Attention: This file has been imported from my previous blog, Background Independence. Comments made at the time, if existent, follow the main post. New comments are welcomed. If you don’t know what is all about, see here.
The Hand of a Master – Parts 1 and 2
Use this space to comment and discuss Lee Smolin´s “Introduction to Quantum Gravity” Lectures, Parts 1 and 2.
Brief outline:
- The basic structure of Loop Quantum Gravity
- Introduce General Relativity as a Gauge Theory
I would probably start off from a doubt I have on the issue of causality. In fact this is a very general doubt. If I got this correctly, Smolin says that if you consider two moves, p and q, in a spin network (and I am not sure whether he was referring specifically to a 3- or 4-valent node graph here), then:
- p > q, if p acts on a subgraph altered by q (this would indicate a “causal” relationship)
- moves are a partially ordered set -> “causal set”
These seem to be just an operational definition. I completely missed the point on how this is related to the meaning of “causality” as understood in GR. I mean, how do you relate “causal” moves in the graph with the “causal light cone” in relativity?
This is a very basic question of course.
Feel free to send your questions, answers, comments, doubts, criticisms, ideas, disscussions and feelings on these lectures, even if they are as basic (or even more basic) than my first doubt above. This is just the first post of a series, and here you will have the chance to post without any censorship (of course, personal attacks and unrelated discussions will be deleted).
Thank you!
Update Feb 15, 2006: System requirements for watching the Lectures:
Sonic Foundry recommends the following system requirements for the best presentation viewing experience with Mediasite Viewer:
-
Microsoft Windows 2000, Windows XP, Windows 2003 or Macintosh OS X,
-
Display resolution of 800 x 600 pixels or greater
-
Windows-compatible sound card
-
Microsoft Internet Explorer 6.0 SP1.
-
Windows Media Player 9.0
-
Broadband Internet connection (256 Kbps & above)
17 Comments:
said…-
That’s a good question, I never really thought about this.
It doesn’t relate directly of course but if you think of the net as having few long range links (that is, it looks at least somewhat classical), then far seperated moves can not be causally dependent. and in fact since each move operates at one vertex the steps needed to the next move that can be causally dependent on both of them (through transitivity) depends linearily on the number of vertices/links between them.
So it is a natural version of local causality, wether or not it is related to GR causality is not clear, since there is no classical limit though. - 2/14/2006 10:50:12 AM
Christine said…-
Hello fh,
Perhaps one possible answer to my question on causality is this paper I have just found, Implementing causality in the spin foam quantum geometry, by Livine & Oriti. - 2/14/2006 07:42:35 PM
said…-
You rule, Christine (this is slang for being excellent)
the blog is royally equipped.
how can you resist spending all your time on it, instead of writing code for satellites and looking after family?
it must be tempting to get totally absorbed in QG.
I will slowly begin to put comments to this thread. But for now just want to say thank you.
hello you to f-h, very glad you are here. Have to run, wife is opening Valentine’s Day chocolates! - 2/14/2006 08:34:49 PM
John G said…-
Christine, here’s a link relating Spin Networks and Feynman Checkerboards. (Seems like the same general idea as the paper you mentioned).
http://www.valdostamuseum.org/hamsmith/cnfGrHg.html#SPINET - 2/15/2006 12:55:41 AM
Nigel said…-
To create gravitational force, energy is being exchanged via all possible pathways between masses. The graphs are abstract representations so don’t need to link directly to causality.
It is like electric and magnetic field “lines” which in QFT merely indicate where gauge bosons go to produce forces; they are convenient for mathematically describing the electromagnetic field, but the lines are just a model.
(There may be too much abstraction in classical electromagnetism and it would be good to forget E and B fields for a while, and just produce a causal version of Maxwell’s equations which describe forces as results of a net energy exchange. This would provide a proper link up between classical EM and QFT, helping to illuminate methods that would help LQG.) - 2/15/2006 10:49:16 AM
said…-
Yesterday evening (Wednesday) I watched Lecture #9. He was talking about BF theory and Chern-Simons. I thought it was one of the especially good ones. I don’t follow it all but some of it always gets through to me and soaks in. There is something special about seeing gestures and hearing tone of voice and sensing how he is responding to questions.
