# Loop-and-allied QG bibliography

• marcus
In summary, Rovelli's program for loop gravity involves coupling the standard model to quantized QG loops, allowing for interactions between eigenvalues of length and momentum. This approach allows for non-perturbative calculations without infinity problems and does not require a continuum limit. The main difference in loop gravity is that the excitations of space are represented by polymers, or ball-and-stick models, that can be labeled with numbers to determine the volume and area of any region or surface. This allows for a more intuitive understanding of the geometry of the universe.
marcus
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Intuitive content of Loop Gravity--Rovelli's program

In the "perche i nostri discorsi" thread, selfAdjoint gave a concise sketch of the direction that Carlo Rovelli sees in loop gravity

Rovelli's vision of how to move forward beyond present day physics boils down to this. Couple the standard model to the set of quantized QG loops so there will be interactions with eigenvalues of length and eigenvalues of momentum, etc. Combined states. Work on the theory non-perturbatively in that way. No infinity problems because no "classical points" - same advantage stringy physics gives, but with perhaps a more basic underpinning. The point for me is, if you could do this you could calculate numbers for the accelerators. The advantage over normal analysis would be non-perturbative calculations without infinities. The advantage over lattice would be, well it's proposed as real physics, no continuum limit required.
...

A few posts later in the thread I suggested trying to map out an intuitive description of what Loop Gravity is basically about and what makes its approach different. Perhaps it would be good to make a separate thread just to do that, without controversy as to comparative merits or any other distractions. Here is my post from the other thread proposing to do that:

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Probably what we need most now is an intuitive sketch of what makes Loop Gravity different from your typical field theory----how is the backgroundlessness implemented?

Things are coming to a head. Rovelli's new book "Quantum Gravity" is major and has IMO material for a bunch of PhD dissertations just expanding on details. It also contains the "Dialog" as a final chapter. You can connect the points made about loop gravity in the Dialog to chapters and sections in the main part of the book. Also Smolin's April 2003 paper lays out what has been accomplished and what remains to do and what the prospects are for getting loop gravity tested---it is a thorough review and comparison: "How far are we from a quantum theory of gravity?"

Plus we have good accounts of loop cosmology by Bojowald
like the recent paper "Quantum Gravity and the Big Bang", and in some of Ashtekar's papers. It appears progress in cosmology has been dramatic of late. New researchers have been getting into LQG at the level of cosmology.

Plus there is this month's Berlin symposium "Strings meets Loops" which will probably generate a series of overview talks
aimed at wider audience----e.g. another cosmology overview by Bojowald, another full theory overview by Ashtekar, a spin foam overview by Rovelli, and so on.

So there is more and more accessible information than there was a year ago, about loop gravity. It looks to me as if new research possibilities are coming into focus. For example, these days I keep seeing papers about the "low energy limit" or "semi-classical limit", another place where newcomers are getting in (like those Argentine people this month---Kozameh, Gleiser, Parisi)

It seems to me to be a good time to try to say what loop gravity is about, in the simplest possible way. I am apt to make several false starts on this. If someone else has been thinking about it and wants to try, go ahead...
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sergioperezf, qinglong.1397 and Telemachus
what a spin network is *NOT*

was practically laughing just now thinking of my trying to give an intuitive explanation of loop gravity. Meteor has a copy of Three Roads, which probably has a perfectly good one, and I am too lazy to go down to the library and get a copy---never seen it.
so meteor or selfAdjoint are probably better equiped to do a sketch of loop gravity

BTW I believe that to give a good conceptual description of something often requires deeper understanding, paradoxically, than to descrbe it technically, so it is a place where one's shortcomings become evident HOWever here goes

I told you before I would make several false starts, it is inevitable, so let's get started

in loop gravity the excitations of space (or geometry or gravity allee same bizness weiss da) are polymers----essentially ball-and-stick models like of very large protein molecules

so the first analogy is a drumhead with sand sprinkled on it. you know that every different way it can vibrate is shown by the lines that appear in the sand when you excite---I hope you did this at the science museum as a kid and know what I mean: nice diagrams of lines appear on the surface and these diagrams CATALOG all the modes of vibration or the excitation modes.

the next analogy is a sink full of soapsuds---you had to wash the dishes as a kid and you can imagine the whole sink or the whole universe (whats the difference) full of soapsuds. Now in the middle of every bubble put a dot, and if one bubble contacts another bubble then connect those two dots with a line.

Now you have a network (a ball-and-stick molecule, a polymer) that fills the whole sink or universe. And we are going to give a number Q to each point (or ball, or dot, or vertex) in the network.
And a number P to each connecting line.

Whatever for? Why label each point and line with a number? (Roger Penrose thought of doing it, he is the one to ask about it) Well, using that additional information the network can tell us the VOLUME of any region----just add up the numbers attached to each point in the region, somehow-----and the AREA of any surface---just add up the numbers, in some fashion, that are attached to every line that punctures the surface.

Furthermore, if this labeled network's sticks become flexy the whole thing can be squashed flat and stuffed into your dresser drawer where you ordinarily keep undershirts.

