Introduction To Loop Quantum Gravity

[marcus]

Introduction to LQG Part II

it isn't good to stay permanently with the naive, non-math story told of LQG in popular accounts, because it can lead to misconceptions.

the main reference cited in the first segment of this thread is a popular wide-audience presentation by Rovelli. It was published in "Physics World" November 2003 IIRC and it is very good for what it is: a non-math story. I have recommended it to beginners as a first exposure to LQG and it seems to work fine. But it is easy to get misled by popularizations and at some point you have to move on.

It came to my attention that there are seriously mistaken ideas going around that appear to come from the impressionistic verbal story (as told by Rovelli or Smolin or whoever) and need to be corrected, so it is probably time to start Part II of this introduction.

 

[marcus]

LQG can mean several things

LQG is either used to refer to a specific definite approach to quantizing Gen Rel, or it is used to refer to a collection of allied approaches that many of the same people (the LQG people) work on.

there should be a recognized collective name like "Loop-and-allied Quantum Gravity", but there isn't yet. People just say LQG for the whole collection.
If we had a collective name for the field like "Loop-and-allied Quantum Gravity", then it would include LQG in the specialized restrictive sense and would also include spinfoam research and some related form-theories of gravity (modified topological field theories) and some allied path-integral approaches like dynamical triangulations. Although most of these approaches are what is called "nonperturbative" it would even include a recently initiated perturbative approach. In a growing field the terminology is necessarily loose and you cannot perfectly delineate things in advance.

BTW I am not an expert, i just watch QG, so I don't speak authoritatively. But I'm pretty sure of what I'm telling you. The field is dynamic and in flux.
It is a creative time in LQG.

So when there is a "LQG conference" people give papers on all these allied lines of research. Then LQG is used as a collective name to refer to a bunch of things. It is what "LQG people" do and it includes a lot of approaches to quantizing the theory of spacetime and its geometry----to quantizing Gen Rel. probabably some of these approaches are eventually going to change and converge and turn out to be connected or even equivalent. but we can't say which, in advance.

So keep in mind that sometimes LQG means a particular canonical nonperturbative approach to quantizing Gen Rel (an approach that Abhay Ashtekar, who helped invent it, likes to call "quantum geometry") and sometimes LQG means a bunch of related lines of investigation that LQG people are currently pursuing.

 

[marcus]

There are textbook-level LQG sources

Rovelli's book Quantum Gravity (Cambridge 2004), also on-line 2003 draft free
Thiemann's Lecture Notes (Springer 2004?) also on-line free
Ashtekar Lewandowski Background Independent Quantum Gravity (2004) online
Rovelli's 1998 Living Reviews introduction online
Smolin 2004 Invitation to Loop Quantum Gravity online

The links to these things are in the surrogate sticky thread where we've been keeping LQG links. You can also get them with an arxiv search or google search using author's name. When time permits i will fetch them from the LQG links thread.

 

[marcus]

Textbook LQG is based on a differentiable manifold

the diff manif is a key concept in mathematics and the main thing separating popular accounts from textbook level.

In Gen Rel, spacetime is represented by a geometry-less floppy limp shapeless thing called a differentiable manifold
this is sometimes called a "continuum"
it is basically a set with a collection of coordinate charts.

you have to realize that a diff-manif is harmless. the good thing about it is that it doesn't have any pre-committment to any particular geometry!

you can impose whatever geometry on a diff-manif by specifying a METRIC or distance function, that will then allow you to calculate areas and volumes and angles and define what corresponds to geodesics or "straight" lines.

the great thing about a diff-manif, before you choose a metric, is that it comes into the world without any preconceptions, innocent of bias in favor of this or that geometry.

Gen Rel has a version which uses a 3D differentiable manifold representing space. It does not always have to be constructed using a 4D diff-manif spacetime. But either way it is based on a limp diff-manif ON WHICH A METRIC IS LATER IMPOSED where the metric arises as a solution of the einstein equation.

you start out with a shapeless continuum and you set up some conditions (which can involve having some matter in the picture) and then you crank out what the geometry (as shown by the distance function) is going to be.

OK, now LQG is characterized by the fact that it tries to imitate Gen Rel very closely. So the first thing you get in textbook LQG is a shapeless differentiable manifold representing space.

