# CDT 2007

###
Virtual Spacetime Geometries:

QUANTA of TETRAHEDRAL SPACETIME VACUUM?

Wildly Fluctuating Geometrical Fabric of Spacetime

**PROBING AND DEFINING UNITS OF SPACETIME**

The shape of units of quantization is assumed to be a sphere simply for convention and convenience. It can be any shape and probably is not a sphere, though we tend to forget that. CDT is an attempt to quantize space without quantizing time, which cant be done because space isnt primary. Rigorously it does not exist but as the precipitate of time. Quantized space-time iiself can't be separated. but the grains would have to be way smaller , (10^-33)^^3.

CDT represents conditioning of spacetime BY sub or other spacetime. The only difference between sub- and real spacetime is in fineness of grain. Regular spacetime cannot condition sub-spacetime without an input of energy. space does not really exist as such but is an artifact of quantized time.

Picture yourself as the creator of the universe: sit down at your easel and paint the universe. There is no canvas; you apply pigment where the canvas would be if there was one. Eventually people assume the canvas exists from the painting, which in a sense it does but only as a mental convenience. But when you try to describe the ghost enlivening the machine, each ghost is a complete universe but of a slightly higher order: that's you. It has to insert itself into another space-time knot -- the body. That 'consciousness' slightly and selectively compresses the preexisting threads of spacetime.With apologies to Blake, each grain of sand is not a universe. Each live spirit -- of whatever species -- is a universe. Your spirit (which is you) is a SELF-CONTAINED universe of a slightly higher order, Whenever it visits the material universe, thats how you are "seen" by the material universe when you are IN it because you *cant *be part of it. That's forbidden. Our natural 'home' is a much better place; this universe is the basement workshop where things can go terribly awry and perhaps we should have stayed out. (Thomas)

**CDT's View of Itself**

Dimensionality is turned into a dynamical quantity. Individual points have no physical significance in empty space. Time really scales with the correct fraction of total spacetime volume. The wavefunction of the universe is a function of the scalar factor. The spectral dimension is really the effective dimension of the carrier space. The space-time dimension is scale-dependent. Deriving 4-d spacetime 'from scratch" is an unprecedented result. More detailed information about the shortscale structure of quantum spacetime remains to be extracted. This approach is opposite to M-theoretical solutions with vast numbers of possible vacua. It eliminates the invocation of additional dimensions and extended objects. It is a model of fractal, scale-invariant lower dimensional structure at the Planck length that is classical at large scale. It is a nonperturbative quantum gravitational propagator. A four-dimensional universe arises from quantum fluctuations -- four-dimensional quantum gravity.

**EXCITATIONS OF SPACETIME GEOMETRY**

**CDT breaks down tiny units of volume and area -- the crucial stuff that makes up any spacetime -- into tiny tetrahedra, a little like a computer graphics chip renders complex surfaces by decomposing them into many itsy bitsy squares and triangles.CDT is a very simplified form of Loop Quantum Gravity. And even if it is not The Ultimate Theory, CDT's practitioners have developed clever solutions and approximation methods that could be used for the real thing. Pros: Classical spacetime, as described by Einstein, does emerge from CDT models. Cons: It's not clear yet if falsifiable predictions can be made that would distinguish CDT from LQG or other theories.**

******

http://www.arxiv.org/abs/hep-th/0509010

## The Universe from Scratch

Authors:**R. Loll, J. Ambjorn, J. Jurkiewicz**

Comments: 31 pages, 5 figures; review paper commissioned by Contemporary Physics and aimed at a wider physics audience; minor beautifications, coincides with journal version

Report-no: SPIN-05/28, ITP-UU-05/34

Journal-ref: Contemp.Phys. 47 (2006) 103-117

A fascinating and deep question about nature is what one would see if one could probe space and time at smaller and smaller distances. Already the 19th-century founders of modern geometry contemplated the possibility that a piece of empty space that looks completely smooth and structureless to the naked eye might have an intricate microstructure at a much smaller scale. Our vastly increased understanding of the physical world acquired during the 20th century has made this a certainty. The laws of quantum theory tell us that looking at spacetime at ever smaller scales requires ever larger energies, and, according to Einstein's theory of general relativity, this will alter spacetime itself: it will acquire structure in the form of "curvature". What we still lack is a definitive Theory of Quantum Gravity to give us a detailed and quantitative description of the highly curved and quantum-fluctuating geometry of spacetime at this so-called Planck scale. - This article outlines a particular approach to constructing such a theory, that of Causal Dynamical Triangulations, and its achievements so far in deriving from first principles why spacetime is what it is, from the tiniest realms of the quantum to the large-scale structure of the universe.