So here is a thing which you can squash any shape and store anywhere which nevertheless tells you everything about the geometry of the universe, or the kitchen sink I forget which.

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Yes, I have a copy of the book, and in it a spin network is defined as a directed graph, jointly with a series of rules that guide the evolution of the graph. To each edge of the graph is assigned a number, and the area of a surface depends on the value of the numbers of the edges that punctures the surface.Btw, the area of a surface can be computed with a formula that includes the Immirzi parameter. The volume of an object is proportional to the number of nodes of the graph thatr are inside the object.
However, in various documents on Arxiv, I've found that the edges of the graph are labeled with group representations of Lie groups, and the vertices with intertwining operators(damned if i now what's an intertwining operator!)I don't know if these spin networks are the same that the cited in the book of Smolin
Spin networks are used in the canonical approach to quantum gravity, but another approach, the sum-over-histories approach, has adopted a particular version of them, called spin foams, that are cell complexes.

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I don't know if these spin networks are the same that the cited in the book of Smolin

Yes they are the same. Smolin said the edges were labelled by a number in order not to confuse his readers. The closer statement would be the edges carry a spinor. This spinor is not just a label, but a genuine bit of physics. The intertwiner functions are like black boxes - deterministically relating spin reps into spin reps out. These again are physics, somewhat like Heisenbeg's S-matrix relating momenta into momenta out.

Yes they are the same. Smolin said the edges were labelled by a number in order not to confuse his readers. The closer statement would be the edges carry a spinor. This spinor is not just a label, but a genuine bit of physics. The intertwiner functions are like black boxes - deterministically relating spin reps into spin reps out. These again are physics, somewhat like Heisenbeg's S-matrix relating momenta into momenta out.

I'm sure you are right---what a spin network is in the literature must be the same as what Smolin describes in his book (allowing for whatever minor naming conventions differ) but I should emphasize that the picture I gave here of a "spin network" (*NOT*) bears only a faint resemblance to what defines a basis for the quantum states of geometry in the theory. I am still trying to see how to introduce the ideas in as intuitive and non-technical way possible----I may have to break down and see how Smolin did it in "Three Roads".

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quantum gravity can't have 4D spacetime

here's another piece of the jigsaw puzzle

in classical mechanics things move along trajectories---curved paths parametrized by time---and when you quantize the trajectories go away.

the curved paths things travel along don't exist any more, you have to erase the trajectories (or in Feynman sum over histories you "integrate" all possible ways of getting from here to there---in any case the clear picture of a path loses reality and dissipates)

in GR, the 4D manifold is not a real thing (individual points are not events and have no physical meaning) because of gauge invariance any point will do----to define an event you need matter, like the event that two particles cross paths (at some point, but which point doesn't make any difference it is an arbitrary choice.

arbitrary choices needed to express something mathematically are called "gauge"----extra physically meaningless information that gets unavoidably mixed in as part of keeping the books.

in GR the 4D manifold is there so that you can write down the trajectory of the gravitational field. GR does not suggest that 4D spacetime exists, it is a mathematical amenity used for defining evolution of the gravitational field and the matter that goes into shaping the field.

but when GR is quantized, the trajectory goes away and one simply has a 3Dmanifold, with a space of all possible geometries defined on it

again, as in classical case, the points of the 3D manifold have no physical meaning---they are just "gauge". One can define surfaces and volumes only using matter---Rovelli and Ashtekar both use examples like by a surface I mean for instance the top of this desk.

The quantum states are functions defined on the space of all possible geometries that the 3D manifold can have. Analogous to quantizing a particle moving on the line by "wave functions" defined on the line.

The curious thing is that no one started out thinking of the "wave functions" defined on the space of geometries as spinnetworks! Nobody was looking for spin networks or asking for them! It just turned out that they appeared as the best way to CATALOG the functions defined on the space of all possible gravitational fields or all possible geometries.

At first they tried defining these functions using loops and they got a hilbertspace of loop-functions, but they couldn't get an orthogonal basis: the loops were too redundant. So they eventually borrowed spinnetworks from Penrose and they turned out to give an orthogonal basis for the space.

Also I even believe that the basis is countable and the hilbertspace is separable----technical conveniences to be sure.

So when I mentioned this polymer network thing that describes the geometry of the whole universe, but that you can stuff into the dresser drawer, it is a quantum state of geometry (a functional like a wavefunction defined on the collection of all classical geometries) and all quantum states look like this or combinations of things like this.

and matter fields must be defined on things like this

and the quantum state can evolve! At noon by some clock it can be this one in the top drawer and then at one o'clock it can be like this other one in the bottom drawer.

But it is a disconnected hopping, and there is no absolute clock you just have to choose some physical PROCESS (essentially something involving matter) to serve as a clock. This clock is part of the world and there is a correlation between what you observe the clock says and where you observe the pendulum is, or how far away the galaxies are, or whatever else. There is no one absolute time that drives the rest only correlations between different processes

processes which include, among other things, the change in the state of the 3D geometry of space.

