Basic LQG questions raised by 1998 LQG Primer article

[marcus]

Basic LQG questions raised by 1998 "LQG Primer" article

B Crowell asked what would be the best way into LQG and then (on looking over the suggested entry-level "LQG Primer" article by Rovelli and Upadhya http://arxiv.org/abs/gr-qc/9806079 ) made a potentially very helpful list of questions.

...
I took a stab at the Rovelli-Upadhya paper...

I hope others here will help answer these questions. They seem to me to be natural ones that could reasonably be asked after taking a look at any LQG introduction. So exceptionally good ones to discuss. Here they are:

==quote==
What was really unclear to me was both the mathematical meaning and the physical significance of the SU(2) connection.

Mathematically, I understand the connection to be a rule for parallel transport, and I've seen it specified using either a metric or Christoffel symbols. The paper seems to be defining it using a smooth vector field, which is unfamiliar to me. Is this covered somewhere in texts like Wald and MTW? The reason for giving it values in su(2) is also unclear to me, and I guess that's a separate issue. Since they're talking about a three-dimensional space, I would think that the tangent space would be R³, not su(2)...??

Physically, I don't understand the motivation for introducing all the spin algebra. Is it basically because they want to be able to describe fundamental particles as existing at different places on this graph? Would there be spin-2 gravitons?

I also didn't understand the physical motivation for introducing the graphs. Is the manifold primary and the graphs secondary? Or vice-versa? What do the graphs represent physically?

Why is this all done in a three-dimensional space? Does this three-dimensional space relate to the quantum state of spacetime on some surface of simultaneity? Or does it not even relate to three dimensions of actual spacetime? I'd thought that the dimensionality of spacetime was an emergent property in LQG, and there was some difficulty in even showing that something like Minkowski space was a solution.
==endquote==

 

[jal]

For a newbie, An understanding of geometry, trig. and symmetry would be pre-requisite.
jal

 

[marcus]

I would welcome it if anyone else wishes to respond, and would answer other questions that I don't, or give alternative responses to the same questions. Also Crowell in some cases I may be repeating stuff you are already familiar with but that might help some other reader, so be patient.

...
Why is this all done in a three-dimensional space?...

I'll mention the ADM (1962) formulation of GR here.
http://en.wikipedia.org/wiki/ADM_formalism
GR has an equivalent formulation that uses a foliation into 3D slices and is independent of the foliation.
The original 1962 article was uploaded to arxiv to make it more accessible:
http://arxiv.org/abs/gr-qc/0405109

The ADM formalism is often preferred for numerical work, where computer is used to generate solutions of GR based on initial conditions given on some 3D hypersurface.

The ADM formalism served as a starting base for Ashtekar (1986) to reformulate GR in a 3D hypersurface way reminiscent of gauge theories and seemingly conducive to canonical quantization à la Dirac. His reformulation is referred to as the Ashtekar (new) variables.

I don't have a free source for Ashtekar's 1986 and 1987 papers. Here are the abstracts.
http://prl.aps.org/abstract/PRL/v57/i18/p2244_1
"A Hamiltonian formulation of general relativity based on certain spinorial variables is introduced. These variables simplify the constraints of general relativity considerably and enable one to imbed the constraint surface in the phase space of Einstein's theory into that of Yang-Mills theory. The imbedding suggests new ways of attacking a number of problems in both classical and quantum gravity. Some illustrative applications are discussed."
http://prd.aps.org/abstract/PRD/v36/i6/p1587_1
"The phase space of general relativity is first extended in a standard manner to incorporate spinors. New coordinates are then introduced on this enlarged phase space to simplify the structure of constraint equations. Now, the basic variables, satisfying the canonical Poisson-brackets relations, are the (density-valued) soldering forms σ^a _A^B and certain spin-connection one-forms A_aA ^B. Constraints of Einstein’s theory simply state that σ^a satisfies the Gauss law constraint with respect to A_a and that the curvature tensor F_abA ^B and A_a satisfies certain purely algebraic conditions (involving σ^a). In particular, the constraints are at worst quadratic in the new variables σ^a and A_a. This is in striking contrast with the situation with traditional variables, where constraints contain nonpolynomial functions of the three-metric. Simplification occurs because Aa has information about both the three-metric and its conjugate momentum. In the four-dimensional space-time picture, A_a turns out to be a potential for the self-dual part of Weyl curvature. An important feature of the new form of constraints is that it provides a natural embedding of the constraint surface of the Einstein phase space into that of Yang-Mills phase space. This embedding provides new tools to analyze a number of issues in both classical and quantum gravity. Some illustrative applications are discussed. Finally, the (Poisson-bracket) algebra of new constraints is computed. The framework sets the stage for another approach to canonical quantum gravity, discussed in forthcoming papers also by Jacobson, Lee, Renteln, and Smolin."

