# Loop Quantum Gravity for newbies

1. Oct 23, 2009

### jokkon

Hi there, I am currently doing a project on LQG and I am having troubles grasping the basics. I am just wondering if anyone here could help answer a couple of questions that I have during my study.

1.) How to use Roger Penrose's original spin-network? I read in his book that they are used to calculate probabilities involving atomic or subatomic processes. How exactly does it work?

2.) Why can't we quantize gravity like we did with QED?

3.) I read somewhere that wheeler dewitt equation is similar in function to schrodinger's equation in quantum mechanics. Does wheeler dewitt gives specifically the time evolution of space-time?

This research project has proven to be difficult since I have not taken any formal courses in GR and QFT. Thank you very much in advance.

2. Oct 23, 2009

### marcus

This is only a partial reply. I hope others will jump in with a fuller response.
I think online video lectures are extremely helpful, and also online audio+slide lectures--I might give you a few links later, in case you'd like to check them out.

First, the main premise of GR is that gravity=geometry.
Spacetime is nothing but 4D geometry, space is nothing but 3D geometry---it is pure geometric relationships, not a substance. There is no surrounding background flat space in which the gravitational field sits or upon which the field is defined.

There is only the dynamic geometry itself, defined internally with the help of metrics, or more correctly the combined system of geometry+matter.

So this is radically different from QED. With QED you assume that gravity does not exist so that spacetime can be the standard flat 4D Minkowski spacetime of special relativity. You start with a given geometry. You build on a fixed rigid platform.
This is typical of all QFT. It is built on a fixed rigid geometry, usually flat Minkowski, but sometimes also a rigid pre-established curved background.

It's good to keep in mind that gravity is not primarily a force, it is above all a causal ordering of events---a lightcone structure. The 4D geometry---or the dynamically changing irregularly curved 3D geometry which is the same thing---determines what is in the lightcone of what. Determines what can cause or influence what.

If you have a flat "background" that determines a fixed causality ordering which becomes wrong as soon as a graviton is superimposed on the picture, as a perturbation. Every perturbation changes (ever so slightly) causal relationships, but the fixed background does not know about this. So any backgrounded approach cannot be fundamental. It may serve as an approximation but it is ultimately unrealistic.

This is a basic lesson we take home from General Relativity. Any type of QFT now in use must be fundamentally wrong because no existing version of QFT is general relativistic. QFTs are at best special relativistic.

So the overall aim of (nonstring) QG is to be able to construct a general relativistic quantum field theory. That means built on a dynamic 4D geometry, which includes the idea of causality ( lightcones) being subject to dynamics.

I have to go. I'll be back later and try to address some specifics about LQG. BTW Loop is not the only nonstring QG research program. Other interesting ones exist as well.

3. Oct 23, 2009

### Naty1

Jokkon....
has been difficult?? no wonder....But good for you for trying.....Why not start with

Wikipedia: http://en.wikipedia.org/wiki/Wheeler-deWitt_equation

Also, you should definitely read THREE ROADS TO QUANTUM GRAVITY by Lee Smolin, 2001
...you should be able to get used copy via Amazon or elsewhere cheap...

Spin networks: spin networks are structures that generate (create) discrete units of space and time at the Planck scale...they flow from loop quantum gravity...each spin network provides a possible quantum state for the geometry of space...integers on the edge (lines) correspond to units of area while the nodes Iintersections) correspond to quantized units of volume...also coded in are rules about how the edges may knot and link with one another...
As I understand it, this very promising area has been on Penrose's menu of work for twenty years but has never fully panned out as intended....am unsure what the original promise was that has not been brought forward...

Last edited: Oct 24, 2009
4. Oct 23, 2009

### Naty1

The solutions provide a discrete spacetime structure as a spin foam network evolves.

Here is some information, relevant I hope, from Lee Smolin's book. Most from Chapter 10.

I'm only going to provide quotes or paraphrases because I have not studied other sources and so lack perspective to feel comfortable offering insights/suggestions and so forth
Smolin notes that he personally, with collaborators, found an infinite number of solutions to these equations in the late 1980's, using new forms of the equations which were dramatically simpler:

Smolin and collaborators plugged into the Wheeler Dewitt equations formulas that described the possible quantum states of the geometry of space and time building these states from expressions that (Wilson and) Polyakov used to describe the quantized loops of electric fields...(loops that do not intersect)...he worked on this with Louis Crane and Carlo Rovelli (and others) ...and found close correlation with Penrose spin networks that had been developed 30 years earlier....and do intersect, the nodal interger values define volume...

