Introduction To Loop Quantum Gravity

marcus

Someone might be curious as to what "Introduction to LQG(Mark II)" is all about, what is the main direction, if one were to look ahead. Someone teaching a course for juniors/seniors in LQG might, by the end of the semester, want to be in sight of this body of work:
http://arxiv.org/find/grp_physics/1/.../0/1/0/all/0/1

and in particular within striking distance of this 2001 trailblazer
http://arxiv.org/abs/gr-qc/0102069
Absence of Singularity in Loop Quantum Cosmology

it is just 4 pages. the classical BB singularity is replaced by a bounce (from a prior gravitational collapse)
later it was discovered that conditions at the bounce automatically trigger a brief episode of inflation (without fine-tuning or elaborate "extras")
see for example http://arxiv.org/abs/gr-qc/0407069, "Genericness of Inflation in LQC" (also just 4 pages) and references thererin.

here are the papers which have cited this key paper:
http://arxiv.org/cits/gr-qc/0102069
there are currently about 75 papers which have cited it, and about half of these appeared after January 2004

marcus

One's approach to LQG, in an introduction, depends a lot on where one wants to be at the end of it and their are several equally valid goals that one could have. For me what stands out is that quantizing General Relativity gets rid of some important singularities where classical GR broke down and allows one to study in more detail what goes on there.

For example it might be good to aim for making contact not only with "Absence of Singularity in LQC" but also
http://arxiv.org/abs/gr-qc/0503041
which treats what emerges when the classical black hole singularity is removed by LQG.

~()

marlon, so are you saying that there is some sort of ''ether'', a actual physical property to space that changes with the interaction of particles on particles? The space itself warps, changes value, and interacts with the particles themself - this being the gravity ?

marcus

It's common knowledge that the three people most responsible for initiating the LQG approach to quantizing General Relativity are Abhay Ashtekar, Carlo Rovelli, and Lee Smolin.

A good way to get a sense of what LQG is about is to keep an eye on major books and survey articles by these people, since they are like the "founding fathers" of the field.

From Smolin we have an excellent recent survey article "An Invitation to LQG" which gives useful information for the trained physicist considering getting into LQG research----main results, experimental tests, and a list of unsolved problems to work on.

From Rovelli we have his book Quantum Gravity which came out November 2004 published by Cambridge Press. the December 2003 draft is still online. He also has some earlier surveys and popular articles.

From Ashtekar there are several valuable surveys. Here are links to a couple of the more recent ones that might be useful.
http://arxiv.org/abs/gr-qc/0410054
http://arxiv.org/abs/gr-qc/0404018
But what is especially interesting right now is a book Ashtekar is preparing, to be published by World Scientific, called
A Hundred Years of Relativity.

this book has a broad scope including all of General Relativity, and it will show how Ashtekar sees LQG and other allied approaches to quantizing Gen Rel in their wider context.

Interestingly, several chapters of this book "100Y.of R." are already online as preprints!

I will get links for some preprint chapters.

Martin Bojowald
[he has contributed an article called "Loop Quantum Cosmology"
which I have not yet seen online]

Larry Ford
http://arxiv.org/abs/gr-qc/0504096

Rodolfo Gambini and Jorge Pullin
http://arxiv.org/abs/gr-qc/0505023

Hermann Nicolai
["Gravitational Billiards, Dualities and Hidden Symmetries" not yet online]

Thanu Padmanabhan
http://arxiv.org/abs/gr-qc/0503107

Alan Rendall
http://arxiv.org/abs/gr-qc/0503112

Clifford Will
http://arxiv.org/abs/gr-qc/0504086

Although Bojowald's article may not be available yet, see
http://edoc.mpg.de/display.epl?mode=...&name=Bojowald

and also
http://edoc.mpg.de/display.epl?mode=...5&col=6&grp=84

marcus

More information has come in, so i will revise parts of the preceding post.
From Ashtekar there are several valuable surveys. Here are links to a couple of recent ones that might be useful.
http://arxiv.org/abs/gr-qc/0410054
Gravity and the Quantum
http://arxiv.org/abs/gr-qc/0404018
Background Independent Quantum Gravity: A Status Report

What is especially interesting right now is a book Ashtekar is preparing, to be published by World Scientific, called
A Hundred Years of Relativity.

