Smolin video LQG online course

[CD]

Anything I come up with is for closed graphs - no open ended vertices. In this case any finite trivalent graph will have an even number of nodes. Two representing a theta. In this case an even n node graph can be reduced to theta with (n-2)/2 contractions.

 

[Mike2]

Notice that \Sigma is a three dimensional manifold, and a graph is just a one dimensional object, so we can just model what happens in our old familiar 3-space.

Imagine that you have a roll of magic string. The magic is this: when you want it to be limp and flexible, it is, but if you command it to stiffen up it will hold any shape you have got it into. It is also weightless so it doesn't sag on you while you're playing with it.

Now cut off a bunch of pieces of this string, which will be the edges of your graph, and knot them together at the ends, any way you want, to make the nodes. Now you have a graph, and it's "embedded in 3-space". Play with it, twist it, without changing the knots, and then command it to stiffen. Obiously you can get all sorts of configurations, and there are even more that you can't get this way (what you are doing are called "isotopies"). For example the graph that looks like reflection of your graph in a mirror is again topologically equivalent to it.

I emphasize that nothing more outre than this is going on. No extra dimensions are involved.

Thanks, that's starting to help. What you are describing seems to be the simplicial complexes of differential geometry, is it?

 

[selfAdjoint]

Thanks, that's starting to help. What you are describing seems to be the simplicial complexes of differential geometry, is it?

Yes, the graph could be used to define two-dimension simplices ("singular chains"), but I don't think Smolin is going to use them for that. I could be wrong though.

BTW I have now watched half of the second video, up to the point where he says he wants to build a real field theory, and notes that he needs fewer field equations, and after discussing counting of degrees of freedom asks the "class", "How can we get fewer field equations?" I'll watch the rest tomorrow. I think my crashing problem is a shaky cable service. Out here in the Wisconsin boonies the cables are not buried but strung on poles, and frequently there are small glitches, some big enough that even the service provider notices them, but others fleeting and no problem as long as I'm not using some streaming source. Last night was our big blizzard with howling winds, and I could hardly get 3 or 4 minutes at a pop. Tonight was much quieter and I got 20 minutes.

On the "spectacles problem" I tried enlarging each of the nodes to a triangle and then exchanging the bar to a vertical one, But to get further I think I need to show that any three connected nodes, not just a triangle, can be shrunk to one. Has anybody looked into that?

 

[marcus]

here is a trivalent graph with two vertices

O-O

it is not a theta graph
using our set of approved moves, how do you change it to a theta?

what is the magic word?
I want to know explicitly how to change it
CD, or anybody?

 

[marcus]

I just saw selfAdjoint response to this "spectacle" problem.

It is probably OK but not the quickest way.
You can do it in ONE MOVE

selfAdjoint, you say to start off by applying expansion moves to each vertex first which makes a bigger graph, now with 6 vertices, and then you have to collapse that down. so that might work but would take several more moves.

one exchange move changes spectacles to theta

=====================

but, like you say, we still have to prove that these moves will take down an arbitrary finite trivalent graph

how would you handle spectacles for a 3-eyed man?

O-O-O

 

[selfAdjoint]

AH! O-O -> "PHI" with one circle with the bar going vertically, which is of course the original graph. Neat! Now the three-specs. On the left two lenses reduce them to a "PHI" using your method. Now we have a PHI linked to a circle by a horizontal bar, The two nodes of the PHI plus the node at its end of the bar make a triangle so do a compression, which reduces the left PHI to just a circle joined to the bar. Now you have two-specs as before. So you have an induction for n-specs; given n=2 the problem is solved, and giving that it is solved for n=k-1, this shows it is also solved for n=k. So it is solved for all n \ge 2.

 

[marcus]

excellent
I just got back to the computer, didn't see your reply 'til now

 

[marcus]

AH! O-O -> "PHI" with one circle with the bar going vertically, which is of course the original graph. Neat! Now the three-specs. ...

I understand your word-pictures. So far we are getting along OK without a blackboard.

what you mean by the PHI graph I understand to be the THETA rotated 90 degrees. (it helps intuition to have it aligned up and down like phi.)

we have two things to think about, then maybe we can move on to Lecture #3 and #4 (or whatever people want)

A. why are moves important, what is this toy model showing us?

Moves have amplitudes. The dynamics of the theory is with path integrals.
A HISTORY is a sequence of moves (a history can be pictured as a foam looking like a spinfoam without labels, in this toy case) that get you from an initial graph to a final graph.
The AMPLITUDE of a particular history is the product of all the amplitudes of all the moves that make it up.
A PATH INTEGRAL is an amplitude-weighted sum of all the histories that get from some initial to some final.

