# Groups in Quantum Mechanics.

1. Mar 5, 2008

### Gigi

I read that the generator of the O(3) group is the angular momentum L and that the generator of the SU(2) group is spin S.

Nevertheless I have some questions.

1. In some books they say that the generator of the SO(3) group is angular momentum L.
SO(3) is the group of proper rotations, i.e. det(Matrix)=1.

Thus is it O(3) or SO(3)?

2. Both O(3) and SO(3) are defined as rotations in Eucledian space, 3-dimensions. Thus I would expect that we are talking about classic angular momentum.

Nevertheless in a quantum mechanics book I read that S0(3) is the generator of the angular momentum operator.
How is that if in Quantum Mechanics we are using the Hilbert space that is a complex function space?

3. I have the similar question regarding SU(2). SU(2) is defined in complex space. Thus it is ok to say that it is more or less the same as saying that this complex space is the Hilbert space?

4. Now in relativistic quantum mechanics, the underlying group is the Lorentz group. Would that mean that the O(3) and SU(2) groups reflect only symmetries in the non-relativistic world? i.e. Schroedinger's equation?

If that is so, how come spin that is a relativistic effect is explained using the SU(2) group?

Many thanks, as I am getting quite confused.
How is that?