OK I’m trying to understand the actual
procedure here.
Garrett may have given me much (in combination with what I had already) of what I need here
We can rotate the coordinate axes of the root system however we wish, describing the same algebra. This just corresponds to a different choice of basis elements for the same Lie algebra -- still E8. I think rntsai has done a good job of explaining this in his previous post. He (or she?) is also correct that the roots alone aren't enough to tell you which Cartan subalgebra particle/field we get when two roots add to give one at the origin. To describe this, we would have to work in a specific representation, or at least write down these structure constants.
First you pick some simpler subalgebra of E8 – say G2 or F4. This will make the problem vastly more tractable. You can ‘see’ things more clearly and the operations will be simpler and/or faster. Next you need to find a basis within that subalgebra where two of the vectors will use the one-and-only Lie algebra operator – the bracket or commutator – and give a result back at the origin. The origin, in 3D, being x=0,y=0,z=0. What does this mean? Linear algebra I and the dot product (so here we’re talking second yr undergraduate mathematics [imo]). The dot product (in 2D – I guess it’s the wedge product in higher dimensions) will act upon two vectors and give a result of 0 when the two vectors are
perpendicular. Perpendicular is the important part. So what does it mean ‘in the real world’ that the vectors are perpendicular? Well that depends. We’ll leave it as an exercise of the reader (but at this point, you
are quite close to ‘the real world’).
Back up. In a (probably special, unitary [except that unitary implies group and not algebra]) Lie subalgebra when the commutator is applied to two (root) vectors) and you end up at 0
you have a particle/field. Yowza! A first big success (you found the right basis). So how do you find the right basis? Random doesn’t seem like a good idea (there are a lot of them). Hunches and intuition, if you’re so provided (meaning a professional, practitioner or very talented amateur), can go a long ways. But better yet, some more systematic way of a) try a new (likely) basis b) compute root vectors and see if any computation lands at the origin. Doing this suggests programming.
Doubtless, there’s much more to be said about finding the right basis. What that is that could be said: I don’t know.
But, as
Garrett said you still have a (probably) very thorny issue before you. You’ve got a particle, but you don’t yet have enough information to figure out which one. Most particles are ‘known’ (outside) the theory, but, according to this theory, there’s a small number (18 I believe) that are not. As regards identifying the hot particle now in your hand, Garrett said:
He (or she?) [rntsai] is also correct that the roots alone aren't enough to tell you which Cartan subalgebra particle/field we get when two roots add to give one at the origin. To describe this, we would have to work in a specific representation, or at least write down these structure constants.
So what procedure is involved here (identifying the particle/field)? With more time, patience (and probably help) I’ll figure it out, but at the moment I don’t know. Although I feel I have, more-or-less, figured out what a representation is. Representation is a keyword and there’s a whole field (of study) corresponding to representation theory. In CS object oriented terms, I’d
like to think: you write an OO class in your text editor. It’s a specification - so far it ‘does’ nothing. Then you run a program (containing that class) and the class, as we say, ‘instantiates’. It’s an object in memory and now it’s actually doing something as a part of the program. Which for many people is still somewhat abstract, but is something I’ve been doing for yrs, so for me it’s intuitive. Anyway the instantiation (or object [running in memory]) is the representation. The analogy breaks down quickly though. For a class there’s basically one instantiation (one might use polymorphism to claim that’s there more than one – it’s certainly the case that you can get different ‘behaviors’ out of the object depending on polymorphism). But there are an infinity of representations for E8. Fortunately though, every member of this infinity can be generated from the basic, unitary representations. There’s an enormous number of these, but that number
is finite.
The explanation of the last procedure (identifying the particle/field that you’ve just computed) may lie somewhere right before our eyes if we look upthread and know how to recognize what we’re looking at. Certainly somewhere out on the web (again you have to know what you’re looking at).
I’m done (for now) but of course ‘the problem’ isn’t. Is it a boson or fermion (well, you’ve sort of determined that already)? More importantly: computing its actions. Again I’m somewhat guessing here, but an example of an action would be how the W and Z bosons combine to produce the weak interaction.
So what was all that? I’m trying to check my understanding of things (so it’s sort of a big, long question).
Berlin told us that he’s using an Excel spreadsheet. Well I know only too well how cumbersome (for me) these become at a point (look up OLAP – extremely cool), so while what I should use popped up as a question in my mind, I think it’s still better that I use the GAP software.
Garrett apparently uses Mathematica. And, it goes
with saying, these are tools. Then there’s understanding. And (according to Einstein) beyond understanding (knowledge) lies imagination.
Sorry too long. A single, specific question. This depends on some small part of the above being correct. Is it always
two roots that combine to produce
one particle/field? But wait I’ve probably gotten something (major) wrong. E8 has 240 symmetries, therefore particle/fields. And so 240 roots also. Well if you could only use each root once, then that would give only 120 particles. Maybe you can use a given root more than once?
slipped just saw Garrett’s latest post as I was writing this up.
Pat,
You belong in New Jersey.
Everyone, obviously, is impressed with your IQ. But what’s also struck me from the time I read your first posts (Backreaction) was your EQ as well. And so I’m puzzled: you might have played the academic politics game quite well. There are secondhand descriptions of academic politics all over the place, but I’ve had the opportunity to watch academic politics (and so the application of EQ in this regard) up close in the form of my physicist brother-in-law who made it onto the tenure track a couple of yrs ago. He’s an astrophysicist (currently one project is the polarization of the CMB) and there are web photos of him somewhere both at ESO, high up in the Chilean mountains, as well as at the South Pole. Oh yes, in your neck of the woods, he's also been to the top of Mauna Kea and the Keck installation (or whatever the whole facility is called).
But as you say yourself: “I didn’t want to do string theory”.
I’m from Boston originally and much of my family has lived in or around New York (but not me). I came to California in 1984 to finish up my undergrad at Berkeley. And have stayed in CA ever since, minus 6 yrs in Tokyo that is. We’re in the Bay Area. I’ve always wondered about this. If there’s a SoCal, do we live in NoCal? I finished at Berkeley Phi Beta Kappa with degrees in Japanese and Computer Science but grad school was sort of foreclosed upon by my having lost several yrs to surviving cancer in my early twenties. So all things considered I’m quite happy to be where I am (as opposed to, say, dead [or in New Jersey]).
pat