a

function of the gravitational constant the mo del turns out to have two phases a

crumpled and an elongated phase

In the crumpled phase distances b etween simplices are very small and the

volume within a xed distance increases exp onentially with that distance In

the elongated phase the system makes narrow tub es that b ehave like a branched

p olymer There is some evidence that the system scales which means that the

b ehaviour of the system b ecomes indep endent of the size of the simplices when

that size b ecomes very small At the phase transition the system lo oks in many

resp ects like a fourdimensional sphere At that phase transition correlations of

the scalar curvature turn out to have a long range and to fall of with the fourth

A Summary

p ower of the distance This could b e taken as evidence for the existence of massless

particles gravitons in the mo del

An interesting question is whether the mo del can repro duce gravitational bind

ing This has b een investigated by studying the b ehaviour of a scalar eld on the

congurations Although the data are still somewhat preliminary there indeed

seems to b e a p ositive binding energy

When doing computer simulations it is imp ortant that the moves used can

reach all the p ossible congurations If this is true we call the moves ergo dic In

our simulations this is indeed the case The number of moves needed however

turns out not to b e b ounded by a computable function of the size of the system

It is not clear what the eects of this are in practice I have lo oked numerically

for such eects but found none This could mean that they are unimp ortant

In most cases one only lo oks at congurations with xed top ology We can

also dene a mo del where the sum over all top ologies is taken This mo del still

has many problems b oth in dening and in simulating it

In the study of simplicial quantum gravity Monte Carlo simulations are very

imp ortant These are done by taking a small starting conguration and changing

it by p erforming a sequence of moves The probabilities with which these moves

have to b e done have to fulll some conditions to make sure that the probability

to reach a conguration is prop ortional to its Boltzmann weight All this has b een

implemented in a C program

Bijlage Twee

Samenvatting

impliciale quantumgravitatie Zo luidt de Nederlandse titel

van dit pro efschrift De quantumgravitatie oftewel het maken van

een quantummechanische b eschrijving van de zwaartekracht is een

van de gro otste problemen in de tegenwoordige hoge energie fysica

Een probleem bij de gewone quantisatie van zwaartekracht is dat S

de algemene relativiteitstheorie in storingsrekening niet renormaliseerbaar is Dit

zegt echter nog weinig over het nietp erturbatieve gedrag

De simpliciale quantumgravitatie is een kandidaat voor een nietp erturbatieve

denitie van de quantumgravitatie Het uitgangspunt is het opb ouwen van een Eu

clidische ruimtetijd do or het aan elkaar plakken van simplices In de dynamische

triangulatie metho de waar dit pro efschrift over handelt wordt de padintegraal

over metrieken dan gedenieerd do or middel van een som over alle mogelijke ma

nieren om de simplices tot een stuksgewijs lineaire varieteit aan elkaar te plakken

waarbij in het algemeen alleen de conguraties met een vaste top ologie in aanmer

king worden genomen

Het mo del b evat twee koppelingsconstanten de gravitatieconstante en de kos

mologische constante Om een continuumlimiet te nemen mo et het aantal sim

plices naar oneindig gaan hetgeen de kale ongerenormaliseerde kosmologische

constante vastlegt Als functie van de gravitatieconstante blijkt het mo del twee

fases te hebb en een gekreukelde fase en een uitgerekte fase

In de gekreukelde fase zijn de afstanden tussen de simplices erg klein en neemt

het volume binnen een b ol exp onentieel to e met de straal ervan In de uitgerekte

fase vormt het systeem nauwe buisjes die zich gedragen als een p olymeer met

vertakkingen Het lijkt er op dat het systeem schaalt hetgeen wil zeggen dat

het gedrag onafhankelijk wordt van de gro otte van de simplices wanneer die erg

klein worden Op de faseovergang lijkt het systeem in een aantal asp ecten op een

B Samenvatting

vierdimensionale b ol Bij die faseovergang blijken de krommingscorrelaties een

lange dracht te hebb en en af te vallen als de vierde macht van de afstand Dit

is een aanwijzing voor het b estaan van massaloze deeltjes gravitonen in het

mo del

Een interessante vraag is of het mo del gravitationele binding kan repro duceren

Dit is onderzo cht do or het gedrag van een scalair veld op de conguraties te

b estuderen Ho ewel de data nog niet geheel sluitend zijn lijkt het erop dat er

inderdaad een p ositieve bindingsenergie is

Bij het uitvoeren van computersimulaties is het b elangrijk of de gebruikte stap

p en in de conguratieruimte alle conguraties kunnen b ereiken We no emen deze

stapp en dan ergo disch In onze simulaties is dit inderdaad het geval Het aantal

b eno digde stapp en blijkt echter niet b egrensd te worden do or een b erekenbare

functie van de gro otte van de conguratie Het is onduidelijk wat dit in de prak

tijk voor eect heeft Ik heb numeriek naar deze eecten gezo cht maar ze niet

gezien Dat zou kunnen b etekenen dat ze in de praktijk niet b elangrijk zijn

In het gro otste deel van de gevallen wordt alleen gekeken naar een som over

conguraties met dezelfde top ologie We kunnen echter o ok een mo del deni

eren waarin over top ologieen gesommeerd wordt Dit mo del heeft echter nog veel

problemen zowel in de denitie als in de simulatie ervan

Bij de b estudering van de simpliciale quantumgravitatie sp elen Monte Carlo

simulaties een grote rol Deze worden uitgevoerd do or een kleine b eginconguratie

