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

• I
• Silviu
In summary: I read what you suggested, but it didn't clarified everything. For example, what is the different between a spinor and a complex...vector?A spinor is a complex vector with an imaginary component.
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!

What is a representation of a group?

George Jones said:
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)

Silviu said:
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.

fresh_42 said:
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

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.

fresh_42 said:
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?

Silviu said:
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?

George Jones said:
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)

George Jones said:
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)\,##?

fresh_42 said:
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)##.

George Jones said:
##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##.

## 1. What is the difference between the Spin group SU(2) and SO(3)?

The Spin group SU(2) is a double cover of the special orthogonal group SO(3). This means that every element in SU(2) corresponds to two elements in SO(3). SU(2) is a group of complex 2x2 unitary matrices, while SO(3) is a group of real 3x3 orthogonal matrices.

## 2. What is the significance of the Spin group SU(2) and SO(3) in physics?

The Spin group SU(2) is important in the study of quantum mechanics and particle physics. It is used to describe the spin of particles, which is a fundamental property of particles that cannot be explained by classical mechanics. SO(3) is important in the study of classical mechanics and is used to describe the rotational symmetry of physical systems.

## 3. How are the Spin group SU(2) and SO(3) related to each other?

As mentioned earlier, SU(2) is a double cover of SO(3). This means that every element in SU(2) corresponds to two elements in SO(3). The two groups are also isomorphic, meaning they have the same algebraic structure.

## 4. Can you explain the concept of spin in the Spin group SU(2)?

In the context of SU(2), spin refers to the intrinsic angular momentum of a particle. It is a quantum mechanical property that describes how a particle behaves under rotations. In SU(2), spin is represented by the Pauli matrices, which are used to construct the unitary matrices that make up the group.

## 5. What are some applications of the Spin group SU(2) and SO(3) in other fields?

The Spin group SU(2) and SO(3) have applications in fields such as computer graphics, robotics, and crystallography. They are also used in the study of symmetries in mathematics and geometry, and have connections to other areas such as Lie groups and Lie algebras.

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