AFAIK he hasn’t mentioned the Kodama state yet, but he has all the necessary symbols on the blackboard now. As of Lecture #9 he is set up to introduce it, it seems to me—–as soon as he gets into talking about quantum states of gravity and not just various simpler field theories.
Pedagogically this course seems to innovate. Instead of doing classical Gen Rel in detail and then applying quantization recipes he comes at it from a different direction.
Instead, he first works out many examples of (constrained) topological qft—-like playing with various attractive educational toys. Like he wants students to get a feel for TQFT and get easy and familiar with the routine and get to kind of LIKE constrained tqft.
And then in Lecture #10, he says, he is going to do some Gen Rel—-but that is being developed occasionally in parallel, sporadically so far: it hasnt been the main focus.
And then, I am guessing, in Lecture #11 he will surprise everybody and show how you can do half a dozen different things with TQFT, including quantum gravity.
I am just guessing, he didnt say. He only said what he was going to do in #10. But it looks like he is ready now, or will be ready soon.
It is a REAL interesting approach, I think. You dont take Gen Rel as a basis and apply quantization recipes. Because, after all, it is the quantum gravity that is the fundamental theory of spacetime and Gen Rel is merely an APPROXIMATION to it. Instead, he says, what is basic is differential forms—-that is, fields that you can define without resorting to a METRIC or a setup geometry. So let’s find out about form theories—-BACKGROUND INDEPENDENT field theories like BF, in other words—and then lets play around……and then (I guess) he pulls a form theory of gravity out of his hat.
I think that is charming. It might be pedagogically the smart way to do the subject. We will see. - 2/16/2006 01:46:28 PM
said…-
Christine, about your question
** how this is related to the meaning of “causality” as understood in GR. I mean, how do you relate “causal” moves in the graph with the “causal light cone” in relativity?**
To begin with the obvious, events in GR form a P.O. set where p > q if p is in the lightcone of q.
And I think what happened is that Sorkin decided to abstract out this P.O. structure from GR and just study the P.O. structure of events and FORGET about the differenntiable manifold and the geometry and the lightcones and all the continuum machinery.
So when Smolin points out that MOVES have a P.O. structure and says the word “causality” he is mostly just connecting to Sorkin’s Causal Sets idea. That is the mental connection, not to geometry and lightcones.
Or it connects the P.O. set of moves to geometry only indirectly by way of Sorkin P.O. set of abstract events.
Sorkin stuff is too abstract for my taste. I like at least some little bit of geometry.
I don’t think Smolin’s point of the formal similarity is very important. He is just pointing out a formal similarity and he is not saying that moves ARE events or anything earthshaking. But Sorkin Causalsets is still a recognized part of the overall QG scene and it is good to point out similarities like that.
Besides the familiar P.O. conditions, what a Causal Set has is this FINITENESS axiom where if P is in the future of Q then there is a finite sequence or chain between them.
Sorkin and his friends like Fay Dowker really milk this finiteness feature for all it is worth and use it to get numbers and deduce stuff.
I think that in the Smolin case with moves, the finiteness condition is satisfied trivially. Because the initial and final networks are finite and the history of moves between them is finite. I may be missing something, or just plain wrong, but I don’t see anything to prove.
Christine, I have two conflicting thoughts:
A. we dont have to spend a lot of time on Lectures 1 and 2 and simple network stuff (or this little sideline about P.O. sets). We can move on to DIFFERENTIAL FORMS FIELD THEORY or constraint TQFT or whatever is the main mountain. If people want.
B. but on the other hand Smolin gave this HOMEWORK in Lecture 1 where he said to show that in the world of TRIVALENT nets the set of moves consisting of the EXPANSION CONTRACTION AND EXCHANGE was an
ERGODIC set of moves.
Maybe we shouldnt get stuck on preliminary combinatorial stuff. but on the other hand maybe we should. maybe we should try to do the homework!
here ergodic is just an impressive codename for being able to get from any finite net to any other finite net by a sequence of those moves
(it has an analog in the continuum theory of dynamical systems but that doesnt matter, here it is a nice simple idea)
I love the idea of an ergodic set of moves. It comes up in Renate Loll CDT where she has a set of moves that alter the triangulated spacetime history and which are a very small set of like only seven or eight moves but if you do them enough then you can get ALL POSSIBLE spacetime histories. she also uses the word “ergodic” for this—it means “completely stirs the soup” or “completely kneads the bread-dough” not leaving any lumps or parts that never get visited. Loll does her computer simulations by applying this ergodic set of MonteCarlo moves to spacetime histories over and over again—millions of times actually—until she has a completely unexpected random spacetime history.
