propagating scalar modes in (2+1)-cdt quantum gravity

8/12/2019 Propagating Scalar Modes in (2+1)-CDT Quantum Gravity

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Propagating Scalar Modes in (2 + 1)-Dimensional

CDT Gravity

A Senior Thesis Presented to the Physics Faculty of the

University of California, Berkeley

Adam Lloyd Bruce


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Contents

1 Why CDT Quantum Gravity? 3

1.1 Quantization and the Fundamental Forces . . . . . . . . . . . . . . . . . . . 3

1.2 Nonrenormalizability of Canonical Quantization. . . . . . . . . . . . . . . . 5

1.3 Causal Dynamical Triangulations . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Classical Gravity in(2 + 1)- Dimensions 9

2.1 general relativity in (2 + 1)- Dimensions . . . . . . . . . . . . . . . . . . . . 10

2.1.1 The Newtonian Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 The [0, 1] Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 The ADM Formalism and Hamiltionian Formulation . . . . . . . . . . . . . 15

2.3.1 The ADM Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2 The Hamiltonian and Canonical Constraints . . . . . . . . . . . . . 17

2.4 (2 + 1)- Path Integrals in the ADM Formalism . . . . . . . . . . . . . . . . 18

2.4.1 Equivalence to 2+1 Canonical Quantization . . . . . . . . . . . . . . 19

3 Nonperturbative Quantum Gravity with(2 + 1)-CDT 24

3.1 The Gravitational Path integral via Triangulation. . . . . . . . . . . . . . . 24

3.1.1 Continuum Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.2 The CDT Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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3.2 Horava-Lifshitz Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Propagating Scalar Modes and Anisotropic Quantization. . . . . . . . . . . 30

4 Computational Approach 34

4.1 The Regge Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1.1 The non-regularized measure space and associated transition amplitude 36

4.2 Monte-Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Markov Chain Stepping . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.2 Sampling an Observable . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 The Metropolis Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.1 Equilibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.2 Monte-Carlo in CDT: The Pachner Moves . . . . . . . . . . . . . . . 42

5 Simulation 44

5.1 Generation of the Seed Manifold . . . . . . . . . . . . . . . . . . . . . . . . 44

5.1.1 Adaptive Remeshing . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1.2 Computation of Simplicial Geodesics . . . . . . . . . . . . . . . . . . 46

5.1.3 Extraction of the First Quadrant Wedge . . . . . . . . . . . . . . . . 48

6 Conclusion 49

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List of Figures

2.1 Diagram of the triangulation used to computeds2. . . . . . . . . . . . . . . 16

3.1 Disallowed Histories in Causal Dynamical Triangulations, such as wormhole

histories and branching points where the light cones are degenerate. . . . . 27

5.1 Triangulation by remeshing as presented in [20] . . . . . . . . . . . . . . . . 46

5.2 The triangulated manifold with simplicial great circles along thexz andyz

planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3 The triangulated manifold with simplicial great circles along thexz andyz

planes, and positive quadrant removed . . . . . . . . . . . . . . . . . . . . . 48

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List of Tables

1.1 The fundamental forces, their strengths, ranges, gauge particles and La-

grangians as in[2]and [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

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Abstract

Causal Dynamical Triangulations (CDT) quantum gravity is a relatively new quantization

scheme which uses simplicial latices to regularize the gravitational path integral. Recently

there has been much conjecture by scholars in the quantum gravity community that such

a quantization procedure may be equivalent to Horava-Lifshitz gravity in at least (2 + 1)-

dimensions. We outline a procedure for testing this numerically and generate a seed mani-

fold to perform such a simulation, using the fact, derived in the text, that the anisotropy of

Horava-Lifshitz gravity leads to at least one propagating scalar mode in (2 + 1)-dimensions

while traditional general relativity does not. We first describe quantization from a historical

perspective, outline relevant results from (2 + 1)-dimensional traditional general relativity,

discuss CDT and Horava-Lifshitz theory, discuss the general computational procedure of

CDT codes and finally describe the method for seed generation, including the adaptive

remeshing technique for generating suitable triangulations.


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Acknowledgements

No one can accomplish such a monolithic task as this in isolation, and I have had the good

fortune to meet and hear the wisdom of many brilliant minds and kind hearts along the

way. First I must thank Dr. Patrick Zulkowski, who approached me with the idea on which

this thesis was written in early 2012, and whose patience, kindness, intelligence and general

good-spiritedness are certainly not surpassed at UC Berkeley and I suspect will be difficult

to surpass for anyone I will ever know. Next I must thank the UC Davis quantum gravity

group, who provided essential input and generously supplied us with their codes, previous

work, and have promised computing time for the remainder of the project. I also thank

Professor Michael DeWeese for agreeing to advice this thesis despite the fact that it was not

in his field, simply because Dr. Zulkowski is a member of his biophysics group. A scientist

who stands so steadfastly and selflessly by his students is by any standard an honorable

and kind man. Finally I must thank Professor Kai Hormann, who for no other reason

but his great generosity and kind-heartedness provided expert opinions, suggestions and

even results from previous work to understand the adaptive remeshing technique, including

supplying us with a basic triangulation to build the seed manifold. Without any of these

great people, this thesis would not be what it is, and might not have been anything at all.

No agency funded this work beyond those funding the afore mentioned scholars. It has

rather been a labor of love on my own part, and I suspect on the part of everyone else.

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Dedication

Aig mAthair,

Bha thu a seolta mi eolas dean sinn saor,

agus seolta mi a leolas cha do chuir e cas am

broig a mise stad. Airson tha eolas rud

cumhachdach s breatha, agus tha mi an

dochas gun gabh sinne am gliocas feum mhath.

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Chapter 1

Why CDT Quantum Gravity?

1.1 Quantization and the Fundamental Forces

Nature, insofar as we currently know, is governed by four fundamental forces: the strong

force, the weak force, the electromagnetic force and the gravitational force. These form the

basis of all interactions which particles and fields engage in. Moreover, it is a reasonable

assumption that these forces should conform to the physics we understand to exist in the

experimental and theoretical regimes which we have studied. Hence we expect each force

to be consistent with special or general relativity and quantum mechanics. While each of

the forces conforms to the requirements of relativity, there is no quantum theory of gravity

which is currently accepted and experimentally tested.

It was Dirac who first studied the quantization of the electromagnetic field, developing

an approach to the theory known as traditional or canonical quantization. In canonical

quantization we assume the system to be governed by a HamiltonianH to which there isassociated a set of observables

Oi which obey Poisson bracket relations. The canonically

quantized version of the theory maps Poisson brackets to canonical commutation relations

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according to

{Oi, Oj} 1i

[Oi, Oj], (1.1)

though this rule is not mathematically rigorous and therefore selection of proper observables

is the subject of ongoing debate, especially in quantum gravity. The electrodynamic field

may for example be quantized via canonical quantization [1].

Another method of quantization which can be shown to be equivalent to the canonical

formalism is the path integral formalism. By considering the action over all possible state

space configurations, it is possible to compute a probability kernel for the particle or field

to propagate to some state from some initial state0. By writing this as a path integral

over a finite Hamiltonian system, this kernel may be expressed as

K(,t,0, t0) =

D[]exp

i

tt0

L( ,,t)dnx

, (1.2)

where the invariant measure on the space of trajectories is defined as

D[] := limN

i2

N/2

N1j=1

dj

, (1.3)

with = it known as a Wick rotation[4], andL( ,,t) is the Lagrangian density of thesystem from which is related to the total Lagrangian via the integral relation

L:=

dtL( ,,t). (1.4)

It is using this method that quantum electrodynamics can be tested to its furthest ex-

tremes, most noticeably to produce an accurate prediction of the electrons gfactor asg 2.00231930419922 (1.5 1012) (see[6]or [7]).