This version of space is infinitely divisible and smooth and continuous like any differentiable manifold has to be. it is the same diff-manif model of space that you get in one of the versions of Gen Rel.

WHAT IS "QUANTUM" ABOUT LQG IS HOW YOU PUT THE GEOMETRY ON

 

[marcus]

"QUANTUM" is a way of handling uncertainty and incomplete information realistically

the main feature of quantum mechanics is that it copes in an apparently realistic way with indeterminacy, uncertainty, the incomplete information that one system or observe has about another system.

in the real world everything depends, literally, on who is observing what.

and no who can ever know everything about any what.

it is not possible to have a realistic description of the world which fails to take this into account

"QUANTUM" IS NOT ABOUT DIVIDING SPACE UP INTO LITTLE BITS
quantum is not about dividing anything up into little bits

It can happen that discrete spectra come out of the mathematics, you quantize a system and you find that a certain measurement has a discrete range of possibilities, like the energy levels of a hydrogen atom

but this discreteness is a byproduct of what quantizing is really about, which is setting up a way to implement uncertainty and incomplete information-----call it fuzziness, call it probability.

the predominant mathematical machine that quantum theories use to hold the uncertainty and deal out the probabilities is called a hilbertspace
The possible states of a system are represented by the hilbertspace and measurements of the realworld system correspond to linear OPERATORS on the hilbertspace.

But if you don't know what a hilberspace is, or what operators on it are don't worry, someday some different language will be invented. What matters now is that

QUANTIZING A CLASSICAL THEORY IS NOT primarily about dividing some stuff into little bits. It is primarily about building a machine which can represent states of the system and measurements on the system but embodies uncertainty.

QUANTIZING GEN REL means to build a machine that can represent STATES OF GEOMETRY and MEASUREMENTS OF GEOMETRIC VARIABLES like area and volume and angle and so on, and which is also more realistic than classical Gen Rel because it implements the inherent uncertainties.

 

[marcus]

Quantizing Gen Rel gets rid of singularities

Gen Rel is an amazingly accurate theory of spacetime geometry whever it is applicable, where it doesn't break down and fail to compute.
Where it is applicable it predicts very fine differences in angles and times out to many decimal places. People have tried for decades to improve on it, or to test it and find it wrong out at the 6th decimal place. But they haven't succeeded yet.

But Gen Rel famously has places where it blows up and predicts infinities, in other words it is flawed. It has singularities.

this has been the case with other classical theories and it has been found that if you can QUANTIZE a classical theory it will often extend the applicability and get rid of places where it breaks down.

So a big aim of quantizing Gen Rel is to get rid of the classical singularities. mainly the "bigbang" and "blackhole" singularities.

The main reason why LQG is so active these days is that it appears to have removed Gen Rel singularities. the main reason Martin Bojowald is a key LQG figure is that he has been in the forefront in this and has gathered a considerable group of people who are working on this.

the first break came in 2001 when MB removed the bigbang classical singularity in a certain case.

To get the history since then, just go to arxiv.org and get the list of all Bojowald papers since 2001. the people active in this field are the people who have co-authored papers with Bojowald, and you can click on their names and find all the papers they have published independently.

the black hole singularity is being removed just now, starting at end 2004 and very much at the present. a bojo paper on that came out this month (March 2005)

When a classical singularity is removed then you can run the model THROUGH where it used to be. the machine no longer blows up or stalls at that point. So you can explore BEYOND the classical singularity and that is interesting. It is expected that one way to check LQG is to look for traces in the cosmic microwave background of what LQG predicts about the bigbang that is different from classical Gen Rel, different because of it having removed the singularity.

So that is a very important feature of LQG, the fact that it doesn't encounter these irritating singularities in Gen Rel that have bothered people so long.

If you want a non-math way to think about it, focus on the uncertainty of a quantum theory. For the universe or a black hole to collapse all the way to a point would just be too certain, wouldn't it? Too definite for real nature to allow . So it doesn't happen. At a certain indeterminate very high density there is a "bounce" according to the math (a time-evolution difference equation model) and contraction turns into expansion. And conditions for inflation are automatically generated.

recent papers
arxiv.org/gr-qc/0503020
Bojo
the early universe in Loop Quantum Cosmology

arxiv.org/gr-qc/0503041
Bojo, Goswami, Maartens, Singh
a black hole mass threshold from non-singular quantum gravitational collapse

if you glance at these papers you will not see anything about thinking of space as divided up into little bits, or grains
because that is not what real LQG is about,
but you will get a taste of what is going on with the overcoming of the Gen Rel singularities at bigbang and black hole.

