How Do SU(2) and SO(3) Relate to Spinors and Vectors in Physics?

[Silviu]

Hello! I want to make sure I understand the relation between this and rotation (mainly between SU(2) and SO(3), but also in general). Also, I am a physics major, so I apologize if my statements are not very rigorous, but I want to make sure I understand the basic underlying concepts. So SU(2) is the double cover of SO(3). Also Spin(3) is the double cover of SO(3). So, SU(2) and Spin(3) are isomorphic. Now I am a bit confused about the objects that these groups act on. If I think of SU(2), they act (in the fundamental representation) on 2 dimensional objects, which are called spinors. Now, if I understand it right, when labeling the representations of SU(2) by j (the value of the angular momentum), if j is half integer it is a spinorial representation, while if j is integer it is a vectorial representation. So for j=1, the object acted upon are 3 dimensional vectors, not spinors? But as we are in SU(2), which are complex matrices, the vectors are complex vectors? So the difference between a complex vector and a tensor, are given by the representation to which they belong to? So a 3 dim object which changes under a 3D representation of SU(2) is a complex vector, while a 2 (or 4, 6 etc) dimensional object changing under a 2D (4, 6 ..) representation of SU(2) is a spinor? Or is it anything deeper that this? Now if we go to higher spin, let's say that a k dimensional object changes under the k-dim representation of Spin(n) and another k-dim object under k-dim representation of Spin(m). Do we decide whether they are spinors or not based on whether that representation is spinorial or vectorial? I.e. a k-dim object on its own can't be called a complex vector or a spinor, unless we know how it transforms? Please let me know if what I said is wrong, and how should I think about all these? Thank you!

 

[George Jones]

What is a representation of a group?

 

[Silviu]

What is a representation of a group?

A homomorphism, between G and D(G), where to each element of G is associated an element of D(G)

 

[fresh_42]

A homomorphism, between G and D(G), where to each element of G is associated an element of D(G)

Whatever D(G) should mean. A representation is a homomorphism $\varphi\, : \,G\longrightarrow GL(V)$ with a representation space, a vector space $V$. This can be a tangent space, a Euclidean space in case $G$ is a matrix group and many other example. $V$ doesn't even have to be of a certain dimension, or even finite dimensional. So whenever you say representation, you should define $\varphi$ in a way, because there are really many possible representations.

 

[Silviu]

Whatever D(G) should mean. A representation is a homomorphism $\varphi\, : \,G\longrightarrow GL(V)$ with a representation space, a vector space $V$. This can be a tangent space, a Euclidean space in case $G$ is a matrix group and many other examples. $V$ doesn't even have to be of a certain dimension, or even finite dimensional. So whenever you say representation, you should define $\varphi$ in a way, because there are really many possible representations.

I understand the general definition, I just want to know if what I stated there is correct and if not, I would like someone to correct me

 

[fresh_42]

I gave up in the middle of the text as it reads as one big confusion, a mixture of meaningless terms. You switch between groups, representations, physical correspondences and don't really define any of them, e.g. fundamental representation. There is a natural representation if we have matrix groups, namely the vector space they apply to as matrices, but I don't know what fundamental should mean. And so on. I already gave you the link where most of the homo(iso-)morphisms you mentioned are defined. You posted this in a mathematical forum, so the physical meanings might not be of help, at least as long as you don't clearly say what you want to know.

 

[Silviu]

I gave up in the middle of the text as it reads as one big confusion, a mixture of meaningless terms. You switch between groups, representations, physical correspondences and don't really define any of them, e.g. fundamental representation. There is a natural representation if we have matrix groups, namely the vector space they apply to as matrices, but I don't know what fundamental should mean. And so on. I already gave you the link where most of the homo(iso-)morphisms you mentioned are defined. You posted this in a mathematical forum, so the physical meanings might not be of help, at least as long as you don't clearly say what you want to know.

I read what you suggested, but it didn't clarified everything. For example, what is the different between a spinor and a complex vector?

 

[George Jones]

For example, what is the different between a spinor and a complex vector?

Every representation of SO(3) give rise naturally to a representation of SU(2). How?

 

[Silviu]

Every representation of SO(3) give rise naturally to a representation of SU(2). How?

Well SO(3) representations are tensorial, so they correspond to integer spin representations of SU(2)

 

[fresh_42]

Every representation of SO(3) give rise naturally to a representation of SU(2). How?

Shouldn't it be the other way around via $SU(2) \stackrel{Ad}{\longrightarrow} SO(3)\,$?

 

[George Jones]

Shouldn't it be the other way around via $SU(2) \stackrel{Ad}{\longrightarrow} SO(3)\,$?

$SU \left(2\right)/\left\{1,-1\right\}$ is isomorphic to $SO \left(3\right)$, so there is a 2-to-1 homomorphism, $\mu$ say, from $SU \left(2\right)$ onto $SO \left(3\right)$. If $\nu: SO \left(3\right) \rightarrow GL\left(V\right)$ is representation of $SO \left(3\right)$, then $\nu \circ \mu$ is a representation of $SU \left(2\right)$.

 

[fresh_42]

$SU \left(2\right)/\left\{1,-1\right\}$ is isomorphic to $SO \left(3\right)$, so there is a 2-to-1 homomorphism, $\mu$ say, from $SU \left(2\right)$ onto $SO \left(3\right)$. If $\nu: SO \left(3\right) \rightarrow GL\left(V\right)$ is representation of $SO \left(3\right)$, then $\nu \circ \mu$ is a representation of $SU \left(2\right)$.

Yes, my fault. I should have drawn the diagram. I thought (not really) given ($\nu \circ \mu$) would define us $\nu$.