Trying to understand spin networks, or at least SU(2)

[Coin]

Hi. I'm a relative layman with a CS/Math background (with more of the former than the latter) and I've been trying to follow some of the recent developments in physics with a hope of eventually understanding at least the outline of the underlying math. I had a couple of questions about some things I'm trying to understand, and if anybody could help point me in the right direction I'd appreciate it greatly. If there's something I should be reading first or somewhere else I should be asking instead of here for any of the following questions, I'd be happy to do so, I am just not sure where to turn for resources at this point.

Specifically at the moment the thing I'm trying to understand is the spin networks used in loop quantum gravity, specifically how they are constructed. I'm eventually hoping to attain some understanding of loop quantum gravity in general, and so I'm kind of curious what I should read if I'm interested in that (I read "Trouble with Physics" in part because I'd heard it covers LQG at some point, but though I found it very interesting and helpful in other ways, it turns out the section on LQG covers about three pages and one diagram), but to begin with I'd just like to get to the point where I can draw a "correct" spin diagram, or look at a spin diagram somewhere and understand what I'm seeing. (For one thing, I figure if I can't figure out what the basic spin diagrams mean then I probably don't have much chance of figuring anything more complicated out. For another thing, the spin diagrams look more accessible to me than most of the stuff going on here-- some of the quantum physics and high-end topology math still kind of intimidates me a bit, but the spin diagrams smell like graph theory, which speaking as a CS person is something I'm a lot more immediately comfortable and familiar with.)

Since I'm not quite sure where to start, what I've been doing is reading through the relevant "week in mathematical physics" columns on John Baez's website, and digging through anything that seems appropriate or accessibly written on ArxiV (in particular, gr-qc/9505006, which I've been working through very slowly and a little bit at a time, seems to repeatedly jump out as both being important and also containing at least one example of basically every bit of graphical notation that confuses me).

What I've gathered from all this is that a spin network is a trivalent graph where each edge has some mathematical object associated with it, and the objects which are associated with each node and/or the entire graph when considered together always satisfy some kind of property or constraint. Beyond that basic description, the details seem to be inconsistent:

Depending on which source I'm looking at, the objects associated with each edge in the spin network seem to be either half-integers, "spinors", or "a number along with some other information". I've also seen two or three differing explanations as to what the property the nodes/graphs have to satisfy is, ranging from "the spin sum must be even", to descriptions of one or more vaguely elaborate norms, where you're supposed to cut off some some portion of the graph and apply the norm to the section you cut out. In every case, it seems like if you drill down on the references enough they all seem to terminate in explaining that you need to somehow find a copy of an unpublished Roger Penrose paper that originally defined the spin network notation and concept and that physics people have been passing around photocopies of for years (muh??).

My main questions here are:

• Do all these differing conceptions of the objects/conditions for a spin network all refer to one single set of rules that are just being described to me in different ways? Or are spin networks such that you're allowed to just declare any edge-associated objects and any condition to be followed, and the different sources with different conceptions of spin networks really are actually using different kinds of spin networks thus defined?
  
• Whether the different kinds of spin networks really are different kinds of networks or just different ways of looking at one set of rules, what would be probably the simplest or most basic while still mathematically rigorous way of defining a spin network, if I was just interested in drawing one to convince myself I understood what was happening? (If there is some conception of spin networks where the edges really are just half-integers and the constraints can be applied on a per-node basis, that would be awesome and I would do a little dance.)
  
• A couple of the norms I saw described in the spin network constraints involved a process of just kind of unceremoniously cutting out part of the graph such that some of the edges are just kind of hanging out in midair. Is there some specific term for a graph section cut out in this fashion? I want to think of these as "subgraph"s, but that term doesn't seem to cut it since (although maybe this is just my limited experience) as far as I'm aware "subgraph" implies every imported edge has both end nodes imported along with it and you're not allowed to just have edges hanging like that.
  
• Incidentally, the gr-qc/9505006 paper and some other stuff I've seen occasionally make use of these little diagrams that look like a series of squares, diamonds and circles that have been cut up and pasted back together vertically at random, and that are usually referred to as something like "loop notation". I've yet to figure out which reference to follow to find them explained. These have NOTHING to do with spin diagrams, right? Where can I find the notation defined, and how was the notation reached at from the starting point of loops (which as far as I can tell are just injections from the unit circle into some manifold, right)?