te nemen en deze steeds te wijzigen do or er zetten op uit te voeren De waarschijn

lijkheden waarmee deze zetten worden uitgevoerd mo eten voldoen aan b epaalde

vergelijkingen die zorgen dat de kans om een conguratie te b ereiken evenredig is

aan het Boltzmann gewicht van die conguratie Dit alles is gemplementeerd in

een C programma

Bijlage Drie

Uitleg voor nietfysici

ijlage drie heb ik geschreven in de ho op de genteresseerde leek

een idee te geven waar dit pro efschrift nu eigenlijk over gaat Aan

de hand van de woorden in de titel zal ik eerst iets vertellen over

het onderwerp van onderzo ek en uiteindelijk over mijn deel daarin

In de voetnoten staat wat extra informatie waarbij geen p oging is B

gedaan deze zonder voorkennis b egrijp elijk te maken

C Zwaartekracht

Zwaartekracht gravity in het Engels kennen we allemaal Het houdt ons op de

grond de maan bij de aarde en de aarde bij de zon

Fysici b eschrijven zwaartekracht als een kromming van de ruimte Een ge

kromde ruimte is helaas niet iets wat we ons go ed kunnen voorstellen Daarom

zullen we in veel voorb eelden do en alsof de ruimte tweedimensionaal is zoals een

vel papier Als we dit papier nu wat buigen of zelfs helemaal in elkaar frommelen



hebb en we een voorstelling van een gekromde ruimte Dat we die kromming van

onze ruimte niet zien komt do ordat deze erg klein is In de ruimte kunnen we

wel voorb eelden zien Sommige sterren zien er met b ehulp van grote telescop en

vervormd uit omdat de ruimte ergens tussen ons en die ster sterk gekromd is

Ongeveer zoals bij een lachspiegel

Ho e kan nu een kracht eigenlijk een kromming van de ruimte zijn Ik zal dit

illustreren met twee varende schepen Kijk eens naar guur C op de pagina

hierna Op een dag b eginnen twee schepen samen aan de lange reis naar het



Strikt genomen hebb en we pas een gekromde ruimte als we het papier bij het vervormen laten

rekken

C Uitleg voor nietfysici

Figuur C Een gekromde ruimte veroorzaakt aantrekking

no orden Om niet het gevaar te lop en tegen elkaar aan te b otsen spreken ze

misschien wat onpraktisch af om evenwijdig te b eginnen en precies rechtdoor te

blijven varen Zo zouden ze dezelfde afstand mo eten houden

Zo gezegd zo gedaan Gelukkig is er op een van de schepen nog een oplettende

matro os die niet veel van evenwijdige lijnen weet en de zaak niet zo vertrouwt Na

enige tijd merkt hij dat de schepen to ch wel gevaarlijk dicht bij elkaar b eginnen

te komen Zoals in de guur te zien is komt dit do or de kromming van de aarde

Ho ewel b eide schepen dachten dezelfde afstand te zullen houden gaan ze naar

elkaar to e Dit gaat steeds sneller Op dezelfde manier werkt zwaartekracht De

ruimte is gekromd en zolang iets niet wordt tegengehouden do or bijvoorb eeld de

vlo er zal het rechtdoor b ewegen in die gekromde ruimte Als ik van het dak af

spring zullen de aarde en ik steeds sneller naar elkaar to e b ewegen omdat we b eide

rechtdoor gaan in de gekromde ruimte

In het voorb eeld met de varende schepen lag de kromming van te voren vast

Dit geldt niet voor de kromming van onze ruimte Die ruimtekromming wordt

veroorzaakt do or alle voorwerpen Ho e zwaarder iets is ho e meer kromming het

in zijn buurt veroorzaakt Dit eect is maar heel klein dus in het dagelijks leven

merken we niets van de kromming die do or gewone voorwerpen wordt veroorzaakt

Pas bij dingen zo gro ot als de aarde wordt dit b elangrijk In guur C op de

rechter pagina is dit getekend Een gekromde ruimte is niet te tekenen Daarom

heb ik de tekening gemaakt alsof er maar twee dimensies zijn en de ruimte dus

een gekromd opp ervlak is In het midden b evindt zich een ster De dikke lijn



is de baan van een ruimteschip Ho ewel het ruimteschip in de gekromde ruimte



Natuurlijk veroorzaakt het ruimteschip zelf o ok een kromming maar omdat de ster zoveel

zwaarder is kunnen we de kromming do or het ruimteschip verwaarlozen

C Ruimtetijd

Figuur C Aantrekking tussen voorwerpen via ruimtekromming

steeds rechtdoor gaat zal het uiteindelijk to ch een andere richting hebb en dan het

mee b egon en dus afgeb ogen zijn

C Ruimtetijd

Als u go ed over het b ovenstaande nadenkt dan zou u tot de conclusie mo eten

komen dat ik onzin sta te verkopen Ik b eweer in de vorige paragraaf dat als u een