This book has a broad scope covering all of General Relativity, including numerical GR and testing. It will show how Ashtekar sees LQG and allied approaches to quantizing Gen Rel in the wider context. Several chapters of this book are already online as preprints:

Martin Bojowald
http://arxiv.org/abs/gr-qc/0505057
Elements of Loop Quantum Cosmology

Larry Ford
http://arxiv.org/abs/gr-qc/0504096

Rodolfo Gambini and Jorge Pullin
http://arxiv.org/abs/gr-qc/0505023
Discrete space-time

Hermann Nicolai
["Gravitational Billiards, Dualities and Hidden Symmetries" not yet online]

Thanu Padmanabhan
http://arxiv.org/abs/gr-qc/0503107
Understanding Our Universe: Current Status and Open Issues

Alan Rendall
http://arxiv.org/abs/gr-qc/0503112

Clifford Will
http://arxiv.org/abs/gr-qc/0504086
Was Einstein Right? Testing Relativity at the Centenary

marcus

Because LQG contains some leading-edge lines of research it cannot be given a fixed definition. The practical definition is that it is what Loop people do-----and in practice that means the research lines that are featured in this year's major Loop conference(s).

So for an operational definition of Loop-and-allied approaches to Quantum Gravity, watch the Programme of the October 10-14 conference at Potsdam AEI. Here's the link and some available details:

http://loops05.aei.mpg.de/index_files/Home.html

http://loops05.aei.mpg.de/index_files/Programme.html

The topics of this conference will include:

Background Independent Algebraic QFT
Causal Sets
Dynamical Triangulations
Loop Quantum Gravity
Non-perturbative Path Integrals
String Theory

A detailed programme will be available in July.

Invited Speakers will include:
Abhay Ashtekar (USA)
John Baez (USA)
John Barrett (UK)
Alejandro Corichi (MEX)
Robbert Dijkgraaf (NL)
Fay Dowker (UK)
Laurent Freidel (FR and CA)
Karel Kuchar (USA)
Jurek Lewandowski (POL)
Renate Loll (NL)
Roy Martens (UK)
Hugo Morales Tecotl (MEX)
Alejandro Perez (FR)
Jorge Pullin (USA)
Martin Reuter (GER)
Carlo Rovelli (FR)
Lee Smolin (CA)
Rafael Sorkin (USA)
Stefan Theisen (GER)
Rainer Verch (GER)
-------------------------------
My comment: because these are fast developing areas of research, it makes sense not to nail down the TITLES of the invited speaker's talks until shortly before the conference (July is 3 months before, plenty of time) but it's nice to know WHO will be giving the plenary talks.

I would say that String and old-style LQG are no longer leading edge, and I dont have a big interest in Causal Sets. So I would narrow the exciting topics down to these:

Dynamical Triangulations
Background Independent Algebraic QFT
Non-perturbative Path Integrals

1. Notice that Renate Loll is on the invited list. She will talk about CDT, causal dynam. triang.
This is currently the deepest part of Loop-and-allied research. Anyone interested in LQG, or quantum gravity in general for that matter, should know about it.

2. What they mean by "Background Independent" QFT is basically that it is done on a (metric-less) differentiable manifold. the way you work on a shapeless continuum without first introducing a prior geometry is you use
DIFFERENTIAL FORMS and stuff like bundles and connections. Cumrun Vafa's term for one case of this is "form theories of gravity". A key invited speaker in this line would be Laurent Freidel.

I am not sure what Background Independent "Algebraic" QFT means. I think the papers of Rainer Verch (which I dont know) could touch on this.

the moment you posit a manifold you have already specified a dimension like D = 4 and you already have patches of coordinates but notice that the CDT of Renate Loll does not have a prior commitment to a dimension and it uses NO COORDINATES AT ALL. the brilliant Tullio Regge figured in 1950 how to do Einstein Gen Rel without coordinates. and, in CDT which is basically a child of Regge, the dimension emerges rather than being specified in advance and the dimension can vary with scale----it can be 4D at macro and run smoothly down to around 2D at micro-scale.

This is why I cannot escape concluding that CDT is deeper-probing. It may be WRONG we dont know about right or wrong. However it seems to have Gen Rel as its classical limit, and integrate out to a simple quantum cosmology associated with Hawking as a kind of semiclassical limit.

3. Non-perturbative Path Integrals might be an improved and more general term for what used to be called Spin Foams, but it also includes CDT because in CDT you get a path integral. Which, however, is evaluated barbarically using Monte Carlo runs on the computer.