There are a few physical insights scattered here and there:
And I take that to be the relationship between Wilson/Polykev loops and Penrose Spin Netwroks...

In THE ROAD TO REALITY, by Roger Penrose, heavy on advanced math, he discusses spin Networks in Chapter 32.6 and notes that loop states don't intersect and spin foam networks are a generalization that do.

If you read Smolin's material first, you should be able to get some value from that Penrose book section, but Penrose operates at a rather sophisticated level...I never got to that level of detail.

In 32.7 Penrose assesses the "Smolin-Rovelli-Ashtekar approach to quantum gravity"...If that might be helpful, let me know and I'll see if I can understand it well enough to post something here.

Smolin also notes (and I had not understood this when I read the book about a year ago) that work of Ambjorn...Loll based their model on a spin foam....and that from time to time the size of the universe jumps suddenly. There are current discussions following this work on PHYSICSFORUMS and I think Marcus' signature has a great reference to the work...yes it's the SciAm item under his signature above...

Last edited: Oct 24, 2009
5. Oct 23, 2009

### Naty1

From Marcus:
So Penrose spin foam represents space-time via edges and verticies, (metrics) ....I think of it as a type of geodesic dome structrue with bumps and valleys, that change values, and warps changing chape, causing spacetime warps...say as a mass passes by and warps(shapes) space time......its a dynamic model that is background independent, unlike string theories which assume a fixed spacetime background....

Last edited: Oct 24, 2009
6. Oct 23, 2009

### jokkon

Hi there thanks for all the replies. I knew that QED is a special relativistic theory but I didn't know that causality would get screwed up in a background dependent theory. Also I did read some review papers written by Carlo Rovelli and understood (I think) that a superposition of spin networks form a picture of space-time. But I am curious as to how they adapted Penrose's original idea to how LQG uses spin networks today. Not only do the spin networks look different, the quantum numbers represent something entirely different either. As far as I know the original spin networks represent total spin of the system but now they represent either area of cross section and volume.

As for the wheeler dewitt equation, i noticed that the equation itself doesnt have a time parameter so i find it weird that it would give the time evolution of spacetime.

7. Oct 23, 2009

### marcus

First let me qualify something: The causality thing might be regarded as a quibble. Because a very small ripple, a tiny perturbation, wouldn't disturb things very much. I take it seriously (I think no theory based on a fixed background can be totally realistic, it can't be fundamentally right.) But not everybody thinks the same way about this.

I'm avoiding the "frozen time" issue for the moment. The issue arises already with classical GR. There is no preferred time coordinate, so the Hamiltonian or "canonical" formulation of classical GR does not give you time-evolution (like a Hamiltonian typically would in other situations.) The "canonical" formulation of GR---called ADM arnowitt, deser, misner---gives you a 3D geometry on a 3D "slice", plus a Hamiltonian constraint, which, if it is satisfied, says yes this metric together with this distribution of matter could indeed have evolved naturally, it is physical. In the "canonical" formulation of classic GR, the situation on the 3D slice somehow contains all the information about the past and the future evolution. You rightly observe that it is a little weird. This carries over to 1990s LQG which is a canonical approach, but does not carry over to the spin foam approach which is a 4D or path integral approach. For now, let's sideline this topic and I'll simply link to the Wikipedia
and avoid further discussion.

BTW compliments. Good questions! And Naty's answers are consistently on target, or seem so to me. It makes for a good thread.

I keep thinking of Lewandowski's brief description of LQG and of spinfoam, and of the relation. Lewandowski is great. There is a recent (September 2009) online audio+slide talk where he gives his account of Loop and then takes questions and comment from Carlo Rovelli, Jon Engle, Abhay Ashtekar, Laurent Freidel. I want to go back and listen to that again this afternoon. It is so concise. Maybe it could be helpful here.

About spin networks. I wouldn't stress the relation to the original Penrose invention. The use is so different. If you think about the canonical formulation, they have to have a way of describing the quantum state of the geometry of a 3D slice.
So at first they said let's represent geometry configuration space by the set of all possible "connections"* on the slice, and as functions on configuration space, in other words as functions of a connection, lets take LOOPS.
A particular loop, evaluated on a connection, will just be whatever the connection does as you run around that loop.
OK, they said, let's build a hilbertspace of quantum states out of these loops. But to make a vector space you have to add loops together. You have to stick them together, and when you do that you get a network.
A convenient basis of the Hilbertspace turned out to be these networks, which turned out to be formally similar to what Penrose had introduced. These networks either live in the 3D slice, embedded, or they exist independently as abstract labeled graphs.

