Lie Algebra

sanctus · 10-23-2004

At the moment I'm studying for quantum mechanics the symmetries and so on. there we speak a lot about Lie Algebras and groups. My question is do you know any easy way to explain these groups or a book which does it? I mean, I already read some books I found in our bibliotheque and they were quite difficult, but if I looked at them for a quite I while I understood the different passages, but never got the whole picture of it, could never see it as whole.

Thanks to anyone who can help

Bo · 10-25-2004

your questions are covered by a mathematical branche called 'group theory', which is the study of symmetries. and indeed this is a very important subject for (advanced) quantum mechanics. But group theory is quite complicated.

A Lie group is a group whereof the product of group elements is differentiable. The nice property of this is that one can define a set of generators for the symmetries of the group.
An example in physics: a spin 1/2 particle. (no calculations, just the results, still this is not a course in group theory, but i assume sanctus has some knowledge).

spin, is a form angular momentum and satisfies a symmetry called SU(2).
spin 1/2 is a doublet representation of this groups, with elements: (+1/2, -1/2) (that is spin up and down)
now SU(2) is a Lie group, so we can calculate the generators of this group. and the nice thing is that for the doublet case these are exactly the pauli matrices (which you have probably encountered). so the Lie algebra of SU(2) is also the same as that of the pauli matrices. [Sx , Sy] = Sz
So the generators of the symmetry group are the same as the operators of the quantum mechanical property. This is of course extremely powerfull. We dont have to do QM, or explicitly calculate Schrodingers equation or whatever; just knowing the symmetry group contains all information.

I hope this clarifies something, but if you have some specific questions, please ask

Bo

sanctus · 10-26-2004

First of all thanks very much.

I already planned to take the option course on groupe theory, but it will be held only in the next semester (when the QM course is already over).

Actually I have got a specific question, if f you could answer to that it would be just great!
What is the connection between the Lie Algebra and the Lie group. We defined the Lie algebra as the {M¦matrix n x n so that exp (tm) belongs to G for all t belonging to R} where G is the lie group. I really don't see the connection with for example SO(3) or SU(2). I mean usaualla an algebra is a vectorial space with an addition and a multiplication, but where there is a multiplication an an addition in this Lie algebra I've got no idea.

Thanks

Bo · 10-26-2004

following a course on group theory is definitly a good idea if you want to continue in theoreticsl physics. i'm not quite sure if it is that important for technical physics though...

Quote:

We defined the Lie algebra as the {M¦matrix n x n so that exp (tm) belongs to G for all t belonging to R} where G is the lie group

this matrix M is not really the algebra; it is the generator of the group. but it is crucial to the algebra.
an algebra -as you noted- is defined by the rules of multiplication. so the algebra of the lie group is defined by the algebra of the generators and these are the (anti)commutation relations of the matrices. (so e.g. the commutation relations of the pauli-spin matrices for SU(2))

for example: the group SO(2). (rotations in the 2d plane) in the 2d representation.
the symmetry transformation is simply:

(x' ; y' ) = (cos(f), -sin(f) ; sin(f), cos(f)) (x ; y) =R(f) (x; y)

now, since SO(2) is a Lie group, we can define a generator. (there is only 1 generator of SO(2)).
the claim is (which you can prove by taking a taylor expansion) is that the matrix R(f) = EXP(-i*f*I), with I = (0, -i ; i, 0).
The algebra of SO(2) is trivial, since it only has 1 generator. but for completeness: the algebra is given by:
[I, I] = 0 ; {I, I} = 2 I^2

some extra terminology (that you see very often): When the elements of the group (or equivalently the generators) commute, we call the group abelian; otherwise non-abelian. so SO(2) is abelian, SU(2) not.

btw: what kind of quantum course are you following that Lie groups are mentioned, but not explained?
Bo

sanctus · 10-27-2004

thanks very much, it's much clearer now! Just to check that I understood right: the matrix I = (0, -i ; i, 0). is the generator of SO(2) and and it generates SO(2) by taking the exp(-i*f*I) where f is the angle of the rotation in the plane.

Quote:

btw: what kind of quantum course are you following that Lie groups are mentioned, but not explained?

They are just badly explained, because the prof once said she has never had problems in understanding the math, so it's difficult for her to be simple. In addition to that we are supposed to have seen the lie groups and algebras elsewhere, but there already the definition are very different.

Bo · 10-27-2004

Quote:

thanks very much, it's much clearer now! Just to check that I understood right: the matrix I = (0, -i ; i, 0). is the generator of SO(2) and and it generates SO(2) by taking the exp(-i*f*I) where f is the angle of the rotation in the plane.

exactly!

Bo