bal go oit dat deze rechtdoor gaat in de gekromde ruimte Maar dat zou b etekenen

dat die bal altijd dezelfde baan in de gekromde ruimte volgt of u hem nu hard of

zacht go oit Snel rechtdoor en langzaam rechtdoor maken tenslotte geen verschil

in de baan die uiteindelijk gevolgd wordt U weet wel b eter een bal die u harder

go oit zal ergens anders terecht komen In deze paragraaf zal ik prob eren uit te

leggen ho e dat komt

In guur C op de pagina hierna heb ik een graek getekend van een etsto cht

van Amsterdam naar Utrecht Horizontaal staat de tijd verticaal de afstand tot

Amsterdam Als de etser pauzeert blijft deze enige tijd op dezelfde plek de

graek lo opt dan horizontaal Als hij etst lo opt de graek scheef en ho e sneller

hij etst ho e steiler de graek lo opt

Zon plaatje stelt de ruimtetijd voor Een punt in de ruimtetijd komt overeen

met een plaats en een tijdstip Omdat we voor het tijdstip een getal extra hebb en

heeft de ruimtetijd een dimensie meer dan de ruimte In het voorb eeld hadden

we twee dimensies no dig voor het plaatje terwijl er maar een ruimtedimensie

was namelijk de afstand tot Amsterdam Onze wereld is driedimensionaal We

C Uitleg voor nietfysici

Figuur C De ruimtetijd

Utrecht

7











"

"

"













#

#

#

#

#

#

#

#

#

Amsterdam

Begin uur uur

leven dus in drie ruimtedimensies en daarom heeft onze ruimtetijd vier dimensies

Eigenlijk is niet alleen de ruimte gekromd maar de hele ruimtetijd

We zien nog iets in het plaatje Wanneer je sneller gaat dan lo opt je graek

steiler en ga je in die ruimtetijd dus een andere kant op Een andere snelheid is

hetzelfde als een andere richting in de ruimtetijd Omdat de ruimtetijd gekromd

is hangt het dus o ok van je snelheid af wat rechtdoor in de gekromde ruimte

precies is en dus welke baan je volgt

In het plaatje van de etsto cht heb ik de schaal eigenlijk niet go ed getekend

De ruimtetijd zit zo in elkaar dat we met graden schuin vooruit gaan in de

ruimtetijd wanneer we b ewegen met de lichtsnelheid die is kilometer p er

seconde Met andere woorden een seconde naar rechts in de tijdrichting is

even ver in de ruimtetijd als kilometer naar b oven Dat b etekent dus dat

we in de tijdrichting zeer grote afstanden aan het aeggen zijn Zon zeer grote

afstand do or een ruimtetijd die overal maar een klein b eetje krom is b etekent

dat het totale eect to ch gro ot kan zijn Daarom ho eft de ruimte maar een heel

klein b eetje gekromd te zijn om de zwaartekracht die we om ons heen zien te

veroorzaken

De conclusie is dat we voor een b eschrijving van de zwaartekracht naar vierdi

C Simplices

Figuur C Een b enadering van een b ol do or middel van platte stukjes

mensionale gekromde ruimtetijden mo eten kijken Ik zal deze voor het gemak weer

ruimtes no emen want de tijddimensie is niet anders dan een ruimtedimensie Een

gekromd opp ervlak is een gekromd opp ervlak of we het nu een tweedimensionale



ruimte no emen of een ruimtetijd met een ruimtedimensie

C Simplices

Om te kijken wat de eecten van de zwaartekracht zijn willen we nu graag deze

gekromde ruimte op de computer simuleren Daarto e zouden we van elk punt van

de ruimte mo eten bijhouden ho e de kromming lo opt Maar zelfs een eindig stukje

ruimte bijvoorb eeld een kubieke centimeter heeft oneindig veel punten Om van

al die punten de kromming bij te houden zouden we een computer no dig hebb en

met oneindig veel geheugen en die b estaat niet Daarom do en we net alsof de

ruimte uit een gro ot aantal bijvoorb eeld tienduizend platte stukjes b estaat We

ho even dan alleen maar bij te houden welk stukje met welk ander stukje verbonden

is In guur C zien we ho e we op die manier met platte stukjes een b ol kunnen

b enaderen

Als platte stukjes gebruiken we simplices vandaar de titel Ho e een simplex



Helaas maakt dit niet alleen eigenlijk to ch uit het is zelfs een van de gro otste problemen van

de quantumgravitatie

C Uitleg voor nietfysici

Figuur C Simplices in verschillende dimensies

a D b D c D

eruit ziet hangt af van het aantal dimensies dat je b ekijkt Enkele voorb eelden

zijn te zien in guur C In twee dimensies is een simplex gewoon een drieho ek

In drie dimensies is het een viervlak o ok wel een driezijdige piramide geno emd

In vier dimensies no emen we het een simplex Dit is niet meer voor te stellen

Desgewenst kunt u in de rest van de tekst voor simplices gewoon drieho ekjes

lezen

Als we nu maar steeds kleinere simplices gebruiken om de ruimte uit op te

b ouwen dan kunnen we steeds b eter de werkelijkheid b enaderen Helaas hebb en

we er dan o ok steeds meer no dig en past het al vlug niet meer in onze computers