About spin foams. The idea was to get away from the canonical approach, with its Hamiltonian constraint equation H=0, and working all the time on a 3D slice. The idea was to describe spacetime as an evolution from an initial 3D geometry to a final 3D geometry. To represent that path from initial to final geometry. Like a Feynman path integral. Maybe there are many paths from initial to final and each has an amplitude, and you can do a "sum over histories" or a path integral to get the overall amplitude of getting from one to the other.
So a spinfoam is the 2-complex describing how a network evolves. The trajectory of an edge becomes a 2-cell.
If a spin network is a good way to represent the quantum state of 3D geometry, then a spinfoam is a good way to represent the path along which that geometry evolves.

It seems to me that Jerzy Lewandowski describes LQG in the most concise and lucid way. Maybe in parts it is so concise as to be incomprehensible. And it is pitched at advanced seminar level. But I like how he does it---the exemplary carefulness. And I like hearing the others respond. So I'll fetch the link. Actually to get it you just have to google "ILQGS".
This stands for international lqg seminar---they do a telephone seminar a couple of times a month that links 5 or 6 locations.
http://relativity.phys.lsu.edu/ilqgs/
Jerzy's talk is the one dated 20 October 2009. Parts of the talk (such as slides 14-18) are way too technical for present purposes. But other parts provide a concentrated definitive treatment that i like very much.

*think of a "connection" as a parallel transport tool defined at every point of the hypersurface slice, that tells how a vector carried along a path thru that point will roll and yaw and come back different if you loop around and return to that point. A connection is a good way to describe the geometry and can serve as an alternative to the metric or distance-function as a way to characterize the internal experience of shape.
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EDIT TO REPLY TO NEXT POST BY ATYY
Please tell me how! I am not asking ironically. Indeed I suspect that AsymSafety is right, there is a UV fixed point somewhere in theory space. But this does not mean one can write a perturbation series. AsymSafe is a non-perturbative approach. The people who do it refer to "nonperturbative renormalizability". So tell me what does "like electromagnetism" mean? What would it mean to "quantize gravity like electromagnetism"? Could you sort of sketch it out for me?
How do you represent the geometry of the universe, in your idea? How do you represent the geometry evolving dynamically along some path? Could you sketch some sort of equations to show the resemblance to QED? I am not talking about gravitons, I'm sure you realize. It's problematical to define particles on anything but a fixed flat background. The concept of particle doesn't work in dynamic geometry---you can't even say how many there are. It is just a useful approximation appropriate to a limited context. So I need a theory not of gravitons but of evolving quantum geometry.
I don't see how you would set up something analogous to QED. So I'm interested to see what you have in mind!

Last edited: Oct 23, 2009
8. Oct 23, 2009

### atyy

But what about Asymptotic Safety? Even if it turns out not to be true, the fact that it is plausible shows that the geometric nature of gravity is not sufficient to rule out quantizing gravity like electromagnetism.

9. Oct 23, 2009

### marcus

Jokkon, I should had said right away, the easy intuitive introduction to LQG is Rovelli's online video
http://cdsweb.cern.ch/record/1121957?ln=en
It is of his summer 2008 talk to the annual Strings conference, which was held at Cern.
The slides PDF can be downloaded separately. Ask if you want a link. It is the obvious place to start---in fact you may have already watched the talk!

Atyy, my reply to your post #8 is two posts back, in the one just before yours.

===================
edit to reply to Atyy's next, post #10
What then are the metric's fundamental degrees of f., and what do you propose to be their symmetry?

Last edited: Oct 23, 2009
10. Oct 23, 2009

### atyy

Well, I'm just thinking that to get a quantum field theory you postulate the fundamental degrees of freedom and their symmetry, write a Lagrangian containing all terms consistent with the postulated symmetries, and interpret it within the path integral formulation. To have only a finite number of terms to determine experimentally, ie renormalizable, you hope the theory has a fixed point, and you search for it. In QED, the fixed point is an infra-red fixed point, while in QCD the fixed point is a UV fixed point. Asymptotic Safety is the scenario in which the metric degrees of freedom have a UV fixed point - so in that sense, it would be more like QCD than QED. A difference from QCD is that the fixed point is not a free quantum field theory, but I think that's a detail (although the calculations are very difficult), not a conceptual difference, at least with respect to being able to say that Asymptotic Safety means gravity can be treated as just another QFT.

I think it's hard to see from the historical route to QED that renormalizability has to do with a fixed point, specifically an IR fixed point in QED, but using this fixed point idea unifies perturbative and non-perturbative renormalizability. IR versus UV fixed points determine whether the theory holds at arbitrarily high energies, while whether the fixed point is free (ie. Gaussian) or non-trivial (ie. non-Gaussian) determines whether one can usefully do perturbations about a free theory.

