Review on Causal Dynamical Triangulation as a Candidate of the Quantum Theory of Gravity

Review on Causal Dynamical Triangulation as a candidate of the quantum theory of gravity

C. W. Lim1 and V. Scarani2

Department of Physics, Faculty of Science, National University of Singapore, 10 Kent Ridge Road, Singapore 11756

Abstract

In this undergraduate research opportunities program in science (UROPS), we study one of the

most successful candidates of the theory of quantum gravity, the Causal Dynamical

Triangulation (CDT). The project was motivated by the publication [1] by Ambjørn, Jurkiewicz

and Loll in the success of CDT producing the correct dimension of our universe in 2004. It

marked the first time that a candidate of the quantum theory of gravity had successfully

delivered the correct dimension from first principles. Here, we will not derive CDT formally or

rigorously but rather we look into the methodologies employed by CDT and try to interpret the

rationality behind the main tools.

Introduction

In the current context of our understanding of the universe, we have quantum physics which

explains our microscopic regime and classical theories which describe our macroscopic world,

demarcating the two “extreme range” of observation. Since ancient times, men are often

motivated and driven by curiosity to explore the dimension of nature and extending our

understanding of physical phenomena to even deeper and wilder territories. What we observe

is what we deserve. Thanks to curiosity, we discovered that material entities can behave like

waves and exhibit interferences patterns. We answered the nature with a mathematical tool

called “quantum mechanics” and extended it to “quantum field theory”. If you take a step back

and look at the development patterns of our sciences, you will realise that our fundamental

theories are often driven by the observations we had or phenomena that we cannot explain, for

example electromagnetism. You can also argue that the theory of classical gravity developed by

Albert Einstein wasn’t observed but was created from “though experiments”, however human’s

thoughts and imaginations are simply manifestation of our interactions and experiences with

the “observable” universe. Gravity in our observable or observed dimension of the universe is

well described by the general theory of relativity (GR), it shows that both matter (mass) and

space‐time structures are correlated and are co‐exist with one and other. However in quantum

physics, the arena for the dynamics of the system is usually formulated on a flat, four 1 Student 2 Supervisor

dimensional Minkowski space. Obviously, there is a conflict of background here as deem by GR

and quantum physics. What follows next is that many distinguish physicists and philosophers

have tried and failed to generalize both the quantum theory and classical theory of gravity.

Popular models like the string theory and loop quantum gravity (LQG) are painstakingly

developed but none have produced convincing results yet. However a life line was thrown in

2004 when causal dynamical triangulation (CDT) managed to produce the correct dimension of

our universe. Now that we have a convincing candidate, we look into the concepts and

methodologies and try to interpret its capabilities and limitations. First, we will re‐visit what

constitute a proper “theory of science” to have a good gauge to review this relatively new model

from the first principles. Secondly, we do a conceptual and process orientated study on CDT and

interpret its features. Lastly, we will discuss about CDT possible future directions and conclude

our view on this candidate of the quantum theory of gravity.

Fundamental features of a scientific theory with CDT in mind

Influential 20th century philosopher of science, Karl Popper stated that any scientific theory

should be falsifiable. Here, falsifiable means that the theory must be able to make definite

predictions that which are capable of being verifiable through our observations or experiences.

He added that any scientific theories that do not satisfy this condition are called pseudo‐science,

delegating them from proper science per se. An example of a falsifiable theory is GR which made

very precise prediction of the classic field when subjected to system boundary conditions. In

contrast, an example of non‐falsifiable theory would be able to fit into any empirical data to give

consistency to its principle ideas. However, pseudo‐sciences do contribute, notably the

discovery of planet Neptune by Adams and Leverrier. Of course, Popper’s criteria are one of the

many definitions of a scientific theory, but here we are studying a candidate that was developed

from the first principles and hence it is absolutely reasonable to apply this qualification to CDT.