Having a finite set of moves that will completely shuffle the cards or stir the soup is an elegant idea.
so Smolin is offering us a kindergarten homework about it. maybe we should do it. can we do it at a blog? or does it require the blackboard? this is what is on my mind. - 2/17/2006 02:38:37 PM
Christine said…-
Dear who:
Thank you for your various comments and clarifications.
I have some reasons to start off only with Parts 1 & 2. First, I am considering that the readers of this blog have very different backgrounds and levels of competence on quantum gravity. So, although many readers would like to discuss more technical aspects later in the lectures, I would tend to believe there are others who would like to understand more basic or fundamental issues in the first lectures. I am myself included in this later group.
It is already a very daunting task to run a blog on quantum gravity even though you are still learning the basic stuff. So if I have obliged myself to do it, I must do it comfortably.
Not only that. I must feel things I write here are (will be) useful not only to me but also to others.
I have learned not to take anything for granted, specially in science. There are several things I do not yet understand and at my present stage of learning LQG I see many leaps of reasoning all around that makes me feel uncomfortable.
Of course that just reflects my present stage of understanding, and I do hope several pieces will fit together opportunely.
This idea of discussing Smolin lectures (which was not my idea — so you see, I like to receive suggestions and try to implement them here if I find them interesting and positive) do not mean we have to go exhaustively through every single equation in order to make progress. Although that would not be bad, it is impracticable: as already mentioned in the “basic curriculum for QG” post, it is inevitable that we learn the subject non-linearly.
Another point is that I am late. At present I can only discuss parts 1 to 4. It is not that I am not in a hurry. I am anxious to learn this subject. But I can only do what I can do.
So my initial thought was to post on parts 1 & 2 and go to the next couple of lectures when I feel there is enough material discussed. This is completely subjective of course.
But I do not intend to be too much rigid in this blog, so the bottom line is: use this space “to discuss Smolin lectures”. If there are readers that are willing to discuss the later parts of the lecture series in the present post, that is all right to me, although things would be more organized if they were discussed separately in the upcoming posts on these lectures.
Concerning the issue of causality in terms of implementing them in a graph, I am still a little unconvinced that it is as trivial as you mention, and of course it must be partially because I must be still too much attached to the continuum manifold. However, I do believe that another part of the problem is the lack of a physical justification, not a mathematical one. I want to think in physical terms on how to understand causality in a discrete, background independent framework. That is why I have said my doubt is a very general one.
But of course, I have a lot to read to make progress in this issue.
Best wishes
Christine - 2/17/2006 09:58:35 PM
said…-
Hi Christine, everything you are doing is fine by me. Probably it is fine with everybody else. Great blog. Do only what you can comfortably do!
Would you like just now to focus on the one homework problem Smolin gave in Lecture 1?
With video lectures one can proceed exactly at the rate one wishes, and that is convenient.
Or one can put the project on the shelf for an indefinite period! I am happy with anything you choose to do.
===================
if you decide that we should talk about the Lecture 1 homework problem, it is to show that any finite trivalent graph can be transformed to any other such graph by some series of moves
where the moves are {expansion, contraction, exchange}
expansion replaces a single trivalent vertex by a triangular triplet of THREE adjacent trivalent vertices.
contaction gets rid of three adjacent vertices and replaces them with one—-it is the opposite of expansion, the “anti-expansion”
exchange operates on an adjacent PAIR of vertices and lets them trade what they connect to.
A B
>-<
C D
here is a pair of adjacent trivalent vertices, the one on the left is connected to A and C (whatever subgraph structures they are) and the one on the right is connected to B and D
If they do a “swap” or “trade” then we can end up with this
A B
\/
|
/\
C D
Now there are two adjacent vertices, but one is above the other.