******

The Triangular Universe; February 2007; Scientific American Magazine; by Mark Alpert; 1 Page(s)

Imagine a landscape composed of microscopic triangular structures that constantly rearrange themselves into new patterns. Seen from afar, the landscape looks perfectly smooth, but up close it is a churning cauldron of strange geometries. This deceptively simple model is at the heart of a new theory called causal dynamical triangulation (CDT), which has emerged as a promising approach to solving the most vexing problem in physics--unifying the laws of gravity with those of quantum mechanics.

For more than 20 years, the leading contender in the quest for unification has been string theory, which posits that the fundamental particles and forces are actually minuscule strings of energy. But some scientists say this theory is misguided because it sets the strings against a fixed background; a better model, they argue, would generate not only particles and forces but also the spacetime they inhabit. In the 1980s and 1990s these researchers developed loop quantum gravity, which describes space as a network of tiny volumes only 10^{-33} centimeter across. Although this approach has achieved some notable successes, such as predicting the properties of black holes, it has yet to pass an essential test: showing that the jumble of volumes always comes together to form the familiar four-dimensional spacetime of our everyday world.

http://en.wikipedia.org/wiki/Unruh_effect

The **Unruh effect **is the prediction that an accelerating observer will observe black-body radiation where an inertial observer would observe none. In other words, the accelerating observer will find themselves in a warm background. The quantum state which is seen as ground state for observers in inertial systems is seen as a thermodynamic equilibrium for the uniformly accelerated observer.

Unruh demonstrated that the very notion of vacuum depends on the path of the observer through spacetime. From the viewpoint of the accelerating observer, the vacuum of the inertial observer will look like a state containing many particles in thermal equilibrium — a warm gas. Although the Unruh effect came as a shock, it makes intuitive sense if the word *vacuum* is interpreted appropriately, as below.

In modern terms, the concept of "vacuum" is not the same as "empty space", as all f space is filled with the quantized fields that make up a universe. Vacuum is simply the lowest *possible* energy state of these fields, a very different concept than "empty". The energy states of any quantized field are defined by the Hamiltonian, based on local conditions, including the time coordinate. According to special relativity, two observers moving relative to each other must use different time coordinates. If those observers are accelerating, there may be no shared coordinate system. Hence, the observers will see different quantum states and thus different vacua.

In some cases, the vacuum of one observer is not even in the space of quantum states of the other. In technical terms, this comes about because the two vacua lead to unitarily inequivalent representations of the quantum field canonical commutation relations. This is because two mutually accelerating observers may not be able to find a globally defined coordinate transformation relating their coordinate choices. In fact, an accelerating observer will perceive an apparent event horizon forming. The existence of Unruh radiation can be linked to this apparent event horizon, putting it in the same conceptual framework as Hawking radiation. On the other hand, the Unruh effect shows that **the definition of what constitutes a "particle" depends on the state of motion of the observer.**

We need to decompose the (free) field into positive and negative frequency components before defining the creation and annihilation operators. This can only be done in spacetimes with a timelike Killing vector field. This decomposition happens to be different in Cartesian and Rindler coordinates (although the two are related by a Bogoliubov transformation). This explains why the "particle numbers", which are defined in terms of the creation and annihilation operators, are different in both coordinates.

*******************

The CDT approach first appeared in 1998 and was at

first "test-driven" by applying it only to lower

spacetime dimensions: 2D and 3D. First results

for 4-D spacetime appeared in 2004.

Main CDT authors are "AJL" standing for Ambjorn,

Jurkiewicz, and Loll. In fact "AJL" is almost

synonymous with CDT.

Here is a short CDT reading list:

The first paper is long and detailed. It is better to

begin with #2 on this list called "Emergence of a 4D

world..."

1.

http://arxiv.org/hep-th/0105267

Dynamically Triangulating Lorentzian Quantum Gravity

J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U.

Krakow), R. Loll (AEI, Golm)

41 pages, 14 figures

Nucl.Phys. B610 (2001) 347-382

"Fruitful ideas on how to quantize gravity are few and

far between. In this paper, we give a complete

description of a recently introduced non-perturbative

gravitational path integral whose continuum limit has

already been investigated extensively in d less than

4, with promising results. It is based on a simplicial

regularization of Lorentzian space-times and, most

importantly, possesses a well-defined,

non-perturbative Wick rotation. We present a detailed

analysis of the geometric and mathematical properties

of the discretized model in d=3,4..."