Hence a quantum theory using path integral quantization may be entirely determined

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Force Relative Strength Range (m) Gauge Boson Lagrangian

Strong 1 1015 gluon LQCD =i(i(

D)ij mij)j

1/4(Ga

Ga

)

Weak 106 1018 W,Z0 LEWElectromagnetic 1/137 photon LQED =icD

mc2 1/(40)FFGravitation 6 1039 G raviton (posited) unknown

Table 1.1: The fundamental forces, their strengths, ranges, gauge particles and Lagrangiansas in[2]and [3].

by a given Lagrangian, or from a more fundamental standpoint a Hamiltonian densityH,

which can be transformed uniquely into a Lagrangian, [2]. Lattice methods, which shall

be of particular importance to the current work, may be used to quantize the strong force

in the form of quantum chromodynamics.

Another important aspect of the quantization of a fundamental force is the production

of gauge bosons, resulting from the gauge group describing the interaction, which are

physically interpreted as the force mediating particles associated with the interaction. We

may therefore summarize the main aspects of the known quantization of the fundamental

forces by the entries in Table 1.1. Similar tables may be found in[2]or [3].

1.2 Nonrenormalizability of Canonical Quantization

There were attempts to quantize gravitational interaction almost immediately. It is known

as a result of classical general relativity that the theory can be derived from an action

known as the Einstein-Hilbert action,

SEH=M2p

d4xgR, (1.5)

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where we have defined Mp 1/

16G, g det(g), and R = gR is the Ricci scalar.Now suppose this is constructed as a typical quantum field theory. We first write the

metric as the Minkowski space metric plus a perturbationh

,g

=

+ h

. Hence the

Einstein-Hilbert action for the quantum field must have the form

S= M2p

d4x(hh + hhh + h2hh + . . . ). (1.6)

From the second term in equation1.6, we see that the second order Feynman term will go

as kkkk/k2k2 = 1, and hence the integral diverges. Taking out two of the momentumterms in the denominator in order to extract the hhdifferential, the action now diverges

as k2. This process is easily seen to diverge more and more rapidly as one moves tohigher and higher order Feynman terms. The third order term will diverge as k4.

We may also include matter in this integral via the term

Matter Interation = 12

d4xhT

, (1.7)

where

T(x) = 2g

SMg(x) ,

is the stress-energy tensor for any non-graviton matter field. This implies that1.6can be

written as

S=

d4x[M2p (hh + hhh + h

2hh + . . . ) + (hT+ . . . )]. (1.8)

We can see that even by inclusion of matter the action still diverges as before. This is the

well-known failure of traditional quantization methods to yield a renormalizable quantum

theory of gravity. The use of Euclidean lattices, such as those successful for quantum

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chromodynamics, also fails to achieve a renormalizable theory. It was shown in simulations

[31], that the seed universes produced by the computational methods could branch off

into a baby universe at each point in the spacetime. The degeneracy introduced by such

baby universe geometries prevents the continuum theory from being well-defined[30], [32].

This has lead to a continuing investigation to determine how to properly quantize the

gravitational field.

1.3 Causal Dynamical Triangulations

Causal Dynamical Triangulations or CDT is a concrete research program which attempts

to renormalize gravity nonperturbatively via the use of a lattice structure, similar to that

employed in quantum chromodynamics but simplicial instead of Euclidean. This is origi-

nally described in [5] [8] as a method inspired by the Wilsonian Renormalization Group.

Using this method, described in detail in chapter 3, the classical Einstein-Hilbert action

can be written in terms of the lattice connectivity data, using the Regge Calculus, which

is discussed in detail in chapter four. The regularized path integral using the Reggeized

action can be computed numerically, using a Monte-Carlo simulation, the details of which

are also discussed in chapter four. After the calculation the lattice length can be taken

to zero1, and a result may be obtained which is invariant under rescaling of the primitive

length. CDT has gained some notoriety for demonstrations that de-Sitter CDT space-

times possesses a well-defined classical limit, and there has been work done claiming that

classical results are in agreement with known physics. It has also been speculated that

simplicial quantization via CDT might be related to Horava-Lifshitz quantization in at

1It should be noted however that this limit cannot be taken arbitrarily, but rather at a second-order

phase transition. At a second order phase transition the discrete structure of the underlying spacetime stopsaffecting the statistical model strongly, due to the rapid divergence of the correlation lengths, implying that

the continuum limit may be taken here safely, where it will be unique and well-defined. This shall be

understood at all other times when we take a 0 in this work.

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least (2 + 1)-dimensions, which we shall provide a method for differentiating from diffeo-

morphism invariant quantization by searching for propagatin scalar modes which are riled

out in general relativity.

We shall see that for the purpose of the present study, it is computationally convenient

to use a (2 + 1)-dimensional spacetime rather than the full 3 + 1-dimensions. It is for this

reason that the next chapter is devoted to developing known classical results from general

relativity in (2 + 1)-dimensions.

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Chapter 2

Classical Gravity in (2 + 1)-

Dimensions

There is no intrinsic nature to CDT gravity which specifies the dimensionality of the

underlying spacetime. Thus we are free to choose any dimensionality we wish so long as such

a dimensionality admits vacuum solutions to the classical Einstein Field Equations. For a

comparison to Horava-Lifshitz gravity however, it is necessary to study general relativity in

(2+1)-dimensions, where it shall be shown that traditional relativity admits no propagating

modes. In order to formulate (2 + 1)-dimensional relativity it is necessary to understand

the ADM decomposition of the metric. This and the former topic are the main subjects

of this chapter, along with several other significant results from traditional relativity in

(2 + 1)-dimensions.

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2.1 general relativity in (2 + 1)- Dimensions

By calculating the Euler-Lagrange equations of the Einstein-Hilbert action we may derive

the Einstein Field Equations

R12

gR+ g= 8GT (2.1)

on the (2 + 1)manifoldM, where may or may not be taken as zero without loss ofgenerality. As in 3 + 1 relativity the field equations remain completely covariant, as can

be verified by showing they are invariant under the action of Diff(M) which is therefore

identified as a gauge group onM.Furthermore we may show that for the special case of (2 + 1)- spacetime the curvature

tensor forms a linear relation with the Ricci tensor:

R =gR + gR gR gR 12

(gg gg)R. (2.2)

It can be shown that due to this decomposition, solutions to the field equations take

two forms: = 0 when = 0 and = constwhen

= 0.

ForN-dimensional classical gravity it can be shown the phase space is characterized by

the spacial metric for a constant-time hypersurface and the conjugate momentum, which

have N(N 1)/2 degrees of freedom. The total dimensionality is reduced however by thefield equations themselves which in general form Nconstraints on the initial conditions,

while a proper choice of gauge can reduce the dimensionality by another N. This gives the

entire dimensionality of the phase space as

D= N(N 1) 2N=N(N 3) (2.3)

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per spacetime point. For N= 3 the phase space has dimension zero at each spacetime point,

implying the theory has no propagating field degrees of freedom in (2 + 1)- dimensions.

Hence the geometry is completely determined by the constraints modulo some finite number

of global degrees of freedom.