For all scientific theories, it is a requirement that its predictions are able to be verified

experimentally. If this is not met, then we will have countless of theory of everything (TOE) or

grand unified theory (GUT). This statement that we are using here must be carefully dealt with,

in a sense that we are not claiming that if a theory is verified experimentally to be true then it

must be the absolute description of the phenomena. It only qualifies to describe the physical

phenomena correctly to a certain order or magnitude; there can be other approaches or

theories that can equally describe the phenomena correctly as well.

For a quantum theory of gravity, in order to observe the activities or behaviour of the

quantum structure of space‐time, we have to look at the Planck length of order∼1035 m. If you

have in mind the large hadron collider (LHC) which is capable of up to 14TeV center of mass

energy for protons‐anti‐protons collision then we are ten plus order off the required resolution.

This only implies that we are still far from being able to verify the any quantum theory of

gravity. Hence it will be very important to set some rules for the qualification of a possible

candidate here. Loll [2] has defined clearly the basic requirement for a possible candidate of

quantum theory of gravity, it is as follows:

“To look for a consistent theory of quantum gravity, which describes the

dynamical behaviour of spacetime geometry on all scales and reproduces

Einstein’s theory of general relativity on large scales. But at the same time, it

should also predict new observable phenomena.” Loll (2008)

From the extract, Loll has included that any candidate must be able to at least provide

observable predictions and cough out Einstein’s theory of general relativity at large scales. We

must re‐emphasis here that the game is to develop a theory from first principles so that it will

have properties of quantum behaviour and classical field behaviour at corresponding regime. It

is not as trivial as to say we modify quantum theories to encompass Einstein’s GR or vice versa.

For a clearer qualification, we propose a generalized set of guide‐lines that allows more freedom

for the candidates but yet without modifications to our current valid understandings.

i. The candidate should be background independent, to say that is to mean that the

candidate’s quantum theory must not assume a fixed metric background. This is due to

the fact that quantum theory of gravity should be able to describe the highly curved3

space‐time at Planck length, which in turn relies on formalism procedures of GR.

ii. The approach should be non‐perturbative. This second guide stems from the first guide

which states that the candidate should be background independent, hence less the need

for any perturbations or approximations around some fixed backgrounds.

iii. When scaled to large distances, the candidate must be able to reproduce classical fields

describing our observed universe. This is to relieve the candidate of strictly reproducing

Einstein’s GR; as mentioned, the GR should be only one of the possible ways to describe

our space‐time structures. Essentially, the fundamentals of GR and results stays but the

methodologies can be different. Hence, numerical expressions must be matched but not

algebraic expressions.

3 The highly curved space‐time structure at Planck Length is simply a prediction due to uncertainty relation.

iv. Since the candidate seeks to explain between two limiting distances (microscopic and

macroscopic), there should be a transition in dimensions and this transition should yield

new observable predictions as this transition range is within our measurable regime.

v. The candidate at appropriate quantum regime should fulfil all quantum physics

postulates and not necessarily background dependent. It should still be able to describe

quantum‐classical transition under the transition of the space‐time to larger scale, for

example the correspondence principle.

Now that we have defined a set of rules for the candidates of quantum theory of gravity that

allows us to verify it in our “observable range”, we can go ahead and study CDT and its

methodologies.

To add on to this section, we want to note that the candidate should be able to approach from

any directions in formulating the quantum theory of gravity. Explicitly, it can choose to build

quantization into GR and use it as a starting point (Planck Length) then scale it to classical

length or it can ignore the idea of highly curved space‐time at Planck length to build GR from

some assumed random curvatures. This is in response to that fact that if we wish to develop the

quantum theory of gravity from first principles rather than to extend either quantum or

classical models. The crucial point we want to highlight here is that the Heisenberg’s

uncertainty relation may depend on the space‐time structure (discrete or smooth) since it is

spatial dependent. This is an area of quantum gravity which hasn’t been explored yet.

Causal Dynamical Triangulation

To remind the readers of our objectives in this UROPS report, we do not derive or follow

rigorously the formalism of the CDT but rather interpret the main ideas, methodologies and

results. For more explicit formalism of CDT, readers can refer to [1], [3], [4], [6] and [7].

