Understanding SU(2) and SO(3) Representations

[Silviu]

Hello! I am reading some representation theory and I am a bit confused about some stuff. I read that SU(2) is the double covering of SO(3), so to each matrix in SO(3) corresponds one in SU(2). I am not sure I understand this. So if we have a 3D representation of SU(2), the 3D object it acts on are complex vectors, while the objects SO(3) acts on are real vectors. At the same time the 3x3 matrices of SO(3) are real while the 3x3 matrices of the 3D representation of SU(2) are complex. How do we make the connection between the 2? Does this means that if I want to rotate a real vector, I have 2 complex matrices that can be transformed both into a real matrix to do the transformation? Any help would be greatly appreciated. Thank you!

 

[fresh_42]

Hello! I am reading some representation theory and I am a bit confused about some stuff. I read that SU(2) is the double covering of SO(3), so to each matrix in SO(3) corresponds one in SU(2). I am not sure I understand this. So if we have a 3D representation of SU(2), the 3D object it acts on are complex vectors, while the objects SO(3) acts on are real vectors. At the same time the 3x3 matrices of SO(3) are real while the 3x3 matrices of the 3D representation of SU(2) are complex. How do we make the connection between the 2? Does this means that if I want to rotate a real vector, I have 2 complex matrices that can be transformed both into a real matrix to do the transformation? Any help would be greatly appreciated. Thank you!

The double cover has at first nothing to do with the action of the group. It's a group theoretical tool, a decomposition. The action, here the natural operation as matrix groups, is simply a certain representation to write the group multiplication and to interpret them geometrically. I don't see the need to combine the two. You can find a lot of explicit homomorphims here:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/ (see eq.(6)).

 

[Silviu]

The double cover has at first nothing to do with the action of the group. It's a group theoretical tool, a decomposition. The action, here the natural operation as matrix groups, is simply a certain representation to write the group multiplication and to interpret them geometrically. I don't see the need to combine the two. You can find a lot of explicit homomorphims here:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/ (see eq.(6)).

Thank you for you reply. What confuses me is the 3D of SU(2) (or in general when SU(2) and SO(3) have the same matrix dimension). SU(2) and SO(3) have the same Lie Algebra does this mean that the 3x3 representations of the 2 are equivalent? I guess not as there is a 2-to-1 correspondence between the 2, while the equivalence implies a 1-to-1 (is this correct?). And if so, how can I get the 3x3 matrices of SO(3) out of the 3x3 matrices of SU(2) and what do the remained ones represent. As I mentioned above, if I want to rotate a real vector by an angle, i.e. I need a 3x3 SO(3) matrix, which are the 2 3x3 matrices of SU(2) that can do the rotation. I read several stuff in abstract form, but I haven't found a given example of this, so an example would definitely help me. For example, for the rotation matrix around the z axis, which looks like $e^{i\theta J_z}$ in the SO(3) case, which are the 2 3x3 matrices of SU(2) able to do this rotation, which is their form and how can I obtain them? Thank you!

 

[fresh_42]

Thank you for you reply. What confuses me is the 3D of SU(2) (or in general when SU(2) and SO(3) have the same matrix dimension). SU(2) and SO(3) have the same Lie Algebra does this mean that the 3x3 representations of the 2 are equivalent? I guess not as there is a 2-to-1 correspondence between the 2, while the equivalence implies a 1-to-1 (is this correct?). And if so, how can I get the 3x3 matrices of SO(3) out of the 3x3 matrices of SU(2) and what do the remained ones represent. As I mentioned above, if I want to rotate a real vector by an angle, i.e. I need a 3x3 SO(3) matrix, which are the 2 3x3 matrices of SU(2) that can do the rotation. I read several stuff in abstract form, but I haven't found a given example of this, so an example would definitely help me. For example, for the rotation matrix around the z axis, which looks like $e^{i\theta J_z}$ in the SO(3) case, which are the 2 3x3 matrices of SU(2) able to do this rotation, which is their form and how can I obtain them? Thank you!

That's why I quoted the link. Look up the homomorphisms. Don't confuse complex and real dimensions here! That's why I placed the scalar field at the notations of the groups in the article. This has also to be considered and noted for the representations. As said, the article contains a lot of different presentations and representations. Maybe you should do some calculations on the certain examples. Isomorphic Lie algebras have isomorphic representations.

 

[stevendaryl]

The relationship between them that I've seen is this: If you have a vector $(x,y,z)$, you can associate a corresponding 2-element complex column matrix $\left( \begin{array} \\ \alpha \\ \beta \end{array} \right)$ via:

$x = \frac{1}{2} (\alpha \beta^* + \alpha^* \beta)$
$y = \frac{i}{2} (\alpha \beta^* - \alpha^* \beta)$
$z = \frac{1}{2} (|\alpha|^2 - |\beta|^2)$

or the inverse relations:

$\alpha = \frac{x-iy}{\sqrt{r-z}} = \sqrt{2r} cos(\frac{\theta}{2}) e^{-i\frac{\phi}{2}}$
$\beta = \frac{x+iy}{\sqrt{r+z}} = \sqrt{2r} sin(\frac{\theta}{2}) e^{+i\frac{\phi}{2}}$

where $r, \theta, \phi$ are the polar coordinates: $r = \sqrt{x^2 + y^2 + z^2}$, $\theta = tan^{-1}(\frac{\sqrt{x^2+y^2}}{z})$, $\phi = tan^{-1}{\frac{y}{x}}$

This is a double-cover because $\alpha, \beta$ corresponds to the same vector $(x,y,z)$ as does $(-\alpha, -\beta)$. If you increase $\phi$ by $2 \pi$, you get the same $(x,y,z)$, but $\alpha \rightarrow -\alpha$ and $\beta \rightarrow -\beta$