Meanwhile-- past here I get into some more general questions, so maybe I'm asking in the wrong forum, if so I'm sorry-- http://relativity.livingreviews.org/open?pubNo=lrr-1998-1&page=node17.html , part of a site by Carlo Rovelli that seems to be a gold mine of information but is mostly a bit over my head, has a section that attempts to explain the spin network diagrammatic notation and really caught my eye. After a rather dense description of how to draw the spin networks and a spin diagram of the more-than-trivalent type explained at the end of gr-qc/9505006, the page says:

These are standard formulae. In fact, it is well known that the tensor algebra of the SU (2) irreducible representations admits a completely graphical notation. This graphical notation has been widely used for instance in nuclear and atomic physics. One can find it presented in detail in books such as [214, 52, 66]. The application of this diagrammatic calculus to quantum gravity is described in detail in [77],

This hits on something I keep running across in everything having to do with physics I've read lately: I'm increasingly becoming convinced that I have no hope of really understanding anything happening in modern physics without first learning about the underlying mathematics of "gauge theories", in particular the SU(n) groups that seem to show up absolutely everywhere. I've so far not had a whole lot of luck figuring out where to go with this. I have a grounding in group/galois theory, but unfortunately no formal education in lie algebras; my understanding of the subject of lie algebras mostly comes from wikipedia plus a friend's attempts to explain the SU(2)/SO(3) connection and infinitesimal generators to me a couple weeks back, and I'm still very fuzzy on how you work with the SU(n) groups and in particular how physicists use them to do all the crazy things they do with them. So I'm furthermore wondering:

• Is there some recommended or expected way to learn some things about the math of gauge groups and the SU(n) matrices?
  
• I'm particularly curious about this idea that SU(2) can be understood through some kind of diagrammatic representation similar to the spin networks-- as I implied earlier, any connection I can find to CS-friendly concepts like graphs allows me to understand things a lot more quickly than I might otherwise. Is this particularly common? Where can I find out more about that?
  
• One particular book that I saw recommended as an explanation of how physicists (and I think in specific the zomg-gauge-groups-everywhere Standard Model) use gauge groups was "Lie Algebras in Particle Physics", by Howard Georgi. The local library system seems to think they have this and I'm hoping to track it down soon, but if anyone's heard of this book I was wondering if there was any specific set of background knowledge I should try to read up on or nail down before trying to read that particular text.

...I mean, I fear the best and most obvious answer to all three of these questions is "take some classes in Lie Algebras from some actual college program", but that's probably not going to be feasible for me for the immediate future so in the short term I'm kind of more curious about resources for independent learning...

Anyway, thanks for reading this far and if you've got any advice on any of the questions above I'd be glad to hear it. :)

 

[william donnelly]

Welcome to the forums!

John Baez is definitely a good place to start. He gives the clearest explanations of this stuff I've been able to find. In addition to TWF, he also puts his lecture notes online here:
http://math.ucr.edu/home/baez/QG.html
In particular, the 2000 notes work up to spin networks, and come in tex form instead of ascii art. It comes in two parallel tracks, so you might want to skip over the part about Lagrangian mechanics:
http://math.ucr.edu/home/baez/qg-fall2000/QGravity/
The notes also contains exercises, which should help.

 

[jal]

Some visuals?

http://www.math.sunysb.edu/~tony/bintet/tetgp.html
For obvious reasons, then, the map L is called the double-covering homomorphism from SU(2) to SO(3). Since the tetrahedral group is a 12-element subgroup of SO(3), the SU(2) matrices which map to elements of the tetrahedral group will form a 24-element subgroup of SU(2). This is the Binary Tetrahedral Group.
----------
some of these might help

http://en.wikipedia.org/wiki/Topological_order
http://en.wikipedia.org/wiki/Kagome_lattice
http://en.wikipedia.org/wiki/Trihexagonal_tiling
http://en.wikipedia.org/wiki/Lattice_%28group%29 [/URL]
[url]http://en.wikipedia.org/wiki/Bravais_lattice[/url]
[url]http://en.wikipedia.org/wiki/Space_group[/url]
[url]http://en.wikipedia.org/wiki/Symmetry_in_physics[/url]
[url]http://en.wikipedia.org/wiki/Isometries#Overview_of_isometries_in_one.2C_two_and_three_dimensions[/url]
-------
Depending on what is causing the problem in getting a part of a concept, try googling for an image with that word.
jal

ps I'm also going to follow the links/suggestions provided by others

 

[marcus]

Effective Loop Quantum Cosmology

there are different paths into the subject---you need to find a path that is in keeping with your own taste and imagination, so it feels right and evokes your best energy.

so I might be going to propose a path which is right for my taste but which is wrong for you and might just deaden your curiosity and enthusiasm. we are all different so maybe there is no one right introduction.

this path is by way of the EFFECTIVE LQC of Bojowald Ashtekar Thiemann and in the Bojo form it is physics-rich and math-simple. You start on the path by googling "Kitp spacetime singularities" and watching 3 seminar talks, by those people, at Santa Barbara KITP in January 2007.

When you google "kitp spacetime singularities" (note the plural) you get this page:
http://online.kitp.ucsb.edu/online/singular_m07/
With the Bojowald and Ashtekar talks you can click on SLIDES first and then click on CAM to see the video. Thiemann doesn't have slides to download but the video includes views of the projection screen.