2.

http://arxiv.org/abs/hep-th/0404156

Emergence of a 4D World from Causal Quantum Gravity

J. Ambjorn (1 and 3), J. Jurkiewicz (2), R. Loll (3)

((1) Niels Bohr Institute, Copenhagen, (2) Jagellonian

University, Krakow, (3) Spinoza Institute, Utrecht)

11 pages, 3 figures; final version to appear in Phys.

Rev. Lett

Phys.Rev.Lett. 93 (2004) 131301

"Causal Dynamical Triangulations in four dimensions

provide a background-independent definition of the sum

over geometries in nonperturbative quantum gravity,

with a positive cosmological constant. We present

evidence that a macroscopic four-dimensional world

emerges from this theory dynamically."

3.

http://arxiv.org/abs/hep-th/0411152

Semiclassical Universe from First Principles

J. Ambjorn, J. Jurkiewicz, R. Loll

15 pages, 4 figures

Phys.Lett. B607 (2005) 205-213

"Causal Dynamical Triangulations in four dimensions

provide a background-independent definition of the sum

over space-time geometries in nonperturbative quantum

gravity. We show that the macroscopic four-dimensional

world which emerges in the Euclidean sector of this

theory is a bounce which satisfies a semiclassical

equation. After integrating out all degrees of freedom

except for a global scale factor, we obtain the ground

state wave function of the universe as a function of

this scale factor."

4.

http://arxiv.org/abs/hep-th/0505113

Spectral Dimension of the Universe

J. Ambjorn (NBI Copenhagen and U. Utrecht), J.

Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)

10 pages, 1 figure

SPIN-05/05, ITP-UU-05/07

"We measure the spectral dimension of universes

emerging from nonperturbative quantum gravity, defined

through state sums of causal triangulated geometries.

While four-dimensional on large scales, the quantum

universe appears two-dimensional at short distances.

We conclude that quantum gravity may be

"self-renormalizing" at the Planck scale, by virtue of

a mechanism of dynamical dimensional reduction."

Strange as it seems, this historical obstacle may now

have been overcome. If you look at

http://arxiv.org/abs/hep-th/0505113

you will see that even though the macroscopic

dimension of spacetime is 4D just as we would expect,

the dimension goes down at very small (planckian)

scales in their computer simulations actually to BELOW

2.

microscopically their computer generated spacetimes

have a kind of fractal character at very tiny scale,

but look normal at large scale

this may be adequate to defeat the "ultraviolet"

divergences of various field theories when they are

moved into the CDT spacetime of Ambjorn Jurkiewicz

Loll.

Another paper by the CDT group at Utrecht has come out

http://arxiv.org/hep-th/0505154

Reconstructing the Universe

J. Ambjorn (NBI Copenhagen and U. Utrecht), J.

Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)

52 pages, 20 postscript figures

SPIN-05/14, ITP-UU-05/18

"We provide detailed evidence for the claim that

nonperturbative quantum gravity, defined through state

sums of causal triangulated geometries, possesses a

large-scale limit in which the dimension of spacetime

is four and the dynamics of the volume of the universe

behaves semiclassically. This is a first step in

reconstructing the universe from a dynamical principle

at the Planck scale, and at the same time provides a

nontrivial consistency check of the method of causal

dynamical triangulations. A closer look at the quantum

geometry reveals a number of highly nonclassical

aspects, including a dynamical reduction of spacetime

to two dimensions on short scales and a fractal

structure of slices of constant time."

William Occam (the guy who said to keep it simple)

would favor this approach to quantizing gravity. By

far the simplest.

Does not require hidden extra dimensions, or various

other string and brane baloney. Is not built in a

fairyland of abstract algebra but right here in

spacetime with ordinary (real and complex) numbers and

ordinary 3 and 4D constructions.

this is an introduction that Renate Loll wrote for her

grad students a couple of years ago.

http://arxiv.org/hep-th/0212340

other approaches have a lot more paraphernalia and

have mostly gotten bogged down (e.g. string/M is a

mess and going no place etc.)

CDT "Simplex" quantum spacetime could conceivably

even be right!