2.1.1 The Newtonian Limit

Since the theory has no propagating degrees of freedom, the Newtonian limit is expected

to be novel. We shall conclude that the Newtonian limit of a classical (2 + 1)- theory has

no force between static point masses.

We make the ansatz that the metric can be separated into a homogeneous Minkowskicomponent plus a perturbation h, making the total metric g = +h. It can be

shown there is some gauge for which the field equations take the form

12

h 1

2

h

8GT

h 1

2

h

= 0.

(2.4)

The Newtonian limit is given by letting T00 , with as the mass density, and weignore all non-00 components of the stress-energy tensor. Then by the first equation of2.4,

h00+

1

2h

= 2

h00+

1

2h

= 16G. (2.5)

We also neglect time derivatives, which shall be suppressed by some power ofv/c so

that 2 2. Furthermore, we recall the Newtonian potential (x) is defined such thatthe Poisson equation

2(x) = 4G(x) (2.6)

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is satisfied. By combining2.5and 2.6we have

2 h00+1

2

h = 2(

4), (2.7)

and thus we may conclude

h00+1

2h= 4. (2.8)

When the assumptions of the Newtonian limit are applied to the Christoffel symbol we

have

i00=1

2(1 + hii)(0h0i+ 0h00 ih00) 1

2ih00, (2.9)

since the time derivatives are ignored and hii


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ForN= 2 the theory is degenerate and for N= 3 the Newtonian potential does not affect

test particles, and therefore renders no force, as was previously claimed.

2.2 The [0, 1] Topology

Any (2 + 1)manifoldM which is relevant to the current study possesses two generalcharacteristics. These are

M must admit time-orientable Lorentzian metrics, viz. a metric for which the metrictensor has signature (++) and there are two spacelike boundaries and + which

may be called past and future respectively.

M must also admit solutions to the Einstein Field Equations in vacuum, including = 0.

It is easily shown that a noncompactMadmits a time-orientable Lorentzian metric ifand only if it also admits a nonvanishing vector field. This makes it possible to use the

Poincare-Hopf index theorem.

Suppose V is a vector field onMand x is a point such that V(x) = 0. Now construct{xi}, an unbounded sequence onM, and let # be a countable set whose elements aresets ofMdiffeomorphisms,{j}i ij #i #, mapping V(xi) V(xi+1). Then

i

([#]iV(xi)) V(x= ), (2.13)

since{xi} is unbounded, where we have defined

[#]i :=

ij#i

ij . (2.14)

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The implicit limit in 2.13will remove the zero ofV in any arbitrary neighborhood ofMwhere we wish to perform computations. Since this may be repeated for any arbitrary zero

ofV,M

must admit a Lorentzian manifold.

IfM is closed then by the Poincare-Hopf index theorem there is a nonvanishing vectorfield if and only if(M) = 0. But since we assumeM to be odd-dimensional the Eulernumber necessarily vanishes, thusMmust admit a time-orientable Lorentzian metric.

Now consider a manifold with boundaries and +. Then we may use a result due

to [9] which states thatMmay admit a time-orientable Lorentzian metric if and only if

(

) =(

+

). (2.15)

We may find which of these are also candidates to admit vacuum solutions of the Field

Equations. Since all closed Lorentzian manifolds contain closed timelike curves, they are

severely limited in their utility from a physical standpoint. Furthermore, noncompact

manifolds are not well understood mathematically.

We make the most use of a theorem by Mess [11] as in[12]. For a compact manifold

with a flat, time-orientable Lorentzian metric and a surely spacelike boundary, the manifold

must necessarily have the topology

TM= [0, 1] , (2.16)

Where is some closed two-surface which is homeomorphic to either or +. This

combined with the previous result implies no solution to the field equations can simultane-

ously admit a topology change. Hence for any manifold under consideration the topology

is fixed by the surface

.

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2.3 The ADM Formalism and Hamiltionian Formulation

In traditional treatments of 3 + 1 relativity, spatial and temporal dimensions are unified

into a single manifold structure in which the spatial and temporal characteristics of a

physical system are formally coupled. Despite this, it is often a matter of practical use

to generate a foliation of the spacetime manifold for which the spatial and temporal di-

rections are mathematically distinct. This may be accomplished by making use of the

Arnowitt-Deser-Miser (ADM) formalism. The ADM formalism is valid in both (3 + 1)

and (2 + 1)- relativity, and is a cornerstone of the theory of CDT gravity and numerical

relativity in general. Classically it leads to the introduction of a natural Hamiltonian for

classical gravity, suggesting a variety of alternative methods for canonical quantization of

the theory, which we shall largely not be concerned with. We shall however make use of

the phase space constraints generated by the Hamiltonian representation to study in depth

the equivalence and uniqueness of path integral quantization to canonical quantization in

(2 + 1)- dimensions. We present the details of the classical theory here.

2.3.1 The ADM Metric

The [0, 1] topology of the last section may be thought of as a continuum existing betweenan initial spacelike surface = {0} and a final spacelike surface += {1}. Usingthese as endpoints we may generate a foliation of the spacetime into discrete time intervals

dt creating a possibly infinite set of constant-time manifolds t on which there exists a

coordinate chart = {xi} and the metric gij(t, xi) naturally accompanying the chart. Weinsist that any set of foliations necessarily recover the topology of the original spacetime if

we choose to reform the space, that is, up to isomorphic reorderings of the foliations, we

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Figure 2.1: Diagram of the triangulation used to compute ds2.

insist that the t obey

[0, 1] = t(0,1)

t

+. (2.17)

In order to ensure this in practice it is necessary to understand how any given foliation

connects to its neighbors in a metric sense, which we develop presently.

Let some point q(x0, . . . , xi) tbe displaced by some infinitesimal distance dq(x0, . . . , xi)with some nontrivial component normal to a two dimensional embedding of t. This change

induces changes on the metric

dq(x0, . . . , xi)

dt d=N dt

xi(t) xi(t + dt) =xi(t) Nidt(2.18)

where we have defined a contravariant vector N(t, xi) = (N(t, xi), Ni(t, xi)) where N0 is

known as the lapse function and Ni is known as the shift vector, preserving informally the

distinction between time and space coordinates.

We may now compute a Lorentzian invariant interval, using a triangulation which

preserves the directionality of the Lie derivative,LXT , for an arbitrary vector field on

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the shift manifold t+dt, as shown in figure 2.3.1. The result of this calculation gives the

metric

ds2 =

N2dt2 + gij

(dxi + Nidt)(dxj + Njdt). (2.19)

Here it is customary to take the raising and lowering convention with respect to the spatial

components alone, so that gijgij =ji , but not necessarily the same for the Greek indices.

2.3.2 The Hamiltonian and Canonical Constraints

We wish to form a Hamiltonian for the ADM decomposition of the metric. This will allow

us to put the ADM version of the path integral on an more rigorous footing. It can be

shown that under the ADM decomposition, the Einstein Hilbert action takes the form

SEH

dt

d2xN

g(R 2 + KijKij K2) (2.20)

where the boundary terms have been dropped and Kij is the spatial component of the

extrinsic curvature. Here this takes the form

Kij

= 1

2N(tgij i

Nj j

Ni) (2.21)

for a surface with a normal vector n = (N, 0, 0) and the full extrinsic curvature K =

n+nnn. We may now compute the generalized canonical momenta in thetypical way,

ij = L

(tgij)=

g(Kij gij) (2.22)

and hence write the Einstein-Hilbert action as

SEH=

dt

d2x(ijtgij NH NiHi), (2.23)

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with constraints

H =

1/

g(ij

ij 2) g(R 2)

2

jij

. (2.24)

The first element, denotedH, is the Hamiltonian constraint while the remainingHi formthe momentum constraints. In a closed universe, viz. a universe for which all boundary

terms in the Einstein-Hilbert action are seen to vanish, these lead to trivial equations of

motion, implying as we would expect that the total energy of a closed universe is zero.

