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## Main Question or Discussion Point

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:

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 [Broken], 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:

...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. :)

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 [Broken], 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:

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: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],

- 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. :)

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