Last edited: Oct 23, 2009
11. Oct 23, 2009

### atyy

12. Oct 23, 2009

### jokkon

marcus thanks for the link, I did watch that video 2 weeks ago when I first started the research. I found it difficult to follow because he seemed very nervous. Not sure if it is because it is a string conference. I will watch it again and see if I can understand more.

Atyy, I read it from Penrose's The Road to Reality. I suppose I can quote it here:
"My own particular goal had been to try to describe physics in terms of discrete combinatorial quantities, since I had, at that time, been rather strongly of the view that physics and spacetime structure should be based, at root, on discreteness, rather than continuity.... I had come to the conclusion that the best prospect for satisfying these requirements was to consider the quantum-mechanical quantity of total spin of a system..." Pg 947 Chapter 32.6

13. Oct 23, 2009

### atyy

Well, spin is an important part of quantum mechanics. Spin is a sort of angular momentum. In everyday life, if you spin 360 degrees, you get back to the same position. In quantum mechanics, there are more bizarre sorts of spin where you need to "spin" at least 720 degrees to get back to the same "position". Despite this difference, the algebra of quantum mechanical spin is similar to that of angular momentum spin. However, this is not directly related to spin networks, so maybe Penrose just meant that was inspired by spin to invent spin networks - but he didn't mean that spin networks are used in quantum mechanics of atoms.

14. Oct 24, 2009

### atyy

Ooops, I missed this till now. The metric is the fundamental degree of freedom, and the symmetry is isometry, and the terms consistent with this symmetry are the curvature invariants.

I haven't used the term "degree of freedom" quite correctly there, but here is a lead to a more accurate statement http://arxiv.org/abs/0709.3851:

"In general it will have the form ... are all possible operators constructed with the fields and their derivatives, which are compatible with the symmetries of the theory."

Eq 1.3.1: "polynomials in the curvature tensor and its derivatives containing 2n derivatives of the metric"

Last edited: Oct 24, 2009
15. Oct 24, 2009

### marcus

Thanks for clarifying for me what you had in mind. I thought you might be going to have the symmetries be the diffeomorphisms (because I think of them as the gauge group for the metrics). Had no idea actually. I'll let what you said simmer for a while and reply later.
I'm glad to have the reference to Percacci's paper, with a pointer to the relevant section!

Last edited: Oct 24, 2009
16. Oct 24, 2009

### atyy

I almost said that - I am perpetually confused about the exact relationship between diffeomorphisms/isometries.

17. Oct 24, 2009

### atyy

http://ocw.mit.edu/OcwWeb/Physics/8-334Spring-2008/LectureNotes/index.htm [Broken]
Kardar's L7, Section III.E has a very good concise summary of Wilsonian renormalization. In L12, Section IV.H, he also notes that "even if some of these terms are left out of the original Hamiltonian,they are generated under coarse graining." Kardar's notes are about statistical physics, but there is a correspondence between statistical field theories and quantum field theories.

http://arxiv.org/abs/0909.0859
Hollowood's second lecture describes Wilsonian renormalization which stresses symmetries and fixed points. His third lecture gives the relationship between Wilsonian renormalization and the historical renormalization approach which seems to be a bizarre procedure of adding counter terms and cancelling infinities. He also gives a useful link to Weinberg's notes where he was teaching himself statistical field theory and finding it very useful for quantum field theory http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?197610218 [Broken]. The Wilsonian point of view in statistical field theory was so powerful that it led Weinberg to propose Asymptotic Safety, although from the point of view of statistical physics, it was nothing new at all in 1976. On the other hand, Wilson was a particle physicist "outsider" when he followed the hints of many statistical physicists including Kadanoff, and solved the problem of critical phenomena in statistical physics. So Asymptotic Safety can be seen both as a "particle physics" approach, as well as a "condensed matter" one

Last edited by a moderator: May 4, 2017
18. Oct 25, 2009

### John86

[QUOTE So Asymptotic Safety can be seen both as a "particle physics" approach, as well as a "condensed matter" one [/QUOTE]

Can anyone give some more details !.

19. Oct 26, 2009

### atyy

Quick pointer, many details are in the references of post #17. Asymptotic Safety is not emergent. However, the ideas of fundamental/emergent, and non-Gaussian fixed points come from Wilsonian renormalization whose first triumph was to solve the problem of universality at a critical point in condensed matter. Universality is the cool experimental observation that close to a second order phase transition, some "critical exponents" are the same for many different materials with completely different microscopic structures, as long as they share the same dimensionality.