So, when you check things out using cosmology, the evolution equation is a finite DIFFERENCE equation! It is not a differential equation. when you do loop cosmology the Friedmann equation that all cosmologists depend on becomes a step by step difference equation----e.g. Bojowald uses the scalefactor as his clock, there must be some physical process to use as clock, and correlates other stuff like curvature and inflation and density with the scalefactor. And so does everybody else that has been doing loop cosmology that I have seen. For example "Quantum Gravity and the Big Bang" the talk Bojowald gave recently.

it is interesting how the concept of time changes.

In Rovelli's textbook "Quantum Gravity" there is a philosophical section on time which I found really interesting---he finds that different branches of physics use ideas of time that are actually different from each other and also from everyday vernacular time.
he is able to distinguish around 8 or 9 different ideas of time.

Quantizing General Relativity seems to exert a strong influence on the ideas of time because both QM and GR bring insights about time which, if you try to put them together, produce something that seems radically new.

(of course one can avoid having to think about it if one throws out GR and replaces it with a lobotomized form or if one is very careful to only use GR and QM in separate situations and never together on the same problem)

Marcus, there are a vast number of papers that continue to reveal a discrete direction, I am really glad that you take the time to post the most interesting ideas from many fields.

You may have this link allready?..but if so others may find it interesting:http://uk.arxiv.org/abs/gr-qc?0306059

Rovelli for me seems to be an architect of new thinking.

Originally posted by ranyart
Marcus, there are a vast number of papers that continue to reveal a discrete direction, I am really glad that you take the time to post the most interesting ideas from many fields.

You may have this link allready?..but if so others may find it interesting:http://uk.arxiv.org/abs/gr-qc/0306059

Rovelli for me seems to be an architect of new thinking.

thanks for calling attention to that paper, Ranyart. For some reason I had just glanced at it earlier. He probably is.

http://arxiv.org/abs/gr-qc/0306059

I will quote the last 5 sentences in (the conclusions part of) this paper and try to say why I think it is interesting

"...We have studied the propagator of our model in detail. We have shown that in the semiclassical limit it has a simple relation with the Hamilton function of the classical theory, but this relation is not a simple exponential, as one might have expected.

Instead, the propagator is real. It is the sum of two exponential
terms complex conjugate to each other, that propagate backward and forward, respectively, along the motions. Accordingly, the physical Hilbert space splits between forward and backward propagating states.

We expect this structure to be the same in quantum general relativity."

this is the kind of simple example (a system like a springbob with only a couple of degrees of freedom) that physics teachers love to use when introducing a new method---try the new approach out on the simplest thing in sight: an harmonic oscillator, a single particle in a potential well, whatever. Then the maths do not obscure the ideas.

So this is Daniele Colosi (a grad student at Marseille) and Rovelli having fun with a toy that moves in a simple ellipse. Is this your reading too? I just looked at the paper. I like it. Maybe we should make a thread about this paper or just look at it in this thread

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Originally posted by marcus
thanks for calling attention to that paper, Ranyart...

http://arxiv.org/abs/gr-qc/0306059

your bringing up this paper got me looking at this and several related ones over the past day or so.

A Simple Background Independent Hamiltonian Quantum Model (Colosi/Rovelli)

Minkowski Vacuum in Background Independent Quantum Gravity

and several related 2003 spin foam papers

recent work involving the hamiltonian seems to connect (in ways I didnt anticipate and don't fully understand) to recent spin foam work

there is the fact that in August 2002 John Baez and some others posted computer results that some spin foam numbers were not what some people expected them to be----this seems to have lead to increased interest in spin foams: something new to understand about them----several new papers with new ideas

then there is the fact that at this months symposium it is Rovelli who is talking about spin foams (and he and his associates have recently, in late 2002 and in 2003) put out several papers on spin foams

then there is the fact that several of these recent papers link up the hamiltonian and spin foam approaches----they are or seem to be trying to cure problems in both the hamiltonian and discover how to use spin foam models properly in a way that suggests some underground connection between the two

i had till now not paid attention much to spin foam quantum gravity but now because of these little hints I've been reading in the past day or so, and because Rovelli has chosen to do the spinfoam presentation, I am beginning to pay more attention and trying to understand a little better.

BTW at the symposium the loop lineup looks like this

Ashtekar: quantum geometry and applications (this means overview and application to big bang, inflation, black holes...)

