Introduction To Loop Quantum Gravity

[marlon]

INTRODUCTION TO LOOP QUANTUM GRAVITY, everything you ever wanted to know...


In Loop Quantum Gravity, also referred to as LQG, the attempt is made to introduce the concept of quantum gravity. This is the unification of the General Theory of Relativity and the Quantummechanics. It is a very well established fact that gravitation and quantummechanics both have totally different foundations, which makes it very difficult to unify them at “first sight”. On the one hand position is uncertain in QM due to the Heisenberg-principle, while this is never the case in GTR. On the other hand, there is no metrical connection between space and time in QM, similar to the space-time-continuum of GTR. This leads to the fact that there is no curvature of space nor time in QM


In order to quantize the GTR we need gauge-fields, curved on a manifold just like in GTR. These gauge fields then need to be quantized just like other fermionic fields are quantized in QFT. When following this procedure, one needs to obey the following two laws at any point and time :

1) diffeomorfism invariance (this is the general covariance of GTR)
2) gauge invariance (like in QFT, invariance of gauge fields under local symmetries)


Basically these two laws ensure us that we have background independence so that we can choose any metric we want in order to describe the manifold. The different possible frames on that manifold must yield the same physical equations at any point on the manifold, that is the covariance (just like in GTR). These diffeomorfisms from one possible metric to another make sure that the physical laws remain the same when metrics are interchanged.


More specifically one needs to describe a manifold. Great mathematicians like Gauss and Riemann have taught us that this is done by the socalled connections. A metric describing a manifold is the most familiar example of a connection, i.e. the socalled metrical connection. There are other options though, like in GTR the connections are not metric-functions but they are gauge fields.


Next question is, how do we study some manifold ? What system can be followed in order to describe how objects behave on some chosen manifold ?
Well, we want background independence, so we must be able to chose any metric or connection we want in order to describe our manifold we are working on. In the early stages of LQG all possible metrics were used in order to implement this concept of back-ground-independence. A certain physical state was then represented as a probability-density containing all these metrics. This way of working was not very practical and in the mid-eighties it was even replaced by a description based upon the set of connections instead of all possible metrics.



Now, how does a connection work ? Well suppose you are on the manifold at a certain point A. Then you want to move in some direction on the manifold along a loop to that starts in A and comes back to A, like a circle. In order to describe this transition in mathematics, one uses the concept of parallel transport of tangent vectors. In order to be able to talk about such things as vectors, we need a reference frame that we can choose as we please because of the two laws mentioned above. Take a frame in A then make a very little step along the loop and look how this chosen frame has changed its position during the movement. Then complete the same procedure until you get back in A after completing the loop. Ofcourse it is not useful to look at the movement of the frame at every intermediate step along the loop. Actually one can integrate out the evolution of the frame over the entire trajectory that is followed from A to A.

When we start in A we actually take a tangent vector. This is an element of the tangent space of the manifold at point A. The transformation that is used to go from a point A on the manifold to the tangential space is called a projection. This tangent space can be turned into a socalled the Lie-Algebra, containing vectors written in terms of differentials, and provides the description for the movement from A to A along the loop. Now the operations that can be executed on the elements of a Lie-Algebra, like the identity or rotations, can be found in the Lie-Group.


As stated in the above paragraph it will be the intention to map elements from the Lie-Algebra to the Lie-Group. To be more specific : suppose we look at some vectors from the Lie-Algebra at A and we parallel transport them along the loop back to A. Now, we see for example that these vectors have rotated 45 degrees during their transport. This 45-degree-rotation is an element from the Lie-Group and the map between these two concepts gives us some idea on how vectors behave when replaced along some chosen loop on the manifold. Thus, yielding in a system to describe the manifold itself. It is proven that if you exponentiate Lie-Algebra-elements, you get the Lie-Group-elements.



More specifically, we take the frame around some loop and integrate all the differential motions of this frame during it’s transport. It is this integral that is exponentiated in order to get the corresponding group-element. In the Lie-Algebra, the group-element has a certain representation like a matrix. It is the trace if this matrix that is considered because the trace is a scalar and it will be the same for all reference frames. The map between the Lie-Algebra element and the Lie-Group element is called a Wilson Loop. Basically it “tries to feel” the metric by parallel transporting a Lie-Algebra element along a loop and “measuring” how this element changes it’s position with respect to the original position, after the loop is completed. Thus yielding a Lie-Group element.