 

[marcus]

So what is the REAL answer to the question "why is all this done in 3D?"

Well I think the reason is in large part historical. People were conditioned to think that
1. You get a quantum theory by quantizing a classical theory e.g. following the Dirac canon.
2. ADM and then Ashtekar reformulation of classical GR set things up in a way that looked ready to quantize.
3. LQG began as a canonical quantization of the classical Ashtekar setup (some other contributors' names omitted for brevity).

But now if you look at a modern treatment of LQG like 1004.1780 it is not all done in 3D! That is more an artifact that the best entry-level paper is still the 1998 one. The best I know anyway.

The formative influence of history has diminished with time.

Now LQG is based about equally on graphs (1-complexes) and foams (2-complexes).
The approach is neither exclusively 3D nor 4D.
They work back and forth in a coordinated way between graphs and foams.

Labeled graphs describe states of knowledge based on a finite number of measurements of volume and area. They can describe initial and final states, or more general boundary conditions.
Labeled foams describe possible histories or trajectories along which graphs can evolve. Metaphorically they are like the particle trajectory paths in a Feynman path integral. Or all the possible diagrams of an interaction with stated initial and final.

LQG is no longer a program of canonical quantization of classical GR. Rovelli makes that explicit on page 1 of the program-defining April paper 1004.1780. They have something now which is beginning to look right--as if it may have the right large and semiclassical limits. The quantization effort can be seen as a heuristic ladder, now pretty much dispensed with.

 

[bcrowell]

Thanks, Marcus, that's very helpful! I really appreciate your taking so much time to give such detailed answers. I think you've pretty much oriented me about one of the two general questions I was asking, about 3-d, the other being about the spins.

 

[marcus]

Here's another good question!"[What is] the physical motivation for introducing the graphs. Is the manifold primary and the graphs secondary? Or vice-versa? What do the graphs represent physically?"

First some terminology that Rovelli uses consistently, and which helps to keep things straight.
A labeled graph is called a spin-network and consists of nodes and links.
A labeled foam, or 2-complex, is called a spinfoam and consists of vertices, edges, faces.
The foam is like a worldline of a graph, a face is the track of a link, an edge is the track of a node, and a vertex is where "something happened" (some kind of geometric event like the creation or annihilation of one or more nodes.)

The good thing about labeled graphs is they are conceptually fairly close to the information (based on a finite number of measurements) which the experimenter has about the state of the world's geometry. Intuitively speaking nodes are labeled with volume and links with area. Maybe it's more accurate to say nodes "carry" volume information and links "carry" area. If you picture the experimenter's knowledge as consisting of chunks of volume combined with information about adjacency and the contact area between neighbors...
if he puts a bag around some number of nodes the volume of that bag will be the sum of the component node volumes...and if he stretches a surface that cuts some number of links the area of that surfacer will be the sum of all the cut link areas.

The good thing about a labeled foam is there is a good way (described in the Lewandowski paper 0909.0939) to calculate its amplitude. A sample calculation, to first order, is done for an S³ cosmology example in the March paper 1003.3483

A graph is a kind of "truncation" or simplification of geometric state.
A foam is a kind of "truncation" of geometric evolution, or historical trajectory.