 

[marcus]

What we have to do, where we have to go

We have to give an introduction to LQG that is reasonably faithful to the textbook version that researchers actually use----not just some possibly misleading verbal imagery. but it has to be understandable as an introduction.

that's hard. it will take several tries and the first ones will fail

It would be so great if people could just go and read a paper like
smolin "An Invitation to LQG" and have that suffice, but it somehow does not work. "Invitation" is too condensed for many people, or not explanatory enough.

Well I will try to get moving on this. I also want to keep those background points handy, from the previous 3 or 4 posts. So here, as a reminder, are the headings from post #31 onwards:

Introduction to LQG Part II

LQG can mean several things

There are textbook-level LQG sources

Textbook LQG is based on a differentiable manifold

QUANTUM" is a way of handling uncertainty and incomplete information realistically

Quantizing Gen Rel gets rid of singularities

 

[selfAdjoint]

Can we get word definitions of some of the technical concepts? Nightcleaner's interest in BF theory suggests that we could show what the Ashtekar variables are, at some honest, non-confusing level. Then Wilson action, and why it takes values in the Lie Algebra of the group, for that matter how the group comes in (ation on the manifold, forget oll the bundle staff), and Circle functions and so on. I go up and down on this; I think it would be boring if it wasn't impossible, and then I think it's a duty to get this across to the bright, self-selected audiance we have here.

 

[marcus]

A differentiable manifold is a shapeless smooth set

Differentiable manifold has 8 or 9 syllables and it is easier to say smooth set, which only has 2 syllables. And that is what one is. It is a set with a bunch of coordinate charts that work smoothly together.
Typically you can't get the whole set on one coordinate chart so you have several overlapping charts

that is like you can't get the whole Earth on one square map, but you can plaster maps all over the Earth so you have overlapping coverage.
On every patch of surface there is some map that is good at least on that local region.

the typical set used to represent space in LQG is the "3-sphere" where the surface of a balloon is a 2-sphere and you have to imagine going up one dimension. a local chart looks like regular 3-D graph paper or familiar euclidean 3-space

Only thing is we ignore the geometry you might have thought we had when I said 3-sphere. If we were thinking of the 2-sphere balloon as an analogy, the air is out of the balloon and it is crumpled up and thrown into your sock drawer. it has no shape. In the same way, by analogy, the 3-sphere has no shape. It is just a set of points, without a boundary, that has been equipped with an adequate bunch of coordinate maps

the "smooth" part is that wherever the charts overlap if you want to start on one map and find the corresponding point on the other map, and do a whole transference thing that remaps you from one to the other, well that
remapping (from one patch of 3-D graph paper to another) is smooth. that is to say differentiable, as in calculus, you can take the derivative as many times as you want. In other words the coordinate charts are COMPATIBLE with each other because whenever you remap between two that overlap you find you can DO CALCULUS at will on the function taking you from one to the other. this is an example of a technical condition that basically doesn't say very much except that we won't have nasty surprises when we get around to using the charts. The charts are smoothly compatible with each other.

The idea of a differentiable manifold was given us by George Riemann in 1854 when he was trying to get a job as lecturer at Göttingen and had to give a sample lecture, and it is actually SIMPLER than euclidean space because it does not have any geometry! Euclidean space has all kinds of rich structure immediately availabe, like you can say what a straight line is and you can measure the angle between two intersecting straight lines!

what we have here is a SHAPELESS SMOOTH SET and you can't do any of that. It has the absolute mininum of structure for something that can serve as a useful model of a CONTINUUM.

this is why it was a good idea of Riemann, because it is simpler and less structured than Euclidean space and so it is more able to adapt to the wonders of the universe. mathematics was changed very much in 1854.
George Riemann lived 1826 to 1866.

Here is his 1854 talk, in full:
http://www.ru.nl/w-en-s/gmfw/bronnen/riemann1.html

I think what he called a "stetige Mannigfaltigkeit" here in this talk we would call a smooth manifold. But thereafter the name "differenzierbare Mannigfaltigkeit" became prevalent and is what we call differentiable manifold.

