# What is the physical implication of a homomorphism between SU(2) and SO(3)?

It can be shown that there exists a homomorphism between $SU(2)$ and $SO(3)$. What is the physical implication of a homomorphism actually? I know that in physics $SU(2)$ acts on a space of spin 1/2 particle whereas $SO(3)$ is a group of rotations in Euclidean space. Now that there is a homomorphism between the two groups, what does it imply?

Moreover, my text only present the case of homomorphism between $SU(2)$ and $SO(3)$, but what about a general $SU(n)$ with $SO(3)$? Is there also homomorphism between them?

• – Qmechanic May 25 '16 at 14:12
• You should specify a theory related to $SU(2)$ or $SO(3)$ you are interested in. Also, there is no genereral homomorphism between $SU(n)$ and $SO(3)$ or $SU(n)$ and $SO(n)$. A homomorphism (with implications to physics) would be one between $Spin(n)$ and $SO(n)$. – Diracology May 25 '16 at 14:24
• There is also analogous homomorphism between $SL(2,C)$ and Lorentzian group. – Blazej May 25 '16 at 14:25
• @Blazej, yes I know that but I am more interested in the application in quantum mechanics. – nougako May 26 '16 at 2:17
• @Diracology, what do you mean by the theory? I am currently considering the 2D space spanned by spin up and spin down of an electron. – nougako May 26 '16 at 2:18