Question about the use of group theory in QM

[AlexChandler]

I am currently in my second undergraduate quantum course and just finished studying the addition of angular momenta. I am also in my third abstract algebra course and am now covering product groups and group actions. In my QM book (griffiths) there was a reference made to group theory. it said " what we are talking about is the decomposition of the direct product of two irreducible representations of the rotation group into a direct sum of irreducible representations" I am familiar with direct products and sums and representations and read a bit about the rotation group on wikipedia, but am still not really understanding what groups we are talking about. for example, for a spin 3 particle and a spin 3/2 particle, we would have

3X3/2=3/2+6/2+9/2

the left hand side representing the spins of the particles and the right side representing the total spin of the system. Now from my understanding, direct products and sums are done on groups, not numbers. So I am guessing that each number here represents a subgroup of the group of rotations SO3. Would this be somewhat accurate? If yes then what subgroups are they, and if no then what groups are they? or am i completely mistaken as to what this notation means? Thanks!

 

[Ben Niehoff]

The numbers denote representations.

For example, the fundamental rep of SO(3) is 3-dimensional, so it is denoted 3. If you combine two spin-1 particles (i.e., particles in the fundamental rep of SO(3)), then you get

$$3 \otimes 3 = 1 \oplus 3 \oplus 5$$
what this means is instead of a 3-component column vector, you now have a 3x3 matrix. It can be decomposed into three parts:

1. The trace, which transforms under the trivial 1 rep of SO(3),
2. The antisymmetric part, which transforms under the 3 rep of SO(3), and
3. The symmetric, traceless part, which transforms under the 5 rep of SO(3) (count that there are 5 components left to make up this part).

In this notation, half-integer spins are even numbers (technically, half-integer spins are reps of SU(2), but SU(2) and SO(3) have the same Lie algebra). So, for example, if we have two spin-1/2 particles, we can combine them via

$$2 \otimes 2 = 1 \oplus 3.$$
That is, two spin-1/2 particles combine to make a singlet (spin-0) state, and a triplet (spin-1) state.

The example you gave, combining spin-3 and spin-3/2, would go like

$$7 \otimes 4 = 4 \oplus 6 \oplus 8 \oplus 10.$$
That is, the combination of spin-3 and spin-3/2 would decompose into the irreps spin-3/2, spin-5/2, spin-7/2, and spin-9/2. (Note that you can't get spin-6/2, because the product of a boson and a fermion is still a fermion).

These things are easy to calculate in SU(2), because all irreps are built out of symmetric tensor products of the fundamental rep. The trivial 1 rep has no spinor index; the 2 rep has 1 spinor index, the 3 rep has 2 spinor indices, and so on. For the example above, spin-3 is the 7 rep, which has 6 spinor indices, and spin-3/2 is the 4 rep with 3 spinor indices. Their product looks like

$$G_{abcdef} \psi_{ghi}.$$
Now you just see which indices you can trace over, and just write down all the possibilties. Since both tensors are fully symmetric, this is easy.

It looks like your quantum book is using a different notation where they write down the weight of the representation rather than its dimension. I think it's easier using the dimension.

 

[AlexChandler]

Thanks so much for your reply! I don't really know anything about Lie algebra yet, but this definitely helps me understand a bit of what I need to learn in order to pursue the topic further. Could you possibly recommend a good book or books that I could buy that would cover these types of things? (tensors, lie algebra, and representations)

 

[Ben Niehoff]

Unfortunately, I don't have the slightest idea what book to recommend.

 

[dextercioby]

Try Wu Ki Tung's group theory text (World Scientific, 1984). It explains the basics pretty well without delving into complicated mathematics.

 

[martinbn]

You can also take a look at "Groups and Symmetries. From Finite Groups to Lie Groups." by Yvette Kosmann-Schwarzbach.

It is very introductory and accessible. Also in the Lie theory part follows the physics notations.

 

[vanhees71]

A classic is

H. Lipkin, Lie groups for pedestrians

 

[AlexChandler]

A classic is

H. Lipkin, Lie groups for pedestrians

Actually I just bought this book. It is very good, however I feel it is a bit above my current level in my understanding of quantum mechanics.