The reason why we can ultimately speak about integrations and so on, is because initially everything is considered to be very very small. We work in terms of differential motions, which add up into the total motion between A and A. We use the Algebra’s in order to talk in terms of differentials d. As we move the frame along some "d(loop)" it experiences some "d(rotation)."


Now, once we have established such a relation, we can calculate the total movement by exponentiating the two differentials of the Lie-Algebra. The d(loop) ofcourse yields a transformation that describes the trajectory of the loop, while the d(rotation) yields the total rotation that has been undergone by the transported vector.









The main consequence of Loop Quantum Gravity is the fact that our space-time-continuum is no longer infinitely divisible. In LQG space has a “granular” structure that represents the fact that space is divided into elementary space-quanta of which the dimensions can be measured in LQG. The main problem of QFT is the fact that it relies on the existence of some physical background. As stated one of the main postulates of LQG is the fact that we need background independence. The diffeomorfisms give us the possibility to go from one metric to another and the physical laws must remain the same. Basically some physical state in LQG is a superposition over all possible backgrounds or in other words a physical state is a wavefunction over all geometries.



In String Theory, the main “competitor” when it comes to quantumgravity starts from the fact that there must be some kind of fictitious background space, thus actually undoing the aspects of general relativity. All calculations can then be made with respect to this background field and in the end the background independence must “somehow” be recovered. LQG starts from a totally different approach, though. We start from the knowledge we have from General Relativity, thus no background field, and we then try to rewrite the entire Quantum Field Theory in a certain way that no background-field is needed.



How to implement this nice background-independence in QFT has already shortly been introduced, i.e. The Wilson Loop and more generally the spin networks :

The map between the Lie-Algebra element and the Lie-Group element is called a Wilson Loop. Basically it “tries to feel” the metric by parallel transporting a Lie-Algebra element along a loop and “measuring” how this element changes it’s position with respect to the original position, after the loop is completed. Thus yielding a Lie-Group element.

The strategy is as follows : in stead of working with one specific metric like in “ordinary” QFT, just sum up over all possible metrics. So QFT should be redefined into somekind of pathintegral over all possible geometries. A wavefunction is then expressed in terms of all these geometries and one can calculate the probability of one specific metric over another. This special LQG-adapted wavefunction must obey the Wheeler-DeWitt equation, which can be viewed at as some kind of Schrödinger-equation for the gravitational field. So just like the dynamics of the EM-field is described by the Maxwell-equations, they dynamics of the gravitational-field are dedeterminedy the above mentioned equation. Now how can we describe the motion of some object or particle in this gravitational field. Or in other words, knowing the Maxwell equations, what will be the variant of the Lorentz-force ?


This is where the loops come in. First questions one must ask is :

Why exactly them loops ?

Well, let’s steal some ideas from particle physics... In QFT we have fermionic matter-fields and bosonic force-fields. The quanta of these force-fields or the socalled force-carrier-particles that mediate forces between matter-particles. Sometimes force-carriers can also interact with each other, like strong-force-mediating gluons for example. These force carriers also have wavelike properties and in this view they are looked as excitations of the bosonic-forcefields. For example some line in a field can start to vibrate (think of a guitar-string) and in QFT one then says that this vibration is a particle. This may sound strange but what is really meant is that the vibration has the properties of some particle with energy, speed, and so on, corresponding to that of the vibration. These lines are also known as Faraday’s lines of force. Photons are "generated" this way in QFT, where they are excitations of the EM-field. Normally these lines go from one matter-particle to another and in the absence of particles or charges they form closed lines, aka loops. Loop Quantum Gravity is the mathematical description of quantum gravity in terms of loops on a manifold. We have already shown how we can work with loops on a manifold and still be assured of background-independence and gauge-invariance for QFT. So we want to quantize the gravitational field by expressing it in terms of loops. These loops are quantum excitations of the Faraday-lines that live in the field and who represent the gravitational force. Gravitons or closed loops that arise as low-energy-excitations of the gravitational field and these particles mediate the gravitational force between objects.


It is important to realize that these loops do not live on some space-time-continuum, they are space-time ! The loops arise as excitations of the gravitational field, which on itself constitutes “space”. Now the problem is how to incorporate the concept of space or to put it more accurately : “how do we define all these different geometries in order to be able to work with a wave function ?”