Bojowald: loop quantum cosmology (a strong run of results by him and about 10 other researchers over past 3 years, giving guidance to development of the full theory by testdriving in the cosmology case)

Rovelli: spin foams (this is the one that I cannot antipate, it will have unexpected things)

Lewandowski: the hamiltonian (this presumably will be profoundly analytic/algebraic as is the way with people from Warsaw. maybe selfAdjoint will help us understand this one ,
it has now been 5 years since Lewandowski found fault with Thiemann's hamiltonian and there has been a great deal of work involving hamiltonians since then! Perhaps L will summarize some of this. As befits a growing theory, the main issue here remains unresoved and people are still discovering how it should look, as for example in the paper you gave the link to )

This is merely by way of saying thanks for the link to that paper. It has given me something to do during spare moments for the past day or so

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Yes they are the same. Smolin said the edges were labelled by a number in order not to confuse his readers. The closer statement would be the edges carry a spinor. This spinor is not just a label, but a genuine bit of physics. The intertwiner functions are like black boxes - deterministically relating spin reps into spin reps out. These again are physics, somewhat like Heisenbeg's S-matrix relating momenta into momenta out.
I've reading the paper where spin networks where introduced in the first time, "Spin networks and quantum gravity"
http://arxiv.org/abs/gr-qc/9505006
In this paper spin networks are defined like trivalent graphs with edges labeled by positive integers.A trivalent graph is a regular graph with 3 edges arriving at each node. The spin network has to follow 2 rules:
-The sum of the 3 edges that converge at a given node has to be an even number
-Each of these 3 edges can't be superior to the sum of the other 2
Do somebody know the paper where the labels passed from being numbers to group representations? Do the Lie groups have to be some specific Lie group? Are actually spin networks continued to be defined as trivalents graphs? Must the group representations be irreducible representations?
Ok, Ok, very much questions but this is interesting stuff

I'm not sure, but I think gr-qc/9707010 is early. See also Baez's TWF #110.

Originally posted by meteor

Do somebody know the paper where the labels passed from being numbers to group representations? Do the Lie groups have to be some specific Lie group? Are actually spin networks continued to be defined as trivalents graphs? Must the group representations be irreducible representations?
Ok, Ok, very much questions but this is interesting stuff

Meteor I hope my replying does not preclude a PF mentor or other knowledgeable person responding.

to say irreducible representation of SU2 is sort of like saying "spin" because there is one for each dimension and so roughly speaking one for each integer (or half integer if you divide each integer by two according to the quaint ancient custom of physicists)

the papers where Penrose made up "spin network" idea are not online!

however I have read about these papers and my understanding is that ALREADY AT THE BEGINNING penrose thought of the graph as labeled by "spins" that is to say either halfintegers or, what is the same, irreducible reps of SU2

as an interesting insight into human, or at least Penrose, nature, he regularly FLIPFLOPPED at the beginning between having the labels be integers which he called "colours" and dividing them all by two and calling them "spins". As a civilized mathematician he wanted to call them colours but as a savage physicist driven by brute instinct and prejudice he needed to divide them by two---as is the custom---and call them spins.

so this ambiguity of labeling has been there from the start

remember also that as children, while others are taught to skip rope and play hopscotch, physicists are taught that SU2 is the "double cover" of the rotation group, which is why an electron can turn around 720 degrees before it looks normal again. the first time it turns around it appears to have pointed teeth and is wearing a Count Dracula costume but then it turns around another 360 degrees and is its old self. But doubtless you know all this already!

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Originally posted by marcus

so this ambiguity of labeling has been there from the start

remember also that as children, while others are taught to skip rope and play hopscotch, physicists are taught that SU2 is the "double cover" of the rotation group, which is why an electron can turn around 720 degrees before it looks normal again. the first time it turns around it appears to have pointed teeth and is wearing a Count Dracula costume but then it turns around another 360 degrees and is its old self. But doubtless you know all this already!

I like it!

The inside of Fort knox has a safe where the rotation of the combination/wheel-number will dictate if one opens the safe or not?

The inside cogs and wheels are dictated by the outside combination wheel, one false turn and CPT kicks in and you are forever going to be turning..and turning..the dials. Yet if one were on the inside and the back of the door had a seethrough covering, one can guide a way through any amount of infinite combinations with ease!

I'm not sure, but I think gr-qc/9707010 is early. See also Baez's TWF #110

Thanks! In Baez 110 put that spin networks can have more than 3 edges meeting at a vertex, so they are not actually considered trivalent graphs (Baez 110 was written in 1997)
I'm trying know to fathom what's the meaning of the Poisson algebra. I will post something about it

Good for you if you post on the Poisson algebra. This is a missing piece in our discussions here.

Good for you if you post on the Poisson algebra. This is a missing piece in our discussions here.

Amen to that!

an intuitive description again

I told you I would make several false starts. Eventually there should be a non-technical description of loop gravity in only one to ten pages. Let's keep this thread going until we have one, or find one in the literature.

the basic picture of any quantum theory is you have a space of possibilities (configurations, might be simply a set of possible positions and momentums) and then you define a kind of "(not)probability" function or wave-function on that space of possibilities.

If the space of all possibilities, of whatever it is (one particle, N particles, a field, a geometry of the universe) is called A, then the the wavefunctions or quantum states or "(not)probability" functions are just complex valued functions on
A

usually there is a measure defined on called A so you can integrate these functions and they are "square integrable" which means they don't run off to infinity too much and have finite integrals

and I have to say that in mathematics this is, in a certain way, as basic as things ever get----a space, some complex-number-valued functions defined on that space---and being able to integrate or sum each of them, so each one has a finite size.

a loaf of bread, a jug of wine, and hilbert space---this is all we ask and it does not seem like a lot----the rest is trimmings.

so in a certain sense if I could just tell you how to build the configuration space called A of loop gravity and then, if I could just explain how to define a function on that space----and get a hilbertspace consisting of all the complex-number-valued functions on called A then I would be done explaining. All the rest----the selfadjoint operators on the hilbertspace, their evolution, their spectra, and all----that all "hatches" from how the original hilberspace of wavefunctions, or quantum states or whatever you call them, is defined.