Terzijde misschien heeft u eens geho ord dat quantummechanica te maken

heeft met het op delen in stapp en van dingen als lading en energie waardoor deze

niet meer traplo os verstelbaar zijn Voor alle duidelijkheid dat heeft hier niets

mee te maken Het op delen van de ruimte in stukjes is hier alleen maar een manier



om die ruimte in de computer te krijgen

C Quantumfysica

In de zogenaamde klassieke fysica dat is nietquantum fysica is gegeven de

situatie op een zeker moment alles precies b epaald Als ik een bal opgo oi zegt

de klassieke fysica mij precies ho e die bal zich zal b ewegen In de quantumfysica



En o ok om het mo del ub erhaupt te denieren

C Simplicial Quantum Gravity

is dit niet zo De bal kan alle mogelijke routes volgen maar sommige routes zijn

waarschijnlijker dan andere De quantumfysica is voor zover we weten de eigenlijke

werkelijkheid De klassieke fysica werkt echter zo go ed omdat de kans dat de bal

meer dan een heel klein b eetje van de klassieke route afwijkt bijzonder klein is

Ik zal die bal dus no oit plotseling heen en weer zien zigzaggen in plaats van zijn

gewone b o og je te zien volgen

Als we echter naar heel kleine dingen kijken zoals atomen dan is een kleine

afwijking to ch erg b elangrijk Het is dan niet meer voldoende om alleen naar het

gedrag volgens de klassieke fysica te kijken We mo eten alle mogelijke manieren

waarop het ato om kan b ewegen b ekijken en daar een gemiddelde van nemen Wil

len we bijvoorb eeld de snelheid weten dan b erekenen we de gemiddelde snelheid

van al die routes Zon gemiddelde is een gewogen gemiddelde Sommige banen

hebb en meer kans dan andere

Om dingen te b erekenen in quantumgravitatie mo eten we dus kijken naar

alle mogelijke gekromde ruimtes en daar een gewogen gemiddelde van nemen

Het grote probleem is echter dat niemand weet ho e je een gemiddelde van alle

mogelijke gekromde ruimtes neemt

Prob eer maar eens het gemiddelde van oneindig veel getallen te nemen Eerst

tellen we ze allemaal op dan krijgen we oneindig en dan delen we de som do or het

aantal o ok oneindig Je kunt echter niet oneindig do or oneindig delen Gelukkig

is het niet zo erg er zijn in veel gevallen wel manieren om zon gemiddelde te

b erekenen Maar een gemiddelde over alle gekromde ruimtes snapt niemand

C Simplicial Quantum Gravity

Het idee van de zogenaamde simplicial quantum gravity waar dit pro efschrift over

gaat is nu om het gewogen gemiddelde te nemen van alle manieren om simplices

aan elkaar te plakken Een zon manier zag u in guur C op pagina maar

er zijn er uiteraard nog veel meer Gelukkig is dit wel een eindig aantal dus nu

weten we ho e we zon gemiddelde mo eten nemen en kunnen we de computer dit

uit laten rekenen

Dit concept was al b ekend to en ik aan mijn onderzo ek b egon Ik heb nader

b ekeken in ho everre er redelijke dingen uit dit mo del tevoorschijn komen In deze

paragraaf zal ik iets zeggen over een deel van de dingen die ik heb gedaan

Ik heb hierb oven gezegd dat ho e kleiner we de simplices maken ho e b eter we

de werkelijkheid b enaderen Wat we echter niet weten is of we o ok het go ede

gemiddelde over alle gekromde ruimtes b enaderen Als alles go ed is dan mo eten

C Uitleg voor nietfysici

Figuur C Steeds gekkere ruimtes

de dingen die we uit de simulatie krijgen als we steeds kleinere simplices gebruiken

naar een b epaalde waarde gaan Dan zeggen we dat die waarde de echte is Het

zou echter b est kunnen geb euren dat wat er uit de computer komt steeds maar

groter of kleiner blijft worden Dan is het mo del gewoon niet go ed

Andere mensen hebb en b eweerd dat als je de simplices steeds kleiner maakt dat

dan de gemiddelde ruimtekromming steeds maar groter blijft worden Ik b eweer

echter dat je daar anders naar mo et kijken In guur C heb ik enkele stukken

golfplaat van de zijkant getekend Eenvoudig gesteld is dit wat er geb eurt als

je steeds kleinere simplices neemt Inderdaad wordt de gemiddelde ruimte die we

met de simplices maken steeds gekker en zou je kunnen concluderen dat hier no oit

een mo oie gladde ruimte lees opp ervlak uit gaat komen Al die golfjes in de



ruimte zijn echter veel kleiner dan een ato om en die zien we dus no oit Als we

van een grote afstand kijken dan zijn al die stukken golfplaat gewoon plat Ik

meen dat het o ok zo werkt met de ruimte die uit de simulaties tevoorschijn komt

Iets anders waar ik naar gekeken heb is of dingen elkaar inderdaad aantrekken

in dit mo del Ook dat is van tevoren niet duidelijk Een mo del van zwaartekracht

waarin we niet kunnen zien dat dingen elkaar aantrekken kunnen we uiteraard net

zo go ed in de prullenmand go oien Gelukkig hebb en we inderdaad aantrekking

gezien

De conclusie is dat het mo del er op dit moment ho opvol uitziet Of het klopt

 - -

Een ato om is ongeveer m gro ot en een proton ongeveer m terwijl de ruimtetijd

-

uctuaties zich afsp elen op de zogenaamde Planck schaal van ongeveer m

C Waarom

dat wil zeggen dat het een go ede b eschrijving van de werkelijkheid geeft weten

we nog niet

C Waarom

Ik heb nu geprob eerd uit te leggen waar dit pro efschrift over gaat maar de vraag