The Wheeler-DeWitt equation has solutions describing excitations of the gravitational field in terms of loops. A great step was taken when Abhay Ashtekar rewrote the General Theory of Relativity in a similar form as the Yang-Mills-Theory of QFT. The main gauge-field was not the gravitational field. No, the gravitational field was replaced by the socalled connection-field that will then be used to work with different metrics. In this model space must be regarded as some kind of fabric weaved together by loops. This fabric contains finite small space-parts that reflect the quantization of space. It is easy to see that there are no infinite small space regions, thus no space-continuum. Quantummechanics teaches us that in order to look at very small distance-scales, an very big amount of energy is needed. But since we also work in General Relativity we must take into account the fact that great amounts of energy concentrated at a very small scale gives rise to black holes that make the space-region disappear. By making the Schwardzschildradius equal to the Comptonradius we can get a number expressing the minimum size of such a space-region. The result is a number that is in the order of the Planck Length.


Now how is space constructed in LQG ? Well, the above mentioned minimal space-regions are denoted by spheres called the nodes. Nodes are connected to each other by lines called the links.



By quantizing a physical theory, operators that calculate physical quantities will acquire a certain set of possible outcomes or values. It can be proven that in our case the area of the surface between two nodes is quantized and the corresponding quantumnumbers can be denoted along a link. These surfaces I am referring are drawn as purple triangles. In this way a three-dimensional space can be constructed.







One can also assign a quantumnumber which each node, that corresponds to the volume of the grain. Finally, a physical state is now represented as a superposition of such spin-networks.


regards
marlon, thanks to marcus for the necessary information and corrections of this text



REFERENCE : maestro Carlo Rovelli “Loop Quantum Gravity”
Physics World, November 2003

 

[marcus]

marlon,
congratulations on a great sticky thread!
though you are the primary author, I'm assuming it is OK
for others to contribute. So I would like to bring in some
bibliography----some more links to online reading, besides
the Rovelli article that you already have

I believe you plan on continuing your essay, when studies permit,
and hope that others' contributions don't interfere with
your writing future chapters.

best regards,

====
hi, I just saw your next message #3 that you posted. I will reply here to save space. That is a good point about keeping the level Introductory.
I will keep that in mind and concentrate on adding just a small amount of bibliography (unless you get around to it before I do) which is the
more accessible sort. Actually that makes sense for several reasons including the fact that more technical articles can have a shorter shelf life!
the technical methods can get old and be replaced while the basic intuitions
stay useful longer. hope your mainstream QFT studies are going well.
BTW this sticky is really nice to have. thanks again!

 

[marlon]

Marcus, thanks for the reply...

It can only be a good thing that others contribute but i am convinced that we need to keep the level basic enough in this sense that i want to move up the "difficulty-scale" gradually. It would be a bad thing if we were to discuss high-level papers because i think most of us (including myself) will not be able to follow this up and we would get discouraged and drop the subject. I will continue this matter and i would suggest that we follow the content of Rovelli's book which is online at his website.You have given the reference to it...

regards
marlon

 

[marcus]

I will continue this matter and i would suggest that we follow the content of Rovelli's book which is online at his website.You have given the reference to it...

regards
marlon

I am very much looking forward to your continuing the essay, marlon!
I will restrain my tendency to talk too much, so as not to crowd.
BTW just yesterday in the mail was delivered the copy of Rovelli "Quantum Gravity" which I ordered from Amazon. I am very happy with the book and
have been reading it instead of being at computer.

I am only sad that it is so expensive----70 dollars. You have to be rich, or be willing to splurge. Or you have to be in graduate school and need it for a class, as textbook. In US the textbooks are all very expensive, so 70 dollars is fairly normal.

Anyway Rovelli is a good writer and Cambridge Press did a good job, with the editing and just the physical production----nice paper, nice binding, nice feel, and printing. So it is a pleasure to own: at least for me.

But to save money it certainly makes sense to print off the free draft copy at Rovelli site. Even just the first 3 or 4 chapters and some appendices---or whatever you find the most accessible parts and most relevant for you.

Marlon, why not give some online bibliography yourself? It would be a refreshing change (I am always doing the librarian work) and I would enjoy seeing your picks and how you organize it. (If you do not want to, I will not shirk the job, but maybe you would like to list intro-level links?)

 

[jeff]

yesterday in the mail was delivered the copy of Rovelli "Quantum Gravity"

I sincerely hope you enjoy your new book, which I know you will. I was leafing through it at the U of T bookstore. I want to point out two things carlo says in the introductory bit.