So moves are basic to dynamics. Also Renate Loll uses moves of a different sort called Monte Carlo moves to randomize triangulated spacetimes and to approximate the CDT path integral. Another reason to get a feel for dynamics based on moves---some of it carries over.

B. Before we move on, if people want to move on and talk more about Smolin Lectures, we could wrap up the homework. I will talk about that in the next post.

 

[f-h]

Ah just now saw this. I just worked out the other half of the problem (reducing everything down to n-specs) on Christine's blog:

Let's pick an arbitrary loop in the spinnetwork with 3 or more links, if it has more then 3 shrink it down to 3 links by exchange moves (which with respect to the loop we picked reduces a link to a node), then eliminate it using a contraction. This strategy eliminates a loop, thus by iteration we can eliminate all loops with 3 or more links, but the n-specs are the only trivalent spinnetworks without loops with 3 links. qed.

 

[f-h]

Anything I come up with is for closed graphs - no open ended vertices. In this case any finite trivalent graph will have an even number of nodes. Two representing a theta. In this case an even n node graph can be reduced to theta with (n-2)/2 contractions.

Nonclosed graphs are not trivalent, furthermore ergodicity is not true for them ;)

Counterexample to your above strategy: The cube. You can not apply a contraction there. You need to apply exchange moves first.

You also clearly can not replace the exchange move by a number of expansion/contractions since these can only add to the number of links around each face, but not reduce them below 4 basically everything you do with these moves is confined to the corners of the cube without changing it's overall shape.

So exchange moves are really quite important for the overall graph structure and not just once you get down to n-specs.

 

[marcus]

selfAdjoint, you and I may be the only folks interested in this but just for completeness

[EDIT: i just now saw F-H post #39 and 40. So F-H also proved it! But more elegantly it looks like (havent read his yet). I will still leave this what I wrote and not erase it, so we have two people doing the same thing different ways and styles.]

I think the rough idea is this (you may have pointed this out in an earlier post already)

it is going to be a proof by induction on the size of the graph and one will imagine the SMALLEST graph that cannot be reduced to a theta---or something like that.

and one can see that it is connected, otherwise one of the components would be a smaller uncollapsible.

I think WLOG one can say that the uncollapsible graph has no vertex that is connected to only one other vertex (and to itself) in this kind of situation

...-O

or one could do an exchange move and have an uncollapsible graph of the same size that makes the vertex be connected to more than one other.

This graph will not have any place where there are THREE vertices adjacent to each other----there are no little "triangles" made of a triplet of vertices. Otherwise one could do a contraction move and have a new graph with two less vertices.

This graph will not have any place that looks like this

...-O-...

(this is 4 vertices with the two in the middle joined by two edges)
or you could do an exchange move and have a "triangle" of 3 adjacent vertices.

so there are no double connections between vertices---it is a very vanilla unimaginative graph. sort of like a TREE but because it is finite and has to end somewhere it has to have connections among the branches

so we know there are no triangles in the graph, and we ask "are there any four-gons? or pentagons? or hexagons?" and we ask what is the SMALLEST gon that there can be in this uncollapsible graph.

How many links can you go before you come round to where you started? What is the smallest cycle?

This is the kicker question that my earlier handwaving considering various cases leads up to. Because if the smallest cycle is N, then you can do an exchange move and get a (still uncollapsible) graph where the smallest cycle is N-1.

I think that is QED, or would be if I could just make pictures illustrate and be very careful and rigorous. I think that is the idea of the proof. If anyone has a better plan, or a clearer way of describing it, please show us.

 

[marcus]

Oh hey f-h, so glad to see you! I was just writing my post trying to prove that thing for selfAdjoint, and I did not see your posts. I guess you proved it some simpler way, I haven't read yours yet. Great!

 

[f-h]

Ah, nice way of getting a contradiction there! We have the same basic observation: n-Loops get turned into n-1 loops by exchange moves for n>2.

Shrink n-loops to n-1 loops using exchange moves. Eliminate 3-loops using contraction, and 1-loops using exchange moves.
You are left with a graph containing only 2-loops. There is only one such graph: Theta.

 

[selfAdjoint]

Very good! Thank you both. Marcus I liked your "infinite descent" approach, and f-h that's so neat and complete. Congratulations.

 

[marcus]

Fine, we seem to have settled the homework problem.
I am wondering about moving on.

Question to f-h and selfAdjoint: have you watched Lecture #2?

the last time we discussed practical matters, selfAdjoint system was crashing and it sounded painful to watch anything

then CD said he had files that could be downloaded, but i am not sure that would work for selfAdjoint or that he wants to go that route

Has anything happened in the meantime? Have you found a way to get around the problem?

Unless someone has another suggestion, let's all watch Lecture #2 and try to summarize what it is about.