So here we are down in the basement and there are not even any "spin networks" or "loops" around. I have to tell you the space called A of loop gravity.

Psssst! It is the space of "connections". A connection is one way to clothe a bare manifold with geometry if it has no geometry. The whole destiny of loop gravity, win or lose, succeed or fail, is in this one choice----the geometry of the universe shall be represented by the possible "connections" on a 3D continuum, a 3D manifold.

Until 1986 the guys like John Wheeler who were trying to quantize GR used the space of "metrics" for their called A and it gave them headaches. After 1986 almost everybody switched over to representing the geometries by connections.

Hey, the whole thing could go into another iteration if some yet other set of variables for GR were found---something that captured the essence of the shape of the world that was not a metric and not a connection---then you could have a new configuration space called A and a new hilbertspace of complex-valued functions defined on it.

It took 50 years to get from a space of metrics to a space of connections---people have tried to quantize GR for a long time. I am not betting they come up with something to replace connections, but they might.

So we are looking at the most basic question---how do you describe the shape of the world, what is a connection, how do you arrange all the possibilites to make a configuration-space, a space of all possible connections, how do you define functions on that space, that have their values in the complex number plane?

what is a connection?

Hi here,
With respect to the Poisson algebra, the only thing that I've discovered is that in 1987, Smolin and Rovelli introduced an infinite set of gauge invariant loop variables on the phase space of the theory (called then the Ashtekar phase space). These variables form a closed Poisson algebra
I don't know if Poisson algebras are anymore important in LQG since loops were substituted by spin networks
Marcus reading your anterior post, there are some ideas that have popped up in my mind, could you clarify, please?
Is it possible that this Ashtekar phase space is really the Hilbert space of LQG?. I mean, is possible that the loop variables introduced by Rovelli and Smolin are functions in this Hilbert space?
Is it possible that before the introduction of the loop variables, the functions in the Hilbert space were the connections?
Best wishes and keep fighting the stringers!

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Originally posted by meteor
Hi here,
With respect to the Poisson algebra, the only thing that I've discovered is that in 1987, Smolin and Rovelli introduced an infinite set of gauge invariant loop variables on the phase space of the theory (called then the Ashtekar phase space). These variables form a closed Poisson algebra
I don't know if Poisson algebras are anymore important in LQG since loops were substituted by spin networks
Marcus reading your anterior post, there are some ideas that have popped up in my mind, could you clarify, please?
Is it possible that this Ashtekar phase space is really the Hilbert space of LQG?. I mean, is possible that the loop variables introduced by Rovelli and Smolin are functions in this Hilbert space?
Is it possible that before the introduction of the loop variables, the functions in the Hilbert space were the connections?
Best wishes and keep fighting the stringers!

Hi there,

I am kind of new in this forum and found this tread about LQG which interests me a lot. I don´t know if I am misleading but I think that poisson algebras are still important in LQG, but they are not so much
talked about. Rovelli and Smolin indeed had a Poisson algebra that was the starting point for quantization. I think that Ashtekar and co-workers wrote a paper showing that the Rovelli-Smolin algebra was not closed and changing the poisson algebra to some other algebra I don't remenber. Recently, Sahlmann, Lewandowski and Thiemann have
taken this proposal and expanded it.
Just a few more comments. The phase space of the theory is not the same as the Hilbert space. Normaly, this is constructed out of functions of the configuration space, in this case, connections.
The loop variables are functions of the connection, but labelled by loops (or graphs in the case of spin networks).

Originally posted by marcus

what is a connection?

Gravity

Well the question to me then raises what the foundation of this whole topic is built upon? Philospohically Smolin was able to unite three roads to form algebraic topology.

The basis of the all the maths including the geometries must also follow the logic? Venn logic, and geometrical Intuitional developement? Category theory and topos(Algebraic Topology) was a integration of Smolins Three Roads?

It had to begin from supersymmetry, and from this, the gravity is understood, as well as, weak field settling to boundaries and defintion in cooling and discrete forms?

But in all of this, there is a exchange between energy/matter and the mobius can see things turn over as well as the klien bottle turning inside out. Where is that? Twisting in differential rotations?

You had to be able to see it from Kaluza and Kliens perspective and how we got there. U(1)=5d...and this includes all the covers?

Kip thorne help us to visualize, and in this great distance Ligo reads and in the quantum world how much more is this energy that continuity says, listen, things seem very smooth. But we have discrete structures, and how is the gravity revealed from the perspective of tangible objects, but in the recogniton of the distances?

Intuitively it must come to the distances? We do not disregard the structure that arises from the movement of the energy into objects(crystalization)

Sol

Sol, it was a rhetorical question. Marcus is going to tell us what a connection is. I'll bet he's working on his metaphors right now.

