CDT 2007

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 blocks

The 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.

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