2.4 (2 + 1)- Path Integrals in the ADM Formalism

The ADM decomposition of the metric leads to a form of the path integral for which

calculations are easier. Since the path integral formalism is not guaranteed to produce a

unique quantization, it is important to check that path integral quantization via an ADM

decomposition leads to the same result as canonical quantization, which is shown in detail

below.

For notational uniformity with[12], define the conventionD[. . . ] =. . . . In this casethe latter symbol will stand as a path integral measure and not an expectation value. In

general we may represent the path integral in a coordinate basis as

qf, tf|qi, ti =dqdp exp{iSEH[q, p]}, (2.25)

where the Einstein-Hilbert action takes the ADM form

SEH=

d3x

gR=

dt

d2x(ij gij NiHi NH). (2.26)

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Then we may write the full ADM path integral as

Z= dij

dgij

dNi

dN

expi dt

d2x(ij gij

Ni

Hi

N

H)[, g] . (2.27)

The shift coefficients appear as Lagrange multipliers in the action, and thus integrated

become delta functionals imposing the Hamiltonian constraints on the phase space. Thus

the path integral is now defined over a measure on the reduced phase space, which is

a simple finite-dimensional space of physical degrees of freedom. Evaluation of the path

integral over such a reduced phase space requires the introduction of a specialized Jacobian,

called the Faddeev-Popov determinant, which shall be treated in the next subsection.

2.4.1 Equivalence to 2+1 Canonical Quantization

The ADM form of the Einstein-Hilbert action has a gauge symmetry, which may be iden-

tified as the diffeomorphism group on the measure space generated by the Hamiltonian

constraints,H H, Hi. If we suppose there is a nondegenerate set of homogeneousgauge equations , the path integral becomes

Z=dijdgijdNidN[]det({H, })

exp

i

dt

d2x(ij gij NiHi NH)[, g]

.

(2.28)

We may simplify the calculation via a procedure outlined in [12]. Choose a family of

constant curvature metrics gij on the moduli space, and thus parametrized by moduli m,

so thatgij(m) =e2fgij(m). Now set

gij = 2()gij+ ()ij+ mT

()ij

ij =1

2gij+

g(P Y)ij +

g pij(),

(2.29)

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where the operator Pmay be defined via the test function

(P )ij =

ij+

ji

gij

kk. (2.30)

T is a multilinear operator, which maps to the orthogonal projection of a given modular

deformation of the spatial metric onto ker(P), and may be written

T = (), ()(), ()1 (2.31)

for which ()ij gij/m, and the () are chosen to form a basis of the kernel ofP.

The inner product in2.31shall be defined shortly.

Supposing = 0, the Hamiltonian constraints become

Hi= g(PP Y)i i

H = 12

ge22 + 2

g

k

2

+

ggikgjle2

(P Y)ij+ p()ij

(P Y)kl+ p

()kl

(2.32)

which is an orthogonal decomposition of the integration variables, implying that all sec-

ondary Jacobians will be equal to unity.

We now wish to define some map, J : gij, ij m,,,p, , Y i, from the nativemeasure space into the reduced phase space of the moduli. For such a map we may define

an inner product from the tangent space to the space of all -metrics,

q, q

:=

d

2x

ggijgklgikgil q= g

d2x

ggijij q=

d

2x

g()2 q= ,

(2.33)

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and consider the integral

d(gij)eig,g

=

d(m)d()d()Jgei,PP e8i,eimmTT (),()

1,

(2.34)

which when evaluated leads to the result

Jg = det |PP|1/2 det |T| det |(), ()|1/2. (2.35)

We may evaluate a similar integral ford(ij) to give the result

J = det |PP|1/2 det |(), ()|1/2. (2.36)

The total Faddeev-Popov determinant is obtained by a convolution of the component parts

J=Jg J = det |PP|1/2 det |(), ()|1/2. (2.37)

Though the map was originally defined from the tangent space to phase space, we may

apply this result to calculate the change of coordinates in the general path integral, which

allows a change of coordinates into the ADM shift coefficients N, and integrate over these.

This allows us to represent the path integral as

Z=

dnpdnm det |(), ()|

d(/

g)ddYd det |PP|[][H/g]det |{H, }|

exp

i

dt

d2x(ij gij NiHi NH)

.

(2.38)

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The spatial component of Y becomes unity when integrated against the delta function

[Hi/g] and the determinant ofPP. The same is true for the integral if we mayseparate the Hamiltonian containing Poisson bracket terms into their spatial and temporal

components, making such a decomposition desirable.

Suppose the gauge condition 0 = /

gT = 0, the so-called York time gauge,is chosen. It may be shown that the determinant det |{H, }| factors evenly into thetemporal and spatial components det |{H, }|= det |{Hi, i}| det |{H0, 0}|. Moreover,by considering the simplicial structure,

=

d2xij

gij

=

d2x(pij gij+ 2

2gij[e2g + k2

Yi+

1

2i(e2)] i

(2.39)

discussed thoroughly in [13], the Poisson brackets in the determinant may be evaluated.

From this it be shown that

det

|{H0,

0

}|= det e2

+T2

2

e2 + e2pp gij gkl()ik()jl . (2.40)

The path integral is now simplified to

Z=

dnpdnm det |(), ()|

[H/g]det |{H0, 0}|

exp

i

dt

d2x(ij gij NiHi NH)

.

(2.41)

The integral is nontrivial but only leaves a term cancelling{H0, 0}, making the

integral

Z=

dnpdnm det |(), ()| exp{iSEH}, (2.42)

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where we now evaluate the Einstein-Hilbert action at the solution of the constraints, SEH.

By using the variable substitution

p =

d2xe2

ij 12

gij

gijm

= (), ()p , (2.43)

we may make such a coordinate substitution p p (), ()p and reduce the pathintegral into the final form

Z=

dnpdnm exp{iSEH[p, m]}, (2.44)

which is an ordinary quantum mechanical path integral of the form 2.25. This form of

standard quantum mechanical path integral can be proven [1] to be equivalent to canonical

quantization of the theory and therefore unique.

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Chapter 3

Nonperturbative Quantum

Gravity with (2 + 1)-CDT

Although we have described classical gravity at length in (2 + 1)- dimensions, we have

yet to describe in detail a quantization. This chapter takes up a general discussion of

CDT quantization in an arbitary number of dimensions. Due to the nature of the CDT

formalism, it is simple to go from a model in an arbitrary number of dimensions to one in

(2 + 1)- dimensions.

3.1 The Gravitational Path integral via Triangulation

CDT quantum gravity is typically introduced in terms of the gravitational path integral,

or sum over histories. The general idea is to use the regularized version of the path integral

on a differentiable manifoldM,

Z(GN, ) =gG(M)

D[g]exp{iSEH}, (3.1)

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whereGNis the Newtonian constant and we have defined the Einstein-Hilbert action,

SEH= 1

GN d3det g(R

2). (3.2)

Here is the cosmological constant. It should also be noted that a similar expression may

be written in 1 + 1 and 3 + 1 dimensions at this point without any loss of generality.