It is still new and testable prediction need to be

worked out. But it is certainly worthwhile learning

about, for anyone with an interest in Quantum Gravity.

http://arxiv.org/hep-th/0505154

Reconstructing the Universe

J. Ambjorn, J. Jurkiewicz, R. Loll

Summary section: This paper describes the currently known

geometric properties of the quantum universe generated

by the method of causal dynamical triangulations, as

well as the general phase structure of the underlying

statistical model of four-dimensional random

geometries. The main results are as follows. An

extended quantum universe exists in one of the three

observed phases of the model, which occurs for

sufficiently large values of the bare Newton’s

constant G and of the asymmetry [tex]Delta[/tex],

which quantifies the finite relative length scale

between the time and spatial directions. In the two

other observed phases, the universe disintegrates into

a rapid succession of spatial slices of vanishing and

nonvanishing spatial volume (small G), or collapses in

the time direction to a universe that only exists for

an infinitesimal moment in time (large G, vanishing or

small [tex]Delta[/tex]). In either of these two

cases, no macroscopically extended spacetime geometry

is obtained. By measuring the (Euclidean) geometry of

the dynamically generated quantum spacetime in the

remaining phase, in which the universe appears to be

extended in space and time, we collected strong

evidence that it behaves as a four-dimensional

quantity on large scales....

In summary, what emerges from our formulation of

nonperturbative quantum gravity as a continuum limit

of causal dynamical triangulations is a compelling and

rather concrete geometric picture of quantum

spacetime. Quantum spacetime possesses a number of

large-scale properties expected of a four-dimensional

classical universe, but at the same time exhibits a

nonclassical and nonsmooth behaviour microscopically,

due to large quantum fluctuations of the geometry at

small scales. These fluctuations “conspire” to create

a quantum geometry that is effectively two-dimensional

at short distances..... they build approximate spacetimes with simple

building blocks called SIMPLEXES.

Simplexes are the key, and a technique for shuffling

simplexes to get random geometries they invented that

makes possible a "Monte Carlo" computation to evaluate

the integral and add up the weighted average.

0 simplex = point

1 simplex = line segment (has 2 endpoints)

2 simplex = triangle (a "pyramid" based on a line

segment, 3 vertex points)

3 simplex = tetrahedron (a trianglebased pyramid, has

4 points)

4 simplex = ? (you think of a name, it has 5 points)

think by analogy what a 4-simplex is, it is the

simplest possible 4D object.

each higher dimension simplex is made by taking the

previous simplex as a base, and putting a new point

above it and forming something like a cone or

pyramid-----with the previous simplex as base.

the usual math name for these CDT buildingblocks is

"4-simplex" but we could use a made-up name instead.

for instance we could call the 4D building blocks

"PENTAMIDS"

because they are 5-pointed pyramids.

the CDT authors build their model of quantum spacetime

out of "PENTAMIDS"

and they build it up in layers or storeys, like a

building

where the floors/ceilings are spacelike sheets and the

numbering of the sheets or layers plays the role of

time.

an interesting wrinkle is that there are TWO TYPES of

pentamid. there is the LEVEL kind that sits

straightandlevel on its tetrahedron base, or else is

the same thing upsidedown with its tetrahedron base on

the ceiling and its point on the floor.

and then there is the TILT kind which is tipped

slightly so that it just sits on one of its triangles,

and at the top it has a horizontal line segment like

the ridgepole of a roof. Or you can also have the same

thing upside down, so it rests on its horizontal line

segment ridge, and has its triangle up in the air..

you need both kinds to make a solid sandwich of

pentamids filling in between two spacelike

floor/ceiling sheets.

All the Level-type pentamids are identical, and all

the Tilt-kind are also identical

In a typical computer run, the CDT people might have a

halfmillion total pentamids, and there would about a

quartermillion identical LEVEL kind and about a

quartermillion identical TILT kind.

the geometry is all in the gluing of the blocks

and the randomization of geometries that makes it a

quantum spacetime is accomplished by a technique they

discovered for SHUFFLING the assemblage of pentamid

blocks. taking a part of the structure apart and

re-arranging the blocksThe geometry is all in the gluing of the

blocks, I mean this assemblage of blocks is not IN

some larger surrounding spacetime, rather it IS

spacetime, and the issue is whether it is curved or

flat (gravity is experienced because of a curvature of

spacetime).

If you COUNT the

pentamid blocks that all come together around some

line and if their angles add up to 360 degrees then

that is what you expect in a spacetime with no

curvature.

BUT IF THEIR ANGLES ADD UP TO SOMETHING YOU DONT

EXPECT because of how they are glued together, then

the spacetime around there is CURVED.

