# History of Loop Quantum Gravity

1. Jan 5, 2010

### JustinLevy

Since there are several people here who intently follow the developements of LQG, I was wondering if someone could give some overview to help understand the 'historical path' of research on this topic better. What methods were developed, which died off, what was learned, where they are going now, etc.

To get things started, my (very rough, and possibly incorrect) understanding is the following:
(Due to the roughness/possibly incorrect info, please don't add to this ... this is more to give an example what kind of history I'm interested in.)

It is my understanding that there is a difficulty when setting out to quantize gravity in choosing the dynamical variables due to the large equivalences given by diffeomorphism invariance. So the starting point was the reformulation of Einstein-Cartan theory (GR extended to allow matter with inherent spin) with Ashtekar variables. This reformulation with the Ashtekar variables as the dynamical variables to be quantized was chosen as the starting point of LQG.

The 'loops' of LQG are wilson loops on the continuous spacetime (despite some claims by non LQG people, it is my understanding that LQG does NOT claim spacetime is discrete). Quantizing this describes the geometry of spacetime with a 'weave' of loops, and it was found at some point that this could be described with spin networks.

I don't understand what happened next, but I got the impression that the spin networks and canonical loop methods somehow diverged? I don't really understand how this was possible. It was found that there was some arbitrariness in how the dynamics was described, is this how the two departed from each other?

At one point (I don't know any dates), the Barrett-Crane LQG description of gravity was shown to be inconsistent with reality. When I first heard about this, it was from a string theorist who didn't like LQG for several philosophical reasons, and so used this as the 'last straw' to consider LQG as thrown in the waste basket. Based on impressions from other people, I also got the impression people were considering LQG 'dead' at this point. Is that a fair characterization of the outlook for LQG at the time?

I don't fully understand what was wrong with the Barrett-Crane model, but then some researchers narrowed down what was wrong and learned from it (what?) to create a better model of dynamics? There appears to be a real push now to get GR in the classical limit back out with this new model.

Some things I don't know how to fit in. How many seriously pursued 'versions' of LQG are there (how many died off, how many are still fighting forward)? How did LQC jump in, since it is somehow separate?

And Marcus sometimes refers to a 'mini-revolution' where spin-networks and the canonical loop methods were 'rejoined' or something. What happened? People here also refer to something as 'covariant loops', and sometimes say even the term 'loop' is just a hold-over from the historical birth of LQG. So what is covariant loops? (Didn't LQG already have poincare symmetry?) And are 'loops' not really involved anymore... how does it make contact with the Ashtekar reformulation of Einstein-Cartan without the wilson loops? I thought that was the whole starting point!?

Sorry for the length of this. Hopefully someone can explain what the current status is, and how we got here.
Thanks.

2. Jan 5, 2010

### marcus

It's good to ask about the history, Justin. But answering completely would be a lot of work. Maybe if everybody with something to contribute were willing to help, we could fill in some of the picture bit by bit.

In your thumbnail sketch, you may have confused spinnetwork with spinfoam.

Spinnetworks are what the canonical approach is based on (rather than loops) so it was never the case that spinnetworks "diverged" from the canonical approach. They are so to speak identical with it.

Nothing ever "diverged" and then "rejoined". A major problem was there were two approaches (canonical and spinfoam) that were never joined, until recently.

A spinfoam is intended to represent the evolution of a spinnetwork. But at first the amplitudes didn't work out right.
In order to calculate the amplitude associated with a spinfoam, you need to have a vertex formula.

The Barrett-Crane was one possible vertex formula. Rovelli et al discovered some problem with it in 2006. In 2007 some new vertex formulas were proposed. Still using the same spinfoam mathematical objects, but calculating different amplitudes for them.

In 2008 it was shown that with the new vertex formula, spinfoams were, as far as anyone could tell, the same theory as the canonical LQG based on spinnetworks. However not all the details were in place. In 2009 Lewandowski et al established a more rigorous thorough equivalence.
========================

Some intuition to put this all in context: classically, a spacetime geometry can be thought of as a trajectory thru the range of possible 3D geometries. A path thru the realm of space geometries.