do et zich voor wat er nu eigenlijk interessant is aan die quantumgravitatie U zou

kunnen denken Als die afwijkingen veel kleiner zijn dan een ato om wat kan ons

die dan schelen en u zou nog gelijk hebb en o ok Toch zijn er enkele redenen

om quantumgravitatie te b estuderen

We zien dat alle melkwegstelsels in het heelal zich van elkaar af b ewegen

Daaruit en uit nog veel meer aanwijzingen concluderen we dat ze dus vro eger

dichter bij elkaar zaten Zo dicht zelfs dat bij het b egin van het heelal alles in

een punt bij elkaar zat Vlak na het b egin de zogenaamde Big Bang was het

heelal dus zo klein dat zelfs die hele kleine afwijkingen die de quantumgravitatie

b eschrijft erg b elangrijk waren Dus om te weten wat er in het b egin geb eurde

hebb en we quantumgravitatie no dig

De tweede reden is dat fysici gewoon grenzelo os nieuwsgierig zijn en graag

willen weten ho e de wereld in elkaar zit zelfs al merken we niets van dat asp ect

ervan

Er zijn vele voorb eelden in de geschiedenis van de natuurkunde waarbij dingen

waar eerst niemand iets nuttigs in zag later to ch op onvoorziene wijze to egepast

werden in alledaagse gebruiksvoorwerpen zoals televisies Velen menen dat de

quantumgravitatie zo obscuur is dat daar echt no oit iets mee gedaan zal kunnen

worden Daarom wil ik dit ho ofdstuk eindigen met een uitermate sp eculatief idee

Aan de ene kant is het b ekend dat men energie uit roterende zwarte gaten kan

halen do or er dingen bijvoorb eeld afval vlak langs te go oien Aan de andere kant

weten we dat vlak bij zwarte gaten quantumgravitatie een b elangrijke rol mo et

sp elen Wie weet ligt hier dus o oit eens een to epassing

Dankwoord

lhoewel het nogal als een cliche klinkt wil ik graag mijn op

rechte dank b etuigen aan Jan Smit zonder wiens b egeleiding van

dit pro efschrift weinig terecht zou zijn gekomen Men ho ort wel

eens verhalen over b egeleiders die hun promovendi nauwelijks zien

en niet weten waar die mee b ezig zijn Niets was minder het geval A

Ho ewel ik de vele malen dat hij met enthousiaste ideeen mijn kamer binnenliep

of zelfs van elders opb elde op het moment in kwestie wel eens als storend ervoer