 

[selfAdjoint]

stetig Maniigfaltigkeit translates literally to continuous manyfoldedness. The manyfoldedness is the number of independent variables, which if you think of them geometrically, become dimensions. Our word manifold is the result of a long history of trying to express the idea of multiplicity of variables in English. The French of course say variete, with a grave accent I am too lazy to supply. Personally I like continuum, which did for Einstein, and which he compared, as to its smoothness, to a marble table top (Soooo nineteenth century!).

 

[marcus]

Can we get word definitions of some of the technical concepts? ...

I would consider it a favor if you stepped in and supplied some. I will not be going very fast so it will be possible to step in at pretty much any point and add or improve definitions. right now I think of this as "Introduction Mark II" and the aim is to get a preliminary description out there which is at least not too misleading. that means there will be topics to expand later in a "Introduction Mark III"
At some point some student of Ashtekar or Rovelli will probably write something that makes all this unnecessary. a real beginner textbook for Loop Gravity. but we can't afford to wait around for that because we don't know when it will happen

 

[selfAdjoint]

Ashtekar varaibles

I would consider it a favor if you stepped in and supplied some. I will not be going very fast so it will be possible to step in at pretty much any point and add or improve definitions. right now I think of this as "Introduction Mark II" and the aim is to get a preliminary description out there which is at least not too misleading. that means there will be topics to expand later in a "Introduction Mark III"
At some point some student of Ashtekar or Rovelli will probably write something that makes all this unnecessary. a real beginner textbook for Loop Gravity. but we can't afford to wait around for that because we don't know when it will happen

I am up for F, but the "densitized dual 2-form" B (or E) has me buffaloed. A dual 2-form maps a pair of vectors multilinearly into the ground field, either reals or complex numbers. Dentsitizing makes it integrable, so far so good, but B or E has values in the Lie Algebra just like F, rather than the ground field. Why?

 

[marcus]

It is suggestive that you mention Ashtekar variables, and also mention the variables of BF theory (which Freidel tries to reform us so that we write EF thinking that it makes better sense than BF). Let me tell you what my sense of direction tells me. I listened to a (January, Toronto?) recorded talk by Vafa and I heard something ring in his voice when he said "form theories of gravity"----and I went back and looked at the current paper Dijkgraaf, Gukov, Neitzke, Vafa just to make sure. there was a sense of relief. it represents a hopeful general idea for him.

from my perspective, Ashtekar variables and BF are foremostly examples of "form theories" and there could be modifications and other "form theories" we don't know about yet. there is a mental compass needle pointing in this general direction.

it we want to play the game of making verbal (non-math) definitions for an intelligent reader, then it is important the ORDER we define the concepts and also the GOAL or where we are going. I think the direction is that we want to get to where we can say what a "form defined on a manifold" is, or to be more official we should always say "differential form" defined on a "differentiable manifold". So we need to say what the "tangent vectors" are at a point in a manifold.

the obstacle here is that these concepts are unmotivated, have too many syllables if you try to speak correctly, and seem kind of arbitrary and technical.

So I am thinking like this. the thing about a tangent vectors and forms is that they are BACKGROUND INDEPENDENT. All that means, basically, is that you don't have to have a metric. A background independent approach to any kind of physics simply means in practice that you start with a manifold as usual (a "continuum" you say Einstein liked to say) and you refrain from giving yourself a metric.

Well, how can you do physics on a manifold that (at least for now at the beginning) has no metric? What kind of useful objects can you define without a metric? Well, you do have infinitesimal directions because you have coordinates and you can take the derivative at any point, so at a microscopic level you do have a vectorspace of directions-----call them TANGENTS. and on any vectorspace one can readily define the dual space of linear functionals of the vectors-----things that eat the vectors up and give a number. The dual space of the tangents is called the FORMS.

and also the forms don't have to be number-valued, they can be "matrix" valued, one form can eat a tangent vector and produce therefrom not simply one number but 3 numbers or 4 numbers, or a matrix of numbers, but that is not quite right let's say it eats the tangent vector and produces not a number but an element of some Lie algebra. then it is a ALGEBRA-VALUED form.

now this already seems disgustingly complicated so let's see why it might appeal to Cumrun Vafa arguably the world's top string theorist still functioning as such.
I think it appeals to Cumrun Vafa because it is a background independent way to do physics. that is essentially what "form theory of gravity" means.