A quantum spacetime geometry is not a distinct path but rather it is a way of assigning amplitudes to all the different paths that get you from some initial geometry A to some final geometry B. Analogous to a Feynman path integral.

A spinnetwork is a quantum state of 3D geometry. (mathematically it is a labeled graph)

If you imagine dragging a spinnetwork thru time, it sweeps out a spinfoam. (mathematically a 2-cell complex, the next higher dimensional thing from a graph).
Also while dragging, imagine the spinnetwork gradually changes, growing a new vertex somewhere, merging two vertices somewhere else. It sweeps out a more interesting spinfoam in that case.

A spinfoam is a 2-cell complex that represents a path thru the realm of possible spinnetworks.

So a spinfoam is like a trajectory thru the realm of quantum states of space geometry (quantum states which spinnetworks represent.)

If you have a good way to assign amplitudes to spinfoams, then you have a quantum spacetime geometry, of the path integral type.

As people often describe it now, LQG exists in both a covariant version and a canonical version.

Neither version seemed to be complete at the time they were proposed. They have each gotten more complete in 2008-2009 as people like Lewandowski worked to show that they described the same theory. During this process of bridging, there actually were minor tweaks and modifications, gaps were filled. I think this is still going on.

Last edited: Jan 6, 2010
3. Jan 5, 2010

### JustinLevy

Yes, completely. I didn't realize they were distinct and it lead me to misread some things. Thank you very much for the detailed description. A lot of previous discussion is making much more sense now.

Hmm.. worded that way I can see now why some people think LQG claims spacetime is discretized. There is a discrete graph, with seemingly discrete steps in time for any changes. But since the start was still wilson loops on a continuous spacetime, am I wrong to state that LQG does NOT claim spacetime is discrete? (Anymore than strings in string theory having a finite size means it describes spacetime as having a minimum length.) The spacetime being described in all cases is still continuous, yes?

I'm still not understanding this statement.
So the 'canonical' version is the spin-networks. Is the 'covariant' version the spin-foam? And why is it called covariant? ... is it just that the poincare symmetry is more readily apparent in that form? (or is the symmetry and/or dynamical variables actually different in one?)

4. Jan 5, 2010

### marcus

Yes.
The basic continuum is a manifold. The Wilson loops have evolved into "wilson networks", so to speak. In the canonical version, the networks form a basis of the hilbertspace of states of geometry.

Some measurements (corresponding to operators on the hilbert space) can have discrete spectrum. This is a kind of discreteness---like discreteness of the energy levels of an atom. It is the discreteness of a measurement, of a quantum observable. So canonical LQG has this double aspect---basically continuous but measurements coming up with discrete spectra.

The underlying mathematical object representing space (for now talking about the canonical case) is continuous, being a differential manifold. Classically, the curvature of that manifold can be described by what is technically called a connection---a gadget for specifying how to do parallel transport on the manifold. The connections can be seen as living in the configuration space. (like the positions of a particle, points on the real line, in elementary QM)

A quantum state of geometry should be complex valued function defined on the configuration space (like a wave function defined on the real line) and these quantum states should form a hilbertspace. What the spinnetwork provides is a function defined on connections. If you have a connection, the network samples it along some pathways embedded in the continuum.

The idea is to have a complex-valued function defined on the configuration space, that is defined on the set of all classical geometries. I'm oversimplifying, but I think that is the gist of it. That is what a spinnetwork is trying to be.

And then the spinnetworks span what is called the "kinematic" hilbertspace. We still have to define the Hamiltonian and other constraints---from them, the "physical" hilbertspace (of states of geometry which satisfy the constraints.)

But as a by-the-way, on the kinematic hilbertspace one can already define some obserservables. Geometric operators such as the area of some physical object. It turns out that the area operator has discrete spectrum. This seems surprising because the underlying objects appear to be continuous---the manifold, the connections. I've already oversimplified enough and shouldn't go further.

Maybe someone else will jump in and give a more rigorous less handwavey development.

