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

Click For Summary

Discussion Overview

The discussion centers on the relationship between the groups SU(2) and SO(3), particularly in the context of their representations and how these relate to spinors and vectors in physics. Participants explore concepts of group representations, the nature of spinors versus complex vectors, and the implications of these relationships in theoretical frameworks.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant states that SU(2) is the double cover of SO(3) and that Spin(3) is also a double cover of SO(3), suggesting an isomorphism between SU(2) and Spin(3).
  • There is a discussion about the representations of SU(2) being labeled by the angular momentum value j, where half-integer j corresponds to spinorial representations and integer j corresponds to vectorial representations.
  • Another participant questions whether a k-dimensional object can be classified as a spinor or complex vector without knowing how it transforms under the respective representations.
  • Several participants seek clarification on the definition of a representation of a group, with one providing a formal definition involving homomorphisms.
  • One participant expresses confusion over the mixing of terms related to groups, representations, and physical correspondences, indicating a lack of clear definitions in the original post.
  • There is a claim that every representation of SO(3) gives rise to a representation of SU(2), while another participant questions whether the relationship should be viewed the other way around.
  • Another participant points out the existence of a 2-to-1 homomorphism from SU(2) to SO(3) and discusses how this affects the representation mappings.

Areas of Agreement / Disagreement

Participants express differing views on the nature of representations and the relationships between SU(2) and SO(3). There is no consensus on the definitions and implications of these concepts, indicating ongoing debate and exploration of the topic.

Contextual Notes

Some participants highlight the need for clearer definitions of terms such as "fundamental representation" and the distinction between spinors and complex vectors. There is also mention of the complexity of representations, which may not be finite-dimensional or of a certain dimension.

Silviu
Messages
612
Reaction score
11
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!
 
Physics news on Phys.org
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)
 
  • #10
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)\,##?
 
  • #11
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)##.
 
  • #12
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##.
 

Similar threads

Undergrad The Lie algebra of ##\frak{so}(3)## without complexification
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Understanding SU(2) and SO(3) Representations
  • · Replies 4 ·
Replies
4
Views
4K
Graduate The colorful world of ##2\times 2## complex matrices
  • · Replies 22 ·
Replies
22
Views
4K
Undergrad Ladder operators and SU(2) representation
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Su(2), so(3) and their representations
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Representation Theory clarification
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Understanding SU(2) Representations and Their Role in Particle Physics
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Understanding 4-Vector Representations in the Lorentz Group
  • · Replies 2 ·
Replies
2
Views
2K
Graduate SU(2) and SU(3) representations to describe spin states
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad What's the relation between spinor space and SO(3) vector space?
  • · Replies 53 ·
2
Replies
53
Views
5K