In causal dynamical triangulation, the main framework mainly consists of two tools. First, the

Feynman’s path integral with each possible evolution of the space‐time structures as the

configurations of virtual paths and the second tool as Regge calculus. While path integral is

highly publicised and employed in many areas, Regge calculus is less popular and is seen as the

most efficient method to obtain numerical solutions to classical field theories, mainly the GR. To

see why CDT chooses to employ path integral as its directing officer and Regge calculus as its

manpower officer, we need to identify the approach that CDT is taken here. It is clear that CDT

assumes that the uncertainty relation is independent of the discretization and seeks to quantize

the space‐time structure from the start. In figure 1, we listed the main methodology of CDT

starting from path integral.

With the flow of methodologies in view now, we can then proceed and examine each process

in details.

Gravitation quantum path integral

The gravitation path integral used in CDT is defined as the partition function of quantum gravity

as a formal integral over all space‐time structures weighted by the complex exponent of the

classical Einstein‐Hilbert action. It is given as

Gravitation Quantum Path Integral ⇒ The integral here mainly serves as a direction but not yet mathematically defined. This is due to the fact that Einstein‐Hilbert classical action serves as complex weights and the space of integration is yet to be defined.

Regge Calculus (Causality built in)

Lattice field Model regularization

Path Integral Monte Carlo (PIMC)

⇒ The geometries that we are computing here are in principles still classical diffeomorphism‐manifolds or smooth mapping space‐times. By using Regge calculus here, we are using discrete building blocks to approximate the smooth manifold. The problem here is how to decide on the effective cut off range for the size of the building blocks. In additional, what makes CDT stands out from the crowd is that it introduces causality from the offset.

Introduces quantization of classical gravitation fields

Impose causality on all building blocks and discretize the smooth manifold from path integral.

⇒ Now our quantum field have been discretized to approximate the classical smooth manifolds, we need to be able to solve it to yield the solutions that we want. Here, lattice field application allows us to make a gradual transition to classical field as the cut off parameter goes to zero.

All the space‐time structures are now represented with discrete forms but with regularization we obtain back our smooth space time at macroscopic scales with finite scaling.

⇒ With the path integration well defined now, we need to compute the integration with large number of events. Using Monte Carlo simulation, one is able to produce a macroscopic world of our universe with the correct dimension. The main idea is to employ the fluctuating lattice field model and use finite scaling to obtain the continuum.

Wick rotation of time

Figure 1. The generic flow for causal dynamical triangulation.

( ) ( )( )( )

( )

0

0 Einstein Hilbert

g TiS g

g

g g T T D g e − ⎡ ⎤⎣ ⎦= ⋅ ⋅∫ Z , ; : (1)

And the Classical Einstein‐Hilbert action as

( )41: = 2Einstein HilbertN

S g d x g RG−

⎡ ⎤ − Λ⎣ ⎦ ∫ det (2)

The GN is defined as the Newton gravitation constant, g as the space‐time metric and Λ as the

cosmological constant. The importance of dependence on Λ is not obvious until when CDT tries

to generate a de Sitter4 universe.

It is important to emphasis here that the path integral is not yet well defined but only serves

as an intention/direction in the formulation of CDT. We have yet to determine and decide on the

integration space or conditions whereby the integral will converges to. Moreover, the integral

do not pre‐determine the background metric like the formalism of quantum field theory and

exhibit diffeomorphism invariance property. Here, the diffeomorphism applies to the class of

space‐time structures that we are considering. This class of diffeomorphism manifolds or range

of integration measures we are computing is causal, Lorentzian space‐time structures in nature

and can maps into another class as long it preserves its structures.

Regge Calculus with causality encoded in each fourdimensional simplex

Now that we have got a smooth or diffeomorphism class of space‐time structures as our

integration space, how are we going to quantify it and gain numerical solutions from such an

integral? The answer will be to use the classical Regge calculus, which is essentially

triangulating classical manifolds. Triangulation is a mathematical method or rather unphysical

mathematical tool to approximate the smooth manifolds. It employs “simplices” or fundamental

building blocks to construct the classical space‐time solutions of Einstein’s GR. These simplices

are objects that can be assigned with restrictions or parameters to achieve certain manifolds.

