Lethe asked for volunteers (rep theory)

[marcus]

Lethe posted this in the group theory thread:

" you know, this thread is turning into a pretty nice lie group/lie algebra thread. there is the differential forms thread. now all we need if for someone to start a representation theory thread, and we ll have all the maths we need to do modern particle physics.

who wants to volunteer?"

 

[marcus]

what should be an easy question

I had a question in that area
that I posted in "theoretical" forum:

"Let φ SU(2) ---> SO(3) be the double covering

Any irreducible representation of SO(3) pulls back by φ
to provide an irreducible representation of SU(2) on the
same finite dimensional Hilbert space.

this seems clear, almost not worth saying:
the pullback is obviously irred. and has the same dimension.

I have a question about the other direction----suppose an
irred. rep. of SU(2) factors thru φ

then it obviously gives a representation of SO(3), same space
same dimension and all, but

under what circumstances is the representation irreducible? "

this may be a dumb question for all I know.

BTW to me, at the Lie algebra level, they look the same
To me at least, so(3) looks like su(2).
I have the feeling that the representation of SO(3)
that you get ought to be irreducible under fairly general
circumstances, but this should be part of time-honored lore
that other people here have. So I ask.

I have a strong preference for online textbooks not only
because I can get them free and quickly but also because
we all share online stuff and can be "on the same page" easily
I would like to know an online "rep theory for dummies" textbook

 

[lethe]

well we should start with some prerequisites.

we need just a little bit of group theory: definition of a group, subgroup, group homomorphism, group isomorphism, normal subgroups too, maybe? but perhaps this is already in/should go in the group theory for dummies thread.

then a little bit of linear algebra and matrix groups. what is an eigenvalue, eigenvector, what are SU(n), SO(n), GL(n) etc. determinants, transposes, hermitian adjoints. inner product spaces. that ought to do it? maybe all our linear algebra needs (e.g. for this thread and for the differential forms thread) should be consolidated into on intro to linear algebra thread? what do you think?

if you ve already got the prerequisites, then you re ready to see the definition of a representation, which should be the natural starting point, eh?


A representation ρ of a group is a group homomorphism from the the group G to GL(V), the group of linear transformations of a vector space V. this means that

ρ(gh)=ρ(g)ρ(h)

so ρ(g) and ρ(h) are vector space isomorphisms of V. the dimension of V is the dimension of the representation. if ρ is an isomorphism (that is, one to one and onto), then we call it a faithful representation.

[/size]

 

[lethe]

perhaps i am being a bit hasty. we should agree on notations, starting points, etc. before we jump right in, eh?

i m going to think about your question marcus. my knee-jerk response is that i think it will always be irreducble. but i haven t thought it through.

but for right now, it s friday night, and i have a need to get sloshy, so that s all you ll get from me tonight.

 

[marcus]

Originally posted by lethe
perhaps i am being a bit hasty. we should agree on notations, starting points, etc. before we jump right in, eh?

i m going to think about your question marcus. my knee-jerk response is that i think it will always be irreducble. but i haven t thought it through.

but for right now, it s friday night, and i have a need to get sloshy, so that s all you ll get from me tonight.

this is the best kind of reaction!
you share your intuition!
I also have the feeling that it should always be irreducible
(and also a terrible sense of my own naivete, because
in a good subject like this there must be surprises)
get good and sloshy
perhaps will see you tomorrow

 

[marcus]

Originally posted by lethe
perhaps i am being a bit hasty. we should agree on notations, starting points, etc. before we jump right in, eh?

i m going to think about your question marcus. my knee-jerk response is that i think it will always be irreducble. but i haven t thought it through.

but for right now, it s friday night, and i have a need to get sloshy, so that s all you ll get from me tonight.

I got an answer to that question!

The induced rep on SO(3) is ALWAYS irreducible.

I will get back in a minute and post in more detail, but
you for sure know what I mean since this was your intuition
about what happened (and mine too) it was almost self-evident
but needed a little time to be sure about.

 

[marcus]

a useful theorem

I found a useful theorem in Brian Hall's book about groups and reps:

Let G be a connected matrix Lie group with Lie algebra {g}

I can't type the usual GOTHIC style lowercase letter g, so just say {g}.