In quantum geometry/gravity one cannot assume that a smooth 4D spacetime really exists, just like one cannot assume that a particle has a smooth continuous trajectory. All we know is that the particle gets from A to B and that the calculation works if we assume it explored all possible paths from A to B, each with a certain amplitude. A foam is one of many many possible trajectories from geometry A to geometry B, each with a certain amplitude.

Intuitively the vertices or "events" in a foam can involve creation/annihilation of chunks, and rearrangement of adjacency relations between them. This is why the "vertex amplitude" formula of the "spinfoam model" is crucial. A new vertex formula came in around 2008 and revolutionized LQG. Turned out to work better and resolve several problems. Lewandowski 2009 completed the definition of the new "spinfoam vertex".

You asked "why all these graphs?" describing the states of geometry. I'm giving some philosophy. Bohr said [microscopic] physics is not about what nature IS, but rather what we can SAY about it---how it responds to measurement in other words, the form and basis of our information. Labeled graphs seem to meet the requirements for a minimalist description of quantum state.

Also look at open problem #17 at the end of 1004.1780 where he mentions that fermions and Y-M fields can be included in the spin-network state by adding some extra labels at nodes and links. There've been some papers about this.

But why are the graph links labeled with group representations?

That's an interesting question and there is an historical reason (a state wave function is a machine to crank numbers out of configurations, the configurations were imagined to be connections which would attribute a group element to each link, how then to crank numbers out of a web of group elements?)

There is also a "group field theory" reason where one simply defines the Hilbert to be square-integrable ("L²") functions on N-tuples of group elements---in other words L² of a "group manifold" cartesian product of N copies of the group.
And then one applies the Peter-Weyl theorem from functional analysis to get a vectorspace basis for L²(G^N) involving group irreps. The "spin-network basis" of the Hilbert.

And there is a third answer to that question (of why graph links are labeled with group reps) and that answer is that it is their destiny to be so labeled. This is the mathematical Darth Vader answer.

I probably need a break. May continue later today or tomorrow. Fun questions! Thanks Crowell.

 

[marcus]

Thanks, Marcus, that's very helpful! I really appreciate your taking so much time to give such detailed answers. I think you've pretty much oriented me about one of the two general questions I was asking, about 3-d, the other being about the spins.

Delighted to hear it! I was writing when you posted and didn't see this.
Some of the last post may be redundant as far as you are concerned, but I assume that's all right. Others might find it helpful. Also it would be great if Tom S. would answer your questions de novo. Make a fresh start. Or maybe reply to stuff I haven't. He has solid expertise.

I expect I'll take a break now and return later today or tomorrow.

 

[marcus]

Crowell something just occurred to me I want to say to TomS and Atyy, so I will put your questions "on hold". If you want just skip this post. Atyy has been persistently reminding about the older embed-in-manifold approach and TomS just said something like: "why should a research program introduce a lot of gauge degrees of freedom that just have to be gotten rid of later?"

And that's the point about embedding graphs in manifolds. You don't need to. The graph already is a truncated (simplified) version of geometry. Originally there was a heuristic reason to have a manifold to embed the graph in. The manifold was a security blanket that constantly reminded people of the classical context and hopefully would guide their thoughts in the right directions for proving graph geometry reproduces classical in this or that limit. Perhaps a bit of superstition there: transference by contact, sympathetic magic.

BTW Crowell and anybody, you should read the March (Spinfoam Cosmology) paper 1003.3483 if you haven't yet. They show how the pumpkin graph ([]) represents possibly inhomogeneous nonisotropic S³ hypersphere cosmology. And they derive something like Friedmann Robertson Walker from the spinfoam transition amplitude between two pumpkin graphs of different sizes.

The hypersphere S³ breaks down into two tets, with their sides paired up and glued. Imagine that we live in one tet, with its center at our house, and the other tet is centered at "the point at infinity"---the other pole of the hypersphere.

The two poles are connected by 4 areas---the 4 triangles where the 4 sides of each tet meet and are glued. The geometry is determined by how much volume you give to each tet, and by how much area you give to each of the glue-joints. So the picture is:
([])
two nodes joined by 4 links.
So you have two labeled pumpkins representing two geometries of the universe, and you set up a transitional spinfoam, and you calculate its transition amplitude. Amazingly you get something resembling the standard cosmology Friedmann equation's Hamiltonian out of this.

