What exactly is loop quantum gravity?

[nomisrosen]

I don't quite understand the properties of loop quantum gravity. I have searched around and have not come up with anything very helpful. I'm pretty good when it comes to understanding quantum mechanics and string theory, but please, NO MATH! I'm only in high school!

thanks

 

[tom.stoer]

It's difficult w/o math.

The problem with quantum gravity is that naive mechanisms to quantize gravity (which have been applied successfully to other fields) fail for gravity. That means that something fundamental has to be changed for quantum gravity.

There are different approaches to solve these problems, e.g.
a) string theory
b) asymptotic safety
c) loop quantum gravity (LQG)

I don't want to comment on a) and b) here.

Essentially LQG does the following: it introduces new variables which replace the (in GR) well-known metric that describes spacetime + curvature. This is pure math, so I don't want to go into details here, but what happens is that these new variables are rather close to fields that we know from gauge theory like QED and QCD. Indeed in a certain sense gravity looks rather similar to QCD, but there is one additional property of gravity that allows one to apply a second mathematical trick which essentially replaces the fundamental fields with something like "fluxes through surfaces" or "fluxes along circles". These surfaces and circles are embedded into spacetime.

The next step is again rather technical and it becomes possible due to so-called diffeomorphism invariance: one can get rid of the the embedding of circles and surfaces into spacetime. Instead one replaces these entities with a so-called spin network, i.e. a graph with nodes and links between nodes where each link and each node carries some numbers which represent abstract entities from which certain properies of spacetime can be reconstructed. You can think about spacetime as made of cells (I will soon tell you that you can't :-); each cell has a certain volume carried by a node; each cell has certain surfaces and the link between different nodes (sitting inside these cells) carry the areas of the surfcaes.

The problem with this picture is that one might think about these cells as sitting in spacetime - but this is fundamentally wrong: this picture is only due to the construction, but basically there is no spacetime anymore; all there is are nodes and links (and certain numbers attributed to nodes and links). Spacetime is no longer fundamental but becomes an entity emerging from the more fundamental graphs with their nodes and links. The graphs are called spin networks b/c the numbers they are carrying have properties well-known from spins. But this is a mathematical property only, it does not means that there are real spinning objects.

Compare this emerging spacetime to a water surface of a lake. We know that it consists of atoms, and as soon as we get this picture it is clear that there is no water between the atoms; the surface is only an emerging phenomenon, the true fundamental objects are the atoms. In the same sense the spin networks are the fundametal entities from which spacetime, surfaces etc. and their properties like volume, area, curvature etc. can be constructed. Dynamics of spacetime (which was curvature, gravitational waves etc. in GR) is replaced by dynamics of spin networks: within a given graph new nodes with new links can appear (there are mathematical rules, but I don't want to go into detail here).

The last puzzle I have for you is the fact that such a spin network is not a mechanical object which "is" spacetime. Instead quantized spacetime is a superposition of (infinitly many) spin networks. This is well-known in quantum mechanics; there is no reason why an atom should be in a certain state; we can achieve that via preparation or measurement, but in principle a single atom can be in an arbitrary complex quantum state which is a superposition of "an atom sitting here, an atom moving in a certain direction over there, an atom moving in this or that direction, ...".

So classical spacetime is recovered by two averaging process: first there seems to be a regime were this superposition of spin networks is peaked around a single classical spacetime, i.e. were one networks dominates the superposition of infinitly many spin networks; second from this single spin network one can reconstruct spacetime in the same sense as one can reconstruct the water surface from the individual atoms. But there may be different regimes (e.g. in black holes or closed to the big bang) where is classical picture and this averaging does no longer work. It may be that in these regimes all there is are spin networks w/o any classical property like smooth spacetime, areas, volume etc. It's like looking at a single atom: there is no water surface anymore.

Eventually this is why one started with this stuff: the classical picture of spacetime seems to become inconsistent when one tries to quantize it, i.e. when one defines these superpositins etc. These inconsistencies do not bother us as long as we talk about spacetime here, in the solar system etc. But they become a pain in the a... when we talk about spacetime near a singularity like a black hole or like the big bang. In order to understand these new non-classical regimes of spacetime a fundamentally new picture is required. This is what LQG (and other approaches) are aiming for: construct a new mathematical model from which well-known classical spacetime (like in GR) can be reconstrcuted, but which does not break down in certain regimes but remains well-defined and consistent.

He must so to speak throw away the ladder, after he has climbed up on it - Ludwig Wittgenstein

 

[marcus]

This is a really good introductory summary. I suggest proposing it for a BTSM forum FAQ. The only minor "typo" is that the usual term is "node" where you have said "knot".
(The two words come from the same etymological root and originally meant the same, but in normal technical language one says that a spin network consists of nodes and links.)
Otherwise this is great. Clearly and concisely giving just the essentials.

 

[tom.stoer]

Thanks a lot - and sorry for the "knots", of course it must read "node"!