Let Π be a representation of G acting on a space V, and let π be the associated Lie algebra representation. Then a subspace W of V is invariant for π if and only if it is invariant for Π.
And in particular, π is irreducible if and only if Π is irreducible.

*****

I think we both had the sense of that in mind, tip of the tongue, but I at least wanted reassurance. It is not a deep or hard theorem---he makes the proof a homework problem---but summarizes one of those things which tho obvious should IMHO always be remembered. What a nice subject this is!

 

[marcus]

the trouble with unicodes on PF

Unicodes just produce squares for some people

When I want to type a psi, I say & psi;

without the space, and that makes ψ

For integral sign:

& int;

∫

When you use those number codes ("unicodes"?)
my browser just writes a box
Maybe a mentor will appear and tell me how to change
the settings. But there is no FAQ that I could find here at PF
about it. And probably there are some other people in the
same boat.

Greg has a sticky about symbols at the top of most forums.
Would prefer if we could just use that vocabulary
so no special numbers need be memorized and so on,
but also could adapt if told how to do it.

 

[Hurkyl]

The online Lie Group text I've been using

"An Elementary Introduction to Groups and Representations"

is at arXiv:math-ph/0005032


However, its main focus is on the matrix lie groups/algebras and not on the general case. I'm not sure how much of a drawback that would be.

 

[marcus]

Originally posted by Hurkyl
The online Lie Group text I've been using

"An Elementary Introduction to Groups and Representations"

is at arXiv:math-ph/0005032


However, its main focus is on the matrix lie groups/algebras and not on the general case. I'm not sure how much of a drawback that would be.

That is Brian Hall's online textbook! I like it a lot.
If there is some sort of ongoing discussion of people
helping each other learn groups and representations
that would be an excellent choice to give something to focus
on.

It is 128 pages, takes a while to print off. But whole book seems
valuable.

Still worried that my browser doesn't see "unicodes" if that is the word for it. In Lethe's post I see some little squares.

 

[lethe]

Originally posted by marcus


When you use those number codes ("unicodes"?)
my browser just writes a box
Maybe a mentor will appear and tell me how to change
the settings. But there is no FAQ that I could find here at PF
about it. And probably there are some other people in the
same boat.

Greg has a sticky about symbols at the top of most forums.
Would prefer if we could just use that vocabulary
so no special numbers need be memorized and so on,
but also could adapt if told how to do it.
[...]

Still worried that my browser doesn't see "unicodes" if that is the word for it. In Lethe's post I see some little squares.

i don t think those codes are unicode. they are ancient html codes. using the token (e.g. psi) and using the number should give the exact same results. so there is no need to memorize numbers.

if your browser can t render the symbol, it will just make a box. which symbols are you unable to see?

a more complete list than what is in the sticky thread can be found here. go there and see how your browser stacks up. i would like to continue using those codes, but if it is a problem, we could look at other solutions (one solution might be for you to use a http://www.mozilla.org/projects/firebird/ !).

how does your browser render the differential forms thread? i used a lot of symbols in that one.

 

[marcus]

I see you are suggesting I switch to Mozilla Firebird.
You asked about the diff forms thread----I don't see any
boxes there! Very strange.
Am used to this browser and tend to be reluctant
to change things about the computer.
I have no trouble reading other people's posts on PF
probably because they use the symbols listed on the sticky.
for some reason those symbols work for my browser.
Dont know if other people on PF are in the same boat
Am wondering how this will work out.

There may be a setting or a preference I can change
on my browser.

Or some other kind of code I can type to get symbols.
Greg may have suggestions.

Originally posted by lethe

how does your browser render the differential forms thread? i used a lot of symbols in that one.

 

[jeff]

Group theory really is at the heart of physics and is needed to describe the relations between symmetry and conservation laws, the geometry of spacetime, and the classification of particles along with their interactions.

When you guys have learned enough group theory, I'd like to help out teaching how all that mathematical technology is used in high energy and elementary particle theory.