Watching Bojo/Asht/Thiem workshop talks (which are interrupted a lot by questions by the string theorists) will give a feel for the CONTEXT of current work in effective Loop Quantum Cosmology and then you could look at two papers:

Asht "Introduction to Loop Quantum Gravity Through Cosmology"
http://arxiv.org/abs/gr-qc/0702030
Bojo "... Physical Solutions of Quantum Cosmological Bounces"
http://arxiv.org/gr-qc/0703144

 

[marcus]

There are several points to starting this way.
One is that LQC has been known for several years to have an extremely good contact with classical cosmology. It has semiclassical states that evolve cleanly and deterministically thru the former bigbang singularity and which converge rapidly to the usual cosmology model shortly after the beginning of expansion.

LQC is a successful effective theory that finally does what people have been wanting a quantum geometrical theory of gravity to do: get rid of the BB and BH singularities.

Moreover it is physically rich and exactly solvable. One can expect PERTURBATION around the LQC core solution to capture information about the full LQG.
(LQC is a simplified, symmetry-reduced version of the full LQG. Present work focuses on removing the simplifying uniformities---weakening the assumptions of homogeneity and isotropy)

Moreover LQC is an area where people are running a lot of computer simulations of the universe. They are running many different cases, corresponding to what cosmologists have as their different possibilities, different sizes, different values for the dark energy or cosmological constant, different matter fields. (there are several Ashtekar et al papers about this, with graphic illustrations of the results, see references in his "Introduction" paper.)

So EFFECTIVE LQC does what people want: it goes smoothly back before the BB former singularity and probes earlier time, it has an excellent largescale limit, it has coherent semiclassical states corresponding to what cosmologists want, it is solvable, variations of it are being studied numerically by computer simulation, and it appears that one can do perturbation theory around it to get a handle on the full LQG

However the recent work of Asht. Thiem. Bojo. is not based on spin networks of Roger Penrose, but on something different. You mentioned spin networks.
They are nifty and good to study and learn about, I guess. But maybe they are not the fastest way to get to the front edge of what the LQG-community is working on. That has changed radically in the past 2 or 3 years. It's a moving target (fortunately or unfortunately )

 

[jal]

I just googled ... quantum geometrical ...for images ... from a phrase by marcus ... just to see what I would get.
Lots of stuff ... use a lot of salt.
jal

 

[marcus]

I just googled ... quantum geometrical ...for images ... from a phrase by marcus ... just to see what I would get.
Lots of stuff ... use a lot of salt.
jal

that's interesting, do you want to share links to any of the prettier ones?

I never image-googled quantum geometry---it is hard to think that you'd get anything with significance for LQG-and-related research.

 

[jal]

Hi marcus!
No, not really. It depends on what is already known and understood by the seeker.

 

[marcus]

Well I don't know the imagery you are talking about so probably should not comment further. But I think your warning to "use a lot of salt" is probably very good advice!

 

[Coin]

Thanks for the references, I'll look through these first.

 

[marcus]

Thanks for the references, I'll look through these first.

I'm afraid they will turn out to be too technical. But at least you can read the first page and the "conclusions" paragraphs at the end, to get an overview.

Sometimes it is good to at least know what recent research is about and what papers exist---even if not following at level of details.

you can also just keep asking questions here

I don't know of adequate popularizations of the more exciting recent research, that go beyond what Smolin has in his book. There is a communication bottleneck and it's a serious problem!

==EDIT==
BTW Coin, I had serious second thoughts about my response. Spin networks are very nice for defining quantum states of geometry .
Maybe I shouldn't say anything to discourage your learning all you can about them. It used to be that the best way into the subject was the Primer paper by Rovelli and Upadya circa 1998.
Maybe that is still the best introduction! Only you may anticipate that the fundamentals change somewhat later on.

 

[ccdantas]

Dear Coin,

The path I've been more or less following is the one I've posted sometime ago in my old blog. Here's a copy:

http://www.geocities.com/christinedantas/basic-curriculum-for-quantum-gravity.html

In particular, read the book by Baez and Muniain, Gauge fields, knots and gravity (excellent book to start, BTW), then go to Frankel and then Nakahara. In parallel, keep on reading the research literature. You may start with LQC, as mentioned by Marcus, see gr-qc/0702030 as a very good starting point.

Good luck!
Christine

 

[jal]

I found something that might help.
http://www.xs4all.nl/~westy31/Geometry/Geometry.html
Riemann surfaces. Complex functions are a bit hard to visualize, because we have 2 components that are a function of 2 components, so we need 4 dimensions. We will use color as a visualisation aid. Suppose we color-code complex numbers, using brightness as a measure of magnitude, and the red/green/blue ratio as a phase.