If you like we can discuss and improve the text, so instead of writing new posts I can change the original text (new color).

 

[marcus]

... (new color).

I would suggest you not bother with new color. It's a minor typo of no consequence. If the mentors like the idea of FAQ for BTSM they will (I expect) do a final edit and move the text to wherever it needs to be.

As far as substantive changes, I can't think of anything to add or alter! We are lucky to have something both careful, relaxed, and readable like this is.

Say! maybe I will move the text to SA forum so people can discuss quietly.

 

[Kevin_Axion]

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?

 

[tom.stoer]

Aren't these two completely different questions?

Of course LQG is fully quantized ... Hilbert space, operator algebra, inner product, path integral ... what is missing?

Yes, there are attempts for semiclassical / smooth / large-j limit and construction of coherent states; there are results regarding the graviton propagator; ...; but this is (afaik) work in progress and (at least for me) it seems to be too early to present physical results.

 

[Kevin_Axion]

Aha, yes they are different questions. That was my fault. Thanks for the response. What else is left for LQG to do?

 

[tom.stoer]

semiclassical limit for reasonable smooth spacetimes like BHs, FRW, ...
n-point functions for gravitons (on top of these spacetimes)
absence of infrared divergence
renormalization group
equivalence of canonical and covariant / spin foam quantization
matter degrees of freedom (gauge fields)
nature of cosmological constant (and Immirzi parameter)
phenomenology, experimental tests (*)

Everything but (*) is currently under investigation and I think for all topics the different research groups make good progress. There seems to be no major obstacle, no road block.

For (*) only very indirect results are to be expected, but I think LQG shares this with all other QG approaches, unfortunately.

 

[Kevin_Axion]

So they're not even close.

 

[tom.stoer]

close to what?

 

[Kevin_Axion]

To having a full and complete description of LQG.

 

[atyy]

semiclassical limit for reasonable smooth spacetimes like BHs, FRW, ...
n-point functions for gravitons (on top of these spacetimes)
absence of infrared divergence
renormalization group
equivalence of canonical and covariant / spin foam quantization
matter degrees of freedom (gauge fields)
nature of cosmological constant (and Immirzi parameter)
phenomenology, experimental tests (*)

Everything but (*) is currently under investigation and I think for all topics the different research groups make good progress. There seems to be no major obstacle, no road block.

For (*) only very indirect results are to be expected, but I think LQG shares this with all other QG approaches, unfortunately.

Also:

consistency of the infinite refinement limit (discussed eg. in http://arxiv.org/abs/1010.5437 and http://arxiv.org/abs/1010.5437 )

 

[atyy]

To having a full and complete description of LQG.

They could be close. String theory says that theories without gravity have gravity, so if LQG doesn't have gravity it could have gravity. (More seriously, some lattice gauge theories have a dual formulation as spin foams, and string theory famously says some gauge theories contain gravity).

However, for LQG as pure gravity, they are far. Eg. even the same authors can't decide how to interpret the semiclassical limit. Contrast http://arxiv.org/abs/1004.4550 and http://arxiv.org/abs/1105.0216v5 - these are not just any authors, but people who established the state of the art on the interpretation of the semiclassical limit http://arxiv.org/abs/0905.4082

 

[Kevin_Axion]

Yes but doesn't AdS/CFT require that the gauge theory be an N = 4 supersymmetric Yang-Mills theory i.e LQG isn't supersymmetric.

 

[tom.stoer]

Do 'they' have a full and complete description of QCD?

I mean they have a complete and consistent theory at hand, though much work has to be done to complete the details, to consolidate the proofs, to work out new tools. They address their open issues - that's what science is.

In QCD color confinement has opposed a theoretical justification, but hints and indications are available; nevertheless we believe in QCD as the correct theory of strong interactions.

The main issue is a lack of (*) b/c afaik no prediction (not post-diction) of LQG is available that could falsify LQG or decide between LQG and other QG approaches.

 

[tom.stoer]

Yes but doesn't AdS/CFT require that the gauge theory be an N = 4 supersymmetric Yang-Mills theory i.e LQG isn't supersymmetric.

In the string / m-theory context this is a rather striking result, but LQG uses a completely different quantization procedure and therefore you can't conclude that AdS/CFT results must apply to LQG; it could very well be that the asymptotic safety program tells us that even standard but non-perturbative quantization of GR is UV complete and fully consistent; it could even be that in a couple of years we have more than obe fully consistent theory of QG at hand.

 

[atyy]

Yes but doesn't AdS/CFT require that the gauge theory be an N = 4 supersymmetric Yang-Mills theory i.e LQG isn't supersymmetric.

AdS/CFT is thought to be braoder than that. However, that is the most celebrated case. Supersymmetric spin foams are studied in http://arxiv.org/abs/0710.3540 and http://arxiv.org/abs/1004.0672 .