Since the pentamid blocks are all IDENTICAL they

have the same angles (all the level kind have the

same, and all the tilt kind have the same) and so you

can figure out the overall amounts of curvature by

COUNTING how many various kind block you have.

There are no coordinate functions or derivatives

you have the pentamids in the computer, and they fit

together, and the computer counts them and that's it

then you do the scramble where they get shuffled about

a million times (they call a pass where the computer

does a million shuffles one SWEEP) so that the

geometry is thoroughly randomized, and then you count

the pentamids again, and so on.

Meanwhile they are measuring the DIMENSIONALITY and

running diffusion processes in these sample spacetimes

and comparing distances and volumes,

experimentally revealing the geometry of

this model quantum spacetime.

### What's New with My Subject?

There is no "Brian Greene-type" popularizer for it;

there are only lecture notes and research papers

but here is a reading list, and the first half of #6

is pretty readable

1.

http://arxiv.org/hep-th/0105267

Dynamically Triangulating Lorentzian Quantum Gravity

J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U.

Krakow), R. Loll (AEI, Golm)

41 pages, 14 figures

Nucl.Phys. B610 (2001) 347-382

"Fruitful ideas on how to quantize gravity are few and

far between. In this paper, we give a complete

description of a recently introduced non-perturbative

gravitational path integral whose continuum limit has

already been investigated extensively in d less than

4, with promising results. It is based on a simplicial

regularization of Lorentzian space-times and, most

importantly, possesses a well-defined,

non-perturbative Wick rotation. We present a detailed

analysis of the geometric and mathematical properties

of the discretized model in d=3,4..."

2.

http://arxiv.org/abs/hep-th/0404156

Emergence of a 4D World from Causal Quantum Gravity

J. Ambjorn (1 and 3), J. Jurkiewicz (2), R. Loll (3)

((1) Niels Bohr Institute, Copenhagen, (2) Jagellonian

University, Krakow, (3) Spinoza Institute, Utrecht)

11 pages, 3 figures; final version to appear in Phys.

Rev. Lett

Phys.Rev.Lett. 93 (2004) 131301

"Causal Dynamical Triangulations in four dimensions

provide a background-independent definition of the sum

over geometries in nonperturbative quantum gravity,

with a positive cosmological constant. We present

evidence that a macroscopic four-dimensional world

emerges from this theory dynamically."

3.

http://arxiv.org/abs/hep-th/0411152

Semiclassical Universe from First Principles

J. Ambjorn, J. Jurkiewicz, R. Loll

15 pages, 4 figures

Phys.Lett. B607 (2005) 205-213

"Causal Dynamical Triangulations in four dimensions

provide a background-independent definition of the sum

over space-time geometries in nonperturbative quantum

gravity. We show that the macroscopic four-dimensional

world which emerges in the Euclidean sector of this

theory is a bounce which satisfies a semiclassical

equation. After integrating out all degrees of freedom

except for a global scale factor, we obtain the ground

state wave function of the universe as a function of

this scale factor."

4.

http://arxiv.org/abs/hep-th/0505113

Spectral Dimension of the Universe

J. Ambjorn (NBI Copenhagen and U. Utrecht), J.

Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)

10 pages, 1 figure

SPIN-05/05, ITP-UU-05/07

"We measure the spectral dimension of universes

emerging from nonperturbative quantum gravity, defined

through state sums of causal triangulated geometries.

While four-dimensional on large scales, the quantum

universe appears two-dimensional at short distances.

We conclude that quantum gravity may be

"self-renormalizing" at the Planck scale, by virtue of

a mechanism of dynamical dimensional reduction."

5.

http://arxiv.org/hep-th/0505154

Reconstructing the Universe

J. Ambjorn (NBI Copenhagen and U. Utrecht), J.

Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)

52 pages, 20 figures

Report-no: SPIN-05/14, ITP-UU-05/18

"We provide detailed evidence for the claim that

nonperturbative quantum gravity, defined through state

sums of causal triangulated geometries, possesses a

large-scale limit in which the dimension of spacetime

is four and the dynamics of the volume of the universe

behaves semiclassically. This is a first step in

reconstructing the universe from a dynamical principle

at the Planck scale, and at the same time provides a

nontrivial consistency check of the method of causal

dynamical triangulations. A closer look at the quantum

geometry reveals a number of highly nonclassical

aspects, including a dynamical reduction of spacetime

to two dimensions on short scales and a fractal

structure of slices of constant time."