For3.1to be well-defined it is necessary to specify the measure over which a finite value

of the integral may be obtained. An obvious choice is the measure of causal Lorentzian

geometries onM, Lor(M), but this space alone is not well-defined as two Lorentzianmanifolds may be diffeomorphic, and should not add separately to the measure. Hence

a measure of Lorentzian manifolds modulo diffeomorphisms may be used as the measure

space, implying

G(M) = Lor(M)Diff(M) . (3.3)

Since the path integral is regularized, so too is the measure spaceG Ga,N whereGa,N isthe regularized version of the measure space ofN-dimensional simplicial manifolds anda

is the atomic edge length. From quantum field theory, we know that this is expected to

be highly singular with the set of smooth, classical configurations corresponding to a set

of measure zero. This is easily shown for the space identified in 3.3.

The atomic edge length a is not seen as fundamental, and in CDT results are only

considered in the case ofa 0, and hence N , which is identical to a continuumsolution. The Regge calculus (discussed in section 4.1) provides for the regularization of

this path integral with respect to the simplicial lattice as

ZCDTa,N

= TGa,N

1

CTexp

{iS

Regge[T]

}, (3.4)

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whereTis a given simplex in the regularized measure andSReggeis the Reggeized Einstein-

Hilbert action. We then will have ZCDT = lim(N,a)(,0) ZCDTa,N , where

ZCDT = lim(N,a)(,0)

TGa,N

1

CTexp{iSRegge[T]}. (3.5)

HereCTis the simplicial analogue of the order of the automorphism group of the Lorentzian

metric g .

3.1.1 Continuum Limit

By taking the discretization to the continuum case, we may postulate the existence of a well-

defined continuum limit. If this exists, it will not depend on details of the regularization

by construction. That is to say, the gluing rules and geometry of the simplices will not

ultimately affect the classical limit. This implies a robustness of Planck-scale physics, as

is discussed in [5]. This robustness is similar to the universality exhibited by statistical

mechanics, whereby the behavior of a system may be independent of the details of the

system, [5], [33].

3.1.2 The CDT Lattice

It was stated before that the simplicial building blocks are 3-simplices for the (2 + 1)-

theory, while the (3 + 1) theory uses four-simplices. By making assumptions about the

temporal component of the spacetime, in particular by introducing periodic boundary

conditions (see[8]), regularizing the spacetime reduces down to the problem of triangulating

a given Lorentzian manifold. This gives a number of so called gluing rules which must be

followed. In general, gluing rules which are too liberal would cause the number of allowed

gluings to grow too fast as a function of the volume of the triangulations to allow the

path integral to be well-defined, while gluing rules which are too restrictive will destroy

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Figure 3.1: Disallowed Histories in Causal Dynamical Triangulations, such as wormhole

histories and branching points where the light cones are degenerate.

local curvature degrees of freedom. Hence we wish|Ga,N| ecN simultaneously withthe existence of curvature degrees of freedom. A conclusion gained from investigating

such models, with gluing rules within this window, is that they do not correspond to a

good classical limit. In these cases the large scale dimensionality of the spacetime becomes

dynamical despite the constant dimensionality of the simplices, a problem discovered during

research into Euclidean Dynamical Triangulation, where there were found Hausdorff phases

ofd= 2 and d= .The only available solution to this comes through CDT. By restricting the histories

allowed in the regularized path integral to those which are causal, forbidding the presence

of branching points or any structure with a classically singular light cone, shown in (b)

of 3.1.2. This is in contrast to purely Euclidean path integral approaches, which are

not necessarily causal. The causality constraint does not generically suppress large-scale

curvature fluctuations, and thus will not suppress curvature degrees of freedom, as desired.

Causality is implemented in the gluing rules by creating a layered geometric structure

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labelled by an integer-valued proper timet. A Wick rotation t it([4]), maps this into asubset of Euclidean piecewise flat geometries. Since Wick rotations map into a strict subset

of these geometries, the sum over geometries is changed, leading to a different continuum

limit than the purely Euclidean theory.

3.2 Horava-Lifshitz Gravity

Horava-Lifshitz gravity is a gravity model which includes with a dynamic spacetime metric

a preferred foliation of the spacetime manifold,M, by spacelike hypersurfaces. The natu-ral mathematical description of this foliation is given by the group of diffeomorphisms of

M preserving the foliation. This leads to an anisotropy between space and time dimen-sions measured with respect to an anisotropic scaling parameter z, such that for a locally

Minkowski metric, t

x

t

x

bzt

bx

, (3.6)

where b > 0 and the ultraviolet divergences of the theory are suppressed for z > 1. In a

general spacetime, tx

t

x

f(t)

(x, t)

, (3.7)

for an arbitrary f and .

The preferred foliation of Horava-Lifzhitz gravity makes it a natural setting to use the

ADM formalism from classical relativity as discussed in chapter 2, with the shift vector

and the lapse function playing the role of gauge fields associated with the diffeomorphism

group. Although not true in general, reparametrizations of the space coordinates may be

independent of the time coordinates, implying that the relative gauge field, N(t, X) might

be only a function of the time coordinate, viz. N(t, X) N(t). This leads to a version of

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the (2 + 1)- theory which is projectable.

By writing the action for such a theory in d spatial dimensions, we can include higher

curvature terms by constructing such an action as the sum of a kinetic term quadratic in

temporal derivatives and a potential term of mass dimension 2d in spatial derivatives. In

this case z= d. [25] give a general kinetic term to be

1

16G

M

dtddx

(t, x)N(t)[Kij(t, x)Kij(t, x) K2(t, x)], (3.8)

where is a parameter of the relevant DeWitt supermetric as described in [26], [27], and

we have defined

Kij(t, x) = 1

2N(t)[tij(t, x) iNj(t, x) jNi(t, x)]. (3.9)

The potential term is given by

1

16G

M

dtddx

(t, x)N(t)V[(t, x)]. (3.10)

HereV[(t, x)] is a scalar functional of the spatial metric tensor and spatial derivatives up

to order 2d. This leads to a total action,

SHL = 1

16G

M

dtddx{

(t, x)N(t)[Kij(t, x)Kij(t, x) K2(t, x) + V[(t, x)]]}, (3.11)

and G GN in the low energy limit.We find that the lapse function, being only a function of time, introduces a Hamiltonian

constraint,H = 0, into the theory with

H =

d2X

(t, x)[Kij(t, x)Kij(t, x) K2(t, x) + R22(t, x) R2(t, x)+2], (3.12)

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where R2 is a total derivative described by the Gauss-Bonnet theorem in the integral of

over .

The final constraint we may make upon the Horava-Lifshitz theory to compare it with

CDT gravity is a choice between noncompact and compact spatial topologies, which dictate

the allowed values of the local Hamiltonian constraint. Noncompact spatial topologies are

typical in many theories of quantum gravity, but in Horava-Lifshitz gravity it is shown by

[25] that theH = 0 constraint impliesH may be thought of as a globally conservedcharge, which is far more natural in CDT than more typical implications for noncompact

spatial topologies.

3.3 Propagating Scalar Modes and Anisotropic Quantiza-

tion

In order to determine the number of propagating scalar modes of the theory, it is necessary

to considerV: the part of the Lagrangian which does not contain any temporal derivatives.