For example, CDT imposed an extra restriction of causality on the simplices as compared to

Euclidean triangulation which ignore this restriction.

4 The de Sitter universe is the exact solution to Einstein’s General relativity with just cosmological constant to define the dynamics of the space‐time structure. No ordinary mass or matters are computed inside this de Sitter model.

Shown in Figure 2, we can see that regularizing a smooth manifold gives us back an

approximation of the smooth manifold. With Regge calculus, we have in hand a method of

unifying all possible diffeomorphism geometries as required of the integration space. That is to

make use of 4‐simplices to build and construct space‐time structures with a glueing system. The

glueing system simply is a set of rules that tells the simplices on how to interact and form bigger

spaces; this is in analogy with swarm intelligence. Swarm intelligence is a unique field of

behavioural science whereby agents possess low degree of freedom (like intelligence or

choices) but yet able to follow simple rules. By conforming to these rules, the lot of agents can

exhibit global intelligence of magnitude ten times more than an individual agent. Surprisingly,

each agent just interacts with surrounding agents locally. They somehow emerge with a

complex system with greater degrees of freedom.

CDT defines its basic building blocks or simplices as 4 dimensional triangles. The four

simplices follow the glueing system’s guides as:

i. The curvatures of manifolds are localized at the vertices of the connected triangles.

ii. The metric is flat or uniform inside each simplex.

iii. Lengths of time connection and space connection are constant. They do not evolve.

iv. A d‐dimensional simplical manifold can be created by gluing pairs of d‐simplices along

their (d1) faces.

v. The notation of causality is defined at the start and the connected simplices follow the

same global time foliation.

Regularize with 4‐simplices

Figure 2. The effects of regularizing a smooth manifold.

Figure 3. Examples of d‐simplices, from left to right: triangle, tetrahedron and 4‐simplex.

From here now, we can do some modification to expression (1) to gain some definition of the

integration. Up to this point, Regge calculus has introduces regularization to our class of

diffeomorphism space‐time structures. So instead of integrating over all space‐time geometries,

we do a summation of all causal triangulations. This gives us

( )

( )( )

0

1 gge

Causal

Einstein Hilbert iS gN

g g

g TiS g

g

Z G eC

D g e − ⎡ ⎤ ⎡ ⎤⎣ ⎦⎣ ⎦

∈

Λ : = ⋅ ⋅ → ∑∫G

Re, (3)

Here, the Cg is the order of the automorphism class of the triangulation. Its presence is to serve

as a correction to the summation of all causal triangulations; in fact it can be seen as an

endomorphism whereby the causal triangulation maps more than once to itself if it possesses

symmetry.

EuclideanRegge Wick rotation

As depicted in figure 1, we have to perform a Wick rotation and compute expression (3). In

evaluating the quantum path integral in QFT, we have to do the Wick rotation by analytically

continuing the time variable to an imaginary time axis. This means we will be converting the

problem from Lorentzian to Euclidean space‐time. This will not preserve the causality notion

that CDT installed in the first place. In the conventional approach of Wick rotation in QFT, we

have

Lorentzian EuclideaniS S→ − (4)

We then sum up over all the Euclidean space‐time structures. What we can do here is to modify

this approach to keep the causality. The CDT’s Wick rotation is then defined as

( ) ( )Lorentzian Lorentzian Euclidean EuclideaniS iS T S iS T= → − = Regge Regge (5)

By invoking the CDT’s Wick rotation, we obtain a statistical physics problem (partition function)

from quantum physics problem.

( ) 1 gge Euclidean

Euclidean

S TN

T g

Z G eC

⎡ ⎤⎣ ⎦Λ : = ∑ Re, (6)

Even as the above expression do not contains any Lorentzian space‐time, the causality notion is

still embedded inside the summation by the corresponding Euclidean‐Regge action.