6.

http://arxiv.org/hep-th/0212340

A discrete history of the Lorentzian path integral

R. Loll (U. Utrecht)

38 pages, 16 figures

SPIN-2002/40

Lect.Notes Phys. 631 (2003) 137-171

"In these lecture notes, I describe the motivation

behind a recent formulation of a non-perturbative

gravitational path integral for Lorentzian (instead of

the usual Euclidean) space-times, and give a

pedagogical introduction to its main features. At the

regularized, discrete level this approach solves the

problems of (i) having a well-defined Wick rotation,

(ii) possessing a coordinate-invariant cutoff, and

(iii) leading to_convergent_ sums over geometries.

Although little is known as yet about the existence

and nature of an underlying continuum theory of

quantum gravity in four dimensions, there are already

a number of beautiful results in d=2 and d=3 where

continuum limits have been found. They include an

explicit example of the inequivalence of the Euclidean

and Lorentzian path integrals, a non-perturbative

mechanism for the cancellation of the conformal

factor, and the discovery that causality can act as an

effective regulator of quantum geometry."

Loll wrote this as an introduction to CDT for Utrecht

graduate students who might want to get into her line

of research. It is a good beginning. It is already 2

years out of date so it does not have the latest

headline results but that is OK

To update the above list, when I posted it, the most

recent Renate Loll paper was

http://arxiv.org/hep-th/0505154

Reconstructing the Universe

J. Ambjorn, J. Jurkiewicz, R. Loll

Another paper Loll paper appeared since then:

http://arxiv.org/gr-qc/0506035

Counting a black hole in Lorentzian product

triangulations

B. Dittrich (AEI, Golm), R. Loll (U. Utrecht)

42 pages, 11 figures

"We take a step toward a nonperturbative gravitational

path integral for black-hole geometries by deriving an

expression for the expansion rate of null geodesic

congruences in the approach of causal dynamical

triangulations. We propose to use the integrated

expansion rate in building a quantum horizon finder in

the sum over spacetime geometries. It takes the form

of a counting formula for various types of discrete

building blocks which differ in how they focus and

defocus light rays. In the course of the derivation,

we introduce the concept of a Lorentzian dynamical

triangulation of product type, whose applicability

goes beyond that of describing black-hole

configurations."

I decided against any kind of made-up name for the CDT

building blocks. Loll calls them by the standard math

term "4-simplex".

I've been looking for some popularized accounts of

CDT, found some by John Baez, Dave Bacon, Adrian Cho

(two professional physicists and a science

journalist).

CDT is developing fast and probably the most important

quantum gravity development at the moment. we really

need popularized account of it. I will get some quotes

and links

physicist Dave Bacon's blog

http://dabacon.org/pontiff/?p=706#comments

<<...One question which plagues theoretical

physicists’ poor little minds is the question of why

we see a macroscopic world of 3+1 dimensions. Mostly

this is because physicists believe that at small

enough length or time scales (large enough energies)

the geometry of spacetime itself can exist in

nontrivial states of connectivity.... “Spacetime foam”

is what we call this strange state of affairs. How do

we get from this spacetime foam up to where our

experiments live and we seem to see a four dimensional

universe?

Concerning this problem, I just today read the paper

“Emergence of a 4D World from Causal Quantum Gravity,”

by J. Ambjorn, J. Jurkiewicz, and R. Loll which was

published in Physical Review Letters, (Volume 93, page

131301, 2004.) This paper attempts the following.

Construct spacetime by glueing together a bunch of

little four dimensional simplical spacetimes. Like I

said earlier, if we glue a bunch of these four

dimensional simplical spacetimes together, we get

something which is not necessarily four dimensional.

Now when we do this glueing we should insist on

maintain causality (i.e. no closed time like curves

and such.) So we can construct these crazy spacetimes,

but what do they mean. Well now we associate with each

of these spacetimes an amplitude. So there is some

notion of an action S for the given simpical spacetime

we have created and we assign to this an amplitude,

Exp[iS]. Now what one would love to do is to sample

over all of these crazy spacetimes and hence calculate

the propogators for different such spacetimes. But

this is hard. This is hard because of the fact that we

have to sample over this crazy oscillating Exp[iS].

But sometimes it is not so hard. Sometimes it is

possible to perform a “Wick” rotation and change

Exp[iS] into Exp[-S]. This means the problem of

calculating the total amplitude looks like adding up a

bunch of different spacetimes with weights Exp[-S]:

this looks just like classical statistical mechanics!

What the authors of the above paper do is they insist

that it is possible to perform such a rotation. They

then perform Monte Carlo simulations of the resulting

statistical mechanical system. And what do they find?