In (2 + 1)- dimensions the following are true: (1) Rabcd = 1/2(gacgdb gadgcb)R, (2) anypotential terms will differ only by total divergences, and (3) ai (as defined below) is the

gradient of a scalar. This allows us to write V as

V =R+ aiai+ g1R2 + g22R+ g3(aiai)2 + g4Raiai+ g5a2( a)

+ g6( a)2 + g7(iaj)(aiaj).(3.13)

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Thegi coupling constants have dimensions ofM2, since the mass scale has been absorbed

into the coupling constants. This gives a general (2 + 1)-dimensional action

S=M2pl

2

d2xdtN

g[KijKij K2 + Raiaig1R2 + g22R

+ g3(aiai)

2 + g4Raiai+ g5a

2( a) + g6( a)2 + g7(iaj)(aiaj)].(3.14)

Field equations can be computed by varying the lapse, shift, and induced metric. This is

shown in[28]. The equations of motion can be stated as follows:

The equation of motion due to variation with respect to the lapse is given by

KijKij K2 + R [2 a + aiai] g1R2 g22R+ g3[4i(ajajai) + 3(aiai)2]

+ g4[2i(Rai) + Raiai] + g5{aiai a + 2 (a[ a]) [2(Naiai)]/N}

g6[3( a)2 + 22( a) + 2a2( a) + 4(a )( a)]

g7[(iaj)(iaj) 2(iaj)aiaj 4i(aj(iaj)) 2i2ai] = 0.

(3.15)

The equation of motion due to variation with respect to the shift is given by

iij i{Kij Kg ij} = 0. (3.16)

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and the equation of motion due to the spatial metric is given by

1

g

t(

gij+ 2N(KliKlij KKij)

1

2N(KijKij K2)gij Nij(mNm)

LN(Nij) + 2NN(ij)mam 1

N[ij gij2]N+

aiaj1

2alalgij

+

1

2g1R

2gij

2g1 1N

[ij gij2](NR) g2 1N

[ij gij2](2N) g2a(ij)R+ g2m(NRam)

2N gij

+ 2g3a2aiaj1

2g3(a

2)2gij+ g4Raiaj g4 1N

[ij gij2](Naiai) 12

g4Ra2gij

+ g5aiaj[( a) a2] 12

g5{(ia2)aj+ (ja2)ai} +12

g5{(a2)2 + amma2}gij 2g6( a)aiaj

g6{(i[ a])aj+ (j [ a]ai} + g6

a2( a) + a ( a) +12

( a)2

gij

12 g7(man)(man)gij 2g7(a a(i)aj) 2g7(2a(i)aj)+ g7a2(iaj) + g7 ([iaj]a) = 0,(3.17)

where LN is the Lie Derivative along the vectorNi. The only one of these equations whichis dynamical is3.17, with the dynamical variablegij . Applying the uniformization theorem

as in[29], and since the theory is invariant under the transformation xi xi(t, xi) we mayset

gij = 2gcij, (3.18)

where gcij is the metric of a constant curvature, spherical, Euclidean or hyperbolic 2-

dimensional space. This makes 3.17 into a dynamical equation for the conformal factor

.

This is a crucial difference between quantizing gravity isotropically or anisotropically.

For isotropic, or a traditional gravity model, we have shown in chapter 2 that there are

no propagating modes of freedom. We have just seen however that quantizing a Horava-

Lifshitz model via a CDT like simulation, it is plausible the quantization might yield at

least one propagating mode of freedom. Hence by simulation we can determine whether

CDT quantization is a numerical quantization of classical or Horava-Lifshitz gravity. This

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requires generating a seed manifold, which is taken up on chapter 5. The next chapter

describes the general computational procedures in CDT quantum gravity and the one after

that the procedures taken to generate the seed manifold.

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Chapter 4

Computational Approach

4.1 The Regge Calculus

The lattice formulation of general relativity was taken up nearly a half century ago, primar-

ily by Regge [14]. The central idea of the Regge procedure is to tessellate the underlying

spacetime with simplices, and in (2 + 1)- dimensions the relevant simplex is the 2-simplex

or triangle for the spatial slice, where the only extrinsic curvature is at the edges. The in-

trinsic curvature is centred entirely at the vertices. These are canonical points idescribed

by the deficit angles i. It is shown in texts with devoted chapters on the Regge calculus,

such as[12] that such an intrinsic angle contributes 2i per vertex to the action. Thus we

may make the finite scale-independent approximation

Sspatial slice=

d2x

gR 2

i(M)

i SRegge, (4.1)

where (M) is the set of all angles with where the curvature is concentrated in the sim-plicial manifold, counting multiplicities.

Ths may be written in terms of the (N 2)-dimensional hinge of the simplex, making

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the Ndimensional Reggeized action now

SRegge

= 2 ViV(M)

iVi, (4.2)

whereV(M) is the set of all simplicial hinges on the manifold. The hinge of a 2-simplexmay be identified as the side of the triangle, making the (2 + 1)- Regge action

S2+1Regge = 2

eV(M)

ee, (4.3)

where e is the length of the eth edge. It is to be noted that alternatively defining units

such that 8G 1 leads to a normalization of the Regge action where the factor of two isabsent.

The Reggeized action may be used to regularize the path integral quantization proce-

dure described in chapter 2. Suppose M has triangulated boundaries 0and 1, such thatM = 0 1. Then a quantum state on 0 or 1 may be defined as some function ofelements ofV(0) orV(1). This implies we may construct a Reggeized representation ofthe transition amplitude between a state on 0 to one on 1 or vice versa, which may be

expressed as path integral

Z(V(0), V(1)) =eV(M)/V(M)

de

V(M)

exp{2ee} . (4.4)

One great benefit of using Causal Dynamical Triangulations is that the regularized path

integral is coordinate independent, implying that the rigorous mathematical issues with

the measure space assumed in4.4are curtailed. The following subsection contains a short

discussion on the mathematical issues that arise when CDT is not used to regularize the

path integral and how to form the transition amplitude in such circumstances. Although

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not of direct significance to this work, it is important to realize the advantages offered by

the CDT approach and hence the author has decided to included it here for completeness

and to satiate the curious reader. It may however be skipped without affecting overall

understanding of this works content.

4.1.1 The non-regularized measure space and associated transition am-

plitude

Note that the invariant measure over the Simplicial manifold is not mathematically clear.

For instance, while such a measure over a Euclidean lattice (QCD) could be deduced from

reducing a 3 + 1 representation of the path integral, the non-uniformity of the simplicialpaths make the choices of appropriate gauge ambiguous and the proper form of the Faddeev-

Popov determinant is not understood.

Suppose we associate with the angular momentum in the full (2 + 1)- dimensions the

geometry of a 3-simplex. Then we may describe the coloring of this simplex via the Wiger-

Racah 6j-symbol[15],

J6

j1 j2 j3

j4

j5

j6

. (4.5)

It can be shown[16] the approximation

exp

i

6i=1

ji

J6 1

6V

exp

i

SRegge+

4

+ exp

i

SRegge+

4

(4.6)

holds for large j, where SRegge is the Regge action for a 3-simplex with side length i =

1/2(ji+ 1/2), given by

S3+1Regge=6

i=1

i ji+12 , (4.7)

with i defined as the angle between the outward normals of any two faces meeting at the

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ith vertex and V is the volume of the 3-simplex with side lengths i. Since4.6holds, we

may conclude that the Reggeized action per tetrahedra can be written as a function Wiger-

Racah 6j-symbols. Furthermore, this implies that the integrand of4.4may be expressed

as some product of Wiger-Racah 6j-symbols. Hence the entire Reggeized action may be

seen as a sum over such 6j-products.