1) That any correct quantum gravity theory must be able to calculate amplitudes for graviton-graviton scattering, and that he hopes that lqg will one day lead to a theory that can.

2) That he knows that GR must almost certainly be an effective field theory that is modified at higher energies so that lqg can't be correct. Thus he says he views lqg basically as a laboratory for investigating certain fundamental issues in quantum gravity.

As far as your sticky goes, would you be bothered if I corrected it?

 

[marcus]

... I want to point out two things carlo says in the introductory bit.

1) That any correct quantum gravity theory must be able to calculate amplitudes for graviton-graviton scattering, and that he hopes that lqg will one day lead to a theory that can.

2) That he knows that GR must almost certainly be an effective field theory that is modified at higher energies so that lqg can't be correct. Thus he says he views lqg basically as a laboratory for investigating certain fundamental issues in quantum gravity.
...

I believe you are mistaken, jeff. Carlo does not say these things in the introductory bit.
At least I looked in the first part of the book, and used the index to search the rest, and could not find any statements of the kind.

It would be nice to have some page references, if you have any more would-be paraphrases from Rovelli----even sweller of you to provide actual quotes. Since a paraphrase can often mislead as to what was said in the original.

Thanks for your kind wish as to the book! Indeed it is surprising me. I was not expecting this much, since I had read much of the last year's draft version.

BTW if you pick up a copy either at library or store and can give me some actual page reference (whether or not in the first 50-or-so pages, anywhere in the book will do) where he says these things 1. and 2. that you state, that would be most helpful of you and I will be very interested to read the actual passages and think about it. If he does say something like that my eye somehow missed it.

 

[jeff]

I believe you are mistaken, jeff. Carlo does not say these things in the introductory bit.

We'll, I don't have the book on hand, but...

In rovelli's dec 30 2003 draft, he says on page ix entitled "PREFACE"

"What we need is not just a technique for computing, say, graviton-graviton scattering amplitudes (although we certainly want to be able to do so, eventually)"

On page 5 of the same draft,

"The einstein-hilbert action might very well be a low energy approximation of something else. But the modification of the notions of space and time has to do with the diffeomorphism invariance and the background independence of the action, not with it's specific form."

Be this as it may, jim bjorken in the forward of carlo's book states quite plainly that effective field theory has taught us that GR must be viewed as just an effective field theory, and it's difficult to believe that carlo would've allowed such a statement if it fundamentally contradicted his position.

Btw, did you notice that carlo writes (probably in the preface) that thiemann is publishing a book on the more mathematical aspects of lqg?

 

[marcus]

the Mexican Loop and String Show (21-27 November)

Oh I see.
I thought you were talking about the actual book. that you said you were browsing in the bookstore.

but you apparently meant the draft, from 2003, which is available online.

there's been considerable up-dating and revision. so one should be specific which
=============

Meanwhile, maybe readers of this thread would be interested in the Loop and String lineup of talks at the conference that just finished in Mexico (at the Quintana Roo beach resort in sight of the island of Cozumel)

A lot of the lectures were by top people both string and loop, and they were rather much introductory. The conference aimed at being a "school" to bring more people in. And to introduce stringies to loop research and viceversa.

I thought the lineup of who the organizers wanted to talk about the various hot topics was enlightening. So since it could be instructive, I will copy it here:

http://www.nuclecu.unam.mx/~gravit/EscuelaVI/courses.html

--quote--
COURSES AND INVITED TALKS

Courses:

A. Ashtekar (PSU, USA): Quantum Geometry

A. P. Balachandran (Syracuse, USA): Quantum Physics with Time-Space Noncommutativity

P. T. Chrusciel (Tours, France): Selected Problems in Classical Gravity

R. Kallosh (Stanford, USA): De Sitter Vacua in String Theory and the String Landscape

A. Peet (Toronto, Canada): Black Holes in String Theory

C. Rovelli (Marseille, France): Loop Quantum Gravity and Spinfoams


Plenary talks:

J. D. Barrow (Cambridge, UK): Cosmological Constants and Variations

M. Bojowald (AEI, Germany): Loop Quantum Cosmology

A. Corichi (ICN-UNAM, Mexico): Black Holes and Quantum Gravity

A. Linde (Stanford, USA): Inflation and String Theory

O. Obregon (U. Guanajuato, Mexico): Noncommutativity in Gravity, Topological Gravity and Cosmology

A. Perez (PSU, USA): Selected Topics on Spin Foams

L. Smolin (PITP, Canada): Loops and Strings

R. Wald (U. Chicago, USA): Topics on Quantum Field Theory


Short talks:

E. Caceres (CINVESTAV, Mexico): Wrapped D-branes and confining gauge theories

A. Guijosa (ICN-UNAM, Mexico): Far-from-Extremal Black Holes from Branes and Antibranes

H. Morales (UAM, Mexico): Semiclassical Aspects and Phenomenology of Loop Quantum Gravity

D. Sudarsky (ICN-UNAM, Mexico): Spacetime Granularity and Lorentz Invariance

L. Urrutia (ICN-UNAM, Mexico): Synchrotron Radiation in Lorentz-Violating Effective Electrodynamics

---endquote---

 

[matthias]

marlon...got any extra info on LQG?

great introduction btw...

lola

 

[jeff]

Oh I see.
I thought you were talking about the actual book. that you said you were browsing in the bookstore.

but you apparently meant the draft, from 2003, which is available online.

there's been considerable up-dating and revision. so one should be specific which

You want to play games? Fine with me.

 

[nightcleaner]

This is a project I've been working on, and I'd very much like to know what the participants on this thread think. Thanks, nc

Abstract and prospectus, Spacetime at the Planck Scale
This is an abstract and prospectus for additional research. The proposal would use computational techniques such as those described in Stephen Wolfram's New Kind of Science as an exploratory probe of events at the Planck scale. Authors are currently recruiting mathematicians and physicists to mentor and contribute to the work. We still need someone who can design the NKS experiments.

In this work in progress, we describe a mechanism by which four space-time dimensions are reduced to the classical view of three space-like dimensions arrayed in the customary orthagonal basis with one time-like dimension which can be thought of as permeating the space-like dimensions. The time-like dimension is shown to appear to be unique to a moving observer, and preserves the appearance of freedom of choice as one perspective in a structure which can also be viewed from other perspectives as competely deterministic.

The Einstein-Minkowski principle of space time equivalence taken in the strongest sense creates a powerful model for investigation of the relationship between general relativity and quantum mechanics. We begin by defining the Planck Sphere (here named to be consistant with the Planck length and Planck time) as a three dimensional volume filled by a radient event at the speed of light in one Planck time. Thus the radius of the Planck Sphere is equal to one Planck length and is equal to one Planck time, making a three dimensional model which can be used in a perspective sense to portray events which occur at the Planck scale in four dimensions.

After describing the features of the model, we go on to propose that computational graphing techniques similar to those used by Stephan Wolfram in his book A New Kind Of Science be developed to explore the evolution of the Planck Sphere in Kepler dense packed space up to the scale of the fine structure constant, thereby showing the geometric origins of mass and charge. The first step in this process is to define a viable space-time lattice structure, which we believe we have done by defining the Planck Sphere as an element in a Kepler stack. The next step in this process is to develop a rational algorithem to simulate events on the Planck scale. This may be accomplished by applying what we know of cosmogeny and of physics near singularities. As a first approximation we advance the conjecture that expansion from the Planck scale will recapitulate cosmogeny. We carry through the first steps in this approximation to demonstrate a mechanism for early inflation in the burgeoning universe.


References:
[PDF] On quantum nature of black hole space-time: A Possible new source of intense radiation DV Ahluwalia - View as HTML - Cited by 11 ... spheres of fluctua- tions. The one that may be called a Schwarzschild sphere, and the other a Planck sphere. The sizes of these ... International Journal of Modern Physics D, 1999 - arxiv.org - ejournals.wspc.com.sg - arxiv.org - adsabs.harvard.edu

[PDF] The Quantum structure of space-time at the Planck scale and quantum fields S Doplicher, K Fredenhagen, JE Roberts, CM Phys - View as HTML - Cited by 242... In the classical limit where the Planck length goes to zero, our Quantum spacetime ...components are homeomorphic to the tangent bundle TS 2 of the 2–sphere. ... Communications in Mathematical Physics, 1995 - arxiv.org - arxiv.org - adsabs.harvard.edu

[PDF] Inflationary theory and alternative cosmology L Kofman, A Linde, V Mukhanov - Cited by 9 ... the large scale structure observed today were generated at an epoch when the energy
density of the hot universe was 10 95 times greater than the Planck density ... The Journal of High Energy Physics (JHEP) - iop.org - arxiv.org - physik.tu-muenchen.de - adsabs.harvard.edu - all 7 versions »