 

[atyy]

Hmmm, I'm not sure about that. In what tom.stoer was saying, I understood the gauge symmetry to be diffeomorphism invariance, and in old LQG some of this was designed in as spatial diffeomorphisms which are reflected in topologically equivalent graphs representing the same state (or something like that , I think).

 

[marcus]

Continuing to keep Crowell's questions on hold, and partly just talking to Atyy and TomS, I want to say that embedding is a liability that saddles you with gauge knot-classes.
And if two strands of a graph cross why shouldn't they pass thru each other by an (even virtual) creation/annihilation event?

If we are including matter in the picture then a strand can break with two fermions at the loose ends, a particle-antiparticle pair, and then rejoin. Something that presumably can happen all the time. But let's not include matter since we don't have dynamics for it yet. Let's just consider pure matterless geometry and imagine that the two strands cross at a node.

Well nodes can break up and merge again, the spinfoam syntax is all about their creation/annihilation. Graph nodes worldlines are spinfoam edges and at any spinfoam vertex some number of edges come in and some (possibly different) number of edges go out.

Intuitively even embedded strands could pass thru each other. So how are they going to stay knotted? Other arguments have referred to the absurdity of doing conventional topology at Planck scale. What Planck-small structure in one strand gets in the way of another?

All that seems silly IMHO because the idea of embedding graph-geometry states is silly, looked at carefully up close. The links in a graph represent adjacency they are not real wires or threads.

So in response to TomS comment about unnecessary gauge degrees of freedom, that is what you get if you embed LQG graphs, and it causes (among other sorts of bother) knot classes.

Two embedded graphs can have the same number N of nodes and the same number L of links, and the same source/target hookup assignments s(k) and t(k) for each link k.
So they describe the same adjacency relations. And yet senselessly enough they may be inequivalent because they belong to different embedding knot classes.

 

[atyy]

marcus, ideologically, I agree. I would prefer to go via GFT, and end up with tom.stoer's question why SU(2) - no reason - and maybe if we don't, we can get matter too, but that's very undeveloped at the moment. I have a feeling Rovelli can't decide whether one should go via KKL's embedded spin foams or via Rivasseau at al's attempt to develop a GFT that will contain EPRL or FK (I think perhaps neither).

 

[marcus]

... a feeling Rovelli can't decide whether one should go via KKL's embedded spin foams or via Rivasseau at al's attempt to develop a GFT that will ...

I don't see any difference. KKL does not require embedding the spin foams. It gives a recipe which works with abstract graph/foam representations of geometry. That recipe conforms with the EPRL vertex obtained in 2008.

So if Rivasseau team develops a GFT that reproduces EPRL, it will be compatible with KKL.

What we have been seeing in the past 3 years is a remarkable convergence of different approaches within the Loop-and-allied community. Simply based on that visible trend I would be inclined to expect that Rovelli has no reason to decide between KKL and GFT. They will be equivalent and merge the way we now see canonical LQG and spinfoam merge.

And certainly Rovelli gives no SIGN of indecision. He shows a complete trust in the KKL spinfoam vertex. It is one of three recent result "pillars" on which the approach presented in the March and April papers is based.

At this point, with everything coming together rapidly, one should be competing KKL and GFT in one's mind not on the basis of "which is more likely right?" but "which will easiest to calculate with?"

An important step was taken with the March 1003.3483 paper because they calculated a spinfoam transition amplitude to first order. Using KKL with, of course, abstract (manifoldless) spin-networks. And the result reflected conventional standard model cosmology.

We don't know if Rivasseau team will succeed in getting a GFT Lagrangian that reproduces KKL and the desired GR limits. But if the Paris team succeeds IMO the relevant question will be about how to calculate, not which is "right".

 

[atyy]

But there is something fundamentally missing isn't there? They need to get the Einstein-Hilbert action, not the Regge action. I don't think Rovelli knows the missing ingredient.