Lattice Regularization

In fact, we have already incorporated Lattice field regularization inside expression (6). By

defining the edges of the 4‐simplices as parameter a, we are designating the field regularization

here. As the limit for 0a → , we get an approximation of our classical fields. Simply put, lattice

regularization is to approximate quantum fields with lattice structures (in CDT’s case, we are

using Regge calculus’s simplices instead of cubes) as finite scaling takes place. Hence much of

the works are achieved in quantum Regge calculus (combination of classical Regge calculus and

Lattice field theory) which we have followed through.

Monte Carlo Simulation guide of 3D universe

The purpose of a Monte Carlo section here is to serve to give completeness to the CDT. For

simplicity reasons, we shall study a publication by Zhang [5] in his numerical method for a 3

dimension universe. From the previous sections, we have an expression that is well‐defined and

computable. The integration space or summation is to be over all Euclidean‐Regge

triangulations (each corresponds to the constructed Lorentzian triangulation) and we will take

Cg to be 15. In addition, the time and space connection lengths are matched6 to 1 for this

simulation.

Starting from Euclidean Regge action, CDT [3] defines the discrete action version as

3

3 3 0 0EucS k N k N= −Reg‐ (7)

Where the dimensionless coupling constants are

0N

akG

= (8)

3

3

3 1 14 348 2 NN

a akGG ππ

⎛ ⎞Λ= + −⎜ ⎟

⎝ ⎠arccos (9)

Here N0 is the number of vertices and N3 is number of tetrahedra.

5 The reason for setting the factor 1/Cg=1 is based on the general observation in the theory of critical phenomena that we do not expect the detailed choice of measure to affect the results. 6 Remember that in the glueing system we defined that both the length of time and space connections must be constant. Here, for simplicity reason we set them to match one and other. This is to say that we set both time and space as equal footing, hence working in Euclidean section first.

The expression (6) is then re‐written as

( ) 3 3 0 0

0 3

1

Euclidean

k N k N

T g

Z k k eC

⎡ ⎤−⎣ ⎦ : = ∑ , (10)

For a 3 dimensional problem, our building blocks or simplices are limited to just 2

configurations but 3 unique tetrahedra. They are (3 ,1), (1, 3) and (2 ,2) tetrahedra.

With the triangulation glueing system as mentioned above, we know that a causal triangulated

space‐time structure is fabricated once a pair of tetrahedron is glued together. From this space‐

time structure, we need to know that how these structures are allowed to move accordingly to

the glueing system. Basically, these moves will be our Monte Carlo moves. First, we know that a

3 dimension triangulated space‐time structure can map onto another structure as long they

belongs to the same proper time slice. It is easy to see this if we slice the evolution of the space‐

time structure at a specific proper time (or global time). Secondly, we want to know how these

movements affect the number of tetrahedra. We define this move by (m, n) whereby the move

will replace the m tetrahedra with the n tetrahedra. In addition, we want to characterize the

tetrahedra with 4‐tuple vertices to keep note of the glueing. Lastly, we do not want to

distinguish moves that are mirror images of themselves in the slices.

Figure 4. The (a) shows a (3, 1) tetrahedron which has another reflection‐symmetry as (1, 3) tetrahedron and (b) is a (2, 2) tetrahedron.

Figure 5. The (2, 6) move.

Hence, we can observe that there will be five legal moves in total from figure 5‐7. These moves

are (2, 6), (2, 3) and (4, 4) with their inverse moves (6, 2) and (3, 2). We have now defined the

Monte Carlo moves and shall proceed to evaluate the summation in expression (10). Let’s define

the expectation value of an observable with N measurements as

( )

( )3 3 0 0

1Euclidean

k N k NEuclidean N

Ti

i

T eC

Z N

⎡ ⎤−⎣ ⎦⋅

= ≈∑

∑O

O O (11)

Where Ci is the Markov chain of configuration sampled during the simulation of random walk