Then, if, in the first picture one identifies A with C and B with D, the first picture will look like a pair of spectacles
O-O
this is two adjacent trivalent vertices which are connected by one horizontal link and, each one is connected to itself by a circular link
If one does an exchange move on that, then one gets
(|)
this is now two adjacent trivalent vertices with one above the other, which are joined by 3 separate edges
To summarize, doing one exchange move can transform this
O-O
into this
(|)
In Lecture 1, Smolin introduces the important idea of a HISTORY and he gives a toy model example with unlabeled graphs instead of the spin networks of regular LQG
and a HISTORY is simply a series of MOVES
(each one with an amplitude, a complex number)
and it is a series of moves that get you from the INITIAL graph to the FINAL graph
and the AMPLITUDE of any given history is just the PRODUCT of the series of amplitudes of each move, step by step.
And the PATH INTEGRAL is simply the sum of all those products—-the sum of the amplitudes of each history that gets you from the initial graph to the final graph.
==============
so understanding the toy model he gives in Lecture 1 really comes down to getting a feel for MOVES
Oh, another thing, a history can be visualized as a foam.
the path of a changing network describes a foam-like complex—–drag a graph through a time-like dimension and it describes a honey-comb looking cellular complex of 2D cells
In regular Quantum Gravity, spin foams are the HISTORIES of evolving spin networks. So a spin foam is what you get if you apply MOVES to get from an initial spin network to a final one.
NO ONE SHOULD HAVE TO UNDERSTAND ALL THIS in one sitting, or in one lecture. So in Lecture 1 what Smolin did was basically make us familiar with the idea of an unlabeled network and the idea of a move.
Probably, we should try to do the homework. - 2/18/2006 02:20:34 AM
Nigel said…-
Wow! Thanks for that, it is very clearly stated. I am a slow learner but am starting to pick up some details, and will have to view Smolin’s lectures again a couple of times to get the finer points! Thank God I’m not attending them in real time, or I’d miss a lot…
- 2/18/2006 12:11:31 PM
Christine said…-
One clarification please.
As I understood it, the operations of contraction (C) and expansion (E) are (by themselves) NOT ERGODIC because you can use infinitely many times these operations without changing the graph and add them to some other finite sequence of operations (whatever they are needed) to get from any trivalent graph G to another G´, for instance:
G -> CECECEC..[infinitely]..ECECECE… [some finite set of moves] -> G´
Exchanges (let us call them X) are other *independent* set of moves that are ALSO NOT ERGODIC by themselves, because they do not change the number of nodes, so you could equaly do something like this:
G -> XXXXXX…[infinitely]….XXXX….[some finite set of moves] -> G´
Also, you cannot start from a trivalent graph, say of one node, and transform it — using C or E as many times as you try to — into a two-node graph (connected by a common edge).
I imagine a set of graphs embedded in some fixed spatial topology, and I suppose you want to represent the dynamics of space (or spacetime) by a series of moves of the embedded graphs. You can imagine that all possible movements in the graph are represented by some sort of phase-space volume of states accessible to the spacetime region. Now if you want to go from one state to another by a finite set of moves, you want to make all states equally probable, just as you assume from one of the pillars of statistical mechanics. In this case the system would be ergodic.
However, what do you do if the phase space is infinitely large in the sense that there is an infinite number of moves that could send a given graph or state to another. I mean, this is a possible path in phase space, however, it would take an exceedingly large time scale for the system to explore its whole phase space, so the system would not be ergodic.
So I really think I am missing something here. - 2/18/2006 05:53:14 PM
said…-
Christine, I see where your question is coming from now, there are some points that justify the term causality for Sorkin/Smolin (I think who is right with the Sorkin connection).
(carefull, terminology ahead)
If you take the points on a manifold with metric their causal structure is actually completely captured by a partially ordered set. This is straightforward, let A => B stand for B is in the future lightcone of A, then:
A => B => C implies A => C
A => A is trivially true
If the spacetime contains no closed timelike curves then: A => B and B => A implies A = B
These are precisely the axioms of a partial order. Clearly if you have the causal partial order on a manifold you know all lightcones, but this is precisely the information contained in the metric up to conformal transformations! Conformal transformations are those where you multiply the whole metric tensor with a positive function, so they are a subset of the diffeomorphisms.