 

[marcus]

But there is something fundamentally missing isn't there? They need to get the Einstein-Hilbert action, not the Regge action. I don't think Rovelli knows the missing ingredient.

I can't keep track of each missing piece and what the current status is on each issue. All I know is the missing pieces have been falling in place, and the issues clearing up, at an amazing pace in the past 3 years or so.

When you say "Rovelli" what you mean is a tribe or extended family. The guy has raised up a number of PhDs and postdocs who are active in QG research. Several are now at Erlangen with Thiemann, or are now faculty with their own incipient groups. It's not easy to tell what they know or don't know, or what they are thinking. You could very well be able to guess what paper, with what kind of result, will come out next, but I not be able to.

What stands out in my mind is it looks like they just got a toe-hold on spinfoam Friedmann model cosmology, but only at first order. There is a great deal of foam calculation work to be done just to extend that, to higher order and to larger graphs than the S³ graph I was affectionately calling "pumpkin". It is the first step towards inhomogenous nonisotropic spinfoam cosmology.

 

[atyy]

the S³ graph I was affectionately calling "pumpkin".

what?

 

[marcus]

Please consult my post #8 in this thread

and for further discussion see post #86 in the "Five QG principles" thread:
https://www.physicsforums.com/showthread.php?p=2867175#post2867175

 

[sheaf]

So what is the REAL answer to the question "why is all this done in 3D?"

Well I think the reason is in large part historical. People were conditioned to think that
1. You get a quantum theory by quantizing a classical theory e.g. following the Dirac canon.
2. ADM and then Ashtekar reformulation of classical GR set things up in a way that looked ready to quantize.
3. LQG began as a canonical quantization of the classical Ashtekar setup (some other contributors' names omitted for brevity).

But now if you look at a modern treatment of LQG like 1004.1780 it is not all done in 3D! That is more an artifact that the best entry-level paper is still the 1998 one. The best I know anyway.

The formative influence of history has diminished with time.

Now LQG is based about equally on graphs (1-complexes) and foams (2-complexes).
The approach is neither exclusively 3D nor 4D.
They work back and forth in a coordinated way between graphs and foams.

Labeled graphs describe states of knowledge based on a finite number of measurements of volume and area. They can describe initial and final states, or more general boundary conditions.
Labeled foams describe possible histories or trajectories along which graphs can evolve. Metaphorically they are like the particle trajectory paths in a Feynman path integral. Or all the possible diagrams of an interaction with stated initial and final.

LQG is no longer a program of canonical quantization of classical GR. Rovelli makes that explicit on page 1 of the program-defining April paper 1004.1780. They have something now which is beginning to look right--as if it may have the right large and semiclassical limits. The quantization effort can be seen as a heuristic ladder, now pretty much dispensed with.

I think one thing to add to the list of reasons for interest in 3D is the General Boundary Formulation from Robert Oeckl:

http://arxiv.org/abs/gr-qc/0312081"

The generalisation of the conventional S matrix style assignment of amplitudes to an in/out state pair of three-geometry states would be an assignment of an amplitude to a boundary three geometry state. The in/out state thing is no longer appropriate when you don't have a background with which to define t=+infinity, t=-infinity.

 

[marcus]

I think one thing to add to the list of reasons for interest in 3D is the General Boundary Formulation from Robert Oeckl:

http://arxiv.org/abs/gr-qc/0312081"

The generalisation of the conventional S matrix style assignment of amplitudes to an in/out state pair of three-geometry states would be an assignment of an amplitude to a boundary three geometry state. The in/out state thing is no longer appropriate when you don't have a background with which to define t=+infinity, t=-infinity.

That's an important point, and I certainly agree about Oeckl's General Boundary idea adding interest. I didn't mention it but was thinking of Oeckl when I wrote this:
"Labeled graphs describe states of knowledge based on a finite number of measurements of volume and area. They can describe initial and final states, or more general boundary conditions."

BTW I need to work some more on understanding how a spin-network can express a general boundary state, or a connected boundary. It's easier for me to understand how two separate graphs can represent initial and final states.