[PDF] Physics, Cosmology and the New Creationism VJ Stenger - View as HTML ... 10. -43 second time interval around t. = 0, if it was confined within a Planck sphere as big bang cosmology implies. The. universe ... colorado.edu


200411290100GTC
Richard T. Harbaugh
Program Director
Society for the Investigation of Prescience

 

[luco_jash]

Hello Marlon

i will thank you for the nice clear introduction on loop quantum gravity. I am planning to do my thesis on this subject and i would like to keep in touch with all the specialists here in order to get more info. I am just starting to know this field...

bye...Luco

 

[Rothiemurchus]

The challenge for string theorists and LQG theorists is to explain why the vacuum energy exists at 10^120 J/m^3 ( there is no reason to think there is anything wrong with the QM calculation) but does not curve space-time.How can
quantum gravity be proved if gravity is not understood on its own yet?

 

[marcus]

( there is no reason to think there is anything wrong with the QM calculation)

!



gotta be something wrong with it

 

[marcus]

explain why the vacuum energy exists at 10^120 J/m^3 ...

beg your pardon Rothie but that is a crazy amount of energy
maybe QFT can come up with a mechanism that cancels all or most of it out, or find some reason to say that it doesn't really exist----maybe QFT already has.

but that density of energy, not canceled out and real enough to cause gravity, is simply incredible (at least to me). commonsense persuades me that there must be something wrong with any theory that predicts it

And there is some reason to be hopeful, because QFT is still formulated in an unrealistic way: using a fixed spacetime framework. Reformulating it in a background independent version might possibly get rid of that huge vacuum energy.

BTW just to have a basis for comparision, the astronomers' dark energy estimate is currently around 0.6 joule per cubic km. In joules per cubic meter (the units you were using) that comes to:

0.6 x 10^-9 joule per cubic meter.

 

[Rothiemurchus]

I am aware of the cosmological evidence.But the problem is this:
the energy that can be experimentally associated with the Casimir force
is greater than the cosmological observation (10^-6 Newtons/m^2 net force
at 10^-7 m plate separation - i think but I'm not sure,that this is at
least 10-7 J/m^3).So, the plates involved in
measurements of the Casimir force must somehow, switch on vacuum energy,locally.And what sort of effect would a galaxy have on the vacuum energy?

 

[marcus]

I am aware of the cosmological evidence.But the problem is this:
the energy that can be experimentally associated with the Casimir force
...

Rothie, I will try to respond---tell me if I am making a mistake. I do not believe that the experimental existence of the Cas. force proves that the
QFT calculation of a huge vacuum energy is correct.
what I think is true is that there is some normal vacuum energy density and that between two conducting plates it is LESS namely

\text{energy density betw. plates = usual vacuum energy density} -\frac{\hbar c \pi^2}{720 d^4}

the QFT calculation of the usual vacuum energy density is bad or dubious, but the Casimir effect does not depend on this, it depends on the fact that the energy density between plates is LESS by the amount shown, which QFT does calculate successfully!, and which depends on the inverse fourth power of the separation distance.

So I say that I believe the QFT calculation of the Casimir effect and I like the Casimir effect, and this is consistent with not believing the huge vacuum energy which QFT calculates, which is roughly 120 OOM wrong---or actually different people try to fix it different ways and say different things, but anyway wrong.

 

[marcus]

\text{energy density betw. plates = usual vacuum energy density} -\frac{\hbar c \pi^2}{720 d^4}

If I calculate right, this is what the energy density has to be in order that the force turn out
what one usually sees for the Casimir effect

\text{force divided by area} = -\frac{\hbar c \pi^2}{240 d^4}

 

[marlon]

https://www.physicsforums.com/journal.php?s=&journalid=13790&action=view#DUALITY%20:%20STRING%20THEORY%20PART%203

check out my journal if you are interested in an introductory text on string theory and dualities


regards
marlon, let me know your comments

 

[marlon]

https://www.physicsforums.com/journal.php?s=&journalid=13790&action=view

Check out my journal. I posted a link to the paper that John Baez will be using for his speech on monday on LQG...very nice introduction...

regards
marlon

 

[marcus]

https://www.physicsforums.com/journal.php?s=&journalid=13790&action=view

Check out my journal. I posted a link to the paper that John Baez will be using for his speech on monday on LQG...very nice introduction...

regards
marlon

marlon thanks for the link!
your journal has become a real trove of information!
I liked the Tensors-made-easy,
and the interesting historical bits

 

[marcus]

Including spinfoam

http://math.ucr.edu/home/baez/acm/

http://math.ucr.edu/home/baez/acm/acm.pdf

the paper Marlon referred to is Baez 24January 2005 talk to the ACM symposium on discrete algorithms, essentially introducing spinfoam research to Computer Science people.