Sol, it was a rhetorical question. Marcus is going to tell us what a connection is. I'll bet he's working on his metaphors right now.

Oh good:) You know how intuition can be sometimes trying to find the right words to explain the essence of things.

Sol

Hi meteor, welcome nonunitary,
everything you two said in your posts seems right to me, and in particular this:

Originally posted by nonunitary
The phase space of the theory is not the same as the Hilbert space. Normaly, this is constructed out of functions of the configuration space, in this case, connections.
The loop variables are functions of the connection, but labelled by loops (or graphs in the case of spin networks).

This is a good thing to be clear about----the configuration space is the set of possible connections (reflecting all the possible geometries there could be on the manifold). In the process of quantizing a classical theory a hilbertspace is constructed consisting of (complex number valued) functions defined on the configuration space.

Spaces of functions are typically convenient to use because they are linear---you can add two functions just like you add two vectors and so on----spaces of functions are typically infinite dimensional vectorspaces and they are handy just the way vectorspaces are. (This is hardly news to you meteor and nonunitary but might as well be said) And a hilbertspace is a type of vectorspace that commonly comes up as a set of real or complex-valued functions defined on something---in this case the classical theory's configuration space: the set of all possible geometries, as represented by the connections associated with them.

A classical theory will have a poisson algebra of various readings off of phase space corresponding to classical deterministic measurements you can make and in the process of quantizing the theory one will want to find an algebra of OPERATORS on the hilbertspace that these things correspond to.

I agree with nonunitary about the role played the poisson algebras and the algebras of quantum observables that people seem able to discover corresponding to them. Also that Sahlmann, Thiemann, Lewandowski and others have been busy with these things recently.

In effect, you are ahead of me right now and there is no need to wait for me to catch up. Anything you know about the theory that you want to explain, you should go ahead! We have no special responsibilities to anyone and no need to follow any special order.

What iterests me right now is this: how would you explain to someone with a minimum of math a way to think about connections.

A manifold is just a set equiped with coordinates around any point so that you CAN use those coordinate patches to give it geometrical shape if you so desire. But a bare manifold is devoid of geometry----there is almost nothing interesting about it unless it has some kinky topological features.

One way to put some starch in your manifold is to define a METRIC on it (which you can do because you have the kind of minimal amenities, namely coordinate patches). Once you have said what the distance between each pair of points in the manifold is, the thing snaps to attention and takes on shape.

But you can also proceed a different way. Because you have this minimal structure of coordinates you can define the tangent "plane" at every point---actually for a 3D manifold it is a tangent 3-space not a tangent plane. And the thing still looks like a ruppled shirt or wet paper bag, it just has tangentspaces stuck to it. But NOW you can decide on the infinitesimal "roll" that happens as you go from one point to the nearest neighboring points. That is, you can pick a "connection". And by continuing along a path and integrating the tiny roll at each point you get
by the end of the path a reall substantive rotation. So a "connection" is basically a contraption that tells how tangent vectors are supposed to rotate as you move along a definite path thru the manifold.

Well that bespeaks geometry too, just like a metric does. Technically it doesn't completely determine it but intuitively it goes a long ways towards defining what shape the thing is.

Now physicists learn Lie groups and Lie algebras long before they put on long pants and start shaving and having dates, so for them it is a kneejerk response that an infinitesimal "roll" is an element of su2, the Lie algebra of SU2. It is like the things you have to learn to pass your drivers test. An element of su2 is a particularly cute kind of 2x2 matrix of complex numbers.

So what is a connection? It is a program I have on my palmpilot that if you show me a point in the manifold and show me a DIRECTION in which to set out from that point, I will tell you the infinitesimal "roll" that is I will tell you a cute 2x2 matrix of complex numbers which is a member of su2.

And in a very rough sense all the possible geometries the manifold can have are reflected usefully in the set of all possible connections that can be defined on it

Originally posted by meteor
Do somebody know the paper where the labels passed from being numbers to group representations? Do the Lie groups have to be some specific Lie group? Are actually spin networks continued to be defined as trivalents graphs? Must the group representations be irreducible representations?

The labels of spin network have always been representations; it's just that in the case of SU(2), representations can be simply labeled by numbers.

The Lie group can be anything, but in the connection variables, an SU(2) spatial connection is typically used, which leads to a kinematical Hilbert space of SU(2) spin networks.

Spin networks don't have to be trivalent. In fact, in LQG, trivalent spin networks have zero volume.

The group representations are irreducible, since the point of spin networks is to form an orthonormal basis of the space of connections (modulo gauge transformations); for that, you want networks labelled with irreps, as follows from the Peter-Weyl theorem; see http://arXiv.org/abs/gr-qc/9504036.

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Originally posted by Ambitwistor

Spin networks don't have to be trivalent. In fact, in LQG, trivalent spin networks have zero volume.

I checked this in Rovelli's textbook and you are right: he says a node must be at least quadrivalent to have nonzero volume. I like everything in your post and look forward to many more. welcome. there is a considerable need for loop-knowledgeable people here.