waren ze zeer stimulerend

Mijn instituutsgenoten dank ik voor het scheppen van een jne plek om te

werken Een paar mensen wil ik in het bijzonder no emen Bernard Nienhuis van

wege de enthousiaste manier waarop hij soms met alledaagse niet direct aan het

onderzo ek gerelateerde fysische onderwerpen kwam Mijn kamergenote Nathalie

Muller vanwege het gezelschap het gebak en het verdragen van mijn onaatend

computergebruik I would like to thank Piotr Bialas for the pleasurable co op era

tion

De systeembeheerders van de vele computers die ik gebruikte wil ik b edanken

voor hun medewerking aan mijn soms b ovengemiddelde wensen voor het terugha

len van de les die ik p er ongeluk verwijderde en voor het niet opmerken wanneer

ik meer deed dan zij mogelijk achtten

Willemien wil ik b edanken voor alles zoals men dat no emt en in het bijzonder

voor de opb ouwende kritiek op de uitleg voor nietfysici

Publications

This thesis is partly based on the following articles

F BV de Bakker and J Smit Euclidean gravity attracts Nucl Phys B Pro c

Suppl

F BV de Bakker and J Smit Volume dep endence of the phase b oundary in

D dynamical triangulation Phys Lett B

F BV de Bakker and J Smit Curvature and scaling in D dynamical trian

gulation Nucl Phys B

F BV de Bakker Dynamical triangulation with uctuating top ology Nucl

Phys B Pro c Suppl

F BV de Bakker and J Smit Exploring curvature and scaling in D dynamical

triangulation Nucl Phys B Pro c Suppl

F BV de Bakker Absence of barriers in dynamical triangulation Phys Lett

B

F BV de Bakker and J Smit Twopoint functions in D dynamical triangu

lation preprint ITFA submitted to Nucl Phys B

References

Agishtein Migdal

ME Agishtein and AA Migdal Geometry of a twodimensional quantum

gravity numerical study Nucl Phys B

Agishtein Migdal a

ME Agishtein and AA Migdal Simulations of fourdimensional simplicial

quantum gravity as dynamical triangulation Mo d Phys Lett A

Agishtein Migdal b

ME Agishtein and AA Migdal Critical b ehavior of dynamically triangu

lated quantum gravity in four dimensions Nucl Phys B

Alessandrini

V Alessandrini Dynamical generation of a Coulomb phase in the meaneld

approach to ZN lattice gauge theories Nucl Phys B

Alessandrini Boucaud

V Alessandrini and Ph Boucaud Meaneld approach to ZN gauge sys

tems with generalized action Nucl Phys B

Alexander

JW Alexander The combinatorial theory of complexes Ann Math

Amb jrn Jurkiewicz

J Amb jrn and J Jurkiewicz Fourdimensional simplicial quantum gravity

Phys Lett B

Amb jrn Jurkiewicz

J Amb jrn and J Jurkiewicz On the exp onential b ound in four dimensional

simplicial gravity Phys Lett B

Amb jrn Jurkiewicz a

J Amb jrn and J Jurkiewicz Computational ergo dicity of S Phys Lett

B

References A

Amb jrn Jurkiewicz b

J Amb jrn and J Jurkiewicz Scaling in four dimensional quantum gravity

preprint NBIHE

Amb jrn et al

J Amb jrn B Durhuus and T Jonsson Threedimensional simplicial quan

tum gravity and generalized matrix mo dels Mo d Phys Lett A

Amb jrn et al a

J Amb jrn Z Burda J Jurkiewicz and CF Kristjansen D quantum

gravity coupled to matter Phys Rev D

Amb jrn et al b

J Amb jrn S Jain J Jurkiewicz and CF Kristjansen Observing d baby

universes in quantum gravity Phys Lett B

Amb jrn et al c

J Amb jrn J Jurkiewicz and CF Kristjansen Quantum gravity dynam

ical triangulations and higher derivative regularization Nucl Phys B

Amb jrn et al

J Amb jrn P Bialas Z Burda J Jurkiewicz and B Petersson Search for

scaling dimensions for random surfaces with c Phys Lett B

Anderson DeWitt

A Anderson and BS DeWitt Do es the top ology of space uctuate in

Between quantum and cosmos eds WH Zurek A van der Merwe and

WA Miller Princeton University Press Princeton p

Antoniadis Mottola

I Antoniadis and E Mottola Fourdimensional quantum gravity in the con

formal sector Phys Rev D

Antoniadis et al

I Antoniadis PO Mazur and E Mottola Conformal symmetry and central

charges in dimensions Nucl Phys B

Ashtekar

A Ashtekar New variables for classical and quantum gravity Phys Rev

Lett

Ashtekar

A Ashtekar A new Hamiltonian formulation of general relativity Phys

Rev D

References B

de Bakker Smit

BV de Bakker and J Smit Volume dep endence of the phase b oundary in

D dynamical triangulation Phys Lett B

de Bakker Smit

BV de Bakker and J Smit Curvature and scaling in D dynamical trian

gulation Nucl Phys B

Barrett Foxon

JW Barrett and TJ Foxon Semiclassical limits of simplicial quantum

gravity Class Quantum Grav

Barto cci et al

C Barto cci U Bruzzo M Carfora A Marzuoli Entropy of random cover

ings and D quantum gravity preprint SISSAFM

Beirl et al a

W Beirl E Gerstenmayer H Markum and J Riedler The well dened

phase of simplicial quantum gravity in fourdimensions Phys Rev D

Beirl et al b

W Beirl H Markum and J Riedler Twopoint functions of fourdimen

sional simplicial quantum gravity Nucl Phys B Pro c Suppl

Berg

BA Berg Entropy versus energy on a uctuating fourdimensional Regge

skeleton Phys Lett B

Bilke et al

S Bilke Z Burda and J Jurkiewicz Simplicial quantum gravity on a com

puter Comput Phys Commun

Birmingham

D Birmingham Cob ordism eects in the Regge calculus approach to quan

tum cosmology preprint ITFA

Bo ck Vink

W Bo ck and JC Vink Failure of the Regge approach in two dimensional

quantum gravity Nucl Phys B

Brezin Kazakov

E Brezinand VA Kazakov Exactly solvable eld theories of closed strings

Phys Lett B

References BD

Brezin et al

E BrezinC Itzykson G Parisi and JB Zub er Planar diagrams Commun

Math Phys

Br ugmann

B Br ugmann Nonuniform measure in fourdimensional simplicial quantum