 

[atyy]

In the string / m-theory context this is a rather striking result, but LQG uses a completely different quantization procedure and therefore you can't conclude that AdS/CFT results must apply to LQG; it could very well be that the asymptotic safety program tells us that even standard but non-perturbative quantization of GR is UV complete and fully consistent; it could even be that in a couple of years we have more than obe fully consistent theory of QG at hand.

Yes, but how about the duality between spin foams and lattice gauge theory? I don't know the exact restrictions for such a duality, but I know there's some correspondence.

Also, LQG is capable of quantizing 3D lattice BF theory (in some cases with additional stuff) , which reminds me of the 3D Chern-Simons (with supersymmetry and matter) ABJM case of AdS/CFT.

 

[Kevin_Axion]

AdS/CFT is thought to be braoder than that. However, that is the most celebrated case. Supersymmetric spin foams are studied in http://arxiv.org/abs/0710.3540 and http://arxiv.org/abs/1004.0672 .

That doesn't seem natural in terms of LQG. LQG is generally celebrated for not containing SUSY and other new parameters i.e extra-dimensions. So what is this Asymptotic Safety you and tom.stoer are talking about, has it had any reasonable progress?

 

[atyy]

That doesn't seem natural in terms of LQG. LQG is generally celebrated for not containing SUSY and other new parameters i.e extra-dimensions. So what is this Asymptotic Safety you and tom.stoer are talking about, has it had any reasonable progress?

It could be natural - eg. http://arxiv.org/abs/1103.6264 mentions emergent gravity.

I don't know about Asymptotic Safety. The state of the art is, I believe, still http://arxiv.org/abs/0805.2909 and the beautiful results of CDT. It is of course unclear what exactly CDT is connected to, except for gut feeling. I think the recent directions that are interesting are http://arxiv.org/abs/1012.3081 and http://arxiv.org/abs/1012.4280

 

[tom.stoer]

I think we are far away from understanding the relation between AdS/CFT / strings and LQG / spin networks / foams. Strings are both broader (matter degrees of freedom, ...) and narrower (gravity requires strings / swampland - no b/c LQG comes w/o strings but with gravity).

Thiemann published some new papers regarding "SUGRA-LQG"; I just started reading his n-dim LQG papers but had no time to check the SUGRA stuff.

All what we can say is that there seems to be a miraculous relation between classical gravity + QFT, LQG and strings. All these theories predict a fundametal relation regarding black holes, but none of them is complete (classical gravity + QFT is inconsistent, LQG cannot fix the Immirz parameter, strings only talk about BPS BHs).

The main problem for all these theories is that they lack experimental input and falsifiability (in practice, not in principle). We live in an age confronted with a paradigm shift in theoretical physics.

 

[tom.stoer]

It seems that both string theory and LQG will turn into theory-producing machineries (instead of theories). There seems to be no reason why not
- construct LQG-like theories in n dimensions (n>2 arbitrary)
- incorporate SUSY / SUGRA / gauge fields
- use higher spin / SU(n) groups
- use qantum deformation

The main difference is that all these theory-producing machines in string theory (they call it 'vacua') are related via dualities and relations like that, whereas in LQG you would do "ordinary construction of theories" w/o any relation between them.

 

[Kevin_Axion]

I see, thanks for the input. I really like LQG for it's simple geometrical solution but dislike it because of the Immirizi parameter (kind of seems like cheating). I also like string theory for it's success but I believe it's tackling way too many problems at once, I mean we don't even know the full extent of physics and String Theory doesn't really go far beyond the SM to be a TOE let alone a QG theory. How does string theory treat space-time anyways? Does it just add new dualities and degrees of freedom to construct a theory of QG?

 

[atyy]

The main problem for all these theories is that they lack experimental input and falsifiability (in practice, not in principle). We live in an age confronted with a paradigm shift in theoretical physics.

Oh, I hope not. That would be the end of physics. I think there's plenty of theory outside of quantum gravity, and also LQG already applies to condensed matter, and there is hope that strings will too. After all the renormalization group came via high energy, was truly understood in condensed matter, and then fed back into HEP for no longer being worried about sweeping infinities under the rug, and ideas of asymptotic safety.

 

[tom.stoer]

OK; I'll change my wording:

The main problem for all these theories is that they lack experimental input and falsifiability (in practice, not in principle). We live in an age confronted with a paradigm shift in (certain areas of) theoretical physics.

 

[nomisrosen]

I didn't understand a thing...thanks for trying!

 

[Kevin_Axion]

We can't leave you unsatisfied! What didn't you understand?

 

[nomisrosen]

the whole thing...im no physicist haha just an average high school student who happens to enjoy reading about physics!

 

[Kevin_Axion]

Well, do you firstly know concepts in Quantum Mechanics/Quantum Field Theory or General Relativity? These are essential to understanding Quantum Gravity i.e loop quantum gravity or string theory

 

[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