However the talk is titled "Loop Quantum Gravity"

this points out an everpresent knotty semantics problem: LQG is used in two senses

A. Restricted sense of LQG proper (program set out in early 1990s) which does not include spinfoams approach and some other allied developments
B. Inclusive sense to mean Loop-and-allied QG approaches, in which case spinfoams is included.

These are mostly approaches which have grown out of the LQG of the 1990s.

I don't know the statistics but I believe that most LQG people actually do their work in spinfoam-related areas: the thrust of the research is towards a path-integral treatment of spacetime geometry.

(path-integral didnt figure in the original LQG program of the early 1990s AFAIK but it obviously is a major part of things now)

Judging from his talk Baez believes in including spinfoams under the LQG rubric because in his talk to the ComputerScience people he titled it LQG and gave an 11 page thumbnail of LQG proper
and then on page 12, without even saying what he was doing, he shifted to talking about the Barrett-Crane (spinfoam) model and related computing problems!

Baez helped initiate the spinfoam approach and invented the term, so if he wants to include it under the LQG heading then I guess he has the right to.

By contrast, Hermann Nicolai had a sad case of talking at cross-purposes recently where he wrote this review paper about LQG (an outsider's view) and didnt even mention what most of the LQG people have been doing for 5 or 10 years! He didnt discuss spinfoams at all!
He took LQG in the narrow (circa 1995) sense and went thru the motions of reviewing it. Didnt even discuss Thomas Thiemann masterconstraint, which is the closest thing to directline development from 1995 strict interpretation LQG. Didnt discuss Loop Cosmology either. So his review looks kind of vacuous: a review of no one's current research.

Somehow we have to get the general classifications straight so that we include in our picture of LQG not just the LQG of 5 or 10 years ago but what LQG people actually do i.e. the models of spacetime and gravity that they actually investigate.

What I'm thinking about is our ADDING ON to this thread whatever it takes to make it more of an introduction to the general field of LQG (including allied approaches that have grown out of the LQG of the past)

Probably the key paper that one wants to prepare to understand is one the authors say they are working on but has not appeared yet! here is a chance to use various embarrassment smilies

that's right it hasnt appeared yet.
Laurent Freidel and Artem Starodubtsev
Perturbative Gravity Via Spin Foam

this was cited in their January 2005 preprint
http://arxiv.org/hep-th/0501191

 

[marcus]

I want to better understand where the spinfoam approach comes from and how it fits together with the original or basic LQG approach. Sometimes it helps to go back to the beginnings of a research line which in this case is not only Baez papers but also this (which does not even use the term "spin foam":

http://arxiv.org/gr-qc/9612035
"Sum over Surfaces'' form of Loop Quantum Gravity
Michael P Reisenberger, Carlo Rovelli
Phys.Rev. D56 (1997) 3490-3508

"We derive a spacetime formulation of quantum general relativity from (hamiltonian) loop quantum gravity. In particular, we study the quantum propagator that evolves the 3-geometry in proper time. We show that the perturbation expansion of this operator is finite and computable order by order. By giving a graphical representation á la Feynman of this expansion, we find that the theory can be expressed as a sum over topologically inequivalent (branched, colored) 2d surfaces in 4d. The contribution of one surface to the sum is given by the product of one factor per branching point of the surface. Therefore branching points play the role of elementary vertices of the theory. Their value is determined by the matrix elements of the hamiltonian constraint, which are known. The formulation we obtain can be viewed as a continuum version of Reisenberger's simplicial quantum gravity. Also, it has the same structure as the Ooguri-Crane-Yetter 4d topological field theory, with a few key differences that illuminate the relation between quantum gravity and TQFT. Finally, we suggests that certain new terms should be added to the hamiltonian constraint in order to implement a "crossing'' symmetry related to 4d diffeomorphism invariance."

that says what spinfoam later came to mean: a branched colored surface in 4D, or an equivalence class of such.