Here is the one that I liked enough to copy from that site.
jal
http://www.geocities.com/j_jall/riem2.jpg

 

[marcus]

I, for one, was distracted by other things and didn't respond very adequately.

Coin, are you still around?
still in a mood to ask questions?
here is a basic question from me, you mentioned "understand at least SU(2)"
are you familiar with the math notion of "double cover" like in an undergrad course in complex variable you might use the squareroot function to construct double cover map onto the complex plane.
either way is fine, it is an easy basic idea so if you don't know you can quickly learn it----and if you know, so much the better!

so then are you familiar with the idea of SU(2) being a double cover of SO(3)?
again it's fine either way, if yes so much the better, if no then going to Wikipedia or Mathworld and learning it will be easy because very basic

maybe someone else has different and better advice? please contribute it.

then you said you want to understand spin networks. One way to think of a spinnet is simple way to "read" the geometry connecting N points. You have N points living in a manifold and think of all possible "connections" defined on that manifold---critters that tell how space rolls as you move along any given path.
You need a way to test-drive or read connections. Two points and a path between them can get you SOME information. You can use that path to try out all the connections and see what they do.

Its a way of feeling the connections---and since a connection rather much defines a geometry, it is a way of feeling out the geometry of the manifolod.

But instead of just two points and a path between them, you allow yourself N points and an arbitrary bunch of paths between them---a spinnet.

That then becomes a great little machine for feeling out geometries, or test-driving or reading them.

You can make it more numerical and quantitative by labeling each path with a REPRESENTATION of the rotation group (or cover of same) because when you go along the path the connection you are reading will give you a rotation that occurs while you do that----and a representation turns that group element into a definite matrix. labeling paths brings the spinnet one step closer to giving out definite numbers---so when it is confronted with a geometry in the form of a connection it can be persuaded to crank out some definite number.

that is one way to think of spinnets. I hope it is not too disappointing or inadequate. It is NOT THE ONLY WAY by far, but it is one window on them.
They also turn out to be candidates for "quantum state of geometry".
The spinnet can itself describe a state of geometry. Also there is a trend toward infinite spinnet, and omitting having a continuum manifold for the spinnet to be embedded in---so there are no more "connections" in the picture---and having the spinnet just BE space. This is radical and like all radical steps it is risky and can lead to unexpected consequences.

So I just gave you a kind of classical minimalist way to think about them for starters.
As a dirt-simple way to EVALUATE connections. (The connection was invented by Elie Cartan in the 1920s IIRC, it is a very nice old classical differential geometry object, as basic as the metric itself just different.)
If you think about devising a system to evaluate Cartan connections and force something like numbers out of them I think you will re-invent spinnets.

Hopefully Wm Donnelly or Christine Dantas will amend or correct what I've said.

Above all, Coin, if you are still around, keep asking questions
Only way to make anything happen.

 

[marcus]

I'll fetch a couple of links in case Coin is interested, here is the 1998
Rovelli Upadhya "Primer" (primer means intro)

http://arxiv.org/abs/gr-qc/9806079
Loop quantum gravity and quanta of space: a primer
Carlo Rovelli, Peush Upadhya
(Submitted on 19 Jun 1998)

Abstract: "We present a straightforward and self-contained introduction to the basics of the loop approach to quantum gravity, and a derivation of what is arguably its key result, namely the spectral analysis of the area operator. We also discuss the arguments supporting the physical prediction following this result: that physical geometrical quantities are quantized in a non-trivial, computable, fashion. These results are not new; we present them here in a simple form that avoids the many non-essential complications of the first derivations."

for many years this short paper was about the best way to start learning background independent QG

there may be other options now, maybe someone else can recommend something
===============
and then there's this recent paper called INTRODUCTION by Ashtekar, aimed at new researchers.
I don't know how this paper actually works as an introduction---it is new and untried. But since it was his intent to provide
an entry level thing to read for beginners, maybe it would be something to try

http://arxiv.org/abs/gr-qc/0702030
An Introduction to Loop Quantum Gravity Through Cosmology
Authors: Abhay Ashtekar
(Submitted on 5 Feb 2007)
20 pages, 4 figures, Introductory Review

Abstract: "This introductory review is addressed to beginning researchers. Some of the distinguishing features of loop quantum gravity are illustrated through loop quantum cosmology of FRW models. In particular, these examples illustrate: i) how `emergent time' can arise; ii) how the technical issue of solving the Hamiltonian constraint and constructing the \emph{physical} sector of the theory can be handled; iii) how questions central to the Planck scale physics can be answered using such a framework; and, iv) how quantum geometry effects can dramatically change physics near singularities and yet naturally turn themselves off and reproduce classical general relativity when space-time curvature is significantly weaker than the Planck scale."