This is going to be an interesting Loops 05 Conference and I guess it is the conference that defines the field (more than the other way round).
So we will see in Potsdam in October what LQG is.

marcus

in case anyone is interested in getting a tast of causal dynamical triangulations (CDT) here is a short reading list.

A new monograph "Reconstructing the Universe" is due to come out this month. It will replace the 2001 paper which I link to here. the 3 short papers from 2004 and 2005 give the highlights of recent research results.
It is better to first read the 3 short recent papers before getting into the details in the 2001 paper IMHO.

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."

marcus

AFAICS a breakthrough form of simplex path-integral gravity called causal dynamical triangulations (CDT) is the most important current development in Quantum Gravity going on. In case anyone is interested in getting a taste of CDT here is a short reading list.

this is an update of what I listed earlier:


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."

this is a landmark paper.
I have been looking also for a reader-friendly introductor paper. there is one that is lecture notes aimed at the graduate student level

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.

marcus

Here is a reminder of the importance of background independence (no prior metric) and diffeomorphism invariance (general covariance) in the case where the spacetime model is a differential manifold.
these are principles are basic to LQG, and to all allied approaches to quantum gravity.

in fact these two features are basic to classical 1915 General Relativity! So any approach that really tries to quantize Gen Rel is going to exhibit these features or the equivalent

Anyway this is sometimes pointed out as one of the troubles with string theory---that it doesnt have background independence etc. And people debate this. I will not take a stand but simply point out that these principles are really important---and implementing them has shaped LQG and some related approaches---and that one can get into trouble if one does not.

this was illustrated by something posted a few minutes ago in "Third Road" sticky-thread,

http://arxiv.org/gr-qc/0505138
Fibered Manifolds, Natural Bundles, Structured Sets, G-Sets and all that: The Hole Story from Space Time to Elementary Particles
J. Stachel, M. Iftime
40 pages

The article had this in the conclusions (partly already quoted in "third road" thread, but we can use them too as emphasizing how crucial background independence is) :

<<...Perturbative string theory fails this test, since the background spacetime (of no matter how many dimensions) is only invariant under a finite parameter Lie subgroup of the group of all possible diffeomorphisms of its elements. This point now seems to be widely acknowledged in the string community. I quote from two recent review articles. Speaking of the original string theory Michael Green[19] notes: “This description of string theory is wedded to a semiclassical perturbative formulation in which the string is viewed as a particle moving through a fixed background geometry .... Although the series of superstring diagrams has an elegant description in terms of two-dimensional surfaces embedded in spacetime, this is only the perturbative approximation to some underlying structure that must include a description of the quantum geometry of the target space as well as the strings propagating through it ( p. A78). ... A conceptually complete theory of quantum gravity cannot be based on a background dependent perturbation theory ...."

"In ... a complete formulation the notion of string-like particles would arise only as an approximation, as would the whole notion of classical spacetime (p. A 86) ” Speaking of the more recent development of M-theory, Green says: “An even worse problem with the present formulation of the matrix model is that the formalism is manifestly background dependent. This may be adequate for understanding M theory in specific backgrounds but is obviously not the fundamental way of describing quantum gravity (p. A 96).”

And in a review of matrix theory, Thomas Banks comments: String theorists have long fantasized about a beautiful new physical principle which will replace Einsteins marriage of Riemannian geometry and gravitation. Matrix theory most emphatically does not provide us with such a principle. Gravity and geometry emerge in a rather awkward fashion, if at all. Surely this is the major defect of the current formulation, and we need to make a further conceptual step in order to overcome it (pp. 181-182). It is my hope that emphasis on the importance of the principle of dynamic individuation of the fundamental entities, with its corollary requirement of invariance of the theory under the entire permutation group acting on these entities, constitutes a small contribution to the taking of that further conceptual step. >>

[19] Green(1999) Superstrings, M-theory and quantum gravity, Classical and Quantum Gravity, 16, A77-A100

Michael Green and Thomas Banks are major figures in string/M research---originators----and speak with authority. They may be wrong (they are the experts on string, not me, so I cannot judge if they are right or not) but in any case these strong words help give adequate emphasis to the issues of background independence and invariance under diffeomorphic mappings.

marcus

somebody might wish to ask how Loll-style "Triangulations" gravity implements background independence and diffeo invariance (or reasonable substitute, since it doesnt have any diffeos)

marcus

the triangulations QG approach of Loll and coworkers looks like the most interesting, and perhaps promising, development being pursued by the people participating in this years "Loops 05" conference.