Last edited: Jan 6, 2010
5. Jan 6, 2010

### tom.stoer

The problem with discreteness is this: when you start with loops you have an Hilbert space with an uncountable basis. You can get rid of this by implementing the diffeomorphism constraint (which means basically coordinate invariance of GR). Doing this you end up with a theory of discrete graphs (spi networks) w/o spacetime! Spacetime has dissappeared completely and will hopefully re-emerge as a low-energy limit.

So the theory elimininates its own roots!

6. Jan 6, 2010

### marcus

In this respect it follows the example of General Relativity.

GR uses the differential manifold to construct a dynamic geometry, and then, in the end, gets rid of the manifold.

In the end, there is no longer a metric but an equivalence class of metrics under diffeomorphism. So points in space and time are no longer represented in the theory and thus have no physical or objective existence. Only the dynamic geometry remains.

It's a tough act to follow.

Referring to the principle of general covariance, Einstein said:

Dadurch verlieren Zeit & Raum den letzter Rest von physikalischer Realität."

“Thereby time and space lose the last vestige of physical reality”.

7. Jan 6, 2010

### tom.stoer

Not really. A space-time point as a "location specified by events" does survive. You can talk about metrics (even if you are aware of the fact that there are different metrics encoding the same physics). So you do not change the mathematical framework.

In LQG you do change the mathematical framework. There are no diffeomorphisms, no metric, no manifold any more. You have discretized spacetime - and you are (at least currently) not able to describe how to reconstruct it!

Last edited: Jan 6, 2010
8. Jan 7, 2010

### xristy

The comment and question was raised:

To which there was the reply:

My reading of some of the work on LQG leaves me confused on these points. It does seem that at least some LQG researchers do claim that the spacetime being described has a discrete character. For example in http://relativity.livingreviews.org/Articles/lrr-2008-5/" [Broken] Rovelli writes on pg. 37:

Also in http://cgpg.gravity.psu.edu/people/Ashtekar/articles/spaceandtime.pdf" [Broken] Ashtekar writes on pg. 8:

and further in http://arxiv.org/abs/0705.2222" [Broken] Ashtekar comments on pg. 7:

Also recently in http://arxiv.org/abs/0910.2936" [Broken] Bojowald writes on pg. 5

I must not be understanding something of this aspect of the discussion in this thread.

Last edited by a moderator: May 4, 2017
9. Jan 7, 2010

### JustinLevy

I may not be fully understanding myself, but I think I understand the basics.
Let me try to explain with an analogy from simple non-relativistic quantum mechanics.

Just as the momentum operator is the generator of linear translations in space, the angular momentum operator is the generator of angular rotations in space.

The angular momentum operator has a discrete spectrum. Does this mean precession can only move in finite angular steps? Does this mean the laws of physics do not have true rotational symmetry? Of course it doesn't mean that.

So returning to the question at hand, does the discrete spectrum (with a lower bound) of the volume operator mean space is discretized? It depends on what you mean by discretized. Can we measure a chunk smaller than the minimum discrete size? No. But, does this mean space is built of blocks and no longer has rotational symmetry? Is angular momentum conservation gone? No, it does not mean that.

Given a spacetime with a metric, the wilson loops can be calculated. But this mapping is lossy in that given the wilson loops, we can't get the spacetime with a metric. But no matter which spacetime it could be, they all are continuous, and have local poincare symmetry (rotational symmetry, translational symmetry, etc.).

Furthermore, since a spin-network is a spatial slice of spacetime, a boost will provide a different spatial slice / a different spin-network. So in this sense, 'observers' can't agree on the 'blocks' that make up space. There are no fundemental blocks of spacetime. It is merely that the volume operator has a discrete spectrum.

I am far from an authority on LQG. So hopefully someone can comment on the veracity of my understanding here. Also, I do not know the outcome of this debate in 2007:
http://arxiv.org/abs/0708.1721
http://arxiv.org/abs/0708.2481
but it sounds like LQG most likely still predicts the volume operator has a discrete spectrum.