gravity Phys Rev D

Br ugmann Marinari

B Br ugmann and E Marinari More on the exp onential b ound of four di

mensional quantum gravity Phys Lett B

Carfora et al

M Carfora M Martellini and A Marzuoli jsymbols and fourdimen

sional quantum gravity Phys Lett B

Catterall

S Catterall Simulations of dynamically triangulated gravity an algorithm

for arbitrary dimension preprint CERNTH

Catterall et al a

S Catterall J Kogut and R Renken Phase structure of fourdimensional

simplicial quantum gravity Phys Lett B

Catterall et al b

S Catterall J Kogut and R Renken Is there an exp onential b ound in

fourdimensional simplicial gravity Phys Rev Lett

Cheeger et al

J Cheeger W M uller and R Schrader On the curvature of piecewise at

spaces Commun Math Phys

Creutz

M Creutz Quarks gluons and lattices Cambridge University Press Cam

bridge

David

F David Planar diagrams twodimensional lattice gravity and surface mo d

els Nucl Phys B FS

David

F David What is the intrinsic geometry of twodimensional quantum grav

ity Nucl Phys B

Douglas Shenker

M Douglas and S Shenker Strings in less than one dimension Nucl Phys

B

References FG

Filk

T Filk Equivalence of massive propagator distance and mathematical dis

tance on graphs Mo d Phys Lett A

Friedberg Lee

R Friedberg and TD Lee Derivation of Regges action from Einsteins

theory of general relativity Nucl Phys B

Frolich Sp encer

J Frolich and T Sp encer Massless phases and symmetry restoration in

ab elian gauge theories and spin systems Commun Math Phys

Garay

LJ Garay Quantum gravity and minimum length Int J Mo d Phys A

Gibb ons et al

GW Gibb ons SW Hawking and MJ Perry Path integrals and the indef

initeness of the gravitational action Nucl Phys B

Ginsparg Mo ore

P Ginsparg and G Mo ore Lectures on D Gravity and D String Theory

in Recent directions in particle theory eds J Harvey and J Polchinski

World Scientic Singap ore p

Goro Sagnotti

MH Goro and A Sagnotti The ultraviolet b ehavior of Einstein gravity

Nucl Phys B

Green et al

MB Green JH Schwarz and E Witten Sup erstring theory volumes

Cambridge University Press Cambridge

Greensite

J Greensite Dynamical origin of the lorentzian signature of spacetime

Phys Lett B

Gross

M Gross Higher dimensional simplicial quantum gravity Nucl Phys B

Pro c Suppl

Gross Migdal

DJ Gross and AA Migdal A nonp erturbative treatment of twodimen

sional quantum gravity Nucl Phys B

Gross Varsted

M Gross and S Varsted Elementary moves and ergo dicity in Ddimensional

simplicial quantum gravity Nucl Phys B

References HK

Hamber

HW Hamber Simplicial quantum gravity in Critical phenomena random

systems gauge theories eds K Osterwalder and R Stora NorthHolland

Amsterdam p

Hamber

HW Hamber Phases of simplicial quantum gravity in four dimensions

Estimates of the critical exp onents Nucl Phys B

Hamber

HW Hamber Invariant correlations in simplicial gravity Phys Rev D

Hartle

JB Hartle Simplicial minisup erspace I I I Integration contours in a ve

simplex mo del J Math Phys

Hawking a

SW Hawking Spacetime foam Nucl Phys B

Hawking b

SW Hawking Euclidean quantum gravity in Recent developments in grav

itation eds M Levy and S Deser Plenum Press New York p

Heller

UM Heller Status of the Higgs mass b ound Nucl Phys B Pro c Suppl

Holm Janke

C Holm and W Janke The critical b ehaviour of Ising spins on D Regge

lattices Phys Lett B

t Ho oft

G t Ho oft On the quantization of space and time in Quantum gravity

pro ceedings of the fourth seminar eds MA Markov VA Berezin and

VP Frolov World Scientic Singap ore p

t Ho oft Veltman

G t Ho oft and M Veltman Onelo op divergencies in the theory of gravita

tion Ann Inst Henri Poincare

Johnston

DA Johnston Sedentary ghost p oles in higher derivative gravity Nucl

Phys B

Kawamoto et al

NK Kawamoto VA Kazakov Y Saeki Y Watabiki Fractal structure

of twodimensional gravity coupled to c matter Phys Rev Lett

References KP

Kazakov

VA Kazakov The app earance of matter elds from quantum uctuations

of D gravity Mo d Phys Lett A

Kazakov et al

VA Kazakov IK Kostov and AA Migdal Critical prop erties of randomly

triangulated planar random surfaces Phys Lett B

Knizhnik et al

VG Knizhnik AM Polyakov and AB Zamolo dchikov Fractal structure

of D quantum gravity Mo d Phys Lett A

L usher Weisz

M L uscher and P Weisz Is there a strong interaction sector in the standard

lattice Higgs mo del Phys Lett B

L usher Weisz

M L uscher and P Weisz Scaling laws and triviality b ounds in the lattice

theory I I I ncomp onent mo del Nucl Phys B

Misner et al

CW Misner KS Thorne and JA Wheeler Gravitation WH Freeman

and company New York

Mo danese

G Mo danese Potential energy in quantum gravity Nucl Phys B

Mottola

E Mottola Functional integration over J Math Phys

Nabutovsky BenAv

A Nabutovsky and R BenAv Noncomputability arising in dynamical trian

gulation mo del of fourdimensional quantum gravity Commun Math Phys

Nieuwenhuizen

P van Nieuwenhuizen Sup ergravity Phys Rep

Pachner

U Pachner Konstruktionsmetho den und das kombinatorische Homoomor

phieproblem f ur Triangulationen kompakter semilinearer Mannigfaltigkei

ten Abh Math Sem Univ Hamburg

Polyakov

AM Polyakov Quantum geometry of b osonic strings Phys Lett B

References PW

Ponzano Regge

G Ponzano and T Regge Semiclassical limit of Racah co ecients in Sp ec

troscopic and group theoretical metho ds in physics eds F Blo ch SG Co

hen A DeShalit S Sambursky and I Talmi NorthHolland Publishing

Company Amsterdam p

Regge

T Regge General relativity without co ordinates Nuovo Cimento

