why group SO(3) is not
any good reference on the relation of SU(2) and SO(3)?
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 [tex]\mathbb R^3[/tex] 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
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.
yes, thanks a lot
your explanation is very good
though the argument is presented in many books, your interpretation tells the inside
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