And string theorists have been held up for two decades by not having a background independent approach. And it JUST HAPPENS that the Ashtekar variables are forms, and the B and F of BF theory are forms, and (no matter what detractors say) Loop has been making a lot of progress lately, and Vafa says "hey, this might be the way to get background independence" and he creates a new fashion called "topological Mtheory" which is a way of focussing on forms and linking up with "form theories of gravity".

So maybe the point is not that this or that particular approach is good or not, but simply that one should work with a manifold sans metric, and do physics with the restricted set of tools that can be defined without a metric. And that means that, painfully abstract as it sounds, nightcleaner has to understand 3 things:

1. the tangent space at a point of a manifold is a vectorspace
2. any vectorspace has a dual space (the things that eat the vectors) and that dual space IS ITSELF a vectorspace.
3. the dual of the tangentspace is the forms and you can do stuff with forms.

Like, you can multiply two forms together (the cute "wedge" symbol), and you can construct more complicate forms that eat two vectors at once or that produce something more jazzy, in place of a number.

The hardest thing in the world to accept is that this is not merely something that mathematicians have invented to do for fun, a genteel and slightly exasperating amusement. The hardest thing to accept is that nature wants us to consider these things because it is practically the only thing you can do with a manifold that doesn't require a metric!

So instead of talking about BF theory or Ashtekar variables in particular, my compass is telling me to wait for a while and see if anyone is interested in "forms on a manifold" that is to say in the clunky polysyllabic language "differential forms defined on the tangent space of a differentiable manifold" UGH.

Also, selfAdjoint, you mentioned the word "bundle". Bundles may be going too far but they are in this general area of discussion, and there is also "connection"
A "connection" is a type of form. So if you understand "form" then you can maybe understand connection.

there is also this extremely disastrous thing that "form" is a misleading term. In real English it means "shape" but a differential form is not a shape at all. Richard being a serious fan of words will insist that it means shape. But no. Some frenchman happened accidentally to call a machine that eats tangent vectors and spits out numbers by the name "form" and so that is what it is called, even tho it is in nowise a shape. It is more like an incometax form, than it is a shape-form. And it is not like an incometax form either.

And as a final ace in the hole we can always say that Gen Rel is an example of a physical theory defined on a manifold without a metric. The metric is a variable that you eventually solve the equation to get. you start without a metric and you do physics and you eventually get a metric.
If there is any useful sense to Kuhntalk then this is a "paradigm". and when Vafa has a good word to say about "form theories of gravity" then this might be the kind of softening that accompanies a shift in perspective.

 

[Cinquero]

Could you please try to give an example of a calculation? I would like to get some feeling of how to handle all up to the mapping from a Lie algebra element to a Lie group element via parallel transport along a loop. Additionally, I'd like to see what a projection from the manifold into the tangent space looks like in practice.

Just to make things more clear: what exactly is the nature of this manifold you are talking of? Is there any physical interpretation?

 

[marcus]

Could you please try to give an example of a calculation? I would like to get some feeling of how to handle all up to the mapping from a Lie algebra element to a Lie group element via parallel transport along a loop. Additionally, I'd like to see what a projection from the manifold into the tangent space looks like in practice.

Just to make things more clear: what exactly is the nature of this manifold you are talking of? Is there any physical interpretation?

hello Cinquero, have you by any chance looked at the beginning treatment of LQG in Rovelli and Upadhya's paper? This was my introduction to the subject back in 2003. Several of us at Physics Forum were reading that paper back then.

It is short (on the order of 10 pages) and shows how a number of things are calculated. If you are interested in learning LQG, then I could review the paper myself, and read some of it with you.

If you do not already have Rovelli/Upadhya and would like the link, please let me know. the date at arxiv is about 1998.

 

[marcus]

meanwhile, a manifold is a topological space locally homeomorphic to R^d by mappings phi, psi,...which have the following differentiability property: where the domains of two maps overlap,
going from R^d to R^d by the composition of one with the inverse of the other is (either continuously differentiable a certain number of times or) infinitely differentiable.

LQG is usually developed in the d=3 case and the manifold that physically represents space is taken to be "smooth"----which means that the mappings from R^d to R^d which I just mentioned are infinitely differentiable.