=================
I EDITED THIS---EARLIER I MISREMEMBERED SOMETHING AND THOUGHT #2 WAS A REVIEW and suggested skipping it, but it isn't the review. IMO is a good one to watch!
(It is lecture #4 which is a review, and which we might skip.)

 

[CD]

Nonclosed graphs are not trivalent, furthermore ergodicity is not true for them ;)

Counterexample to your above strategy: The cube. You can not apply a contraction there. You need to apply exchange moves first.

You also clearly can not replace the exchange move by a number of expansion/contractions since these can only add to the number of links around each face, but not reduce them below 4 basically everything you do with these moves is confined to the corners of the cube without changing it's overall shape.

So exchange moves are really quite important for the overall graph structure and not just once you get down to n-specs.

Yes you are correct, thank you. I think I may be missing something here. So, these are just the pachner moves 1-3, 3-1 and exchange?

 

[f-h]

Yes you are correct, thank you. I think I may be missing something here. So, these are just the pachner moves 1-3, 3-1 and exchange?

Yep. Precisely. Thanks for pointing that out.

I am up to lecture 7 or so, so whatever you want to look at next is fine by me. I might not have much time to contribute over the next week anyways unfortunately.

 

[marcus]

while we are deciding what to do and whether to move on, I will tie up a loose end where there was some disagreement about terminology at Christine's blog. Yesterday I posted this where I used the word "ergodic"

I just watched Lecture 1 again and I think the only main homework or thing to check is the fact about a certain set of moves being ergodic (in this case meaning that you can get from any trivalent graph to any other by repeatedly applying just those moves)
...

In this situation that is simply what ergodic means. that you can get from any graph to any other graph by doing enough of these. We just showed that in the homework. Because if you can reduce any graph to a particular one, say theta, then you can get back from theta TO that graph by reversing the moves.

So you can get from A to B by collapsing A down to theta and then expanding theta out to B. There will certainly be OTHER ways to do it but that shows there is at least one.

The mental image I have of ergodic transformations is shuffling a deck of cards---which you do by repeating elementary moves. If the elementary shuffle moves are really ERGODIC then that means that if you do elementary shuffle moves enough times you can get ANY ordering of the deck.

Sorry, have to go, be back to finish later
=================
my wife is reading a book about Medieval monks and the relics they had at their monasteries. and the book is about other things too, but Chapter 8 is about things like
FURTA SACRA which is adventure stories about how bold and crafty monks from monastery A were sent to steal the relics at monastery B. quite a lot of that happened. there were spies and moles, sometimes a heist operation took years to prepare and put into effect.
and she tells me that at one point they were confronted with the problem that there were two heads of John the Baptist, at two different competing monasteries, and they resolved the logical dilemma by declaring that both heads were authentic, simply that one was his head when he was a young man and the other was when he was older. Of course we all know the story of how Salome danced for Herod and got him to have John the Baptist's head chopped off and given her as a present---which would have been the origin of the second of the two relics. this book is "Off the Road" by an editor of Harper's magazine, named Jack Hitt.
====================

Well we still have to draw the connection between THIS kind of ergodic----moves which effectively mix things around by getting from any configuration to any other configuration---and the OTHER kind of ergodic that people are used to where there is a probability measure on a set of points and a transformation of the set.

http://www.cscs.umich.edu/~crshalizi/notebooks/ergodic-theory.html

According to that defintion the transformation is ergodic, for that probability measure, if any invariant sets have either probability 1 or zero. In effect that means that the only way a nontrivial set can be invariant is that (up to sets of measure zero) it is the WHOLE THING.

But if you think about it that just means the transformation thoroughly moves things around because you can start at any point and applying the transformation over and over will eventually get you anywhere else. If it didnt, you could have an invariant set which was only a part of the whole. an ergodic transformation, if you keep repeating it, explores the whole set.

==========
those University of Michigan notes say that it was Ludwig Boltzmann who coined the term ergodic and what he originally meant by it was what SMOLIN means---our meaning here! Boltzmann thought ergodic meant that it will take you from any point to any other point if you do it enough times.

funny, going by the Greek root ergos, work energy, ergodic should mean energetic, vigorous, hard-working. I wonder if that was what the mighty Ludwig meant---I would call it THOROUGHLY MIXING.

 

[CD]

Three was just classical dynamics. Developing the dynamics of a time reparameterization invariant system. I haven't watched four yet but it sounds like its just a review of various topics.

 

[marcus]

Three was just classical dynamics. Developing the dynamics of a time reparameterization invariant system. I haven't watched four yet but it sounds like its just a review of various topics.

You are right, it is FOUR that is the review.
I misremembered and thought that TWO was the inessential one that could be skipped.
Sorry sorry sorry.