(Monte Carlo moves). By applying the Metropolis algorithm, Zhang simulated the 3 dimension

universe conditioned on the acceptance or rejection of the Monte Carlo moves based on the

change in Euclidean action. What if left now is to run the Monte Carlo moves to explore the

phase space of the system. This is achieved by say varying the dimensionless coupling constant

k0 from 1 to 6. We then choose the observable as the order parameter7 as the ratio between the

total (2, 2) simplices and total space time volume N3. In other hand, we can also choose the ratio

(3, 1) or (1, 3) simplices to N3 as our order parameter, but we have to use larger numbers of

events/moves to run the simulation due to the anti‐symmetry of its geometry. Zhang [5] used

two test configurations to probe the phase change in the system, the two test space‐time

configurations are:

7 Here, the purpose to choose the order parameter to observe the change in the phase space with two test configurations below and after the identified critical point.



( )2 2

38 8000

1

T V

NTriangulation

N= =

=

,

,

(12)

( )2 2

332 32000T V

NTriangulation

N= =

=

2,

,

(13)

With these two test configurations, the plot for the expectation value of the order parameter to

the dimensionless coupling constant k0 from [2] is shown in figure 8.

The critical point is observed to be around k0 = 3.3 and from this value, the before and after

phase changes were plotted for spatial volume N2 against time.

Figure 8. Plot of expectation values of order parameter N22/N3 as a function of k0.

Figure 9. The left plot is the phase space for K0 < 3.3 and the right plot is K0 > 3.3.

From figure 9, we see that the left plot for the phase space k0 < 3.3 shows a well behaved

geometry. What is important here is that the successive spatial slices are strongly coupled and

exhibit near classical universe structure. In fact the left plot is well described by the Friedmann

universe which is an exact solution of Einstein’s GR for a homogeneous and isotropic expanding

universe. To get more information out, we measure the Spectral dimension by doing a diffusion

process on the discretized geometric ensemble. We will not discuss this in depth but readers

can refer to [7] for a more detailed method on how to conduct a diffusion process for quantum

gravity.

The basic idea for diffusion process is to do a random walk in a specified space‐time structure

after we have determined the correct phase in the system. For Zhang’s example, we take the K0 =

2.0 as shown in the left plot of figure 9. We simply pick any random tetrahedron and do a

random walk to any of the four surrounding neighbouring tetrahedra. Each move is counted as a

step, and with a large number of successive steps, we want the walk to return to the original or

starting tetrahedron. So let us define the probability of starting and returning to the same

tetrahedron with n steps as P(n). It is then important to give equal chances to all existing

tetrahedron and those that are in the extended spatial slices. It is known that the return

probability for diffusion on a fractal or discrete geometry is

( ) 2DP n n−≈ / (14)

And the diffusion time is taken to be much smaller than total number of tetrahedra to the power

of 2/D, where D is the spectral dimension that we are interested in. The spectral dimension can

be extracted out by taking the logarithmic derivative and approximate as

( ) ( )( )( )

2d P n

D n finite size correctionsd n

= − + − log

log (15)

Where the finite size corrections are due to the scaling method in Lattice field model.

We should expect then the expression (15) to converge to 3 dimensions when dealing with

tetrahedra and 4 dimensions when applied to 4‐simplices.

Discussions and Conclusion

In the first section of this report, we discussed about the validity of a scientific theory with the

orientation aligned with a quantum theory of gravity and the guide to achieve such a model. We

also highlighted some of the assumptions like the uncertainty relation’s dependency on space‐

time structures discretization and the possible approaches that any candidate can take.

Following this was the conceptual study of CDT’s methodologies and an example of 3 dimension

universe simulation by Zhang. With these two materials in hands now, we can examine the

rationality that CDT have taken.

So far with all the numerical experiments conducted with CDT till now, it has produced

i. Classical behaviour of the quantum universe at the macroscopic regime; the emergence

of a stable and extended quantum geometry which is correctly 4 dimensional at large

scales and well described by the Friedmann universe. The result also agrees with de

Sitter universe in the case of employing the cosmological constant Λ only.

ii. The CDT observes strong deviations from the classical behaviour at microscopic scales,

with extrapolation to very short distances, the space‐time structures are fractural and

effectively 2 dimensional.