IfM has boundary M with a triangulation (M), then some triangulation (M/M)may be chosen such that (M) := (M/M) (M) forms a simplicial manifold. Ifthe interior and exterior edges of some 3-simplex Tare labelled with discreet coordinates

xi and ji respectively andji(T) is the spin-coloring ofT, then[16][17] claim the transition

amplitude may be expressed according to

Z(M)[{ji}] = limL

xeL

(

iL(M)

(1)2ji2ji+ 1 ViV(M/M)

(L)1

lL(M/M)

(2xl+ 1)

TT(M)

(1)6

i=1ji(T)J6(T)),

(4.8)

whereLis the set of edges on some manifold,Vthe set of vertices on some manifold,T isthe set of 3-simplices on some manifold, and

(L) :=jL

(2j+ 1)2. (4.9)

It can be shown that4.8 is invariant under a subdivision or rearrangement of the 3-

simplices inT, and moreover it can be proven that the Reggeized quantization procedurein (2 + 1)- dimensions forms a topological field theory. This is further suggested by the

Newtonian limit of the theory described in chapter 2.

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4.2 Monte-Carlo Method

Monte-Carlo simulation may be reduced to the problem of approximately evaluating an

integral

I=

V

f(x)dnx, (4.10)

where V is a vector space of dimension n and f : V is a function on V. Whiletraditional quadrature methods use deterministic schemes to discretize the subspace form-

ing the domain off based on either geometry or interpolating polynomials, the basis of

Monte-Carlo simulation is to use a randomly sampled set of vectors{xi}Mi=1 to esti-mate the function atMpoints within the domain. The integral may then be approximated

according to the formula

I 1M

Mi=1

f(xi). (4.11)

In general a Monte-Carlo estimate is improved for a larger number of trials, however this

may not always be the case. From elementary statistics it follows that the error on a

Monte-Carlo estimate is given by

E(I) =f2 f2M 1 , (4.12)

where it can be simply shown that

fk 1M

Mi=1

f(xi)k. (4.13)

4.2.1 Markov Chain Stepping

It is typically desirable to choose a sampling set ergodically but not necessarily completely

at random. The concept of Markov Chain stepping is that the outcome of a step can affect

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its successor. Let S ={si} be a set of distinct states of a system. Then, if the stepsare assumed to be ergodic, there is in principle a probability pij of the system in state

si

transitioning into a state sj

. From such transition probabilities we may construct a

transition matrix P whose elements are pij.

In itself this operator is not clearly useful. However it may be shown that determining

the probability a state si has transitioned into a state sj after n iterations is just the ijth

entry of the matrix Pn. This allows a certain degree of determinism for a chain of steps

such as those which might constitute a Monte-Carlo sampling.

We may also define a stepping length, p, as the radius of the neighborhood from which

the sampling vector is taken. For a smaller sampling length, the approximation is improvedat the cost of a larger convergence time for the simulation.

Although Markov Chain sampling has the advantage of a lower degree of randomisation,

it also introduces an unwanted autocorrelation, which causes many moves to be discarded.

Markov Chain stepping is thus significantly more desirable for the generation of complex

states, which are far more difficult to generate completely at random, but not necessarily

always the most appropriate option.

4.2.2 Sampling an Observable

It is a result from statistical physics that the expectation value of any observable, O, is

given by

O = 1Z(H, )

s

O(s)exp{H(s)/}, (4.14)

where H is the Hamiltonian of the system, kT and Z(H, )sexp{H(s)/} isthe equilibrium partition function for the system. We may write this using the Boltzmann

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distribution,P as

O

= sO(s)exp{H(s)/}

sexp{H(s)/} = s[O(s)exp{H(s)/}/P(s)]P(s)

s[exp{H(s)/}/P(s)]P(s) . (4.15)

It is shown in [18]that the far right side of equation4.15may be written in a condensed

form as

O 1M

i

O(si), (4.16)

where the si are points sampled according to the Boltzmann distribution.

4.3 The Metropolis Algorithm

The Metropolis algorithm was developed in 1953 at Los Alamos National Laboratory. The

fundamental idea is to compute the transition amplitudeT(s s) from state s to s.The probability of a transition is compared to a randomly generated uniformly distributed

number between 0 and 1, and if the probability exceeds this number, it is accepted and if

it does not, it is rejected.

First construct a Markov Chain, s1

s2

s3

. . . , with probability P(si

si+ 1)

for any given state to transition into its successor. Coupling this with the transition

probability we can compute the total probability of the (k+ 1)th step from the kth as

Pk+1(s) = Pk(s) +s

{T(s s)Pk(s) T(s s)Pk(s)}. (4.17)

As k the Markov Chain approaches an equilibrium value andPk(s) Peq(s). Thisimplies

T(s

s)Peq(s

) = T(s s

)Peq(s), (4.18)

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which may be rewritten in a more suggestive form as

T(s s)T(s

s)=eH(s,s

)/, (4.19)

where we have defined H(s, s) =H(s) H(s). Now it is necessary to formulate somemodel forT(s s). Although such a model is not unique, we will use the Metropolismodel, described by[18] to be

T(s s) =

for H 0

eH(s,s)/ for H

0

(4.20)

where 1 is the Monte-Carlo time. This allows us to state formally the Metropolis Algo-

rithm, which is shown schematically below.

Algorithm (Metropolis):

1. Generate a starting state,s0.

2. O= 0.

3. while iterations < total iterations:

(a) generate trial state,s,

(b) computeP(s s, ),

(c) compute randomx [0, 1]:

(d) ifP> x: accept trial state, else: continue.

(e) O+ = O(s),

4. returnO/steps.

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As before, in the limit as steps , the observable approaches its exact value. We mayextend this basic algorithm to take into account the equilibration of the simulation, which

is discussed below. Other effects not extremely pertinent to this study are found in the

vast literature on Monte-Carlo methods.

4.3.1 Equilibration

It is not necessarily true that a given sampling of the system will be in equilibrium. Since

initial states may not necessarily be in equilibrium, achieving such a configuration takes

a given time, known as the equilibrium time, eq which is a function of the system time

N = Ld

. The equilibrium time also varies inversely with the temperature of the system.It typically takes a given number of Monte-Carlo sweeps, where we define 1 MCS = N

spin updates. In any practical Monte-Carlo simulation observables are monitored as a

function of MCS to determine that the system is in an equilibrium state. Although some

observables have a small equilibrium time, others do not, and each variable it typically

sampled to determine whether the entire simulation has equilibrated.

4.3.2 Monte-Carlo in CDT: The Pachner Moves

The connection of the Monte-Carlo method to the actual simulation is crucial, but may

not be immediately clear at this point. Synoptically, we may say that the evaluation of the

regularized Einstein-Hilbert action over a suitable simplicial manifold is perfomed via the

means of a Monte-Carlo simulation with a particular set of moves, known in the literature

as the Pachner moves, and acting the same way as including something like the chain

stepping previously considered might act on the Monte-Carlo. The Pachner moves can be

found in nearly any review article on CDT gravity and many specific articles containing

new work (such as[10]). In (2 + 1)-dimensions the Pachner moves correspond to drawing

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and replacing various diagonal elements on the unit 3-simplices constituting the simplicial

spacetime. The final requirement of such a set of moves is that they be ergodic, meaning

that they are random but that the average state of each individual slice will also be that of

the entire ensemble. Although the Pachner moves have not yet been proven to be ergodic,

it is assumed in nearly all CDT literature that this is the case.