I will go back and correct my post, and propose (we don't have to do it but this is just my suggestion) that we just move on to Lecture #2.

(the skipping idea was based on misremembering, I take it back)

 

[marcus]

I went back and edited post #45 and got rid of the error

Fine, we seem to have settled the homework problem.
I am wondering about moving on.

Question to f-h and selfAdjoint: have you watched Lecture #2?

...
...
Unless someone has another suggestion, let's all watch Lecture #2 and try to summarize what it is about.

...
...
(It is lecture #4 which is a review, and which we might skip.)

sorry about the mixup earlier today.

 

[selfAdjoint]

Fine, we seem to have settled the homework problem.
I am wondering about moving on.

Question to f-h and selfAdjoint: have you watched Lecture #2?

the last time we discussed practical matters, selfAdjoint system was crashing and it sounded painful to watch anything

then CD said he had files that could be downloaded, but i am not sure that would work for selfAdjoint or that he wants to go that route

Has anything happened in the meantime? Have you found a way to get around the problem?

Unless someone has another suggestion, let's all watch Lecture #2 and try to summarize what it is about.

I watched all of Part 2, but with the same difficulties; during the last half hour of the video yesterday I was only getting 2 or 3 minutes of viewing time between crashes. Also I had difficulty understanding Professor Smolin during this part of his talk, he often had his head turned away from the mike, and I didn't get all those equations he put on the three boards copied fully, so all I have is some very high level notes on why Euclidean QG is easier than Lorentzian QG and how the new equations are polynomial while the old naive way of quantizing produced very messy hard-to-handle equations. I would appreciate if anyone has better notes or a source for the official notes on that talk.

I am very open to CD's download procedure if he will indicate how I should go about it. I don't want to continue watching without trying to get something better.

 

[CD]

I sent you a pm containing the access information. Anyone else that would like to download a copy just pm me. If you have any problems let me know.

 

[marcus]

I sent you a pm containing the access information. Anyone else that would like to download a copy just pm me. If you have any problems let me know.

I would like to try it, if the memory requirement is not too large.
CD, does the file that we can download from you have only the sequence of blackboard still-frames, or is it the audio/video, or the whole thing?

It's not that I need to go this route, so if you only want people who don't have a good alternative, don't bother to send me the URL. The thing is, I am curious to try how it works.

 

[selfAdjoint]

does the file that we can download from you have only the sequence of blackboard still-frames, or is it the audio/video, or the whole thing?

It apparently has the video & audio, although I haven't tried it yet.

CD says he watches the video from his site and also has the perimeterstreaming site up to click through the slides.

 

[marcus]

It apparently has the video & audio, although I haven't tried it yet.

CD says he watches the video from his site and also has the perimeterstreaming site up to click through the slides.

Great! hope it works. Let's see if we can say anything about Lecture 2, then.

 

[selfAdjoint]

Successfully watched the first hour of Lecture Two, without slides (just didn't bother with them). I watched much of it twice and am now much clearer on what he was saying. My lack continues to be reproducing his board work in my notes, especially the index gymnastics that he went through so fast. From my memories of watching it at the Perimeter site I don't expect the slides to be much help.

One issue I would like to discuss is the degree-of-freedom counting which he used to show that SU(2) BF theory has no local degrees of freedom. I got the point that 18 equations in 18 unknowns determines a single solution (if nonsingular), and thereby uses up all the degrees of freedom available. It was the details of the counting that went by me, first three Fs and three Es and then... His audience seemed not only to be with him, but some of them were ahead of him. So this is apparently a prime way of reasoning about conjectured theories. Can we get up to it ourselves?

 

[CD]

I got 18 degrees of freedom from the 9 A's and 9 E's
then 3 equations from D_{a}E^a=0
and 9 equations from E^{i}_{ab}=\frac{1}{\lambda}F^{i}_{ab}
3 equations fixed by the gauge in A
3 equations from diffeomorphism invariance

Sorry about the equations here but I'm in a rush at the moment.

edit - updated to tex

 

[selfAdjoint]

I got 18 degrees of freedom from the 9 A's and 9 E's
then 3 equations from D_[a]E^a=0
and 9 equations from E^i_[ab]=1/@*F^i_[ab]
3 equations fixed by the gauge in A
3 equations from diffeomorphism invariance

Sorry about the equations here but I'm in a rush at the moment.

No prob about the equations, and thank you. What does the @ represent?

 

[CD]

I believe it was lambda in his equations. It was just introduced as a free parameter but eventually gets recoined as the cosmological constant.

It was something like
F_{ab}^{i} = -\frac{\Lambda}{3}\epsilon_{abc}E^{ci}
Then by breaking the gauge and setting the A's proportional to a delta we end up with a deSitter spacetime with cosmological constant lambda.