I guess I need to look again at the volume formula, for some reason until I checked just now, I thought that a trivalent vertex contributed a unit of volume. I know that any vertex can be broken down by a series of surgical steps into a collection of trivalent one, I must go back and try to understand what happens to the volume.

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Originally posted by marcus
In the process of quantizing a classical theory a hilbertspace is constructed consisting of (complex number valued) functions defined on the configuration space.

At this level, it might be worth emphasizing that these "complex-valued functions on configuration space" are what some people might know better as "wavefunctions": the configuration space describes the system (like the position of a particle), and the value of the wavefunction (a complex number) is the probability amplitude for finding the system in that particular configuration (like the probability of finding a particle in a particular location).

So a "connection" is basically a contraption that tells how tangent vectors are supposed to rotate as you move along a definite path thru the manifold.

Well that bespeaks geometry too, just like a metric does. Technically it doesn't completely determine it but intuitively it goes a long ways towards defining what shape the thing is.

I'm not sure that I understand this side remark correctly, but if you want to speak technically, a connection does completely determine the geometry: if you have a Levi-Civita connection, then that is equivalent to having a metric. If you have some other kind of connection, then it defines a more general kind of geometry (not Riemannian) that does not arise from a metric.

Originally posted by Ambitwistor
At this level, it might be worth emphasizing that these "complex-valued functions on configuration space" are what some people might know better as "wavefunctions": the configuration space describes the system (like the position of a particle), and the value of the wavefunction (a complex number) is the probability amplitude for finding the system in that particular configuration (like the probability of finding a particle in a particular location).

absolutely right! In another thread I stressed the term "wavefunction" for this. I should always mention that as a synonym. I like to try a few alternative ways of saying things to allow for people coming to the subject from different backgrounds.

...a connection does completely determine the geometry: if you have a Levi-Civita connection, then that is equivalent to having a metric. If you have some other kind of connection, then it defines a more general kind of geometry (not Riemannian) that does not arise from a metric.

I am so glad you are on hand, ambitwistor! I will try (if the board permits it) to edit some of my posts to remove the vagueness about that in accordance with what you say.

My intuitive feel is that the connection describes the geometry, and I am puzzled that the Ashtekar variables are not simply A (the connection) but are various pairs, like [A, E] where E is the densitized triad or "electric field"(a term sometimes used depending on a possibly confusing analogy). A and E are presented as "conjugate" variables. Yet the configuration space is just the collection of all possible A's. what is the essential additional information given by E? You are probably familiar with the notation I'm using, for brevity omiting subscripts and so on.

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Originally posted by marcus
I guess I need to look again at the volume formula, for some reason I thought that a trivalent vertex contributed a unit of volume. And that any vertex could be broken down by a series of surgical steps into a collection of trivalent one.

As mentioned in Baez's Week 55, Loll showed that trivalent vertices do not contribute volume (http://arXiv.org/abs/gr-qc/9511030).

As for breaking down vertices into trivalent vertices, you're probably thinking of those tangle diagrams, where the spin network edges are decomposed into "virtual" nodes and edges, wired up according to recoupling theory. You can decompose spin networks into trivalent diagrams of that sort, but they aren't spin networks -- though they are equivalent to (not-necessarily trivalent) spin networks.

See http://relativity.livingreviews.org/Articles/lrr-1998-1/node17.html#TheVirtualNode

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I'm impressed. In 5 minutes Ambitwistor came up with two
precisely-on-target online references---one to original work by a remarkable woman named Renata Loll and one to the exact place in Rovelli's classic LivingReviews exposition. That means Ambitwistor is a pro. Probably his/her time is too valuable to hang around much. Came thru like a big train through a small station and that was it.

Well well

I'd stay up and see what else happens tonight, but its after midnight here and time to turn in

(today selfAdjoint plugged PF, and the loop gravity threads in particular, on SPR Usenet. this could explain unexpected visits.
nice if some of these people stayed around)

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Marcus,
My intuitive feel is that the connection describes the geometry, and I am puzzled that the Ashtekar variables are not simply A (the connection) but are various pairs, like [A, E] where E is the densitized triad or "electric field"(a term sometimes used depending on a possibly confusing analogy). A and E are presented as "conjugate" variables. Yet the configuration space is just the collection of all possible A's. what is the essential additional information given by E? You are probably familiar with the notation I'm using, for brevity omiting subscripts and so on.

I have two thoughts on this. One, Ashtekar doesn't use all of his connection, but only the "anti-self-dual" part. Two, the Ashtekar variables specify not only a geometry but a kinematics. Thiemann's intro does a lot of degree-of-freedom counting, maybe that would be a reference on this issue.

Marcus,

I have two thoughts on this. One, Ashtekar doesn't use all of his connection, but only the "anti-self-dual" part. Two, the Ashtekar variables specify not only a geometry but a kinematics. Thiemann's intro does a lot of degree-of-freedom counting, maybe that would be a reference on this issue.