It is one of the broadly defined "Loop-and-allied" approaches that Loop people do----not narrowly defined core LQG. there are a bunch of approaches that deal with similar stuff but differ in details.

this thread can serve a useful purpose as an INTRODUCTION to more than just one of the Loop-and-allied approaches. Probably the most timely to consider at the moment is CDT-style Triangulations.

As a point of departure here is how the abstract of a recent landmark CDT paper starts off:

"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,..."

this is from
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

here is a short reading list
http://www.physicsforums.com/showpos...4&postcount=59

Now what I want to do is describe the CDT "Triangulations" method as simply as I can.

marcus

First there is a kind of birdseye view illustrated by a talk that Renate Loll gave in 2002 called
"Quantum gravity IS counting geometries"

It is a possible approach---sometimes called "state sum" or "path integral"
Its roots go back to the Feynman path integral for a particle where you add up all the possible (approximate piecewise straight) paths the particle might take to get from here to there----with complex weights to make a kind of weighted average. It is a way to get probability amplitudes and calculate things about the quantum path the particle takes. This turns out to be 100 percent of the time very ROUGH, nowhere is it even differentiable, but nevertheless intuitively it kind of blurs or fuzzes out to resemble a smooth continuously differentiable classical path that you might expect from freshman calculus.

In the "state sum" approach to a particle's path the calculation is in effect counting lots of different (piecewise straight) paths. And adding them all together, with a system of weighting that embodies the microscopic dynamics, to get answers. It has been a very successful method. the CDT authors found out how to apply it to spacetimes.

A spacetime is like a path, from space being this way to space having evolved to be that way, or more grandly from the beginning of a universe to its end. In QUANTUM gravity, that is in quantum spacetime dynamics, one is not certain exactly which path it took. One only has amplitudes of various ways of evolving from this shape to that shape. It is very much analogous to the particle path. You can even think of the universe wandering around in the space of all geometries and its evolution an actual path, but I can see no compelling reason to think so abstractly as that about it.

to put it simply, the CDT authors found a way to approximate (by piecewise flat geometries, made of flat Minkowski building blocks) all the possible spacetimes that get you from here to there, or from the beginning to the end. And they found a way to compute experimental answers from the STATE SUM of all these geometries.

After that it almost seems obvious and really straightforward. They can generate random spacetimes, random histories of the universe, as 4D worlds living in the computer memory, and they can HAVE LITTLE IMAGINARY MEN RUN AROUND IN THEM TO EXPLORE THEM, by taking random walks----a so called diffusion process---which is a way of finding out about the geometry, like what dimension it really is in there.

then after studying each random example they can add everything up with the usual weights (there are actually two sets of weights connected by the Wick trick, one set is simply real numbers like ordinary probabilities and the other set of weights is complex amplitude-type numbers but this doesnt matter to the overall picture) So they add everything up in a weighted average and get the state sum report (from the little imaginary men) on how it is in there.

Now having done this, Loll and co-workers are catching results like the fish are running. They are just pulling them in hand over hand. throw in the line and hook one every time.
This is a big change from the Nineties when many people worked the state sum triangulations approach but didnt catch anything edible. everything they got was the wrong dimension.

so this is part of an overview.

what I have to EXPLAIN is how they set up one of these layered triangulated geometries----and how they then shuffle the cards so as to get a series of random geometries. this is the nutsandbolts part.

a 4-simplex is the 4D analog of a triangle and they build these appoximate piecewise flat geometries out of two TYPES of 4-simples, the
"level"-kind and the "tilt"-kind

they call them the (4,1) kind and the (3,2) kind. it is how the vertices are destributed between two causal layers

I have to balance giving an overview with giving some introductory nutsandbolts.

marcus

Here is some more overview. It elucidates the "state sum" idea of adding up all possible geometries. and the essential business of reducing the calculation to COUNTING.


In this case what we have is something from an American Physical Society publiscation

http://focus.aps.org/story/v14/st13

The American Physical Society sponsors the major peer-review journals series Phys. Rev. and Physical Review Letters. And they pick out articles for highlighting journalistically in the accompanying publication Physical Review Focus.

This is from Adrian Cho's Focus article on a paper by Loll and co-workers.

<<The researchers added up all the possible spacetimes to see if something like a large-scale four-dimensional spacetime would emerge from the sum. That was not guaranteed, even though the tiny bits of spacetime were four-dimensional. On larger scales the spacetime could curve in ways that would effectively change its dimension, just as a two-dimensional sheet of paper can be wadded into a three-dimensional ball or rolled into a nearly one-dimensional tube. This time the researchers found that they could achieve something that appeared to have one time dimension and three space dimensions--like the universe we know and love.

