# How to probe the group SU(2) is simply connected?

1. Jun 23, 2010

### wdlang

why group SO(3) is not

any good reference on the relation of SU(2) and SO(3)?

2. Jun 23, 2010

### Fredrik

Staff Emeritus
You can prove that SU(2) is homeomorphic to a 3-sphere, like this:
The relationship between SO(3) and SU(2) can be found by first noting that $$\mathbb R^3$$ is isomorphic to the 3-dimensional real vector space of complex 2×2 traceless self-adjoint matrices, and then showing that if X is a member of that space, and U is a member of SU(2), then

$$X\mapsto UXU^\dagger$$

is a proper rotation, i.e. a member of SO(3). Since you can change the sign of the U without changing the result, there are two members of SU(2) for each member of SO(3).
Not sure what the best reference is if you don't want to figure out the details for yourself. I think Weinberg's QFT book covers this pretty well (vol. 1, chapter 2), but he's actually doing it to find the relationship between SO(3,1) and SL(2,C), so he's doing essentially the same thing with the "traceless" condition dropped, and U not necessarily unitary. This brings a fourth basis vector into the picture: the 2×2 identity matrix.

3. Jun 23, 2010

### wdlang

yes, thanks a lot