What exactly is loop quantum gravity?

[tom.stoer]

@nomisrosen: I reported this thread (my last post) b/c I think we hijacked this thread and lost you from the very beginning; I hope they can split this thread such that we can focus on your initial question and my response, post #2. Forget about everything else.

So what is unclear about #2? And what could be a good starting point based on your existing knowledge?

 

[marcus]

Tom, I agree the thread got off track. Here is post #6 by Kevin for example:

Nice introduction but what I really want to know is how close LQG is to becoming fully quantized? Does it even have a semi-classical description yet?

We got away from concentrating on the most basic explanation of what LQG is---we veered off into more technical evaluations of it.

So suppose we re-focus on math-less explanation for Rosen or some hypothetical high school student.

I will make an attempt similar to yours.

I will say to Rosen that that Einstein showed us 100 years ago that gravity is really geometry. Gravity is caused by the changing shape of space.

"Quantum" carries the idea of uncertainty, indefiniteness. Nature's resistance to being precisely pinned down. Nature's ability to be in a mixture or "superposition" of different states----so to speak one on top of the other.

So "quantum gravity" = uncertain geometry.

LOOP quantum gravity is basically an approach using spin networks to describe the uncertain geometry of the universe.

They started out using loops but soon found that a kind of spiderweb network worked better than simple loops. So loop QG is a bad name---they don't use loops. they should call it spin network QG. The name refers to the historical beginnings.

To understand LQG you have to concentrate on understanding spin networks.

A network is made of nodes and links. Nodes are the junctions where links meet. Nodes are labeled to represent a bit of volume and links are labeled by area numbers. Networks can be mixed or superimposed, so we can get vague chunks of volume glued together at blurry bits of area.

A network with enough nodes and links can represent quite a lot of geometry---the nodes provide places where particles can be located and the links indicate possible moves the particles can make. A labeled network represents a kind of simplified world, or the geometry thereof. Nodes and links have no specified location in some conventional space. They ARE location themselves. There is no conventional standardized space.

Another name for network is "graph". A spin network is a labeled graph where the node labels refer to vol and the link labels refer to area where adjacent volumes meet.

I think if I had to explain LQG to a high schooler I would begin by drawing examples of graphs on the blackboard, or scratch paper.
I'm not sure this explanatory tactic would work. It may be a dud (failure). I'll see what NomisRosen says.

 

[nomisrosen]

Marcus, thank you so much! That really helped. The uncertain geometry did it for me. But now, what are these nodes and spin networks made of..? Do they operate at the Planck scale?

Also, is there some sort of wave function of probability to know how this geometry might behave in a certain situation?

What determines how much space a node can give rise too? And of course, what is "outside the node"

...sorry for all the questions, answer what you

 

[atyy]

In quantum mechanics, the basic wavefunction is a position. An arbitary wavefunction is a superposition of positions. Additionally, observable quantities like position or momentum correspond to position or momentum operators. The probability of observing a particular momentum is given by the "product" of the wavefunction (superposition of positions) and the operator (momentum operator).

In LQG, the wavefunction is the spin-network, and the spin-network is in some sense geometry. So an arbitrary wavefunction is a superposition of geometries. An observable quantity like volume corresponds to a volume operator. The probability of observing a particular geometrical quantity like volume is given by a "product" of the wavefunction (superposition of geometries) and the operator (volume operator).

Here geometry means spatial geometry. However, gravity is spacetime geometry or the time evolution of spatial geometry. It is unknown how to describe the time evolution of spatial geometry in LQG, which remains a major problem.

I think the lesson of string theory is that not every geometrical object in the theory is necessarily spatial or spacetime geometry. So it has from time to time been suggested that the geometry of LQG represents not just space or spacetime, but possibly also matter. It is unknown which, if any, interpretations of the same mathematics as physical quantities will eventually work.

 

[tom.stoer]

However, gravity is spacetime geometry or the time evolution of spatial geometry. It is unknown how to describe the time evolution of spatial geometry in LQG, which remains a major problem.

That's only partially true. There are proposals for a time-evolution operator or Hamiltonian H which look consistent. In addition the spin foam formulation seems to avoid this problem completely; in addition they are working on a harmonization of these two formulations, canonical spin networks and spin foams.

So it has from time to time been suggested that the geometry of LQG represents not just space or spacetime, but possibly also matter. It is unknown which, if any, interpretations of the same mathematics as physical quantities will eventually work.

This is still a rather speculative idea - but of course it would be a highly appreciated major breakthrough

 

[ensabah6]

That's only partially true. There are proposals for a time-evolution operator or Hamiltonian H which look consistent. In addition the spin foam formulation seems to avoid this problem completely; in addition they are working on a harmonization of these two formulations, canonical spin networks and spin foams.