They argue that what they find is that the resulting

spacetime is indeed dominated by a spacetime of

dimension “3+1!”

So starting out from something which had only a

totally local sense of dimension (the original

building blocks are “3+1?) you glue them together in

pseudo-arbitrary (preserve causality, able to Wick

rotate) ways (this is what is called “background

independence”) and yet, you find, at the end of the

day, that you have effectively a global “3+1?

spacetime! Amazing, no?>>

If that is confusing, maybe Baez will be clearer. This

is an exerpt from John Baez TWF #206:

<<...In other words, can we do quantum physics without

choosing some fixed spacetime geometry from the start,

a "background" on which small perturbations move like

tiny quantum ripples on a calm pre-established lake? A

background geometry is convenient: it lets us keep

track of times and distances. It's like having a fixed

stage on which the actors - gravitons, strings,

branes, or whatever - cavort and dance. But, the main

lesson of general relativity is that spacetime is not

a fixed stage: it's a lively, dynamical entity!

There's no good way to separate the ripples from the

lake. This distinction is no more than a convenient

approximation - and a dangerous one at that.

So, we should learn to make do without a background

when studying quantum gravity. But it's tough! There

are knotty conceptual issues like the "problem of

time": how do we describe time evolution without using

a fixed background to measure the passage of time?

There are also practical problems: in most attempts to

describe spacetime from the ground up in a quantum

way, all hell breaks loose!

We can easily get spacetimes that crumple up into a

tiny blob... or spacetimes that form endlessly

branching fractal "polymers" of Hausdorff dimension

2... but it seems hard to get reasonably smooth

spacetimes of dimension 4. It's even hard to get

spacetimes of dimension 10 or 11... or anything

remotely interesting!

It almost seems as if we need a solid background as a

bed frame to keep the mattress of spacetime from

rolling up, getting all lumpy, or otherwise

misbehaving. Unfortunately, even with a background

there are serious problems: we can use perturbation

theory to write the answers to physics questions as

power series, but these series diverge and nobody

knows how to resum them.

String theorists are pragmatic in a certain sense:

they don't mind using a background, and they don't

mind doing what physicists always do: approximating a

divergent series by the sum of the first couple of

terms. But this attitude doesn't solve everything,

because right now in string theory there is an

enormous "landscape" of different backgrounds, with no

firm principle for choosing one. Some estimates guess

there are over 10100. Leonard Susskind guesses there

are 10500, and argues that we'll need the anthropic

principle to choose the one describing our world:

3) Jan Ambjorn, Jerzy Jurkiewicz and Renate Loll,

Emergence of a 4d world from causal quantum gravity,

available as hep-th/0404156.

This trio of researchers have revitalized an approach

called "dynamical triangulations" where we calculate

path integrals in quantum gravity by summing over

different ways of building spacetime out of little

4-simplices. They showed that if we restrict this sum

to spacetimes with a well-behaved concept of

causality, we get good results. This is a bit

startling, because after decades of work, most

researchers had despaired of getting general

relativity to emerge at large distances starting from

the dynamical triangulations approach. But, these

people hadn't noticed a certain flaw in the

approach... a flaw which Loll and collaborators

noticed and fixed!

If you don't know what a path integral is, don't

worry: it's pretty simple. Basically, in quantum

physics we can calculate the expected value of any

physical quantity by doing an average over all

possible histories of the system in question, with

each history weighted by a complex number called its

"amplitude". For a particle, a history is just a path

in space; to average over all histories is to

integrate over all paths - hence the term "path

integral". But in quantum gravity, a history is

nothing other than a SPACETIME.

Mathematically, a "spacetime" is something like a

4-dimensional manifold equipped with a Lorentzian

metric. But it's hard to integrate over all of these -

there are just too darn many. So, sometimes people

instead treat spacetime as made of little discrete

building blocks, turning the path integral into a sum.

You can either take this seriously or treat it as a

kind of approximation. Luckily, the calculations work

the same either way!

If you're looking to build spacetime out of some sort

of discrete building block, a handy candidate is the

"4-simplex": the 4-dimensional analogue of a

tetrahedron. This shape is rigid once you fix the

lengths of its 10 edges, which correspond to the 10

components of the metric tensor in general relativity.

There are lots of approaches to the path integrals in

quantum gravity that start by chopping spacetime into

4-simplices. The weird special thing about dynamical

triangulations is that here we usually assume every

4-simplex in spacetime has the same shape. The

different spacetimes arise solely from different ways

of sticking the 4-simplices together.