10. Jan 7, 2010

### marcus

"The central physical result obtained from loop quantum gravity is the evidence for a physical quantum discreteness of space at the Planck scale. This is manifested in the fact that certain operators corresponding to the measurement of geometrical quantities, in particular area and volume, have discrete spectra. According to the standard interpretation of quantum mechanics (which we adopt), this means that the theory predicts that a physical measurement of an area or a volume will yield quantized results. In particular, since the smallest eigenvalues are of Planck scale, this implies that there is no way of observing areas smaller than the Planck scale. Space comes, therefore, in “quanta” in the same manner as the energy of an oscillator. The spectra of the area and volume operators have been computed in detail in loop quantum gravity. These spectra have a complicated structure, and they constitute detailed quantitative physical predictions of loop quantum gravity on Planck-scale physics."

Rovelli's 2008 Living Reviews article, which you quote is the closest thing we have to a careful detailed overview of the subject. The other things you cite are aimed at non-specialists and the language is more impressionistic.

One way to sort out the apparent double-nature is to say geometry at certain places where Rovelli says "space" but it talking about geometrical measurement.

In canonical LQG the math object representing space is a differential manifold, so as a set, space is a continuum.
However, what we measure is not space, we measure geometry. And in LQG geometry has a discrete character.

See if it makes it clearer if we substitute a word and rewrite Rovelli's passage this way:

"...this implies that there is no way of observing areas smaller than the Planck scale. [Geometry] comes, therefore, in “quanta” in the same manner as the energy of an oscillator."

What Justin said about the discreteness of angular momentum is an apt comparison!
Rovelli mentions a harmonic oscillator. It is helpful to think of cases where the underlying object can be represented by a continuum, say a rotating volume of a certain density, but measurement of some observable gives results which are in principle discrete.

Space itself, in LQG, is not discrete. Discreteness is not "put in" to the mathematical object representing space. However in geometric measurements discreteness comes out. So there is this dual nature that I mentioned in my earlier post.

Last edited by a moderator: May 4, 2017
11. Jan 7, 2010

### qsa

please,can somebody tell me if my understanding is correct or not:

1- LQG tries to reproduce GR(geometry of the metric) using QM but no such luck
2- CDT tries the same using discrete QFT but also limited results.
3- Shouldn't a fundemantal theory produce both the lagrangian plus symmetries(their effect or results) and space-time configuration from basic principles. And none of the so called QG theories try to actually do that,i.e. stuck in the conventional methods (even the highly sophisticated non-commutitive geometry).

12. Jan 7, 2010

### Finbar

There seems to be some confusion here about discrete space time.

This is my understanding of spin foam models http://arxiv.org/pdf/0911.0543: [Broken]

1) First we write down a classical action.

2) We then use a spin foam model where spacetime is discrete derived from discretizing the (classical)theory on a Regge geometry.

3) Then we quantize; from which we find that volumes have a discrete spectrum.

Whats important is not to confuse 2) and 3). In 2) the classical action is written as a regge action which is analogous to putting a theory on a lattice. Step 3) is the quantization where we go from the classical too the quantum theory.

From what I understand one still needs to take the continuum limit such that the theory really is defined on a manifold. Moreover this limit is equivalent to taking the renormalisation group scale k to infinity.

http://arxiv.org/pdf/gr-qc/0401087

But my main point is not to confuse 2) and 3).

Last edited by a moderator: May 4, 2017
13. Jan 7, 2010

### marcus

Justin, you asked about the history of LQG. In particular, you could ask when did the spinfoam idea show up?
Here is what I think is the original paper. I think it is of historical interest.

http://arxiv.org/abs/gr-qc/9612035
"Sum over Surfaces'' form of Loop Quantum Gravity
Michael P Reisenberger, Carlo Rovelli
(Submitted on 13 Dec 1996 (v1), last revised 24 May 1997 (this version, v3))
"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 a' 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."

You can see that the core idea of spinfoam is "sum over surfaces" in 4D. These surfaces were later named "foams".
The important idea is sum over histories of evolving 3D geometry.

It is analogous to "sum over paths" in a Feynman integral. In a Feynman path integral the path is regularized by making it discrete in the sense of being piecewise linear. It consists of line segments. This reduces the problem and makes it calculable.

In analogous way the spinfoam is piecewise linear, a regularization. One imagines all the possible spinfoam that lead from initial network A to final network B. One gives each of them an amplitude and sums.