This is still a rather speculative idea - but of course it would be a highly appreciated major breakthrough

what do you think of the current work of including a spectral triple with LQG quantization? i.e NCG+LQG?

 

[tom.stoer]

what do you think of the current work of including a spectral triple with LQG quantization? i.e NCG+LQG?

Do you really mean NCG a la Connes or simply q-Deformation?

The latter one seems to be natural (forget about the reason for the SL(2,C) = 4-dim spacetime, take any symmetry group and study its spin foams w/o any reference to dimension; this includes SU, SO, SP?, E? and q-deformation)

Regarding NCG: I do not know enough about it, but it seems that it could spoil the simple picture of LQG; the Connes approach is rather special and seems to explain nothing (it simply replaces the standard model with a special NC geometry, but it can't explain why THIS gemometry, not something else); I would prefer to see matter and the cc emering from q-defomed / framed spin networks - but this could be wishful thinking ...

 

[marcus]

... The uncertain geometry did it for me. But now, what are these nodes and spin networks made of..? Do they operate at the Planck scale?

Also, is there some sort of wave function of probability to know how this geometry might behave in a certain situation?

What determines how much space a node can give rise too? And of course, what is "outside the node"

...

I think of spin networks as descriptors used to describe simplified geometry. So a spin network is analogous to a word, or a number. We don't need to ask "what is the number 3 made of?" or "what is the word mass made of?" I guess adding more and more nodes and links to the network is in some way analogous to adding more decimal places to a number---making the description more refined/accurate/realistic.

The bottom line is not "what is it made of?" but rather: does it work as a description? Is it the right way to diagram the uncertain geometric reality?

In answer to your first main question, YES the spin network description is supposed to work at planck scale!
This is, in fact, one of the principal goals of LQG research! To find a description of the world's uncertain geometry that continues to work in extreme circumstances (like extreme density, where classical geometry suffers a "singularity" and fails to make sense.)

Beyond that, and equally important, the aim is to have a description that predicts enough about the early universe to be TESTABLE. To be science (and not just myth or fairytale) it has to predict features that people can look for in the ancient light (the so-called microwave background or CMB). A good description must risk falsification by predicting some observable footprint in the oldest light. Or traces in something else, say neutrinos?, which might have been left over from the extreme density Planck era.

Your second main question was about describing behavior.

We can think of a spin network as describing an instantaneous state of geometry, so then we want to know how that evolves. Eventually we want to be able to talk about how geometry interacts with matter---so there is this general issue of behavior, spelled out in transition probabilities.

The descriptive tool used in LQG to get transition probabilities (from one spin network state to another) is called a spin foam.

A foam is like the moving picture of a changing network. If you picture a network as a spider web, then a foam is sort of like a honeycomb. Both are mathematical objects. In LQG, both have labels.

So your second question points in that direction: is there some sort of mathematical machinery to calculate transition probabilities, from one geometric state to the next? The answer is YES. There are spin foams which are the paths of evolution of spin networks, and techniques have been developed for calculating probability amplitudes.

Also I think at this point you find disagreement. Different LQG researchers have proposed different procedures for calculating transition amplitudes. There are unresolved issues about infinities that have to be ironed out. And some way must be found to TEST. If a theory does not risk falsification by making predictions about something that you can reasonably expect might be observed, then it is empty. Ways to test are beginning to be proposed, but are still controversial. There is a lot of work to do.

This week the biannual LQG conference is being held. To see the list of talks, and get an idea of the topics being researched, go here
http://www.iem.csic.es/loops11/
and click on the menu where it says "scientific program".

 

[atyy]

That's only partially true. There are proposals for a time-evolution operator or Hamiltonian H which look consistent. In addition the spin foam formulation seems to avoid this problem completely; in addition they are working on a harmonization of these two formulations, canonical spin networks and spin foams.


This is still a rather speculative idea - but of course it would be a highly appreciated major breakthrough

Yes indeed to both points. Maybe the problem has already been solved by Thiemann's old proposal, and it just isn't understood how to extract the right classical limit. I've often read that it's thought the Thiemann solution had the wrong classical limit, but I don't know the literature apart from isolated statements here and there in other papers. Do you know any papers that examine the classical/semiclassical limit of Thiemann's solution?

 

[nomisrosen]

I think of spin networks as descriptors used to describe simplified geometry. So a spin network is analogous to a word, or a number. We don't need to ask "what is the number 3 made of?" or "what is the word mass made of?" I guess adding more and more nodes and links to the network is in some way analogous to adding more decimal places to a number---making the description more refined/accurate/realistic.

The bottom line is not "what is it made of?" but rather: does it work as a description? Is it the right way to diagram the uncertain geometric reality?

I don't really understand how the universe's geometry can be something real and yet not be made of anything...