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Chapter 5

Simulation

In order to run simulations it is necessary first to evaluate a seed manifold. For the problem

at hand, considering the propagation of gravitons across the space time and from that the

number of propagating modes, the best seed manifold to use is one which resembles (an

American) football shape, since this will most closely resemble the initial state of the

process we wish to simulate.

5.1 Generation of the Seed Manifold

Properly evaluating the regularized path integral entails the generation of a seed manifold to

be used in the Monte-Carlo simulation. The typical starting place for a (2+ 1)-dimensional

CDT simulation is a 2-sphere, tessellated by equilateral triangles (simplices) of order 6. It

is a mathematical result given by [19] that there exists no such perfect triangulation of a

sphere a priori, so it is commonly considered equivalent to use a sphere triangulated by

simplices mostly of order 6 and a much smaller number of orders 5 or even 7. It has been

a major problem in previous work in CDT quantum gravity to successfully and efficiently

generate a spherical triangulation with the desired characteristics. During the course of

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this work we successfully resolved this issue by the use of a graphics technique known as

adaptive remeshing, discussed in detail presently.

5.1.1 Adaptive Remeshing

Adaptive remeshing is a common technique in computer graphics, and has been in common

use for decades in that field. [20] provides an excellent introduction to the general tech-

niques, as well as several advanced results which we shall not be concerned with here. We

shall follow this exposition, though it should be known that[21] [22] [23][24] also provide

references on the technique of remeshing and adaptive remeshing.

A complex triangulation may be considered as the result of an iterative subdivision al-gorithm starting from a less complex triangulation. In this case it is said that the complex

triangulation, which for this chapter shall be called a meshfor consistency with the com-

puter graphics terminology, exhibits subdivision connectivity with respect to the simpler

mesh. To be perfectly clear about the terminology, we shall call the simplest mesh in this

iterative procedure the base mesh, and the most complex mesh the final mesh.

Given a base mesh, S0, and topological manifold, the general problem of remeshing may

be stated as follows: ifM

is the set of all meshes andS

is the set of all meshes exhibitingsubdivision connectivity, we wish to find some operator U : M S subject to the con-straint that the resulting final mesh, S S, is the closest possible to its preimage U1[S].This map can be approximated by a series of maps terminating in a map U, whose image

corresponds to the final subdivision connectivity mesh of the desired remeshing.

The operator U is difficult to construct computationally. This can be resolved by

first projecting the mesh without subdivision connectivity, M, into some planar domain

R2

. Consider projection operator f : , for which M := f(M) is the meshdefined by vertices pi , viz. the projected verticesPi such that the connectivity

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Figure 5.1: Triangulation by remeshing as presented in[20]

of the set{Pi} is preserved in the projection. It is obvious that an inverse map may beapplied F := f1, which shall be called the inflationof the mesh. Now the problem has

been reduced to that of finding an R2-morphism from the planar mesh without subdivision

connectivity to one with such a connectivity, subject to the closeness condition as before.

Then the projection and inflation operators may be used to construct the equivalent map

U = FUf1. The general idea of the mathematical structure of the adaptive remeshingalgorithm may be summarized by the commutative diagram below.

MU Sf F

MU S

Details of the procedure and several algorithms are described at length in [ 20]. This

group also had generated a triangulation of a sphere following the adaptive meshing pro-

cedure which was used with permission for the purposes of this work. The first few steps

presented in their work is shown in figure 5.1.1.

5.1.2 Computation of Simplicial Geodesics

Once the triangulation was accomplished by the process described above, it was necessary

to compute the geodesics of the triangulation in order to pinpoint the region to be extracted

in the construction of the seed manifold. For a continuous sphere the geodesics are easily

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Figure 5.2: The triangulated manifold with simplicial great circles along the xz and yzplanes

shown to be the great circles along a given equatorial plane. An algorithm was developed

to generalize this concept to the triangulated manifold. First a given number of discrete

points were calculated along the great circle in the continuum case, via the simple geometric

formula for an effective radius along the circle:

Reff,i(gi) = ||gi|| sin

arccos

1

||gi||

, (5.1)

wheregi is the vector from the origin to the specified point on the continuum great circle.

The algorithm then found the vertex Vm S subject to the condition that

gE(Vm, gi) = minVS

{gE(V, gi)}, (5.2)

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wheregEis the canonical euclidean metric in R2. This was repeated for each point in the

continuum great circle. The set of all{Vm,i} was considered to constitute the simplicialgreat circle along the chosen axis. This procedure was carried out for great circles along

the orthogonal axis corresponding to the xz and y z planes in the euclidean coordinates of

the remeshed sphere. These are shown in the figure 5.1.2.

5.1.3 Extraction of the First Quadrant Wedge

The second step of the generation of the seed manifold involved the extraction of a given

wedge, isolated by the boundaries of the great circles. The manifold and great circles gen-

erated in this case were well geometrized enough to make coarse cuts and replace individualpoint. The extraction is shown in the figure 5.1.3.

Figure 5.3: The triangulated manifold with simplicial great circles along the xz and yzplanes, and positive quadrant removed

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Chapter 6

Conclusion

We have outlined a concrete investigation into the possibility of using Causal Dynamical

Triangulations (CDT) to distinguish between Horava-Lifshitz quantization and quantiza-

tion of traditional general relativity via the generation of a football shaped seed manifold

and use of such a manifold in a (2 + 1)-dimensional CDT simulation, looking for evi-

dence of propagating scalar modes. We showed analytically in chapters two and four these

quantization procedures can be distinguished by the fact that while quantizing traditional

gravity in (2 + 1)-dimensions yields no propagating scalar modes, (2 + 1)-quantization ofHorava-Lifshitz gravity predicts at least one, dynamical in the spatial components of the

metric. We saw that this is intimately related to the fact that Horava-Lifshitz gravity

scales anisotropically in the temporal dimension, and hence the difference can be thought

of as the difference between isotropic and anisotropic quantizations of gravity.

After a cursory description of quantization in a historical context, we described in detail

the derivation and scientific content of results from traditional (2 + 1)-dimensional general

relativity. This was followed by an in depth description of the theory of CDT, as well asHorava-Lifshitz gravity and the derivation of propagating scalar modes. We then presented

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a description of the computational process, outlining the general computational theory of

CDT simulations, and finally described the method by which a suitable seed manifold

was generated, where in particular we introduced a heretofore unused computer graphics

techniqueadaptive remeshingas a method of creating a high quality approximation of the

triangulation for any seed manifold, implying that in the future many other seed universes

may be generated easily by employing this technique. The UC Davis CDT code, on which

simulations will take place, is currently in preparation for handling specific topological

aspects of the seed as generated, and are expected to be functional in the near future.

Results, when available, will give insight into whether or not Horava-Lifshitz gravity

has deep underlying similarities to CDT-gravity. This will provide important evidenceto be considered in further studies of Horava-Lifshitz and CDT gravity, and will thus

help advance the understanding of various theories of quantum gravity, arguably the most

important problem in all of science today.

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propagating scalar modes in (2+1)-cdt quantum gravity

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