 

[marcus]

In Baez talk to the ACM symposium which Marlon just flagged a couple of posts back he concludes:

"Moral for physicists: In a regime where analytical methods don’t work well (yet), we need computer simulations to test our models.

Moral for computer scientists: In loop quantum gravity, the geometry of space is built from qubits. Spacetime is like a parallel-processing quantum computer that constantly modifies its own topology. "

he is definitely talking about spinfoams here, and indeed has just been discussing his own and others' computer simulations of spinfoam models (Barrett-Crane in particular IIRC). the dynamics of a spinfoam is step by step modification of topology and there are amplitudes of each kind of "move"----to put it crudely and imprecisely, moves like inserting a vertex, removing a vertex, replacing one sort of edge by another, disconnecting things and reconnecting them differently, bit by bit. Maybe we could say it portrays space as a sort of glittering blur of never-certain and ever-shifting relationships.

anyway the point is that each elementary change or move, in the path that is a spinfoam, can have an amplitude number calculated for it.
the researchers want to integrate or average over lots and lots of paths (spinfoams are pathways that the geometry of space can take) from one shape of space to another. they need to calculate an amplitude for each, and sum.

 

[marcus]

I am going to try to get another piece of the jigsaw out of this February2005 paper of Freidel and Livine. For now I'll just get it out on the table.
http://arxiv.org/hep-th/0502106
"1 Introduction

Spin Foam models offer a rigorous framework implementing a path integral for quantum gravity [1]. They provide a definition of a quantum spacetime in purely algebraic and combinatorial terms and describe it as generalized two-dimensional Feynman diagrams with degrees of freedom propagating along surfaces. Since these models were introduced, the most pressing issue has been to understand their semi-classical limit, in order to check whether we effectively recover general relativity and quantum field theory as low energy regimes and in order to make physical and experimental predictions carrying a quantum gravity signature. A necessary ingredient of such an analysis is the inclusion of matter and particles in a setting which has been primarily constructed for pure gravity. On one hand, matter degrees of freedom allow to write physically relevant diffeomorphism invariant observables, which are needed to fully build and interpret the theory. On the other hand, ultimately, we would like to derive an effective theory describing the propagation of matter within a quantum geometry and extract quantum gravity corrections to scattering amplitudes and cross-sections..."

 

[katelynndevere]

D'Oh! i just found your beautiful introduction, after I posted a thread asking if anyone had one. it's very close to being simple enough for me to understand it. I will persevere, dictionary at hand...

 

[marcus]

D'Oh! i just found your beautiful introduction, after I posted a thread asking if anyone had one. it's very close to being simple enough for me to understand it. I will persevere, dictionary at hand...

I think you must mean Marlon's introduction, which begins this thread. He will be very pleased that you are interested and finding it useful. For something that is very introductory, conceptual and non-math, I think a magazine article by Carlo Rovelli is pretty good. I will get a link or two.
Here is Rovelli's homepage:
http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html

Here is the Rovelli magazine article:
http://cgpg.gravity.psu.edu/people/Ashtekar/articles/rovelli03.pdf

this is a general audience intro to LQG from Physics World November 2003 issue.

this link to Rovelli's article is actually at the website of Abhay Ashtekar at Penn State. Abhay has links to an interesting collection of other popular and semipopular articles, as well as to technical writings about LQG. Here is the main Ashtekar link
http://cgpg.gravity.psu.edu/people/Ashtekar/articles.html
and his home page
http://cgpg.gravity.psu.edu/people/Ashtekar/index.html

Ashtekar and Rovelli are two of the original pioneers of the LQG approach, so its worth checking out what they they think is a good introduction.

 

[katelynndevere]

Thanks, Marcus!

The Rovelli article is superb; I can actually understand it! I really need to sort out my maths, though.
Thanks again,
Kate.

 

[Tom McCurdy]

Yes kately marcus and marlon are the sources to go to for loopquantum questions and most other questions. I hope to get back to my studying of loop quantum gravity over the summer... I know I know.. I have been saying this for months but I am redesigning the site in my spare time to get ready for a relaunch...

omg... this is my 1,000 post... I have moved out of the triple digit range.

 

[marlon]

I hope to get back to my studying of loop quantum gravity over the summer... I know I know.. I have been saying this for months but

I can relate, really i can...i have not done any LQG related studies over the past months because i was too busy with applying and preparing for my phd...maybe in the future...

regards
marlon

ps : thanks for the nice complements, glad to see the intro to LQG is usefull

 

[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?