These two results with the assumptions of CDT have fulfilled all but one of our proposed guide‐

lines. That is to say how can we observe or measure the transition of the dimensions. Without

this requirement, the candidate may not be a valid one yet. Adding on, we want to note that

there are some assumptions that CDT made but yet contradicts with its interpretations. Firstly,

the geometrical basic building blocks are nothing more than mathematical approximation or

unphysical entities that serve to regularize our classical smooth manifolds. What comes next is

that these entities obey a glueing system and when accumulated, they produce classical space‐

time structure behaviours. That is to say, we have to assume that they are indeed real and

physical entities if we were to move on to lattice field approximation and phase transition

simulation. Analogously, atoms and particles in phase transitions are real and they interact with

one and other, hence the possibility of critical phase and a transition in phase. What CDT is

doing is to map the behaviour of phase transition over to these simplices and based on a set of

glueing system emerged a well behaved and stable classical universe. Maybe it is not that

important to say if the simplices are real or unphysical explicitly as they are indeed not

observable at very short distances (they are shown to be 2 dimensional and fractal). But from

the assumptions, these microscopic discrete structures must have interaction so that a classical

field is produced. In addition, we have yet to know if the uncertainty relation is dependent on

the discreteness of the background structure or not. The general idea is that uncertainty

relation is a fundamental property of nature and is universal at all scales that are smooth. By

applying this relation to Planck length, we will have huge fluctuation in energy and by the

understanding of GR, this means that we will have highly curved fluctuation in space‐time. But

will our uncertainty relation still holds for a discretized manifold like it does on the smooth

manifold? This deserves some attention if any candidate wishes to state that uncertainty

relation will emerge due to the transition of dimensions, which is reasonable. This is also in

accordance with our proposed guidelines for a quantum theory of gravity.

As a summary, we would like to say that the Causal Dynamical Triangulation has successfully

shown that it can retrieve back the observed universe at large scales. It is developed from first

principles based on independent of background and non‐perturbative approach. However, in its

manner of progression, it employs techniques that are only computable through Monte Carlo

simulations. We must be careful here because of the infinitely free parameters that we can play

with. It is like introducing more and more degrees of freedom along the way to achieve some

desired effects. Nevertheless, unless CDT can predict some observable phenomena that can be

measured, this candidate can only serves as a theory to pseudo quantum theory of gravity.

References

[1] J. Ambjørn, J. Jurkiewicz and R. Loll, Emergence of a 4D world from causal quantum gravity,

Phys. Rev. Lett. 93 (2004) 131301 [arXiv: hep‐th/0404156].

[2] R. Loll, Quantum gravity on your desktop, Classical and Quantum gravity 2 (2008) 114006

[arXiv: 0711.0273, gr‐qc].

[3] J. Ambjørn, J. Jurkiewicz and R. Loll, Dynamically triangulating Lorentzian quantum gravity,

Nucl. Phys. B 610 347‐382, (2001) [arXiv: hep‐th/0105267].

[4] J. Ambjørn, J. Jurkiewicz and R. Loll, Nonperturbative 3d Lorentzian quantum gravity, Phys.

Rev. D64 044011 (2001) [arXiv: hep‐th/0105267].

[5] J. Z. Zhang, Causal Dynamical Triangulations of Quantum Gravity in Three dimensions, Cornell

University (2007).

[6] J. Ambjørn, J. Jurkiewicz and R. Loll, The universe from scratch, Contemp. Phys. 47 (2006)

103‐117, (2001) [arXiv: hep‐th/0509010].

[7] J. Ambjørn, J. Jurkiewicz and R. Loll, Reconstructing the universe, Phys. Rev. D 72 (2005)

064014 [arXiv: hep‐th/0505154].

Review on Causal Dynamical Triangulation as a Candidate of the Quantum Theory of Gravity

Documents

Transcript of Review on Causal Dynamical Triangulation as a Candidate of the Quantum Theory of Gravity