LQG can be defined in any dimension d. It is not limited to the d = 3 case, and indeed has been studied in some other cases besides d = 3. But typically the manifold representing space is a compact smooth 3-manifold, a "continuum", denoted by the letter M.

You can get all this from any beginning treatment of LQG like, e.g. Rovelli/Upadhya, or Rovelli/Gaul. Again, if you need links, let me know.

All I have done to supplement the standard treatment that you find there is to define a differentiable manifold. I assume this is very familiar to you Cinquero but some other reader might conceivably want it defined.

 

[marcus]

As a background note, the classical "ADM" treatment of General Relativity has been posted on arxiv!

http://arxiv.org/gr-qc/0405109

this is a reprint of something published in 1962! You might say this is where the manifold M, that Cinquero is asking about, comes from:
the 1962 Arnowitt, Deser, Misner treatment of classical Gen Rel. Instead of a purely spacetime development, ADM looked at the metric restricted to an embedded 3-d spatial hypersurface.

Around 1986, two other people, Sen and Ashtekar, adapted the ADM approach by shifting attention to connections defined on the 3-d manifold. The connection then, rather than the metric, represented the variable geometry on space.

In the 1990s, when LQG started to develop, much of the context (concepts and notation) was already in place because of this prior work in classical Gen Rel. It was a matter of quantizing the ADM/Ashtekar version of Gen Rel, which had already been established for some time and was familiar to relativists. Here is some of that context (notation has not been fully standardized)

M smooth compact manifold represent space
A connections on M, representing the set of all geometries
K complex-valued functions on A, quantum states of geometry

K is too big and needs to be collapsed down (by applying constraints and equivalences) to a separable Hilbert space----the physical state space---of quantum states of geometry.

But already, with this bare minimum of concepts, one can begin to get oriented. K is a linear space of functions defined on A, the set of connections. One can think of A as the "configurations" and K as "wave functions" familiar from common QM. It is interesting to look for a BASIS of K----a minimal spanning set of complex-valued functions defined on connections.

Already, even with this abbreviated roadmap of the subject, I am touching on concepts that would be a lot of work to define and are better to read about. So if there is further interest I will get some links.

this little sketch is typed from memory, I haven't reviewed the definitions and history for quite a while. And I'm not omniscient either! So suggestions and improvements, including links to articles that develop LQG formalism, are welcome.

 

[marcus]

hi Cinquero, I got the link for Rov/Upad in case you want it

http://arxiv.org/gr-qc/9806079

I checked and they are using notation L where I wrote K, but otherwise no change

BTW, I see you are a new member, welcome!

 

[Cinquero]

Thx! That article is very helpful.

Question:

in II.B, the very first sentence: A is defined on M, right? But then, why does AV make sense? V maps from M to SU(2), but A is defined on M! What am I missing?

 

[marcus]

Thx! That article is very helpful.

Question:

in II.B, the very first sentence: A is defined on M, right? But then, why does AV make sense? V maps from M to SU(2), but A is defined on M! What am I missing?

Hi Cinquero, I just saw your post. Sorry for not replying earlier!

you remember on page 1, section II A, they say

"Let A be an SU(2) connection on M; that is, A is a smooth 1-form with values in su(2), the Lie algebra of SU(2)."

that means if you specify a point and a direction you get a matrix

(lets imagine that a basis has been chosen so that things are less abstract and all the SU(2) things and su(2) things are actually just 2x2 matrices )

but at every point of M, the function V also gives a matrix! so we can conjugate A by V and have
the new matrix V^-1A V
what is meant by writing them together this way is just matrix multiplication

this is how to interpret the first sentence of II B, where the notation
A_V is defined

==============
do I need to be more rigorous and formal, and spell this out in more detail?
or is this OK?

 

[Cinquero]

do I need to be more rigorous and formal, and spell this out in more detail?
or is this OK?

No, it's ok. Thx!

 

[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/au:+bojowald/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=people&fname=Martin&svir=0&name=Bojowald

and also
http://edoc.mpg.de/display.epl?mode=doc&id=213885&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]

LQG means what comes to the Loops 05 Conference in October

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 don't 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 don't 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 don't 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]

short reading list for CDT

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 doesn't 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.