Why such a drastic simplifying assumption? To make

calculations quick and easy! The goal is get models

where you can simulate quantum geometry on your laptop

- or at least a supercomputer. The hope is that

simplifying assumptions about physics at the Planck

scale will wash out and not make much difference on

large length scales.

Computations using the so-called "renormalization

group flow" suggest that this hope is true if the path

integral is dominated by spacetimes that look, when

viewed from afar, almost like 4d manifolds with smooth

metrics. Given this, it seems we're bound to get

general relativity at large distance scales - perhaps

with a nonzero cosmological constant, and perhaps

including various forms of matter.

Unfortunately, in all previous dynamical triangulation

models, the path integral was not dominated by

spacetimes that look like nice 4d manifolds from afar!

Depending on the details, one either got a "crumpled

phase" dominated by spacetimes where almost all the

4-simplices touch each other, or a "branched polymer

phase" dominated by spacetimes where the 4-simplices

form treelike structures. There's a transition between

these two phases, but unfortunately it seems to be a

1st-order phase transition - not the sort we can get

anything useful out of. For a nice review of these

calculations, see:

4) Renate Loll, Discrete approaches to quantum gravity

in four dimensions, available as gr-qc/9805049 or as a

website at Living Reviews in Relativity,

http://www.livingreviews.org/Article...1/1998-13loll/

Luckily, all these calculations shared a common flaw!

Computer calculations of path integrals become a lot

easier if instead of assigning a complex "amplitude"

to each history, we assign it a positive real number:

a "relative probability". The basic reason is that

unlike positive real numbers, complex numbers can

cancel out when you sum them!

When we have relative probabilities, it's the highly

probable histories that contribute most to the

expected value of any physical quantity. We can use

something called the "Metropolis algorithm" to spot

these highly probable histories and spend most of our

time focusing on them.

This doesn't work when we have complex amplitudes,

since even a history with a big amplitude can be

canceled out by a nearby history with the opposite big

amplitude! Indeed, this happens all the time. So,

instead of histories with big amplitudes, it's the

bunches of histories that happen not to completely

cancel out that really matter. Nobody knows an

efficient general-purpose algorithm to deal with this!

For this reason, physicists often use a trick called

"Wick rotation" that converts amplitudes to relative

probabilities. To do this trick, we just replace time

by imaginary time! In other words, wherever we see the

variable "t" for time in any formula, we replace it by

"it". Magically, this often does the job: our

amplitudes turn into relative probabilities! We then

go ahead and calculate stuff. Then we take this stuff

and go back and replace "it" everywhere by "t" to get

our final answers.

While the deep inner meaning of this trick is

mysterious, it can be justified in a wide variety of

contexts using the "Osterwalder-Schrader theorem".

Here's a pretty general version of this theorem,

suitable for quantum gravity:

5) Abhay Ashtekar, Donald Marolf, Jose Mourao and

Thomas Thiemann, Constructing Hamiltonian quantum

theories from path integrals in a diffeomorphism

invariant context, Class. Quant. Grav. 17 (2000)

4919-4940. Also available as quant-ph/9904094.

People use Wick rotation in all work on dynamical

triangulations. Unfortunately, this is not a context

where you can justify this trick by appealing to the

Osterwalder-Schrader theorem. The problem is that

there's no good notion of a time coordinate "t" on

your typical spacetime built by sticking together a

bunch of 4-simplices!

The new work by Ambjorn, Jurkiewiecz and Loll deals

with this by restricting to spacetimes that do have a

time coordinate. More precisely, they fix a

3-dimensional manifold and consider all possible

triangulations of this manifold by regular tetrahedra.

These are the allowed "slices" of spacetime - they

represent different possible geometries of space at a

given time. They then consider spacetimes having

slices of this form joined together by 4-simplices in

a few simple ways.

The slicing gives a preferred time parameter "t". On

the one hand this goes against our desire in general

relativity to avoid a preferred time coordinate - but

on the other hand, it allows Wick rotation. So, they

can use the Metropolis algorithm to compute things to

their hearts' content and then replace "it" by "t" at

the end.

When they do this, they get convincing good evidence

that the spacetimes which dominate the path integral

look approximately like nice smooth 4-dimensional

manifolds at large distances! Take a look at their

graphs and pictures - a picture is worth a thousand

words...>>

here is the link to Baez column "This Week's Finds in

Mathematical Physics #206" from which I'm quoting

http://math.ucr.edu/home/baez/week206.html