I can discuss quantum field theory, string theory, LQG, or virtually any other theory people here want to learn about and in as much depth as they can stand.

Who else around here wants to help out with this stuff?

 

[bhall]

Lie groups text

Originally posted by Hurkyl
The online Lie Group text I've been using

"An Elementary Introduction to Groups and Representations"

is at arXiv:math-ph/0005032

--

I (Brian Hall) am the author of this set of notes, and the version above is freely available on the web. I just want to mention that a substantially expanded and improved version has now appeared in book form, published by Springer. For more information, including several free sample chapters, see:

http://www.springer-ny.com/detail.tpl?isbn=0387401229

or my home page (currently under construction, but should be up soon) at

http://www.nd.edu/~bhall

 

[chroot]

Originally posted by bhall
I (Brian Hall) am the author of this set of notes, and the version above is freely available on the web.

Hi Dr. Hall,

Welcome to physicsforums! Thank you for the update, your book looks very informative indeed.

- Warren

 

[selfAdjoint]

rebirth anybody?

Hey let's get the thread going again. Now we have Latex to talk about instead of HTML codes, and two choices for a common text (I vote for the online one). Let's dig into it!

 

[Lonewolf]

I'm in once my workload diminishes (about a week's time from now).

 

[marcus]

here's an area of confusion I have---relating to reps

in highbrow papers something like this is
sometimes said in an offhand manner (presuming that everyone
understands):

"quantizing a classical theory is nothing but finding
representations in hilbertspace of the algebra of observables"

well they never say it quite that simply, but I get the impression
that some people EQUATE quantum theories with hilbertspace
representations of algebras, or groups since groups and algebras are associated so closely

can someone clarify? to what extent or under what limitations are quantum theories subsumed under the heading of representations?

if one takes that attitude, what does having the right classical limit mean--and what role does the inner product of the hilbertspace play?

(perhaps I know some of this but it might be good to get the relation between QM and representation theory established at the getgo)

 

[lethe]

Originally posted by marcus
well they never say it quite that simply, but I get the impression
that some people EQUATE quantum theories with hilbertspace
representations of algebras, or groups since groups and algebras are associated so closely

yeah, i do, for example.

can someone clarify? to what extent or under what limitations are quantum theories subsumed under the heading of representations?

well, i would say that the act of creating a quantum theory is completely subsumed under the idea of finding a representation on a Hilbert space of the classical Poisson algebra.

if one takes that attitude, what does having the right classical limit mean--and what role does the inner product of the hilbertspace play?

i think the inner product on the Hilbert space contains the symplectic form of the classical theory

 

[rick1138]

Group theory really is at the heart of physics and is needed to describe the relations between symmetry and conservation laws, the geometry of spacetime, and the classification of particles along with their interactions.

When you guys have learned enough group theory, I'd like to help out teaching how all that mathematical technology is used in high energy and elementary particle theory.

I can discuss quantum field theory, string theory, LQG, or virtually any other theory people here want to learn about and in as much depth as they can stand.

I'm in. The questions I have are about how the decomposition of a continuous group or algebra into its irreducible representations is related to the physical structures seen in particle physics. I've been reading a lot on representation theory and Lie groups and algebras, learning to multiply reps using Young tableaux. - The book sitting in front of me right now is Georgi's Lie Algebras in Particle Physics. I was wondering if anyone thought it was possible to use genetic algorithms to breed a rep that would solve the puzzle of the universe.

 

[mathwonk]

in response to posts 3,4,5,6 it seems it is always irreducible. I am already sloshy but the point would seem to be that the induced rep pulls back to the originalk rep so if the induced rep is reducible so is the pullback?

 

[mathwonk]

This very appealing thread seems to have died after all the initial good intentions. I have just downloaded Brian's book and read the first two chapters.

Anybody want to suggest what I should have come away with? There seem to be some beautiful continuous groups out there, and for group homomorphisms, amazingly continuity implies differentiability, although this is not proved. Is that the main idea? (Is that Hilbert's 5th problem?)

The book seems to begin in earnest only in chapter 5 however.

Maybe we could try the homework problems, but likely I will not have time in an other week.