# Questions aboug Special Groups SO(n) and SU(n)

1. Apr 13, 2006

### Plott029

Dear Friends,

I have many questions about the special Orthogonal Group SO(n) and the Special Unitary Group SU(n). The first, SO(n) has $$\frac {n (n-1)}{n}$$ parameters or degrees of freedom, and the second, SU(n) has $$n^2 -1$$.

If I take for example the group SO(3), this has 3 degrees of freedom, and SU(2) has too 3 degrees of freedom, and theres a relation 2->1 from SO to SU. The question is, if SU(n) can describe the same that SO(n) or if there's a lose of information in using SU(n).

Best Reggards.

2. Apr 13, 2006

### HallsofIvy

Staff Emeritus
Excuse me? Didn't you just say that SO(3) has$\frac{3(3-1)}{3}= 2$ degrees of freedom? And that SU(3) has 32- 1= 8 degrees of freedom?

3. Apr 14, 2006

### Plott029

There's a mistake. SO(n) has [n(n-1)/2] and SU(n) [n2-1], thus, S(3) has 3 degrees of freedom and SU(2) has 3 degrees of freedom. The question is if SU(2) has the same information that SO(3).

Best reggards.

Last edited: Apr 14, 2006
4. Apr 14, 2006

### matt grime

What on earth does 'information' mean?

5. Apr 15, 2006

### HallsofIvy

Staff Emeritus
Ah, I misread SU(2) as SU(3)! SU(2) and SO(3) both have dimension (degrees of freedom) 3(3-1)/2= 22- 1= 3. That does not mean that the are isomorphic in which "knowing one tells us everything about the other". There exist a surjective homomorphism from SU(3) to SO(2) with kernel {I, -I}.

6. Apr 17, 2006

### Plott029

:D

Usually, in Quantum Mechanics, SO(3) and SU(2) are utilized to describe, first, rotations in space, and SU(2) to describe and calculate rotations on an abstract space. My question is on about the description that SO(3) and SU(2) can do on earth... if SU(2) has "less information" or is less descriptive than SO(3) or are isomorphic. And this leaves to 2nd answer...

This surjective homomorphism, means that are equal?

7. Apr 17, 2006

### matt grime

Until you can accurately state what the 'information content' is in mathematical terms then we really can't help.

And equal is not the same as isomorphic. No SU group is equal to any SO group. One is a complex lie group the other is a real lie group. They are never equal, though some of them might be isomorphic (as groups).