I was counting on being able to edit some of this expository writing, but discovered yesterday that the PF rules have changed. there is a time limit of 30 minutes afterwhich I cannot edit a post
It makes it easier to write if you can put in placeholder stuff at some point, then go research it and fix it up if necessary.

If you have more extensive editing capability (a mentor perk?) than I do, you are welcome to fix vague points in my discussion, correct errors, improve style or whatever. editing is half of writing

I was a bit sleepy when Ambitwistor passed through---not sure but I got the impression of someone who may actually do research in quantum gravity----knows work of Baez and of Renate Loll with quick exactitude.

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My intuitive feel is that the connection describes the geometry, and I am puzzled that the Ashtekar variables are not simply A (the connection) but are various pairs, like [A, E] where E is the densitized triad or "electric field"(a term sometimes used depending on a possibly confusing analogy). A and E are presented as "conjugate" variables. Yet the configuration space is just the collection of all possible A's. what is the essential additional information given by E? You are probably familiar with the notation I'm using, for brevity omiting subscripts and so on.
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Hi there,

I think there is a confusion here. It is true that a metric and its (Levi Civita) connection carry almost the same information (up to constant re-scaling of the metric). In the basic variables of LQG the metric information of the manifold is in the triad E. The connection "A" that is known as the Ashtekar-Barbero connection has more infromation than just the metric. It also knoes about the ADM conjugate variable, namely the extrinsic curvature $K_{ab}$. Then the connection $A$ is given by $A_a^i=\Gamma^i_a- (const) K^i_a$, where the constant in the formula is the infamous Immirzi parameter.
Thus, even when the connection knows about the metric, it also has information about the extrinsic curvature, and that is why it serves as a conjugate variable for the $E$'s (that is, if the Immirzi parameter were zero, the variables would be all "configuration variables", and its Poisson bracket would vanish).

Another comment. Te self dual connections were the original variables introduced in 1986 by Ashtekar, but they were replaced in the 90's by the Ashtekar-Barbero connection with a "real" Immirzi parameter (instead of $i$ for the original self-dual case). The nice geometrical interpretation is however, lost.

Originally posted by marcus
My intuitive feel is that the connection describes the geometry, and I am puzzled that the Ashtekar variables are not simply A (the connection) but are various pairs, like [A, E] where E is the densitized triad or "electric field"(a term sometimes used depending on a possibly confusing analogy). A and E are presented as "conjugate" variables. Yet the configuration space is just the collection of all possible A's. what is the essential additional information given by E? You are probably familiar with the notation I'm using, for brevity omiting subscripts and so on.

(A,E) are a conjugate pair, i.e., a point in phase space (not configuration space). It's analogous to how in QM, position and momentum (x,p) are a conjugate pair (but you only pick one of them them to form your quantum Hilbert space). In the ADM geometrodynamic variables, the 3-metric and the extrinsic curvature form the conjugate phase space variables, and you generally form your Hilbert space over the 3-metric. In the Ashtekar variables, you pick the Ashtekar connection and the densitized 3-triad.

Classically, you use the configuration space variable to describe the kinematics, and the conjugate momentum to describe the dynamics. (e.g., position in QM or the 3-metric in geometrodynamics describes the system at a given instant of time, and momentum or the extrinsic curvature describes how that state will evolve).

You can reconstruct the (densitized) 3-geometry of a spatial slice from E, not A (since the triad is basically just the "square root" of the 3-metric, as nonunitary mentioned). A itself determines a "geometry" on space, just like any connection does, but it's not the kind of metric geometry that a Levi-Civita connection defines. A actually carries information that one can use to reconstruct the spacetime geometry -- like extrinsic curvature does, and in fact A involves the extrinsic curvature (as nonunitary also pointed out).

In Thiemann's derivation of the Ashtekar variables he first enlarges the phase space of the Palatini action, spanning this larger space with canonical variables K and E, K will go away but E will remain in the Ashtekar variables. He shows that the new (K,E) coincide with the Palatini (p,q) variables when a constraint is satisfied; this constraint is satisfied identically in the Palatini geometry. Only then is the connection A introduced, and it replaces the nonce variable K, and the new variables (A,E) are canonical and span the big phase space.

In general is it really true that a connection by itself specifies a geometry? Recall that in traditional Riemann you have first a metric - specified by a symmetric tensor, which restricts your choice of geometries, and then define the connection as a function of your metric (through the Christoffel symbols). This then gives you the curvature tensor and all the rest. But the contribution of the symmetric metric was important.

In general is it really true that a connection by itself specifies a geometry? Recall that in traditional Riemann you have first a metric - specified by a symmetric tensor, which restricts your choice of geometries, and then define the connection as a function of your metric (through the Christoffel symbols). This then gives you the curvature tensor and all the rest. But the contribution of the symmetric metric was important.

Only a Levi-Civita connection specifies a Riemannian geometry, because only L-V connections are compatible with metrics. But starting at least with Klein, and certainly since Cartan, the notion of "geometry" has been expanded to include geometries other than Riemann's. You can think of a connections as giving a generalized kind of geometry, a special case of which are the Riemannian (metric) geometries.

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