Schleich

K Schleich Conformal rotation in p erturbative gravity Phys Rev D

Shamir

Y Shamir Quantum gravity via random triangulations of R and gravitons

as Goldstone b osons of SLO preprint WISPH

Sharp e

SR Sharp e Phenomenology from the lattice preprint UWPT

Stelle

KS Stelle Renormalization of higherderivative quantum gravity Phys

Rev D

Varsted

S Varsted Fourdimensional quantum gravity by dynamical triangulations

Nucl Phys B

van de Ven

AEM van de Ven Twoloop quantum gravity Nucl Phys B

Weingarten

D Weingarten Euclidean quantum gravity on a lattice Nucl Phys B

FS

Wheeler

JA Wheeler Geometro dynamics and the issue of the nal state in Rela

tivity Groups Topology eds BS DeWitt and CM DeWitt Blackie and

Son Ltd Glasgow p

Williams Tuckey

RM Williams and PA Tuckey Regge calculus a bibliography and brief

review Class Quantum Grav

Index

Binding energy A

Branched p olymer Action

discrete

C

of scalar eld

C programming language

eective

Canonical partition function

EinsteinHilb ert

Clusters

Euclidean

COBE

ve dimensions

Cob ordism

of scalar eld

Co ding

ReggeEinstein

maximal

with scalar eld

minimal

Alexander moves

Computability

Ashtekar variables

uctuating top ology

Attraction

Conguration

Auto correlation time

smallest

average distance

Connected

baby universe surgery

comp onents

denition

correlation function

Average

Connection

cluster size

Continuum limit

curvature

curvature

phase transition

uctuating top ology

distance

scaling

triviality B

Convergence Baby universe surgery

continuum path integral computability

discrete partition function Baby universes

Co oling Barriers

times Big Bang

Index CG

Distance Correlation function

average connected

uctuations curvature

Double scaling limit

volume

Double subsimplices

Critical

Dynamical triangulation

exp onent

E

Eective

action

Critical slowing down

curvature

Crumpled phase

volume

Curvature

EinsteinHilb ert action

average

Elongated phase

phase transition

curvature correlation

correlation

illustration

denition

Ergo dicity

discrete

uctuating top ology

eective

Euclidean

uctuations

action

integrated

gravity

running

Euler number

scaling

Exp onential b ound

susceptibility

F

D

Faces of simplex

DehnSommerville relations

Five dimensional action

ve dimensions

Fluctuating top ology

Detailed balance

Fluctuations

uctuating top ology

curvature

Dimension

distance

fractal

volume

global

FORTRAN

scaling

Fractal dimension

Discrete

action

G

General relativity of scalar eld

Geometry test curvature

Global dimension volume

Index GP

Measuring Gravitons

Metastable state Gravity

Metrop olis Euclidean

Minimal co ding quantum

Monte Carlo simulation R

uctuating top ology Riemannian

Moves

H

Alexander

Hinge

baby universe surgery

uctuating top ology

I

k l

Ination

Integrated curvature

N

N dep endencies

i

J

ve dimensions

Jackknife metho d

Noncomputability

K

Nonlinear mo del

O

critical

Orientation

reversal

critical

Outlo ok

k l moves

KPZscaling

P

Partition function

L

canonical

Lattice eld theory

Percolation

Lo cal volume

Phase

crumpled M

Manifold

elongated

condition

curvature correlation

illustration pseudo

transition simplicial

average curvature Mass renormalization

curvature correlation Matrix mo del

welldened symmetry

Planckian regime Maximal co ding

Pro duction notes Measure

Programming language continuum

Index PW

Susceptibility Propagator

curvature matrix mo del

string scalar eld

Sweep tensor mo del

Symmetry Pseudomanifold

breaking

Q

of triangulations

Quantum gravity

T

Quenched approximation

Tensor mo del

R

Thermalization

R gravity

ve dimensions

Recognizability

Transition phase

S

average curvature

References

curvature correlation

Regge calculus

Triangulation

correlation functions

dynamical

ReggeEinstein action

Triviality

Renormalizability

Two dimensions

Riemannian gravity

attraction

Rob ertsonWalker

Twopoint function

Running curvature

U

scaling

Universality

S

Unrecognizability

Scaling

S

dimension

V

KPZ

Volume

running curvature

correlation

Simplex

discrete

picture

eective

volume

uctuations

Simplicial gravity

lo cal

Simplicial manifold

simplex

String susceptibility

small sphere in curved space

String theory

W Strong coupling

Weak coupling Subsimplex

Welldened phase Sup ergravity

Pro duction notes

This thesis was edited using GNU Emacs and typeset with the help of T X and

E

A A

many macro packages in particular L T X and A SL T X The typeface used

E M E

in the main text is Concrete while that in the formulas is AMS Euler The initials

made by Yannis Haralambous are simply called yinit The b oldface used in

headings and on the front cover is my own adaptation to Concrete of Computer

Mo dern Bold Extended The cyrillic typeface is my own adaptation to Concrete

of Computer Mo dern Cyrillic All these were made with the help of METAFONT

Drawings were made with xg and converted to p ostscript by gdev Plots

were made using gnuplot together with the enhp ost driver In b oth cases text and

formulas were put into them using psfrag Sp ecial thanks go to GNU make which

A

I nally co erced into running L T X exactly as many times as needed to get the

E

cross references stable

All computer programs mentioned ab ove as well as all fonts used in this thesis

are free software

There is a danger that authors

who are now able to typeset their own b o oks with T X

E

will attempt to do their own designs

Donald E Knuth The T Xb o ok

E