Why do these preserve the causal structure? Because if you think about a lightlike vector, it has length 0, so if you multiply the metric in the definition of length with a positive function it’s still 0. Same for timelike and spacelike vectors.
Of course it’s a lot less clear that given the Poset we automatically have a metric up to a conformal transformation, but it’s actually true.
If in addition to the causal poset structure you know the volume of an arbitrary region, then this fixes the conformal factor, and you have your metric back.
This is the point of Sorkins program, take a set, impose a poset structure and take the volume to be proportional to the number of points (set elements) in the region (subset), and you get a metric manifold!
Except…
That Sorkin is taking the set to be countable (discreet) as opposed to noncountable (smooth), so you end up with something that in principle corresponds ot a discretization of a manifold + metric. Which you have no control over at all, and might be everything but smooth.
Clearly the moves in Spinfoams form a poset.
Now Spinfoams can be constructed by discretizing a manifold. I don’t know if the causal/poset structure Smolin mentioned is the same as you get from the discretization in some appropriate sense?
But that’s reaching significantly ahead now. Maybe we’ll see this in a later lecture.
—
who,
any ideas how to prove that we can get *any* trivalent spinnetwork from these moves? The obvious way to go about this is to show that we can reduce every spinnetwork to a simple minimal spinnetwork like for example the theta one (|). Note that expansion/contraction changes the number of nodes by 2. A Spinnetwork that is trivalent always has an even number of nodes. That this is the case can be seen by cutting one link at each trivalent node at a time. Each cut leaves two nodes with 2 links, therefore the number of nodes is 2*the number of cuts and hence even.
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 theta network is the only trivalent spinnetwork (up to the one possible exchange move) without loops with 3 links. qed.
Right? - 2/18/2006 06:41:19 PM
said…-
Ah, I missed your comment while writing mine Christine.
I don’t really understand your question there though… I think the term ergodic here is used in analogy with the term in statistical mechanics, but it’s not neccesarily the same concept precisely.
If we couldn’t go from any spinnetwork to any other in finitely many moves then no dynamics you could write down with local moves could be ergodic in the classical sense. However the no exchange moves/no contractions case is a special case in the general framework, so it’s clear that not every dynamics in this framework is ergodic in the classical sense… - 2/18/2006 06:49:33 PM
said…-
Addendum, as you point out on physicsforums I overlooked one thing (there was a loop hole in my strategy, har har har. Sorry), the graph I get might have links that are not part of any loop, eliminate those with exchangemoves along the way, and the statement holds:
Theta is the only graph with no loops containing 3 or more links/nodes, and no links not in any loop. - 2/18/2006 07:13:31 PM
Christine said…-
Hi fh,
Thanks for your comment on causality. I´ll think about it.
Concerning the homework problem, I do not think the question was to prove that we can get any trivalent spin network from those moves.
(Yes, expansion always increases the number of nodes by 2 and the number of edges by 3, contraction, decreases these numbers by the same amount respectively, whereas exchange does not alter these numbers. So we can see that any trivalent graph will always continue to be trivalent under that set of moves).
The question was (I think): moving from a given trivalent graph to some another, given the only allowed set of moves is {E,C,X}, is the system ergodic or not?
I am also confused here, but I have replayed Smolin lecture 1 and that is what I understand.
Notice that there are always several possible alternative paths or move sequences (or histories) from a given graph to some other. I guess those histories which would take a very large set of moves (or an infinite set of moves) would be much less probable, hence the amplitude would be vanishingly small. So they are not important when you sum them up.
But beware: I could be in complete error here. - 2/18/2006 07:28:21 PM
Christine said…-
as you point out on physicsforums
No, it wasn´t me. ‘who’, perhaps??? :) - 2/18/2006 07:30:14 PM
said…-
Yes, I meant who, ;)
And no, I think the point of the exercise was precisely that, wether it is possible to go from any trivalent network to any other trivalent network using these local moves.
The question was pure combinatorics in this sense, I think, but then it’s been a while since I watched that lecture…
If that is possible, and none of the fundamental amplitudes vanish, then we have a finite probability for the transition to occur… But I don’t know how to define ergodicity for quantum systems to begin with… - 2/18/2006 08:03:26 PM
0 Responses to “Smolin’s Introduction to Quantum Gravity Parts 1 and 2 (re-posted)”