"It's exceedingly important" work, says Lee Smolin of the Perimeter Institute for Theoretical Physics in Waterloo, Canada. "Now at least we know one way to do this." Des Johnston of Heriot-Watt University in Edinburgh, Scotland, agrees the work is "very exciting" and says it underlines the importance of causality. "The other neat thing about this work is that you're essentially reducing general relativity to a counting problem," Johnston says. "It's a very minimalist approach to looking at gravity.">>

marcus

the best source on the basics is http://arxiv.org/hep-th/0105267

we have to COUNT THE CAUSAL GEOMETRIES of spacetime, it sounds terribly hard but it isnt and they managed to program it, and it's the basic job we cant get around

causal means LAYERED, each model of spacetime gets laid down in sheets or slices, like a book with pages or a tree-trunk with rings, an event in one layer can only be caused by something from a deeper layer----or think of it like a many-storied building.

so we have to BUILD ALL POSSIBLE LAYERED SPACETIME GEOMETRIES in such a way that we can COUNT THEM or anyway explore to find what are the most numerous kind or the most likely kind, or somehow average them.

Maybe in the end we wont be able to count them exactly but we will have statistics and averages and random samples about them just as if we could actually count them. We will take the census of these layered spacetime geometries.

The technique will be to learn how to build layered geometries using "triangular" building blocks cut out of the txyz space of special relativity

these blocks will all be the same size----their spatial edges will be a fixed length 'a' that we will successively make smaller and smaller---and they will be of two kinds. the LEVEL kind and the TILT kind. The level kind is like a pyramid which has a 3D spatial tetrahedron as its base, on one floor of the building, and its 5-th vertex on the floor above or, the upsidedown version, the floor below.

the authors write the level kind as either (4,1) or (1,4), because it has 4 vertexes (the 4 vertices of the tetrahedron) on this floor and 1 vertex on the floor above, or viceversa one vertex on this floor and 4 on the floor above

intuitively one layer is all of 3D space, and the spacetime history of the universe is being built 3D layer by 3D layer, so it is like a book except the pages are 3D.

a LEVEL kind of building block has 4 timelike edges going from each of the four corners of its spatial tetrahedron base up to the vertex on the floor above, or else going down to the solitary vertex on the floor below. the other kind of buildingblock is like the LEVEL kind but tilted over so that now one of those timelike edges becomes a ridge and is entirely in the floor above, and instead of sitting on a full tetrahedron base it is now only sitting on a triangle side of it.

the authors write the TILT kind of buildingblock as either (3,2) or (2,3)
because it has 3 vertices on one floor, that make its spatial triangle base, and it has 2 vertices on the floor above or below, that make this ridge I mentioned. Like the ridge of a roof or the keel of a boat, depending it is up or down.

the TILT kind has 4 spacelike edges (three for the triangle and one for the base) and it has 6 timelike edges, whereas the LEVEL kind had 6 spacelike (that you need to make a tetrahedron) and 4 timelike.

the quickest way to understand this business is to follow through the analogous 3D case which is spelled out in
http://arxiv.org/hep-th/0105267

there, the building blocks are tetrahedrons---spatial layers are intuitively 2D, like the pages of a book---everything is easy to imagine, and they have a lot of drawings

but I am trying to discuss modeling 4D spacetime geometry without first going thru the 3D case.

marcus

after that it is not too hard to say, in general terms, how the method works

you want to get the EFFECT of building a layered geometry with, say, a halfmillion identical LEVEL kind buildingblocks
and howevermany you need (which will also be about halfmillion) TILT kind blocks to fill in.

because when you try to build layers with the LEVEL kind it always turns out that you get gaps which are just right to fit the other kind into, so it turns out that to build up layer by layer you need approximately the same number of the other kind.

