SU(2) as representation of SO(3)

In summary: I think we should stop here.In summary, the SU(2) and SO(3) groups are homomorphic groups, with a natural group homomorphism mapping SU(2) to SO(3). However, not all representations of SU(2) are also representations of SO(3). The fundamental representation of SU(2) is the group itself, and there is a double cover relationship between SU(2) and SO(3). A representation is a homomorphism from one group to another, and care must be taken when discussing "double-valued representations".
  • #1
maxverywell
197
2
The SU(2) and SO(3) groups are homomorphic groups. Can we say that the SU(2) group is representation of SO(3) and vice versa (SU(2) representation of SO(3))?

Is a representation R of some group G a group too? If so, is it true that G is representation of R?
 
Physics news on Phys.org
  • #2
In the case of matrix Lie groups, the fundamental representation is the group itself, so then you could say, in an abuse of language, that "the representation is a group". But in general the group is an abstract set with four rules, and can be represented in many ways.

But be careful: not every representation of SU(2) is a representation of SO(3)! SU(2) is the double cover of SO(3), and SU(2) is isomorphic to the coset SO(3)/Z2.
 
  • #3
haushofer said:
In the case of matrix Lie groups, the fundamental representation is the group itself, so then you could say, in an abuse of language, that "the representation is a group". But in general the group is an abstract set with four rules, and can be represented in many ways.

But be careful: not every representation of SU(2) is a representation of SO(3)! SU(2) is the double cover of SO(3), and SU(2) is isomorphic to the coset SO(3)/Z2.

Oh I see... I think I get the point now, thnx!

Btw, what is a fundamental representation and a double cover?
 
  • #4
maxverywell said:
Btw, what is a fundamental representation and a double cover?

Elements of [tex]SU(2)[/tex] are 2x2 complex matrices. If to each matrix [tex]A\in SU(2)[/tex] you assing the transformation [tex]x\mapsto Ax[/tex] of [tex]\mathbf{C}^2[/tex] - then you have the fundamental represantation of [tex]SU(2)[/tex]

There is a very nice a natural group homomorphism, call it [tex]\rho[/tex],

[tex]\rho: SU(2)\rightarrow SO(3)[/tex].

It has the property

[tex]\rho(A)=\rho(-A)[/tex].

Matrices [tex]A[/tex] and [tex]-A[/tex] are mapped to the same element of [tex]SO(3)[/tex]. Thus the name "double cover".
 
  • #5
maxverywell said:
The SU(2) and SO(3) groups are homomorphic groups. Can we say that the SU(2) group is representation of SO(3) and vice versa (SU(2) representation of SO(3))?

Is a representation R of some group G a group too? If so, is it true that G is representation of R?
R isn't a group. It's a group homomorphism from G into GL(V) (the group of invertible linear operators on a vector space V).
 
  • #6
Fredrik said:
R isn't a group. It's a group homomorphism from G into GL(V) (the group of invertible linear operators on a vector space V).

R is a set of matrices which with matrix multiplication forms a group.
 
  • #7
maxverywell said:
R is a set of matrices which with matrix multiplication forms a group.

Set is not a representation. Representation is a map from one set to another, with particular properties.
 
  • #8
arkajad said:
Set is not a representation. Representation is a map from one set to another, with particular properties.

I said set with multiplication. So a group representation is a group whose elements are matrices.
 
Last edited:
  • #9
No. Please, check the definition from a good book. Well, I will do it for you:

From H. Jones, "Groups, Representations and Physics", p. 37:

Definition

A representation of dimension n of the abstract group G is defined as a
homomorphism D: G -> GL(n, C), the group of non-singular nxn
matrices with complex entries.

More generally, you can replace GL(n,C) by L(V,K). But the important thing is that it is homomorphism, that is a map with appropriate properties, not a "set with appropriate properties".
 
  • #10
I know that it's a homomorphism etc. but it's a group as I said. Take for example the group of order 2: G={e,a} and its regular representation D={D(e),D(a)}
were

[tex]D(e)=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}[/tex] and [tex]D(a)=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}[/tex]

Now, this two matrices form a group (wich is actually the same as G, they are isomorphic because it's regular representation).
 
Last edited:
  • #11
maxverywell said:
I know that it's a homomorphism etc. but it's a group as I said.
Well, you may like to learn how to distinguish between objects and arrows that connect objects. You never know, one day this ability may come handy...

You may find some info here: http://en.wikipedia.org/wiki/Category_theory" [Broken]
 
Last edited by a moderator:
  • #12
You are right, sorry. Indeed, a representation is an arrow from one group to another (group of matrices).
 
  • #13
So, for instance, we have representations:

[tex]SU(2)\rightarrow GL(2,C)[/tex]

[tex]SU(2)\rightarrow SO(3)\subset GL(3,R)[/tex]

Physicists also sometimes say that there is a "double-valued representation"
[tex]SO(3)\rightarrow SU(2)[/tex]
but this must considered with great care, and mathematicians do not like it all. Instead mathematicians prefer to discuss "projective representations", "multipliers", "cocycles" etc. And they are right. But this is an "advanced subject".
 

What is SU(2)?

SU(2) is a mathematical group known as the special unitary group of degree 2. It is a set of 2x2 complex matrices with determinant equal to 1, and it is often used to represent the rotation group SO(3) in physics and mathematics.

How is SU(2) related to SO(3)?

SU(2) is a double cover of SO(3), meaning that every element in SO(3) has exactly two corresponding elements in SU(2). This relationship is important because it allows us to use the simpler and more convenient algebra of SU(2) to represent the more complex rotations of SO(3).

Why do we use SU(2) to represent SO(3)?

SU(2) is used to represent SO(3) because it is a simpler and more compact representation. By using only 2x2 matrices instead of 3x3 matrices, we can reduce the number of parameters needed to describe a rotation from 9 to 3. This makes calculations and theoretical analysis much more efficient.

What are the physical applications of SU(2) as a representation of SO(3)?

SU(2) is used in many areas of physics, including quantum mechanics, particle physics, and solid-state physics. In quantum mechanics, SU(2) is used to describe the spin of particles, while in particle physics, it is used to describe the symmetry of fundamental interactions. In solid-state physics, SU(2) is used to describe the symmetry of crystalline materials.

Can SU(2) be used to represent other groups?

Yes, SU(2) can be used to represent other groups, such as SU(N) and SO(N). In general, SU(N) is used to represent the rotation group SO(N+1), and it has important applications in quantum mechanics and particle physics. SU(2) is also a subgroup of the special orthogonal group SO(4), meaning that it can be used to represent 4-dimensional rotations in addition to 3-dimensional rotations.

Similar threads

  • Linear and Abstract Algebra
Replies
1
Views
463
  • Linear and Abstract Algebra
Replies
1
Views
864
  • Linear and Abstract Algebra
Replies
11
Views
2K
  • Linear and Abstract Algebra
Replies
7
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
3K
  • Linear and Abstract Algebra
Replies
1
Views
913
  • Linear and Abstract Algebra
Replies
3
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
886
  • Linear and Abstract Algebra
Replies
12
Views
3K
  • Linear and Abstract Algebra
Replies
1
Views
1K
Back
Top