Are the links basically like guidelines of space that nothing can cross (as in the only way to get from one point in space to another)?

Is this a good visualization of how matter interacts with spin foam?


Thanks again for all your clarifications.

 

[marcus]

I don't really understand how the universe's geometry can be something real and yet not be made of anything...

This is something in our culture, Simon. It leads to a problem. If something is only real if it is made of something else, where does it stop?

Is this a good visualization of how matter interacts with spin foam?

That is a thoughtprovoking visualization. I wouldn't think of it as literal fact, but as suggestive (a simplified 2D toy universe.) Plus I personally can't vouch. I'm an observer from the sidelines. I am not an expert. I don't do LQG research. It is exciting and interesting so I watch it. Thanks for the link.

Ultimately what matters is HOW NATURE RESPONDS TO MEASUREMENT. How we interact with it. Ultimately you cannot continue to explain nature by saying what it is "made of". There are limits to what we can measure. Certain things lose their operational meaning past Planck scale, if you cannot measure them. Anyway that is what I think.

So I want fundamental descriptions (of interaction and geometric relationship), I do not expect "this made of that" answers.

 

[marcus]

A spin network is not made of anything. It describes a situation you might think of discovering by making measurements. Put a bag around this and these and this and these nodes and your bag will now contain just that much volume. Add up the volume labels contained in the bag---there is one associated with each node.

Plus you can determine the area surrounding the region by adding up the cut links. The bag surrounding that bunch of nodes has to pass through a bunch of links that connect the inside with the outside. Or think of the links as pins puncturing the bag. All those links are labeled, so add up the labels to determine area. I know this seems a bit vague but it actually works pretty well to specify the "skeleton" of a geometry. You can even get angles (as well as areas and vols) from the labels.

The labels are simple enough: whole numbers and half-integers like 1/2, 1, 3/2, 2, 5/2,... They encode the geometry that lives on a particular network or skeleton.

 

[negru]

Only Marcus would find a made up animation based on absolutely nothing to be "thought provoking" ;)

Look he also has a video of color confinement!
http://www.youtube.com/user/DavidMTRutten#p/a/u/0/dvmE1XKZiwI

 

[marcus]

I notice folks still go to this thread, so it might be helpful to bring it up to date. Particularly as regards the new formulation and the OPEN PROBLEMS relating to it that various people have listed.
When I say LQG I mean of course the new formulation that uses spin foam to calculate transition amplitudes between quantum states of geometry.

A quantum state of boundary geometry (e.g. initial and final quantum states) is determined by a network of measurements (e.g. angles, distances, areas...) represented by a labeled graph.

The probability of the implied transition between boundary states is given by an amplitude calculated from the foam (a honeycomb-like "cell complex") enclosed by the boundary. Mathematically a foam is analogous to a graph but at one higher dimension. Instead of merely having nodes and links, it has vertices, edges and faces. Labeled with quantum numbers, a foam describes a possible way that geometry can evolve from one 3D geometric quantum state to another.

I'll refer to the new LQG formulation as the Zakopane formulation. It appeared in a series of 4 papers:
A New Look at LQG... (April 2010) 1004.1780
Geometry of LQG... (May 2010) 1005.2927.
Simple Model... (October 2010) 1010.1939
Zakopane Lectures (February 2011) http://arxiv.org/abs/1102.3660

A remarkable thing about the new formulation developed in these papers is its concise easy-to-understand presentation. A clear definite description of the theory can be given in one page. The fourth paper also gives the math prep needed to appreciate the one--age formulation, making it self-contained. The fourth is basically an improved version of the first and these two papers present a list of OPEN PROBLEMS for researchers to tackle. Several of these are areas where progress is currently being made. The listed problems are primarily conceptual in nature, having to do with the theory itself. There is also current activity in cosmology, investigating ways to test the theory by comparing its predictions with observation.

Another window on interesting conceptual problems, for someone getting into LQG research at graduate or postdoc level, is the October 2011 paper by Freidel Geiller and Ziprick which reveals the classical continuum phase space discretization that Zakopane LQG is the quantization of.
http://arxiv.org/abs/1110.4833
Continuous formulation of the Loop Quantum Gravity phase space
Laurent Freidel, Marc Geiller, Jonathan Ziprick
(Submitted on 21 Oct 2011)

In this paper, we study the discrete classical phase space of loop gravity, which is expressed in terms of the holonomy-flux variables, and show how it is related to the continuous phase space of general relativity. In particular, we prove an isomorphism between the loop gravity discrete phase space and the symplectic reduction of the continuous phase space with respect to a flatness constraint. This gives for the first time a precise relationship between the continuum and holonomy-flux variables...
​

For various reasons, the conceptual importance of this paper is hard to overestimate.