NOW YOU DONT ACTUALLY BUILD EVERY POSSIBLE LAYERED SPACETIME GEOMETRY with these million virtually identical blocks

it is like taking an opinion poll where you dont talk to everybody, you take a random sample.

you want the EFFECT of having built all of them, and studied each one, and counted and made statistics about how they all are. you dont want to actually do it. you want the effect as if you did it.

this is where "shuffling the deck of cards" comes in. the CDT authors call it "thermalizing" the geometry. you set up a very simple plain geometry to start with, in computer memory, and then you do RANDOMIZING PASSES thru it, until it gradually becomes totally unrecognizable.


like, have a look at Figures 4,5 and 6 in "Reconstructing the Universe"
http://arxiv.org/hep-th/0505154
they are all three quite different-looking but they all come from starting with a simple initial geometry and doing randomizing passes.

the authors call each pass "making a sweep", and each sweep involves doing a million or so "Monte Carlo moves" which are individual shuffles that change some of the building blocks around.

they use a lot of computer time. thermalizing (thoroughly randomizing) a geometry can take a week on a workstation. then you study it and measure things

when you have a random geometry you can run random walks in it, or diffusion process, and you can measure distances and volumes and see how they relate...

marcus

here is an example.
say you have built a layered spacetime geometry in your computer and you pick just one spatial layer and you want to explore that by a random walk.

well a spacelike slice is just made of the tetrahedrons which were the bases of the LEVEL buildingblocks!
So you have a set of things in your computer which are little 4-face pyramids (equilateral triangle bottom, so 3 side faces and a bottom face, except no way to distinguish the bottom from the other faces). And this bunch of tetrahedrons are fitted together some way so that every face of one is up against the face of some other.

So you can pick a random block to start in, and then TOSS A FOURSIDED COIN to select which face to go out of

and when you go out one face you are now in a new tetrahedron and you can toss the foursided coin to choose which face to pass thru, and again and again.

it will be a clue to the actual dimensionality of the spatial slice to see if you get completely lost by doing this random walk, or if you now and then get back home to where you started. the authors determine the probabilities EMPIRICALLY by actually running the random walks in the computer, and this tells them about the dimensionality of the spatial slices

the nice thing is the answers gotten this way are weird and quite Alice-in-Wonderland. at microscopic level the continuum (as pictured by CDT) is a non-classical, unexpected world which Lewis Carroll would have loved.

marcus

in the path integral or state sum approach you make a WEIGHTED sum using a badness or handicap function S(path) which is large for really kooky unphysical paths

this is how you introduce the classical dynamics into the quantum picture.

Feynmann has an essay about the "least action principle" in his Lectures, it is one of the core things in the Feynmann physics textbook. It is terribly important and he suddenly takes a serious tone of voice when he gets to it.

the way you do classical physics is you consider all the possible paths and you DIFFERENTIATE S(path) and set to zero so as pick the one and only one path that MINIMIZES S(path)-------you pick the unique path that minimizes the "action", which is a word for badness or silliness of paths.

we have inherited our "action" functions from great old classical guys like Lagrange and Einstein and Hamilton, directly or indirectly, whom we revere, and they have the feature that minimizing them gets you the expected classical equations of motion

the way you do quantum physics is when you consider all possible paths you dont try to pick the unique winner, you ADD THEM ALL TOGETHER, but you dont do this in a completely indiscriminate way! You handicap each one by putting a little real or complex number "weight" on it. this is incidentally how they used to handicap racehorses, with a little weight, but you can do it at a different level with the betting odds too.

one weight you might consider putting on is exp( - S(path))
that is:
"eee to the minus badness"

e^-badness

so if the badness is large it make the weight exponentially very small and then the path tagged with that weight will not count for very much in the sum or weighted average of all paths

YOU DO NOT JUST PICK ONE HORSE THAT IS YOUR FAVORITE, you add together all the horses, but you weight each one so your favorites count for more and the bad ones count for less-------you get a "composite" horse.

another kind of weight you might consider putting on is exp(iS(path))

"eee to the eye badness"

e^{i badness}

As you may know from elementary complex numbers "eee to the eye theta" is complex numbers going around and around the unit radius circle.
and this is very clever because if you go around the circle, around zero, very fast it will average out to ZERO ITSELF by simple vector addition.

taking a step N and S and E and W adds up to going nowhere

so if you are averaging things with rapidly increasing badness and tagging them with "eee to the eye badness" numbers then these things with lots of badness will CANCEL EACH OTHER OUT in the sum and not have much influence on the sum

this is the two kinds of handicaps, the real number weights and the complex number weights from around the unit radius circle.

YOU CAN GET FROM ONE SET OF WEIGHTS TO THE OTHER SET OF WEIGHTS by the simple expedient of changing the eye into a minus sign, or the minus into an eye. this is called the WICK ROTATION, in honor of Joe Wick born in Torino around 1906.